Acß^Al,

y

f^yANALYTICAL TOOLS

for Studying

em and an

Price Structure

Agriculture Handbook No. 146

UNITED STATES DEPARTMENT OF AGRICULTURE

Agricultural Marketing Service

Washington, D. (!

U.S. DEFT. OF AGRICUtTURE

NATIONAL AGRICULTURAL LIBRARY

JAN30

CATALOGING PREP
For sale by the Superintendent of Docioments, U. S. Government Printing Office

Washington 25, D, C. - Price $1.00
August 1958

ANALYTICAL TOOLS

for Studying

Demand and

Price Structures

by Richard J. Foote

Head, Price and Trade Research Section,

Agricultural Marketing Service

Agriculture Handbook No. 146

UNITED STATES DEPARTMENT OF AGRICULTURE

Washington, D. C.
PEEFACE

Considerable progress has been made in recent years in

developing methods to he used in analyzing the factors that

affect prices and consumption of individual commodities and

in studying their demand and price structures- This handhook discusses certain methods which appear to be of value

for this purpose. Some of them are relatively standardized;

others were developed only recently. These latter have been

applied in only a few cases. In most instances^ examples

are included which indicate specific ways in which these

techniques can be used.

The handbook is designed mainly to acqxiaint research

workers in agricultural economics and related subjects with

some of the recent developments in the field. No attempt is

made to cover all new developments that apply, although many

of the more inrportant elements in analysis of demand are

touched upon. Use of the handbook presumes a general knowledge of the theories of price and demand and of the standard

techniques of regression analysis. Some of the sections also

presume a knowledge of college algebra, matrix algebra, and

calculus. But the conclusions are presented in nonmathematical terms, so that, except for certain developmental or

explanatory sections, the handbook as a whole can be used by

those not acqxaainted with higher mathematics.

Some of this material was included in Agriculture

Handbook No. 61^-, entitled "Analytical Tools for Measuring

Demand" by Richard J. Foote and Kçtrl A. Fox, published in

195^* This handbook supersedes Handbook 64. Sources for

previously published material and sections prepared by

/Staff members are indicated in the text or by footnote in

each section. Helpful suggestions were received from

various members of the staff.
CONTENTS

Page

Diagrajas of sujpply-demand-prlce structures 1

Systems of equations 7

Economic models of increasing complexity 10

Considerations in form\ilating statistical models > 19

Choice of the time xmit 19

Years to be included 20

Use of prices at the local market, wholesale, or retail level

in equations that relate to demand • 23

Choice and handling of variables that cause shifts

in the demand c\i3rve 27

Use of actual data or of first differences 29

Measuring relationships among prices under alternative formulations •.. 33

Deriving estimates of quantity or price from value aggregates • 36

Use of arithmetic or logarithmic variables and other

transformations that relate to functional form 37

Problems related to the use of time as a variable 39

Deciding whether certain variables are endogenous or predetermined •... kh

Choice of the dependent variable in least squares analyses

designed to estimate structural coefficients ... • 51

Considerations in choosing a statistical procedure for fitting the

equations when we wish to estimate the structural coefficients 53

Some economic considerations •. • 53

Some statistical considerations 57

Some econometric considerations 60

Alternative methods ;.., 6k

Reasons why a least squares fit may be satisfactory even though

the estimates of the structural coefficients are subject to

some statistical bias 67

An exaoople that illustrates some fitting procedures 69

Equations and varisibles in the statistical model 70

Fitting the linearized model 72

Computations on an electronic computer 76

Special econometric topics 78

Elasticity of demand 78

Problems of aggregation 8k

Equation systems for competing commodities 87

Measiiring relationships among complementary products 98

Derived demand eqiiations, partially reduced form equations,

and the estimation of demand at different market, levels 100

Allowing for systematic shifts in the regression

coefficients over time 110

Use of distributed lags in demand analysis Ill

Measuring cyclical factors that affect demand II6

Obtaining additive weights for the components of variables to

be used in a logarithaiic analysis 122
CONTENTS

Page

An experiment to test the relative merits of least squares and

limited information coefficients for forecasting imder

specified conditions , 128

Allowance for errors in the data lij.3

Effects of serial correlation in the error terms 1I+8

Spatial equilibriimi analyses , I69

Statistical considerations in interpreting the completed analysis 17I

Considering the signs and magnitudes of the statistical

coefficients • » ,. ^ I7I

Considering the unexplained residuals 172

Testing for serial correlation in the residuals 173

Testing for nonlinearity in an equation • • 17Í1.

Charts that indicate the degree of partial correlation 17Í+

The use and interpretation of tests of significance 177

Application of the F-test to some problems in regression analysis 179

Validity of an estimate from a multiple regression equation I83

Conditions required for valid forecasts I83

A chi-square test to estimate the degree of extrapolation

in a m-ultivariate scatter , 18^1

Use of systems of equations for analytical purposes • I86

Assigning values to specified parameters I86

Modifying structural systems I89

Literature cited I95

Appendix ., 203

Glossary , 207

Index 2l4
ANALYTICAL TOOLS FOR STUDYING DEMAND AND PRICE STRUCTURES

By

Richard J. Foote^ head. Price and Trade Research Section,

Agricult\aral Economics Division

AgricTxltural Marketing Service

The measurement of demand is a complicated subject. Competent analysis

requires three things. |:4lÄt# the economi§t.muat.iiaxe...a^^

'^ISê.^SomAo iMS^^mß ttet affect the commodity a^d obtain adeqimte.,4âta. on

^êiH^JtoJoa^^ fet.J9M^.t.ma§XStai3iâ_^^^^^^

general • JQaSëLy he must; heahle tp^ us§..saQâern techniques ot .^i^ysis.

This report is mainly concerned with techniques of analysis. But the

first few sections discuss ways to make a preliminary survey of the economic

relationships to be expected. Here is where the researcher brings to bear his

knowledge of the commodity and his understanding of theory. A number of

alternative ways to formulate and fit statistical relationships then are discussed in detail. The report next considers some of the techniques that can

be used to test the assumed model and to make quantitative estimates, or forecasts. Specific examples in each case are cited when needed to indicate how a

particular technique should be used.

DIAGRAMS OF SUPPLY-DEKAISD-PRICE STRUCTURES

Diagrams that show the flow of commodities from producer to consumer in

terms either of physical products or marketing channels, or both, have been

in use for many years. Such diagrams are useful in showing the relative

importance of specified sources of supply or kinds of outlets and, in some

instances, the sequence of marketing channels or processing operations.

Figure 1 shows three such diagramis taken from Gerra lk2) l/, Rojko (8l), and

Poote, Klein, and Clough (30). ""

Similar diagrams that show the economic forces or relationships that

affect a given commodity or group of commodities have been developed during

the last several years by staff members of the Agricultural Marketing Service.

Figure 2 shows four diagrams of this sort taken from Fox (33), Meinken, Rojko,

and King (72), Hermie (Vf), and Armore (5). Similar charts have been prepared

for many other commodities. Such diagrams usually are most helpfia when they

are relatively simple, thoi;igh, at times, complex diagrams serve a useful

purpose. Diagrams of this kind may be used in the following ways: (l) To help

the analyst think through basic factors and relationships involved, (2) to aid

in the preparation of a logical writeup of the economic structure of the industry, and (3) to assist the reader in following fairly complex relationships

and discussions. Any statistical analyses that are run shoiild be consistent

with relationships indicated in the diagram though, as with any research work,

modifications may be made as the study develops.

1/ Underlined numbers in parentheses refer to Literature Cited, page 196,
DIAGRAMS THAT SHOW COMMODITY FLOWS

^^ UTILIZATION

-iJT, OF

¡Sr SOYBEANS

(beans, oil, cake & meal)

Nonfood

» Military

FOR THE PERIOD 1*47-4*. DATA DERIVED 6/ táULTIPLYIHG

QUAHTITIES FOR EACH CATEGORY Sr 1947-4« FRICES ANO ADJUSTING

VALUES OF PROCESSED PRODUCTS TO FARM-LCVEL EOUIVALEMTS

NEG. 1753-55(8) AMS

STATISTICAL FLOW CHART FOR MILK AND DAIRY PRODUCTS, 1955

NEG. 3624-56(10) AMS

MOVEMENT OF CORN

Oat», barUy, torghum

groin, wh*at and ry»

f«d, byproduct f««dt

4r

Corn Harvest

icvsrag*«, fitáuetrtat proávctt, mte.

Com maol. hominy, corn flak«», «te.

Storch, tugar, lirup, oil

Farm and Nenfarm Consuntptien—i^

Export

Ù

*GKAINS AND tYfROÙUCT FÍEOS fBO AS SUCH OK IN COMMBUCIAUy-nEfAKBD FEEùS

^THE ENTIRE ÍOX MEASUKES THE TOTAl FEED FED, THE SHADED A«EAS THE AMOUNT Of COIIN fCO

NEG. 48511-X BAE

U. S. DEPARTMENT OF AGRICULTURE NEG. 4428-57(8)/ AGRICULTURAL MARKETING SERVICE

Figure le—Diagrams that show the flow of commodities from producer to consiomer are useful in illustrating the relative importance of specified sources of supply or outlets and in showing marketing or

processing sequences.
- 3 -

The two.diagrams in the upper part of figure 2 indicate the kind of statistical questions that can be discussed in teims of the diagrams. The first

shows the supply-demand structure for a certain type of perishable crop, for

example, early-season fresh asparagus. If (l) practically none of the crop

normally is unharvested for economic reasons, as suggested by the narrow

dotted lines around this factor in the diagram, (2) all of the production

moves directly into a single outlet, and (3) no close substitute or complementary commodities exist, then the elasticity of demand in retail markets can

be measured directly from certain coefficients in a single equation fitted by

least squares. 2/ In the diagram this point is brought out by the fact that

no arrows are double headed and, if the dotted arrows are omitted, no circular

chains are shown. Elasticity of demand at the farm or local market level can

be derived from that at the retail level by appropriate allowance for the

assximed nature of the marketing structure (see page 100).

The second diagram as siamés that conditions (l) and (2) referred to in the

preceding paragraph apply approximately to beef and pork but it shows that the

quantity of each affects both its own price and that of its competitor. In

the words of Meinken, Rojko, and King (72, pp. 71^-715)^ "a given combination

of production of beef and pork resiilts in a imique set of market prices that

is simultaneously detennined. To obtain estimates of the elasticities of

demand that are statistically consistent (see page 58)^ the parameters in the

structxiral demand equations ... must be estimated by a statistical method that

allows for the simultaneity." Methods for doing this are discussed in detail

beginning on page 87.

The third diagram is designed to illustrate several features of the

demand and price structure for wool in domestic markets. Domestic prices tend

to exceed world prices by approximately the amount of our tariff, but prices

rise and fall with changes in world supply and demand. Demand for wool at

local markets is derived from the combined demands of the many processors and

users of wool, but the level of demand at each step in processing is affected

by stocks of raw, semifinished, and finished products. A system of equations

that allows for the many aspects of simultaneity is needed to describe adequately the domestic wool economy and to measure elasticities at the several

marketing levels but, imfortxmately, data are not available to fit such a

system.

The fourth diagram shows the two channels through which the effects of an

increase in the yield of cottonseed oil per ton crushed on total returns received by farmers take place. One channel affects the quantity of oil

obtained per ton of seed crushed and therefore directly affects its value.

This channel is illustrated by the arrow toward the top of the diagram running

from the box entitled "Cottonseed oil yield per ton of cottonseed processed,"

to the box entitled "Value of cottonseed oil yield." The second channel

affects the price of the oil by increasing its total supply. These effects

are 'shown by the second column of boxes and the arrow rimning diagonally upward from the last box in this column to the box labeled "Value of cottonseed

2/ The reasoning here is developed in detail beginning on page 53.
- 4 -.

DIAGRAMS THAT SHOW ECONOMIC RELATIONSHIPS

DEMAND AND SUPPLY STRUCTURES

FOR PERISHABLE CROPS

Supply Pr9dmt»rminmd: Singlm Markmt

\

CONSUMPTION \

1 MARKETINC 1

1 SYSTEM J

DISPOSAILE

CONSUMER INCOME

^

^

X'

PRODUCTION

PRODUCTION

~ ~ 1 ~ ~

1

1 .

HARVESTING

>.

1 FARM FRKE 1

WEATHER; ECONOMIC

INFLUENCE PRIOR

TO HARVEST

A««OWS SHOW OIKCCriON OF INFlUINCf. HfAVT AIIIOWS INOICATf mA/Ot PATHS Or INFlUf NCI

WMfCH ACCOUNT FO* THE iUlIC OF THf VAflATION IN CUHtENF MICIS. IICHI SOIIO A*IOWS INDICATC DCFINirC lUF IFSS IMFOtFANI PATHS. OASHtO AII*OWS INOICATt PATHS OF NtCHCISlf,

OOUtTFUl.OI OCCASIONAl IMPORTANCf

NEC. 48930-X BAE

SIMPLIFIED DEMAND AND PRICE

STRUCTURE FOR BEEF AND PORK*

ARROWS SHOW DIRECT/ON

OF INFLUENCE

(SUPPLIES PReoeTERfilNEDI

NEC. 3091-56 (2) , AMS

STRUCTURE OF PRICE-MAKING FORCES FOR WOOL

~:::t

4

Pruc

Wi )1 Fabrics

Wholesale Prico

of

Finished Goods

1 Demand bv Consumers

Retail Price

of

Finished Goods

NEC. 48015 BAE

U. S. DEPAPTMENT OF AGRICULTURE

ECONOMIC FACTORS AFFECTE!

CHANGE IN THE YIELD OF COTTON

> BY A

SEED Ol L

COTTOHlfeDOa.yiILOPÏ»

TOM OP COTTOHSitO PPOCÏJItO \^

OF COTTONSEED /\ CRUSHED

y X

SUPPLY OP

COTTOMÍfíB

OIL ; 1

COTTONSCCD

OIL TKLD

^

/

SS¿

IN POOO OTHÏ«

THAMCOTTOH.

Sf «BOII., iUTTe«,

' '

-

SUPPLY OP

OILS OTMt«

IMAM tUTTl«

AND LAPO ; Ifffl £rS

W \

\

LINTER YIELD OKPOSAILt

IHCOMt V

\ r '

SUPPLY OP "7

AVEPACI PPICt

OP POoe PATS

AND OILS OTHER

THAN »UTTE«

AND LARD

COTTONS* ED

RiCIIYÍDaY

PAPMEM

-PPROCESSING AND

MARKETING

.,/

' suPPtroP /

1 tUTTm« 1

\ 1 MEAL

» 1 /

COTTONSEtO 1 '

PERCENTAGE OP

COTTONSEED

CROP, LESS USE

FOR PLANTIHO.

SOLD TO MILLS

/

r

MERCIAL PERTI.

LIIER USED ON

\

\

PRICE

OP

CORN

USDA NEC. 48872- X BAE

NEC. 4429-57(8) AGRICULTURAL MARKETING S&RViCE

Figxire 2•—Diagrams that show economic forces or relationships are useful in

thinking through complex relationships^ in preparing a logical vriteup of

them^ and in helping readers to follow the discussion.
- 5 -

oil yield," Single equations fitted by least squares could be used to measure

the relationships suggested by the first and second coliJinns^ although this

part of the diagram actually is an over-simplification of the fats and oils

economy^ particularly for the period following World War II. Effects on the

total value of cottonseed oil throiigh the first cheuinel can be computed

directly by arithmetic. If the dotted line running diagonally upward from the

last box in the third colxmin to the box, entitled "Supply of cottonseed oil,"

was of greater importance, a system of eq\xations would be needed to measure

the relationships in the diagram because of the implied circular chain.

Diagrams shown in figure 3 are designed to illustrate a demarcation

which, in the opinion of the author, exists between those that are too complex

to be meaningful and complex diagrams that are useful to the analyst or

reader. These are taken from Meinken (7l), Rojko (8l), and Sh\affett (84),

respectively. Although the author assisted in the development of the first

diagram, he now feels that it is too complex to be of much value either to

careful or casual readers. The important relationships shown can be-discussed

more easily and clearly by a direct referral to the system of six equations

developed to represent the domestic wheat economy than by trying to follow

through the many double-headed arrows and circular chains suggested by the

diagram. On the other hand, the diagram developed for the domestic dairy

economy, which is at least as complicated, clearly shows that portion of the

dairy industry which is included within the overall feed-livestock economy and

that part of the feed-livestock economy which is outside of the dairy economy

—one of its principal piarposes. It also shows certain income and service

flows that are discussed in detail in the original text. In this diagram,

boxes relating directly to the dairy economy are given in a simplified form,

as they relate to aggregates for all milk and dairy products. When relationships among dairy products are studied, a system of equations is required to

obtain estimates of the structural coefficients, such as elasticities, which

are statistically consistent.

The diagram that relates to the demand and price structure for commercial

peas is useful in showing the extent to which fresh peas and peas for processing can be considered separately and the extent to which they must be considered simultaneously. By the same reasoning as developed with respect to

the demand structure for beef and pork (see page 3)^ a system of simultaneous

eq-uations is needed to obtain a statistically consistent estimate of the

elasticity of demand for either fresh (and frozen) 3/ or canned peas. If the

competitive relationships could be ignored, the important relationships indicated by heavy lines in the diagram could be derived from certain coefficients

from single equations fitted by least squares.

3/ Shuffett {&4;, p. 85) shows that by adding frozen consumption to fresh

supply, an analysis based on data prior to World War II (when frozen peas were

of negligible importance) can be used in the post-War period.
- 6 -

SOME

COMPLEX

DIAGRAMS

MAJOR RELATIONSHIPS IN

THE WHEAT ECONOMY

NEC, 1356-55 ¡1)

MAJOR ECONOMIC INFLUENCES

AFFECTING THE DAIRY INDUSTRY

DtSrOSAtlf

CONSUMII

INCOMIS

Domestic

«,Dairy Economy

ssasr /: ; sw

miViJSmiJi»

m.^

i^r^

Fe«d-Livystp'ck Economy: Farm LeveP

JNCOMt

INCOMC X^'" —- - - —

V MOM ^ ^^*f "^ MOM y,

N^ tlVtSTOCK-llVtSTOCK MODUCTS- HtO ^J%,

..o> ^^\w^= T^^ur zr^.INCOMiy /

NEC. 37I«-5<(I1) AMS

U. S. DEPARTMENT OF AGRICULTURE

DEMAND AND PRICE STRUCTURE

FOR COMMERCIAL PEAS

FRESH PEAS

ECONOMIC

FORCES PRIOR

TO HARVEST

ACREAGE

AVAILABLE

FOR HARVEST

DEMAND & SUPPLY

CONDITIONS IN

IMPORTING

COUNTRY

' UNHARVESTED |

I PRODUCTION I

« CONSUMPTION

PEAS

\t

FOR PROCESSING

AIIROWI SHOW OMCCriON or IHFLUtHCt. llt*V1

«COIU« LmtS P*TM W «OOCKATC IHH-iltHCe, ÄMO o

MteUeltLC. OK OCCASIONAL IMPOMTANCC

NEC. 4430-57(8) AGRICULTURAL MARKETING SERVICE

Figure 3.---Some diagrams that show economic relationships are so complex as to

materially reduce their value. The diagram for wheat may fall in that category, but those for dairy products and peas are believed to be helpful to

readers of the publications in which they were initially published.
- 7 -

Use of heavy lines to show the more important relationships is an aid to

readers in following diagrams that are even moderately complex. In the chart

for wheats however^ too many relationships are important, so this device is of

less value than, for example, in the diagram for peas.

SYSTEMS OF EQUATIONS k/

In the language that has been developed to consider statistical analysis

of economic relations, the process by which a set of economic variables is

believed to be generated is called a structure. 5/ ^^j^m^m^Mlß^^^^ß

are explained by the structxg^e^^ , whereas those

whose valueOgF*8^^

^^.ñÍfflÍE.tagE±he3:..Aí^ .,,,Tbe sgji^f

struc?tÄi§,. that^^ ar£^_com^ati^^^^ with_ the írw^^||a3^^i,, ^riyai3^i^^iMMn»iinp.fctons

aljout the statis1¿cMIBaÍ^rj¿e.Jfr^ wM are drawn is called a

igodel. _jrhus, within a model, we specify which structural relations are

assumed to hold exactly and which include an "unexplained residual. The fonner

are referred to as identities and the latter, as relations. At times, it is

useful to distinguish between economic and statistical models.

The diagrams which have been described are based essentially on economic

theory and knowledge of the workings of that part of the economy being

studied. The term economic model is applied to the set of structures consistent with the assumptions developed by the investigator from economic theory

and knowledge of existing factors that relate to a particular commodity area.

Thus an economic model is a set of equations that is consistent with the relationships and identities implied by the diagram.

In the present state of economic and statistical theory, a research

worker typically finds it necessary to make additional assiamptions for which

economic and commodity considerations offer little if any guide. For example,

he must specify the algebraic form of the relations and the specific way in

which the relations are affected by imobserved influences. Although economic

considerations sometimes may exclude certain possibilities, they usually do

not provide strong groimds for preferring a particular set of assxmiptions. The

specifications made about these aspects of the structure often are chosen

partly to simplify the statistical analysis and are to a considerable extent

arbitrary. Here we refer to the set of structures consistent with all the

specifications of the investigator (both economic and statistical) as the

statistical model.

k/ Material in the first four paragraphs for the most part is adapted from

Hildreth and Jarrett (^9, pp. 6-7).

5/ Marschak (66, pp. 1-8) gives an easily \mderstood example that illustrates the basic nature of struct\iral relations and parameters. This example

is discussed briefly beginning on page 56.
. 8 ^

Althoiigh the distinction between economic and statistical models is not

always a sharp one^ it frequently is useful. If the research worker makes

explicit the basis for the a^suniptions underlying the economic model, these

can be evaluated by economists and by persons well informed about the particular commodity area. The analyst may wish to experiment with alternative sets

of specifications that apply to the statistical model, such as whether to use

actual data or logarithms, whether to use first differences, and similar

questions. Considerations that help in choosing among the several alternatives are given beginning on page 19.

After preparing diagrams of the sort described in the first section, the

analyst will wish, in general, to write down the equations that describe at

least the more important economic relations. As mentioned on page 3> two

equations are needed to describe the relationships indicated by the chart in

the upper left corner of figure 2, and we may wish to use also an identity to

remind us of a basic assumption on which this chart is based. In the discussion that follows, we use the following symbols:

Q - Production

Qç - ConsTomption

P^ - Retail price

Ff - Local market or farm price

D - Disposable income

M - Factors associated with niarketing costs

A convenient way to simmiarize the relations implied by the economic model

is as follows: ^ j ^,

^^'""^^ Pj. I Qc^ ^ (priçe-coç^iamption relation) (l)

^r^^^f ^' ** (price level relation) (2)

^c ~ %• (production-consumption identity) (3)

.^. p^ h]

A colon may be read "depends on;" a comma may be read "and;" and a semicolon

may be read "appear in a relation with." The variables to the left of the

colon or semicolon are current endogenous variables and those to the right are

regarded as predetermined within the particular model. A colon is used only

when a single variable appears on the left and the relation is believed to be

of a causal nature.

If production is a predetermined variable and consumption and production

are assumed to be identical for practical piorposes, as implied by eqxaation

(3), consimiption is assiomed to be a predetermined variable. In'this model,

disposable income also is tsiken as predetermined. We then are interested in
- 9 -

finding the retail price that is consistent, in an economic sense, with this

consumption and consumer income» We do this by statistically fitting equation

(l). Once we have that price, we can find the local market price from the

relationship implied hy equation (2).

An advantage in using colons and semicolons rather than a more conventional equation form is that we need not, at this stage, become involved with

the essentially statistical problems of whether to use actual data or logarithms or similar matters for which a decision might be implied by a conventional equation. Analysts sometimes have used the form

Pr = il(Qc> D) (1.1)

Pf = f2(Pr> M) (2.1)

but this fails to distinguish between endogenous and predetermined variables

and may suggest with respect to equation (2.iJ^ that a third equation with P^,

dependent is needed to go from P^» to Pf, whereas our model assumes that the

relation is nearly enough a functional one so that only a single statistical

relation is needed. If relation (2) is fitted by a simxxltaneous equations

approach, rather than by least squares, results that are algebraic equivalents

are obtained regardless of which endogenous variable ts treated as the one

having a coefficient of one.

Let us now go through a similar formulation for the economic model imr

plied by the diagram in the upper right corner of figure 2. The diagram does

not show production of beef and pork, but identities like equation (3) are

implied for each item. We omit these identities in what follows. This

diagram also omits local market prices and, in the following, we omit the two

implied price level equations. Thus the diagram implies the following equations, where the subscript b relates to beef and the subscript p relates to

pork:

Pb • %^ %> ^ (Price relation for beef) (k)

Pp : Qb> Qp^ '^* (Price relation for pork) (5)

As discussed beginning on page 87, if we wish to derive the direct- and

cross-elasticities of demand for beef and pork, as normally defined in

economic literature, we need to write these equations in another way; but if

we think of them chiefly as price determining relations, then the form shown

here is correct.

lem equations to be used for analytical purposes or prediction. If we are

interested only in ascertaining the probable magnitude of certain coefficients

then, at times, a complete model is not required. The author has found by

experience that it is generally advisable to at least formulate a complete

economic model in terms of the sort of relations given here. Balancing of the
- 10 -

total number of endogenous variables with the total number of relations frequently helps in the basic formulation of the model. After the complete model

has been written down in symbolic form, a decision may be reached at a later

stage to fit statistically only part of the equations.

Our first example might be thought of as a 1-, 2-, or S-equation model,

depending on whether we consider as endogenous variables (l) only P^, (2) P^

and ?f, or (3) P^^ Pf^ and Qe- In models of this sort, Qp always is considered as predetermined, as no .supply eqxaation is shown. Factors which tend

to make production for many agricultural commodities essentially a predetermined variable are discussed beginning on page kk. Since consumption is

assumed to equal production, it normally is considered directly as a predetermined variable; but if equation (3) is considered as an integral part of the

model rather than a definitional identity, then Q^» would be considered as

endogenous. The analyst should always regard the second model as at least a

2-equation one, with both P^ and Pp endogenous; the reasons for this are given

on page 3, in the discussion of the related diagram. The equations shown can

be used to forecast prices from a given production of beef or pork, but other

equations are needed to show how consumption of each depends on the respective

prices.

Each of these models is on a national aggregate basis and each assumes

production, marketing cost, and disposable income to be predetermined variables and production and consumption to be nearly identical. In some cases,

prices at the end of or diiring the season may decline to a point where, only

part of the crop is harvested. If this is frequently true, then a harvesting

equation must be introduced to show the economic factors that determine the

ainount of unharvested production. An example of this sort for watermelons is

discussed in the ^next section. For some commodities, marketing charges may

depend in part on the quantity moving through the market; if this is so, an

additional eqviation showing, factors that affect marketing charges must be

added.

ECONOIIC M0DEI5 OF INCREASING COMPLEXITY

The best way to gain facility in formulating economic models is to

examine soitó that have been constructed by other investigators. The examples

given here are designed to demonstrate how the number of relations tends to

increase as certain simplifying assumptions are relaxed. Only enough detail

is given to indicate the basic model-building principles that are involved;

the reader is referred to the original sources for complete details.

We Indicated on page 3 that if (l) production can be assumed to be determined in advance of the marketing season, (2) essentially all of the production moves directly into a single outlet, and (3) no close substitute or

compleiiientary commodities exist, then our model can be thought of as containing oiay a single structural demand or price equation. But, as noted on page

3, if close siJbstitutes exist, then we must have as a minimum one equation for
- 11 -

each of the several substitutes. If in some years part of the crop is iinharvested for economic reasons, then we must add to our model an equation for

each such item showing the economic factors that determine the quantityharvested.

Suits (88) made use of an eqxiation of the last-named sort in developing

an econometric model of the watermelon market. We translate his notation into

that used here for our first example. He points out that in no event can the

harvest of melons exceed the crop. Thus the harvest equation has the general

form

^ ^ ) (harvest relation) (6)

or Qc = Qp ^ )

whichever gives the smaller value of Q^, where Qp is production, Qc is the

number of watermelons harvested, W is an index of southera wage rates on

farms, and P is price. In addition, we have a demand equation similar to that

indicated by relation (l) except that it contains two endogenous variables. 6/

Thus, if no close substitutes or complements exist, we have at a minimum a

2-equation model, one equation of which is made up of two alternative forms.

The two endogenous variables are, of course, Q^ and P. The composite relation

(6) is part of a statistical rather than an economic model, but in this instance the pure economic model does not bring out the nature of the assxmied

relationship.

We next turn to a commodity for which production is assumed to be a predetermined variable and for which no close substitutes are assumed to exist,

but which moves into four price-determined outlets. The commodity is wheat,

and we make use of the system of equations developed by Meinken (71). A simplifying assumption is that a single price for wheat can be used in each of

the demand equations. The model permits us simultaneously to estimate

domestic and world prices of wheat and utilization in each of the four outlets. The following symbols are used: C^^ use for food products; Cf, use for

feed; Ce^ use for export; Cg, use for commercial storage; P^, price in world

markets; P¿, price in domestic markets; Q¿, domestic supply less use in

certain outlets not affected by price; Q-^, world supply; and Z±, a predetermined variable other than supply. The following structural relations are

involved:

P^ : Q^, Zi (world price relation) (7)

Cj^, P¿ ; some Z*s (domestic demand for food) (8)

Of, P¿L ; some Z's (domestic demand for feed) (9)

6/ Suits also gives a supply relation showing the economic factors that

affect Qp but, as only prices in a prior year are used, we need not concern

ourselves with this aspect.
- 12 -

Ce> Pd> Pw ^ some Z's (domestic net export

relation) (lO)

Cg, P¿ ; some Z's (domestic storage relation) (ll)

C^ + Cf + Cç + Cg = Qd- (domestic supply-utilization

identity) (12)

The reader will note that the above system of equations contains no domestic

price relation^ but no such relation is required because domestic prices must

be at a level such that the identity implied by relation (12) holds exactly.

Formulation of the complete model verifies this fact, as ve have the same

niomber of economic relations as endogenous variables, 7/

This example shows another advantage of writing our economic model in

this form; at this stage we need not specify which Z*s come into each equation. The nimiber of Z^s affects the amount of work involved in fitting and to

some extent has an effect on the statistical reliability of the results, but

the number included in each relation has no effect on the basic structure of

the model unless an identification problem is involved. 8/ As we see later,

some equations may have no variables on the right of the semicolon.

We next turn to a model for dairy products developed by Rojko (8l) which

is similar to that for wheat except that a price relation is used for each of

the several dairy products. The retail price of butter, for example, differs

from the retail price of milk or ice cream for a variety of reasons, but the

value of milk used for making butter at the farm on the average must be equivalently priced with that used for making any other dairy product or no milk

will be channeled into the lower-priced outlet. This economic fact is allowed

for in the model by showing the price in each outlet as a function of the sajne

set of predetermined variables. Using a notation similar to that in other

examples, we have the following symbols in a model that relates to a period

following World War II: Qf, use for fluid milk and cream; Q-^, use for butter;

Qç>, use for American cheese; Q^, use for other manufactured dairy products;

Qjjj, quantity of margarine sold; Pf, price of fluid milk and cream; P^, price

of butter; Pe, price of American cheese; PQ^ price of other manufactured dairy

products; Pjjj, price of margarine; Pg, price of substitutes for cheese; Q,

total quantity of milk available for cons\xmption; and D, disposable consiomer

income. The price of margarine, Pjjj, is assumed to be determined by the fats

and oils economy and thus to be exogenous to this model, but the quantity of

margarine sold is determined in part by the dairy economy. Pg is assumed to

7/ The model coxild have been written to include a domestic price relation

but, had this been done, one of the other equations would have been omitted.

8/ Most equations that relate to supply-demand structures for individual

commodities are highly overidentified (see page 63). For certain other kinds

of models, problems of identification are important, and we may need to

specify in the economic model which Z's appear in which equations.
- 13 -

berexogenous to this models as is D^ Q is assxmed to be a predetermined variable, although a research project currently underway is attempting to ascertain whether this is justified. The model thus consists of h endogenous

quantities and h endogenous prices of individual groups of dairy products, and

the endogenous quantity of margarine sold—or 9 endogenous variables in all.

The following structural relations are involved:

Qf^ Pf J D

Qb. Pb •> D, Pm

Qe. Pc > ^' Ps

Qo^ Po ' D

«im> Pb ' ^> Pm

Pf ' D. Pni> Q

Pi. : D> Pm> Q

Pc : ^> Pm^ Q

Po : D. Pm^ Q.

(demand for fluid milk and cream) (13)

(demand for butter) (1^)

(demand for American cheese) (15)

(demand for other dairy products) (16)

(demand for margarine) (17)

(price relation for fluid milk

and cream) (18)

(price relation for butter) (19)

(price relation for Americeui

cheese) (20)

(price relation for other dairy

products) (21)

Since we have 9 equations corresponding to our 9 endogenous variables, this is

a complete model* However, the following identity always holds:

Qf + Qb + Qc "*" ^o ~ ^* (production-utilization

identity) (22)

An alternative way to formulate a model of this type is as follows. We

retain the 5 demand relations, but in place of the k price-relations, we use a

single price relation for all milk, and then relate each of the individual

prices to this overall price through a price level relation similar to that

used in relation (2). This gives us 10 relations in all, but our balance is

retained as this model has an extra endogenous variable, namely the overall

price of milk. Estimates of prices from the alternative formulation in

general differ somewhat from those obtained from the original formulation

because a different sort of market structure is assumed. Rojko (8l) presents

results from both fonnulations for several models that relate to dairy

products.

In the models discussed so far, production is assumed to be determined

chiefly by economic or other factors that exert their influence before the

start of the marketing year. Allowance is made in the watermelon example,

however, for partial harvesting of the crop for economic reasons. For a
- Ik -

number of livestock product3 that are produced on a continuous basis^ production can be modified during the marketing period if this period is taken to be

as long as a year, as is frequently done in statistical analysis. This is

true for eggs, because annual production can be affected by current price

through both (l) varying the culling rate and (2) changing the number of

replacement chicks raised. Our next model, which relates to eggs/ allows for

such factors by providing relations that permit supply to be partially endogenous. This example was fo3rmx4ated and fitted by Gerra. 9/

The model consists basically of two endogenous supply relations, three

identities that relate to supply, a demand equation, and a farm-retail price

relation. Four more relations are required, however, to explain endogenous

variables that appear in other relations. In a notation similar to that used

for other examples in this handbook, the following endogenous variables are

involved: Qp, production of eg^s; Q^ ^^^ ^c> consumption of eggs; P^ and F¡.^

retail price of eggs; Pf and Pf, local market price of eggs; Na, average

nxmiber of layers during the year; Np, nimiber of replacements started during

January-June; N^^ number of layers culled during the year; and S^, eggs placed

in storage during spring. A symbol with a prime (') indicates that the variable relates to the period January-Jiuie; lack of a prime indicates that the

variable relates to the calendar year. A prime is not used on Np because,

althoxigh this variable relates to replacements started in spring, the replacements do not enter the flock until fall. We have 11 endogenous variables. The

following predetermined variables need to be specified to indicate the nature

of the economic model: R, rate of lay per hen in each year; Q^^^ and Q' miscellaneous uses for eggs other than as food by civilians, including net change in

stocks as of January 1 and net exports; N-^, mmber of layers on hand at the

start of the year; %, number of layers dying during the year; P^ and P¿,

price of feed grains, which is assumed exogenous for this model; and Q*, production of eggs during January to June, which is assumed to be determined by

economic factors in operation prior to the start of the marketing year. The

symbol Z± is used for other predetermined variables. R is taken as predetermined because price relations normally are of such a nature that it pays to

feed hens as much as they will eat, so that,the rate of lay depends chiefly on

developments relating to breeding, methods of feeding and housing, and so

forth, rather than on the level of economic factors during the marketing

period.

The model consists of the following 11 relations: 10/

Np> ^f ; Pg (replacement relation) (23)

^c^ ^f ^ ^g (culling relation) {2k)

9/ Gerra, Martin J. The Supply, Demand, and Price Structure for Eggs.

Unpublished manuscript. 195?.

10/ The model as xised by Gerra differs sli^tly from this in that Q^^^ is defined so as to use a plus sign rather than a Mnus sign in relation (27) and

QjQ is omitted due to a lack of data.
- 15 -

Ng^ = N|^ + Np - Nç. - % (average nimber of layers

identity) (25)

Qp = R Na (production identity) (26)

Qc = Qp - Qja (consumption identity) (27)

Q^^ ^r ^ ^^^® ^^^ (price-consumption relation) (28)

P|.^ P^^ ; some Z's. (price level relation) (29)

The following relations apply to the January-June period:

S% -^ j some Z's (storage in spring relation) (30)

Qp

F^, Qç., S' ; some Z^s (price-consumption relation) (31)

Pf> Pp ; some Z's (price level relation) (32)

Qc = Qp - Q¿ - S'. (consimiption identity) (33)

The economic reasoning behind these relations should he clear to the

reader except for the following: (l) In the statistical model for relation

(30)^ storage is assumed to depend on (a) the ratio between production in

spring (assxmed to be predetermined) and production for the entire year

(assumed to be partially endogenous) and (b) profits made from the storing

operation in the preceding year. (2) Qj^ and Q¿ are in fact at least partially

endogenous^ but in most years these net uses for export or storage represent

only a small percentage of production or consumption and, when they have

represented more than this, they have essentially reflected Governmental

activity of an institutional nature. (3) The last four relations are needed in

the model to get Pf for use in relation (23).

The last model that we consider in this section is one relating to asparagus developed by Carstensen. 11/ Here production is assumed to be a predetermined variable, but we consider three producing and consuming regions for

fresh asparagus and also allow for production and consumption in terms of

national aggregates for frozen and canned asparagus. The regions referred to

are (1) the West, (2) the central part of the country, and (3) the East. In

terms of volimie of production, California is the chief State in the West,

Michigan and Illinois in the Midwest, and New Jersey in the East. As data on

retail prices are not readily available, market prices are taken at wholesale

and for fresh asparagus relate to prices, respectively, in (l) San Francisco

and Los Angeles, (2) Chicago, and (3) New York. A national average price must

be used for canned and frozen asparagus, as prices by areas are not available

11/ Carstensen, Hans. The Demand and Price Structure for Asparagus.

Unpublished manuscript. 1958.
- 16 -

in published form. No allowance is made in the model for foreign trade in the

processed items• In effect, net trade is assiamed to be zero* This simplifi-.

cation reflects in part a lack of knowledge as to the.nature of the economic

relations involved. The model relates to the late spring season of heavy production, and each region is assimaed to produce enoxxgh asparagus at least to

take care of its needs for fresh use. The surplus in each area is used for

processing.

The notation used for other examples in this handbook is not satisfactory

here because of the need to distinguish between the three regions. The symbol

Q is retained for production, but the letters A, B, C, and so. forth, are used

for other variables in the order in which they are defined. The subscript w

is used for the West, c for the central part of the country, and e for the

East. The following endogenous variables are included in the model: A^^ AQ,

and Ae^ consumption of fresh asparagus in each area; B^^ BQ, and BQ, wholesale

price of fresh asparagus in each area; B, a weighted average national aggregate price for fresh asparagus; C, consumption of canned asparagus; ,D, wholesale price of canned asparagus;^ E, consximption of frozen asparagus; F, wholesale price of frozen asparagus; G^, GQ, and Ge^ local market price for fresh

use; H^, H^, and H^^ local market price for processing; I^, I^, and I^, use

for processing in each area. We have 20 endogenous variables in all. The

following predetermined variables need to be specified to illustrate the

nature of the economic model: Q^, Q^^^ and Q^, production of asparagus in each

area; J^, Jç>, and J^, economic factors that affect the relative cost of preparing asparagus for sale for fresh use or for processing in each area; K^,

Kç, and K^, economic factors that affect marketing costs for fresh asparagus;

Lv^, LQ, and Le, economic factors that affect marketing costs, including transportation, for canned asparagus; M^, M^ and Me^ economic factors that affect

marketing costs, including transportation, for frozen asparagus, N, year-toyear net increase in stocks of canned and frozen asparagus. As in other

examples, the symbol Z^ is used to represent other predetermined variables.

The model consists of the following 20 relations:

A^, B^, D, F ; some Z's (demand for fresh asparagus

in the West) (3^)

Ac> BQ, D, F ; some Z*s (demand for fresh asparagus

in the Midwest) (35)

Ag, Bg, D, F ; some Z*s (demand for fresh asparagus

in the East) (36)

C, D, B, F ; some Z*s (demand for canned asparagus) (37)

E, F, B, D ; some Z's (demand for frozen asparagus) (38)

Qv = A^ + Iv (production-utilization identity

in the West) (39)
- 17 -

Qç = A^ + le (production-utilization identity

in the Midwest) {ko)

Qe ~ -^e + ^e (production-utilization identity

in the East) (4l)

G^, Ey j J^ (price relation "between use for

fresh and for processing in

the West) (^2)

G^, HQ ; JQ (price relation between use for

fresh and for processing in

the Midwest) (kS)

Gç> Hg ; JQ (price relation "between use for

fresh and for processing in

the East) (hk)

^w> B^ ; K^ (price relation between local

market and wholesale price of

fresh asparagus in the West) (^5)

Gç> B^ ; K^ (price relation between local

market and wholesale price of

fresh asparagus in the Midwest) (kS)

Gg^ Bg j KQ (price relation between local

market and wholesale price of

fresh asparagus in the East) (hj)

D^ H^ ; L^ (price relation between local

market price for processing

in the West and wholesale

price of canned asparagus) (kQ)

B, H^ ; Lç (price relation between local

market price for processing

in the Midwest and wholesale

price of canned asparagus) (^9)

D, Hg ; LQ (price relation between local

market price for processing

in the East and wholesale

price of canned asparagus) (50)

F^ H^ ; M^ (price relation between local

market price for processing

in the West and wholesale

price of frozen asparagus) (51)
- 18 -

F, HQ j Me (price relation between local

market price for processing

in the Midwest and wholesale

price of frozen asparagus) (52)

F^ Hg j MQ. (price relation "between local

market price for processing

in the East and wholesale

price of frozen asparagus) (53)

The following identities are implied by the model:

V^w + ^c^c + ^e^e

B = AITTA—TU (weighted average national

^ ^ price for fresh asparagus) (54)

C + E = I^ + Iç. + Ig - N. (national aggregate processing

identity) (55)

As these examples illustrate, decisions frequently must be made by the

analyst regarding the complexity of the model to be used and the degree of

aggregation- Klein (58, pp, 185-200) has a useful suggestion here in connection with his discussion of sector models. He suggests that a master model,

for the entire national economy be formulated; he and others have made some

progress in formulating models of this type. 12/ Models for individual

industries, individual commodities, or individual regions then can be formulated and fitted as separate entities in such a way that they can be "grafted"

onto the master model. In the examples discussed so far, variables that

relate to the national economy, such as disposable income, have been treated

as exogenous, but we did not mean to imply that this was necessarily an

approved method. With Klein's approach, either of the following methods may

be used for variables of this sort: (l) The computed value of disposable

income for each observation, based on the master model, is used as a predetermined variable in fitting equations for a particular sector or (2) disposable

income is treated as an endogenous variable and the predetermined variables on

which it depends are brought in as predetermined variables in the system for

the sector.

The same general approach can be used in narrower fields. Hildreth and

Jarrett (^9) developed a model that relates to the feed-livestock economy but

for which all livestock and livestock products are aggregated. We have considered in this section a model for eggs for which the price of feed grains is

considered as determined outside the egg economy. Instead, we co"uld have used

as a price for feed grains the calculated price from the Hildreth-Jarrett

model. In later research studies, feed-livestock models might be fitted for

separate regions, with separate relations for each major type of livestock.

12/ See Klein and Goldberger (60) and Suits and Goldberger (89). For criticisms of these models, see Christ (lA) and Fox (37).
- 19 -

and these regional models then cotxld he grafted onto a master feed-livestock

economy models which in turn coiild he grafted onto a master model of the

entire economy. Thus, over time, a series of studies could he huilt up, each

of which would he manageahle as a research unit, hut all of which would tie

together to make a united whole. 13/

CONSIDERATIONS IN FORMULATING STATISTICAL MODELS

After writing down the economic model, we are faced with the difficult

task of locating or developing the necessary data and of deciding upon the

exact nature of the variahles and relations to he used. As noted on page 1,

decisions in this area frequently must he hased chiefly on experimentation or

judgment. In many instances, alternative methods exist, hut methodological

experts sometimes do not agree as to the hest procedure to follow. In certain

cases, particular conditions that are known to prevail with respect to the

commodity area to he studied indicate a preference for one method over

another. In this section, the more important decisions that must he made are

listed, and some of the considerations that indicate a preference for one

method over another are discussed.

Choice of the Time Unit ik/

Most analyses of factors that affect the price or consumption of a given

commodity are hased on annual data for either calendar or crop years. This is

satisfactory if conditions within the period are sufficiently homogeneous. For

products that are produced continuously throughout the year, such as dairy

products or eggs, availahle puhlished data frequently relate to a calendar

year and this is the most convenient time xmit to use. Data for crops, on the

other hand, normally are puhlished hy marketing years, so that this is the

most convenient time imit. For some items, however, the economic structure

differs considerahly in one part of the year from that in other parts. For

asparagus, for example, production in early spring is concentrated almost

entirely in California and practically all of the crop is shipped for use in

fresh form. Thus the economic model that relates to early spring differs

materially from the one that relates to late spring--descrihed on page 15—and

two economic and statistical models for asparagus are required, one relating

to early spring and one relating to late spring. A similar sitxmtion exists

for corn. During the period November to May, the supply variahle that affects

price in years for which price is determined chiefly hy free-market influences

is the total supply of feed concentrates for the Octoher-September marketing

13/ A start in this direction, for major sectors of agriculture in relation

to the general economy, has heen made hy Cromarty, William A., Economic Structure In American Agriculture. Ph.D. thesis, Michigan State University, 195?.

ih/ Decisions with respect to the time imit are based more on economic considerations than on statistical ones and frequently must he made before

formulating the economic model.
- 20 -

year. Diiring the period June to September, however, allowance must be made

also for the effects on price of new-crop supplies of oats and harley and for

the expected size of the new corn crop. [See Foote (26).]

For some crops, the marketing season covers only a few weeks or months.

Here, naturally, the several variables used in the analysis must relate to

this period. At times, interest is centered on the prediction of sales or

prices during specific weeks or months. A study of this sort for apples was

made by Pubols (80). He considered factors that affect price in each month of

the fall-winter marketing season from September through May, using as variables in each, items that are known in advance of the month for which predictions are made. Foytik (39) made an analysis of the weekly price-demand

structure for California plums, and Harrington and Gislason (kS) have in

process a study of factors that affect daily sales of fruit from retail

stores. Shorter periods tend to be.more homogenous than longer ones but the

effect of this on the analysis may be offset, at least in part, by the fact

that irregular or nonmeasurable factors become more important. The period

chosen should be long enough to average out the effect of irregular or nonmeasurable factors and short enoxxgh to insure that a relatively homogeneous set

of factors are operating.

Years To Be Included

Now that enough data have accumulated since World War II to permit running analyses for the postwar years, we generally run analyses for (l) the

years between World War I and World War II, (2) the years following World

War II, and (3) the entire period. If the differences between the coefficients

for the three analyses are not statistically significant (see page 180), results for the entire period are used as the best predictor for the future,

since this analysis contains the largest nuniber of observations. Otherwise,

the second analysis normally is used as the best predictor for the futvire. For

some commodities, neither price ceilings nor rationing were in effect during

World War II and, unless the war is believed to have had other abnormal

effects on the demand and price structure, the war years can be retained in

the analysis.

At times, other years may be so abnormal, owing to special conditions,

that these also should be omitted from the analysis. For example, a linear

relation may hold except for \musually large or unusxially small crops. If,

because of considerations to be discussed later, a linear relation is desired

in the fitting process, data for these extreme years may be omitted from the

analysis or the actual data may be replaced by figures that are consistent

with a linear relation. 15/ Except in extreme cases, the analyst should

decide which years to include before he runs the analysis. Calciliated values

for these years should be obtained and checked to see that they are in line

with expectations.

15/ This procedure was used by Meinken (71, pp. 24-25) in deriving figures

on use for feed in his 6-equation model of the wheat economy. A scatter diagram between wheat fed and the price spread between wheat and corn indicated
- 21 •

For some studies, the basic structure of the analysis may have changed

from one period to another. Under these circumstances, the analysis might he

confined to the latest homogeneous period or, if interest exists in the nature

and extent of the structural change, "before" and "after" analyses can be run.

The model for dairy products described on page 12 relates to the period following World War II when margarine had become a major competitor of butter. In

this instance, analyses also were r\m for the pre-World War II (and pre-margarine) period to show how the increasing competition from margarine has

affected the entire demand and price structiire for the dairy industry. [See

Rojko (§!)•] If siifficient data were available for the period during which

frozen asparagus has been an important factor in total asparagus consumption,

only the model described on page 15 might have been fitted. As data on canned

asparagus are available for a considerably longer period, a model which allows

only for competition between fresh and total processed asparagus was fitted

also. [See Carstensen, j^. cit.]

Sometimes the nature of the structttral change is such that it can be

allowed for by basing the analysis on one homogeneous period and then algebraically modifying the model to take accoimt of the structxaral change. This

was done by Meinken (71) in the case of wheat. The model was fitted for the

marketing years beginning in 1921-29 and 1931-38. These were years for which

average prices and utilization for the entire marketing year are believed not

to have been affected significantly by Government action other than processing

taxes, tariffs, and export subsidies which are allowed for directly in the

model. In years for which domestic prices are maintained above their freemarket level by a Government price-support program, the system can be used to

estimate the use in each outlet and the probable accumulation of stocks by the

Government in the following way: (l) The domestic price, which otherwise woxild

be predicted from the system, is set at the expected season-average level

\mder the support program. (2) This price is used to determine the freemarket utilization in each of the four price-determined outlets. (3) When

these amounts are subtracted fron supplies available for use in these outlets,

the remainder is expected to show up as end-of-year stocks held by the

Government under the loan program. 16/

that, for the years used in the study, wheat fed in 3 years was higher than

indicated by a linear regression. Ihe apparently linear relationship for the

other years was extended to cover these years, and net exports were increased

by the difference between the actual amount of wheat fed and that read from

the linear regression. The value read from the line was substituted for the

actual amotxnt fed in fitting the 6-equation model and the adjusted data on

exports were used.

Exports were picked as the outlet on which a compensating adjustment should

be made because the elasticity with respect to price for exports was believed

to be of approximately the same magnitude as the elasticity with respect to

price for quantity, fed. Subsequent analysis confirmed this belief.

16/ Other ways of algebraically modifying Meinken's wheat model to allow for

specified changes in structure are discussed in Meinken (71, pp* -^9-50, 89-93)

and Foote and Weingarten (31) • These techniques are siammarized beginning on

page 189.
- 22 -

We referred earlier (footnote 3) to an analysis for peas by Shiaffett (8^)

in which a similar procedure was used. The statistical analysis was hased on

data for 1921-4l^ when frozen peas were of negligible importance. By 19^8-52^

consumption of frozen peas on a shelled-weight equivalent basis exceeded that

of fresh peas on the same basis and, during the last few years, fresh peas

have declined to a negligible part of the total consumption. Prices estimated

for 19ÍÍ-2-52, based on the 1921-^^1 relationships, differed from actual prices

by more than did the estimates in the earlier period and, from 19^6 to 1952

(the last year for which data were available at the time his study was published), actual prices were below the estimated price in all years, reflecting

increased competition from frozen peas during these years. The equivalent

consxamption of frozen peas was added to the supply variable for 1937-52^ and

new estimates of price were computed. Prior to 19^^^ when consumption of

frozen peas per capita first exceeded 0.5 pound, residuals for the two sets of

estimates were nearly equal on the average; for all years from 19^6 to 1952,

estimates based on the supply of fresh peas plus consumption of frozen peas

are closer to the actual price than are estimates based on supply of fresh

peas alone. From this, Sh\iffett concluded that fresh and frozen peas are

close substitutes and that the analysis based on prewar data can be used in

the postwar period by adding frozen consumption to the fresh supply before

obtaining the calculated price.

Sometimes analysts combine data from periods that are believed to be nonhomogeneous into a single analysis by using the so-called 0-1 variable. Such

a variable has a value of 0 in one period and a value of 1 in another period.

In a least squares analysis, the regression coefficient on the 0-1 variable

indicates the extent to which the dependent variable is larger or smaller in

the second period than in the first, after allowing for the net effect of all

of the factors specified in the analysis. This approach is satisfactory if

the only effect of the change in structure is to affect the level of the dependent variable and if the entire adjustment occurs within a single year., If

the change in structure affects the magnitude of the coefficients or the basic

nature of the relationships, or the change in structure occurs gradually over

time, use of a 0-1 variable is unsatisfactory. Changes that occur gradually

over time sometimes can be allowed for by use of a time trend as described on

pages 39 "to k-S*

In some analyses, deviations from one average for certain variables

appear relevant for some years and deviations from a different average are

relevant for other years. This appeared to be true in an analysis of factors

that affect mill consumption of cotton run by Lowenstein and Simon (65 ). One

of the independent variables represented the departure from normal in the

ratio of stocks to \mfilled orders. The average level of this ratio shifted

from about 1'in the period 1926-40 to about one-third in 19^7-52. Deviations

from each of these averages were used to represent deviations from normal in

the respective periods, but these deviations from the two periods then were

combined for use in a single analysis.
- 23 -

In working with least sqxiares analyses, we have a method that we can use

to test whether a structiaral change has taken place. If only two variables

are involved, or the change is "believed to affect the relationship only

between the dependent variable and one of the independent variables, say X2,

we follow this method: (l) Create a new variable X^* X2 takes on actual

values for the first period and zeros for the second period. X2 takes on zero

values for the first period and actual values of X2 for the second period.

(2) R\m an analysis using both X2 ^nd X2 and the other variables for the^

entire period. (3) Test whether the regression coefficients on X2 and X2 differ by a statistically significant amount by a one-tail test of the type

described on page 123. If the coefficients differ significantly, a change in

structure is assumed to have taken place, axiä. analyses should be run separately for each period. Presumably, a similar procedure could be used if

structural changes were believed to have taken place for the relationship of

the dependent variable to more than one of the independent variable. If this

were done, a multiple test on the regression coefficients would be made.

At times, the analyst may know that many changes in structure have taken

place but still wish to use a long series of years in the study. This was

true for coffee in an analysis mn by Hopp and Foote (¿0). They say, "Because

of known inaccuracies in the data on supply and disappearance and the many

extraneous factors that have affected the coffee economy, many years are required to obtain statistically-significant results for the more important

causal factors." The analysis was based on marketing years from 1882 through

1914.9^ omitting 1890-91, 191^1-17, and 19i|-0-lf6. The equation was constructed by

expressing each of the factors relating to supply as a ratio to an appropriate

factor relating to demand. For example, the variable relating to available

world stocks at the start of the marketing year was divided by average world

inrports or deliveries for the« preceding 5 years to allow for the long-term

gradiml increase in coffee consumption. They comment, "Thus the analysis became basically a relationship between two types of variables—(l) price

(dependent) and (2) supply deflated by demand—but because there is more than

one supply factor, a multiple regression equation is required. A time factor

also was included and found significant." Coefficients on two of the supply

factors differed from zero by a statistically significant amoxmt, and the

analysis, wheri adjusted for certain cyclical effects (see page 117)^ explained

prices satisfactorily during 1949-5ÍI-.

Use of Prices at the Local Market, Wholes^ale, or Retail Level

in Equations that Relate to Demand

In the section on economic models, we suggest that equations designed to

measure the elasticity of demand for consumer goods should be based preferably

on^prices at the retail level or, if these are not available, then at the

wholesale rather than the local market level. In measuring domestic demand

for items used to a large extent by industry, a wholesale price may be preferred to a retail price. In measuring the demand for livestock feeds or
- 2U -

fertilizer, a local market price is preferred. If a price at the desired

level is not available, it is not, in general, possible to measure the elasticity of demand at this level, but, as shown in the section beginning on page

100, it is possible, under relatively unrestrictive assumptions, to set a

lower or upper limit to the elasticity of demand at this level. 17/

If the available price relates to the local market or wholesale level and

an elasticity at retail is desired, then a lower limit to the elasticity of

demand at retail is given. If the available price relates to the wholesale or

retail level and an elasticity at the local market level is desired, then an

upper limit to the local market elasticity is given. In the discussion that

follows, we consider problems involved in obtaining a lower limit for the

elasticity at retail when only a wholesale or- local market price is available.

The reader shovdd remember that an upper limit for the elasticity at the local

market level is given if we consider a derived measurement at this level based

on retail or wholesale prices.

When marketing margins do not depend on the gixantity moving through the

marketT^If a wholesale or local market price is used in an equation designed

to set lower limits to the income and price elasticities of demand at retail,

we will generally, but not necessarily, wish to include a marketing cost

factor as a variable in the equation. The reason for this can be seen by

considering the following model. Let Qc ^® domestic consumption; D, disposable income; P;^, price at retail; P^, price at wholesale; and M, a factor

relating to marketing costs, such as industrial wage rates, and assume that

the following consumer demand and price level relations hold:

Qc = n + ^ll^r +,^12^ 4^^ (56)

pp « ag + bgiPv + "bggM. (57;

If data for P^. are not available, we can substitute eq-uation (57) for Pp in

equation (56) and obtain the following partially-reduced form or derived

demand equation: 18/

Qç = a3 + b3iPv + b32M + ^^330* (58)

Equation (58) replaces equations (56) and (57) in the statistical model. Inasmuch as M appears in equation (57), it must, if no further equations are

specified, appear in equation (58).

An estimate of bo^ can be used to set a lower limit to the value of bj^l

and, hence, to the elasticity of demand with respect to price at the retail

17/ Where the elasticity of demand is defined as a positive quantity.

Ï5/ A partially-reduced form equation is any equation that results from a

substitution of this sort. In this case, it is related to what Marshall (67)

calls a derived demand curve.
. 25 -

level. 19/ However^ when the quantity passing through the marketing sector

does not influence the difference between wholesale and retail prices^ b32

equals big and we can obtain an exact estimate of the elasticity of demand

with respect to income at retail from the corresponding coefficient in equation (58). The same reasoning applies to both coefficients if only local

market prices are available except that P^ is replaced by Pf.

In the case considered above it can be shown^ under Tonrestrictive assumptions, that the elasticity of demand with respect to the wholesale or local

market price is a lower limit to the elasticity of demand with respect to the

retail price; that is, the elasticity obtained from the coefficient b3i in

equation (58) is a lower limit to the elasticity that would have been obtained

from bxi in eqimtion (56) had data been available to estimate that eqimtion.

This proposition is intuitively obvious, since the existence of marketing

groups (processors, dealers, and so forth) between the producer and the consumer should tend to speed up adjustments in the market. This notion is

further elaborated and a proof given in the section beginning on page 103-

When marketing margins depend on the quantity moving through the market.

—If marketing margins themselves depend in part on the quantity moving

through the market, as is true for meats 20/ and may be true for many other

commodities, 21/ derivation of the reduced-form or derived demand relationship

becomes more complicated. Here, in addition to equations (56) and (57)^ we

may have an equation of the form

M = ai^ + 'bli.iQc + '^k2^ (59)

where W represents a factor similar to that represented by M in equation (57)-

W might be industrial wage rates, or it might be a weighted average of wage

rates and per unit transportation and container costs. If data are not available on retail prices, then our partially-reduced form equation must be

derived thro\agh the following steps:

(1) Substitute equation (59) for M in equation (57) to get

Pp = a^ -f b^iPw + ''^5^c "*" ^53"^- (^°^

(2) Substitute equation (60) for P^^ in equation (56) to get

Qc = ^6 + b6iPw + ^6^c + ^63^ + ^6i^D. (61)

19/ The reader may easily verify that it is impossible to obtain an estimate

of~ii in equation (56) from h^2,, b32/or b33 of equation (58). Hence, it is

impossible to estimate exactly the elasticity of demand with respect to price

at retail from estimates of the coefficients of equation (58).

20/ See Breimyer (ll, pp. 69I-693).

21/ See page lOT.
. 26 ^

(3) Transpose equation (6l) to give

(1 - T:>62)Qc = ^6 + ^6lPw + ^63^ + Hk^^ (6l.l)

(h) Rewrite equation (6l»l) as

Q^ = a^ + ^71^w "^ ^72^ * '^73^- (6l,2)

Comparison of equation (6l.2) with equation (58) indicates that they are

of exactly the same foim, as W in equation (61.2) is the same variable as M in

equation (58)- Although equations (61.2) and (58) are of the same form^ the

parameters in equation (6l,2) are combinations of the parameters in equations

(56)^ (5T)> and (59), whereas the parameters in equation (58) are combinations

of those in equations (56) and (57) only; hence, the meaning of the two equations (61.2) and (58) is different. 22/ Estimates of the coefficients in

equation (6l-2) cannot be used to derive exactly the elasticities of demand

with respect to price and income at retail, although they can be used to set

appropriate lower limits.

In contrast to the previous case, the income elasticity obtained from the

coefficient on income in equation (6l.2) is not the same that would be obtained from the corresponding coefficient in equation (56) if data w;ere

available to fit this equation. Instead, only a lower limit to this coefficient is given. A proof that the coefficient obtained is a lower limit is

given in the section beginning on page 106.

If dataware available on prices at the several marketing levels, we may

wish to include^in our model an equation like (59). If bi^^ does not differ

significantly from zero, then we may wish to assiome that M in equation (57)

can be replaced by a predetermined variable W. If bi|.i does differ significantly from zero and we adopt the framework developed above, our structural

model should contain equations like (56), (57), and (59)- Under such circumstances, equation (57) becomes an identity. Exact estimates of the elasticities of demand at the several market levels can, of course, be obtained»

A frequently used model for certain agricultural commodities is one that

contains an equation at the retail level that relates to consumption and an

equation that would be primarily applicable at the farm level that relates to

use for livestock feed. If data are not available to permit use of the respective price series in the respective equations, a partially-reduced form

equation can be used for one of the two consiitnption levels. Availability of

data on prices in general will determine which consumption equation is expressed in a partially-reduced form. If price data at a wholesale level are

used, a partially-reduced form equation might be needed in each case, with a

variable added in each equation that relates to marketing costs to and from

the wholesale level, respectively.

22/ In the section beginning on page 100 we develop a simpler model, which

does not require such involved substitutions to reach the reduced form equation.
. 27 -

If different sorts of structural equations are assumed and partiallyreduced form equations are to be used, the research worker should make an

algebraic analysis of the sort given in preceding paragraphs to check on the

exact form of the partially-reduced form equation and the variables involved.

A discussion of methods that can be used to obtain a derived elasticity

of demand for use in partic\ilar outlets at the local market or farm level or

of the elasticity of total demand in all outlets at the farm level is given in

the section beginning on page 108.

Choice and Handling of Variables That Cause

Shifts in the Demand Curve

In order to isolate price-quantity relationships or demand curves by statistical means, variables that cause such curves to shift back and forth must

be included in the analysis. In equations that relate to the measurement of

consumer demand from time series data, it is convenient to divide such variables into four categories: Those that relate to (l) consumer income or other

measures of the general level of demand on a national basis, (2) the general

price level, (3) supplies or prices of competing products, and (k) population.

Each of these is mentioned briefly in the paragraphs that follow. In certain

demand equations, other types of variables are included. Comments on some of

these are given at the end of the section.

Variables that relate to the general lev^l of demand.—Personal disposable income commonly is used to represent the general level of demand in equations that relate to consumer goods. For products that are used chiefly as

industrial raw materials, the Federal Reserve Board index of industrial production might be used instead. Friedman (^) suggests that instead of

responding to their current income, consxmaers respond to their long-run

expectations of income. He shows a derived series on a national aggregate

basis which he feels represents consumer expectations of income; experiments

currently are underway within the Agricultural Marketing Service to learn

whether use of this series resijilts in improved analyses.

Allowing for effects of changes in the general price level.—If we have

reason to believe that a doubling of all price and income variables has no

effect on consimiption, effects of the general price level should be allowed

for by deflation, that is, by dividing each price, income, or marketing margin

variable by a variable such as the Bureau of Labor Statistics Consumers* Price

Index. This seems a reasonable assumption with respect to most perishable

items and perhaps some of semidurable or durable goods. Changes in the value

of money may affect the nature of the demand for certain items that involve

large initial expenditures, such as automobiles or refrigerators, owing to the

effect of such changes on the relative value of fixed assets or liabilities.

In such cases we might (l) allow for changes in the price level by including

the Consumers * Price Index as a separate variable or (2) deflate and include

variables that relate to fixed net assets in the equation. In the section
- 28 .

"beginning on page 111^ a discussion is given of the use of distributed lags to

study demand for items that require substantial initial expenditures by consumers •

If our price variables relate to a level other than at retail^ some

research workers might feel that the Consumers* Price Index is not a satisfactory series to use as an indicator of the general price level when dealing

with analyses of consumer demand. However, if a variable relating to marketing costs is included in the analysis, then we can show that the Consimiers*

Price Index is an appropriate series to use to represent the general price

level by a technique similar to that used on page 24.

When we deflate by an index number that contains the price of the commodity with which the analysis is concerned, we tend statistically to bias the

regression coefficients downward. A similar tendency holds when such an index

nxmiber is included as a separate variable. This has led some analysts to construct special index nimibers that eliminate the price of the commodity or

commodities under study and to use these to deflate the price variables that

appear in each analysis. This is theoretically desirable; whether the extent

of the bias is sufficient to warrant the work involved is for the analyst to

decide. The unadjusted measure of the general price level, however, should be

used to deflate variables that relate to consi;imer income, and for all variables when studying equations that relate to derived demand.

Allowing for competing or complementary products.--Methods for fitting

demand equations that involve competing or complementaiy products are discussed in detail beginning on page 87. Systems of equations frequently are

required to measure the direct and cross elasticities in such cases.

Use of per capita data.—To avoid confusing the time trend for population

with one that might reflect other effects, per capita data probably should be

used whenever applicable.

As the proportion of people of various ages in the population differs

from time to time, specially weighted population aggregates may be needed for

certain items. For cigarettes, for example, cons\jmption varies by age groups

and differently for people in urban or rural areas, and for men or women. [See

Sackrin and Conover (83)*] In analyzing time series data, a carefully constructed population aggregate is required to avoid confusing trends in

consumption with trends in the proportion of men and women in rural and urban

areas and in specified age groups. A similar, thoi;igh possibly less serious,

problem exists for textile fibers, hide and leather products, and certain

foods. Research currently underway in the Agricultural Marketing Service

deals with the development of a population series adjusted for age composition

by sex, to be known as clothing expenditure units, based on survey information

covering I6 age breakdowns. If consimiption differs greatly among income

groups, weighted pop\ilation aggregates that allow for shifts in the proportion

of people in the various income groups may improve results from the analysis

of time series data.
- 29 -

Measuring changes in demand for export or storage.—Attempts to develop

satisfactory shift variables for use in equations that relate to storage or

export have been less successful than others^ though some limited progress has

been made. An equation relating to exports is included in the model for wheat

described on page 11^ and equations that relate to storage are included in

that model and in the one for eggs described on page 1^.

Brandow (lO^ pp. 7-8)^ in a study relating to the demand for apples^ comments, ^'Export demand could not be satisfactorily represented by actual

exports because normally they reflect both foreign countries* willingness to

^buy and prices and production in the United States. To isolate the former for

the years before the war, a regression of fresh apple exports on production

was computed and differences between actual and * expected' exports were taken

as a measure of changes in export demand. For the postwar period, when exports were much lower and in some years considerably influenced by shortages

of dollar exchange or by U. S. government export subsidies, simple deviations

from average net exports were used.^'

In some analyses, measures of consumer income or the general price level

in one or more foreign countries may be- used to measure export demand, just as

similar variables are used with respect to the domestic market; at other times

the level of stocks or supply in the foreign market may be used as a factor

causing the demand for domestic products to shift.

Use of Actual Data or of First Differences

Whenvariable^are^^exgr^s^

^BXj ^m^'W^^^XMt7,w^, ^%¥.^^^^^ If we are primarily

interested in the percentage change from one year to the next, we may express

the change in this way or, as is more common, use is made of the first differences of logarithms. First difference analyses also may be run in terms of

the actual year-to-year change in the variables. In some analyses, certain

variables are expressed as first differences and other variables are used in

an actual, or non-first difference, form.

First-difference equations have been used to some extent by price

analysts for many years, but they have come into prominence only recently.

In equations designed primarily for use in forecasting, first differences are

used when emphasis is placed on measuring the factors that affect year-to-year

change rather than on deviations from a long-term average. The recent

interest in the use of first-difference equations has arisen partly because

of the obvious inapplicability of a pre-World War II average to the postwar

period. This reason generally is of less importance when we work with deflated data than when actual data are used. First differences may be hard to

use in connection with long-range forecasts and in some types of analyses of

relative benefits to be obtained from specified Government programs, as the

forecast is in terms of a change in level from a preceding year. In certain

cases, first differences might be used to derive statistical coefficients that
- 30 -

are assumed to apply in future to non-first difference emulations; such would

iDe the case if the statistical conditions^ described in the next paragraph^ in

the period for which data are available are consistent with the use of first

differences^ hut in the period for which forecasts are made are believed to be

more nearly consistent with the use of actual data.

From a statistical standpoint^ first differences should be used in preference to actioal data when the successive imexplained residuals from singleequation analyses based on actual data are almost perfectly serially correlated with a positive sign, A transformation to first differences will

eliminate most of the serial correlation in the residuals if the analysis is

rer\Jin in terms of the transformed variables, g^^serlal or auto-correlation

we mean ç9ji,^,atially-4hA^.,ao^^^ a series of observations, and the

same'series lagged by,.,<;3Ae, or.more units of time. If the unexplained residual

in one year on the average equals a fixed proportion of the imexplain'fed

residual in the preceding year plus a random variable^ resulting in some

degree of positive serial correlation, then a transformation to first differences may remove some of the serial correlation in the residuals. If the

serial correlation is less than O.5 or negative, a conversion to first

differences tends to make the degree of serial correlation in the residuals

greater in the transformed than in the original analysis, and first differences should not be used.

Many analyses that relate to economic data tend to give serial correlations in the imexplained residuals that are positive when the analysis is

based on actual data. The following considerations have a bearing on the

extent to which this is likely to be true. The unexplained residuals may

represent essentially random errors in the data 23/; if so, we would expect

the serial correlation in the residuals to be close to zero. Unexplained

residuals also represent the influence of variables excluded from the equation

because (l) we have no data with which they can be measured or (2) the influence of each, on the average, is believed to be too small to warrant their

inclusion. If we have an idea as to the nature of these excluded variables,

we may be able to make assumptions as to whether, on the average, their combined effect on the dependent variable is likely to be similar from one time

period to the next. If it is, we woxild expect positive serial correlation in

the residuals; if not, we would expect the serial correlation in the residuals

to be small. Thus, our final appraisal as to the likelihood of positive

serial correlation in the residuals depends on (l) the extent to which the

residuals reflect errors in the data and the extent to which they reflect

omitted variables and (2) the nature of these omitted variables. Another

possible cause of serial correlation in the residuals is an incorrect specification of the form of the relation, particularly if one or more of the independent variables tends to follow a time trend.

23/ Errors in the data in most statistical analyses, if allowed for at all,

are assumed to relate only to the dependent variable. See page 1^3.
- 31 -

If^ when working with actual data, we find positive serial correlation in

the residuals and our appraisal suggests the likelihood of some positive

serial correlation but not a serial correlation coefficient of one when "based

on a time lag of a single year, we might find it desirable to make a first

difference transformation. We can then determine by means of the DurbinWatson test 2^/ whether we have reduced the serial correlation to a point

where it doe~not differ from zero by a statistically significant amount* If

so, we might assume that the transformation is satisfactory and that the usual

tests of significance apply to the regression coefficients obtained.

Our interest in serial correlation stems from the fact that most of the

statistical methods described in this handbook are based on the assumption of

mutual independence among the successive unexplained residuals. As discussed

in detail in the section beginning on page 57^ if we are interested in estimating the coefficients of a structural equation that contains only a single

endogenous variable, so that the least squares approach is the recommended

fitting procedure, then the least squares estimates of the coefficients are

both statistically unbiased and consistent, 25/ This will be the case even if

the independent variables and the true residuals of the equation are serially

correlated [see Wold (103) and Wold and Jureen (106, pp, 208-213)]. However,

the usual tests of significance of the regression coefficients do not apply,

and other types of estimators might give coefficients with a smaller variance

[see Watson (97)^ Watson and Hannan (98), and Gurland (hk)] if we knew how to

compute them. Wold (103^ pp. 283-284y"gives a large sample test of significance for the least squares estimates, but we rarely deal with a large sample

in economic research.

Cochrane and Orcutt (l¿, pp. 5^-55) suggest that the use of first differences, rather than the original data, tends .to eliminate most of the serial

correlation in the residuals of economic relationships. If this is the case,

ordinary least squares estimates of the coefficients based on the use of first

differences are statistically efficient (see page 58) and the usual tests of

significance apply. However, the validity of the Cochrane-Orcutt approach

rests on very strict assumptions concerning the form of the serial correlation

in the residuals (see page 167). Wold (103) and Gurland (hk) point out that

only rarely are these assumptions likely to apply. Relaxation of the assumptions even a little leads to great computational complexity. Thus statisticians wish to assume mutual independence in the residuals for two reasons:

(1) If they don't, they must make specific assumptions about the form of the

nonindependence in order to obtain valid standard errors and estimates of the

2k/ This test for serial correlation in the calculated residuals of a least

squares regression is described on page 173* As pointed out by Cochrane and

Orcutt (15, p. 45)^ a test of this sort, rather than a conventional test of

whether the (serial) correlation coefficient differs significantly from zero,

is required because the calculated residuals of any least squares regression

tend to be biased towards randomness.

25/ These terms are defined on pages 57-58.
- 32 -

coefficients that are statistically efficient^ and this is difficult in

applied research; and (2) even with the simplest forms of nonindependence^

computations to obtain such coefficients and their standard errors "become

relatively complex.

Based on an experimental study by Orcutt and Cochrane (78)^ the situation

appears to be even more complex for systeds of equations. Here we are concerned not only with the degree of serial correlation in the residuals in a

single equation but also the degree of correlation among the residuals in the

several structural equations. If there is no serial correlation in the residuals^ correlation among the residuals in the several equations presents no

problem^ as the method of fitting used with respect to such equations automatically allows for this. However, when both types of correlation prevail, a

shift to first differences appears to be of less value with respect to systems

of equations than when we deal with a single equation. Orcutt and Cochrane

(78^ P» 356)^ who also consider the necessity of working with relatively small

samples in applied research, conclude, "Unless it is possible to specify something about the intercorrelation of the error tenus in a set of relations and

to choose approximately the correct autoregressive transformations, a certain

amount of scepticism is justified concerning the possibility of estimating

structural parameters from aggregative time series of only twenty observations . "

We now turn to some considerations of the use of first differences that

relate more directly to applied research. In essence, however, they are

special applications of the statistical considerations discussed in preceding

paragraphs.

In some cases, strong trend factors tend to overshadow the effect of

economic variables, but the trend factors cannot be allowed for directly in

the analysis due to a lack of data or knowledge. If the trend factors can be

allowed for by a time trend (see page 39)^ this might be preferred to a shift

to first differences, but if this does not appear feasible, a study based on

year-to-year changes may give meaningful results whereas one based on actual

data may result in multiple correlations close to zero. This has been found

true, for example, in certain studies that relate to factors that affect milk

production per cow and to consumption of tobacco products.

Use of first differences has been advocated in the past to avoid obtaining an unduly high correlation between two or more variables when each of them

is more closely correlated with an unrealized third factor than with each

other. Here it is true that, in certain cases, improved results are obtained

by the use of first differences, but more reliable results are obtained when

the third variable is included in the analysis and its effects are allowed for

by statistical means, either directly or by its use as a deflator.

At times, available data may not be strictly comparable from one year to

the next, but the comparability is greater from year to year than over longer

periods of time. This is true of some statistics that relate to commercial
- 33 -

vegetables^ for example. Here the niamber of States included in the total has

heen increased gradually over time. Under such circimstances^ use of first

differences generally improves the reliability of the results.

Meastiring Relationships Among Prices Under

Alternative Formulations

Armore (¿, pp. 52-55) ^as concerned with measuring price relationships

among certain fats and oils or^ more correctly^ with demonstrating from market

data that the price of cottonseed oil is more closely related to the price of

certain fats and oils used in food products than to the price of lard^ and

that it bears little statistical relationship to the price of butter. As indicated in his bulletin^ he had strong theoretical and empirical reasons for

believing this to be true.

Results from a number of alternative analyses of these data are given in

table 1^ but only those based on first differences of undeflated data were

included in the bulletin by Armore. In each case^ the analysis is based on

data for the years 1922-40. Correlations shown in the first three columns are

those that are directly relevant to the price relationships among the several

fats and oils. Within each row, the expected ranking is found, but the magnitude of the coefficients differs greatly from row to row. Correlations shown

in the third row are those that would have been used had he relied on first

differences to remove the common effect of the general price level on prices

of cottonseed oil and the related item.

Lower correlations for butter are obtained when the effects of the

general price level on each series are removed statistically by the use of

partial relationships, regardless of whether first differences or actual data

are used. (See second and fourth rows of table 1.) For lard and the miscellaneous group of fats and oils, however, the correlation is higher when actual

data are used and the effects of the general price level are removed by the

use of partial relationships than when first differences alone are used, and

much higher than when the effects of the general price level are removed by

the use of partial relationships in the analysis based on first differences.

Simple correlations for the analyses based on deflated data (see last two

rows) in each case are higher than the partial correlations for the corresponding analyses based on imdeflated data, and in each case are higher when

based on actual data than when based on first differences.

Statistical problems involved in attempting to decide whether to deflate

by the general price level or to use the general price level as a separate

variable differ for analyses of this type from those involved in reaching a

similar decision when working with demand equations. Our economic theory with

respect to demand equations applies only in terms of relative prices, whereas

we may be interested in price relationships that prevail among the undeflated

variables when considering relations among prices per se. In order to determine whether a real correlation exists among prices of the individual commodities, we must correct each in some manner for the effects of the general price
- 3h -

Table 1.—Cottonseed oil: Relation of wholesale price to that of specified

other items, 1922-40 l/

Correlation between

cottonseed oil and

related item

Related item

Other fats

and oils

used in

food k/

All

commodities

5/

Based on actual data:

Simple

Partial 6/ .'

Based on first differences:

Simple

Partial 6/

Based on deflated data

when using—

Actual values .•

First differences

0.83

I/. 15

7/-34

2/8/

• 52

0.93

.75

.66

l/.hl

.88

• 83

0.97

.88

.80

.66

• 9^

.88

0.88

.k6

l/ When all prices are in cents per pound. Cottonseed oil is crude, tanks,

southeastern mills. Data on which these analyses are based are given in

Armore (¿, p. 54),

2/ 92-score, creamery. New York.

3/ Prime steam, loose, Chicago.

%/ Average price of coconut, corn, oleo, palm, peanut and soybean oils,

oleostearine, and edible tallow weighted by their average domestic disappearance in 1931-^0.

5/ Bureau of Labor Statistics index, 1935-39 = 100.

%/ After allowing for the effect of prices of all commodities.

7/ Does not differ from zero by a statistically significant amount.

"5/ Less than O.OO5.

level, but we cannot say, without some thought, whether this should be done by

deflation or by the inclusion of the general price level as a separate variable .

Kuh and Meyer (62) show that if each of the variables to be deflated is a

linear homogeneous function of the deflating variable, then (l) the simple

correlation between the deflated series equals the partial correlation between

the \jndeflated series with the deflator held constant and (2) if we study

relationships among prices of only two commodities, the simple correlation

between the deflated variables equals the multiple correlation based on the

deflator and the two -undeflated prices.
- 35 -

When we consider linear relationships between two variables, a linear

homogeneous fiinction is one for which one variable is a constant proportion of

the second variable except perhaps for a random error term, that is, the regression line passes through the zero intercept. In relation to the data

shown in table 1, the findings of Kuh and Meyer suggest that if the price of

cottonseed oil and the price of butter, for example, each are linear homogeneous functions of the general, price level, then we would expect the correlation

coefficient in the fifth row of the first column to equal that in the second

row, and the correlation in the sixth row to equal that in the fourth row.

These coefficients obviously are not equal, a fact which suggests that a

linear homogeneous function does not prevail between each of the individual

prices and the general price level. Furthermore, we have no reason to expect

that this function is one of constant proportionality. In this case it

appears likely that, if we are interested in relationships among the \mdeflated variables, we should obtain the partial correlation coefficients rather

than using the computationally-simpler technique of getting the simple correlation between the deflated variables. This is what Armore did in his study.

On the other hand, if we are working with a system of equations that involves

deflated price variables in certain demand equations and wish to measure

relations among these deflated variables, we would use the deflated variables

directly in our analysis. Considerations involved in deciding whether to rxm

the analysis in first differences or based on actual data are those discussed

on pages 29-33-

Kuh and Meyer give another criterion that may be used in deciding whether

to deflate or to include the deflator as a separate variable in the analysis

in cases of this sort. This depends on whether the -unexplained residuals are

more nearly unifonn over the range of the independent variable when working

with deflated or undeflated data. All of the statistical methods considered

in this handbook are designed to be used with data for which the variance of

the unexplained residuals about the regression line is uniform over this

range. When this variance is uniform, the residioals are said to be homoscedastic. Sometimes the degree of homoscedasticity in the residuals can be

increased by deflation. If this appears to be true, and if each of the variables to be deflated appears to be a linear homogeneous function of the

deflating variable, then we should deflate. If the variables that might be

deflated are not linear homogeneous functions of the deflating variable, and

the residuals are expected to be heteroscedastic unless we deflate, we might

use a logarithmic or some other transformation to increase the degree of

homoscedasticity. If the residuals are expected to be homoscedastic when the

variables are used in their undeflated form, then the deflator should be

included as a separate variable in the analysis. If interest is centered

primarily on the regression rather than the correlation coefficients, Kuh and

Meyer suggest that, even if the zero-intercept (or linear homogeneous function) concept is not met, deflation or some similar transformation may often

be appropriate. They conclude (62, p. 4l3) as follows: "The discussion and

the cited illustrations point up the fact that each application is likely to

have its own special characteristics and that careful consideration must be

given to a wide range of alternatives if appropriate estimates are to be

obtained."
- 36 -

Deriving Estimates of Quantity or Price

From Value Aggregates 26/^

A somewhat different problem, though one often identified with that of

deflation, concerns the isolation of changes in quantity from data representing value through division of the latter hy an appropriate price series. 2?/

Obviously, for any individual commodity, the division of value by price should

exactly represent quantity. This generally is not true when we divide an

aggregate value series by a price index with fixed weights, the usual procedure. But we may be able to regard changes in the derived series as

reasonably approximating changes in quantities. For example, consumer expenditures for clothing in constant prices are often derived by dividing

dollar expenditures by the Consumers* Price Index for Apparel Products (19^7-

^9 = 100). The result presimiably shows the expenditure for clothing in terais

of average 19^7-^9 prices. With prices thus held constant, changes in the

derived series are assimaed to represent changes in quantities purchased. The

same procedure frequently is used to estimate real or unit changes in income,

retail sales, wholesale inventories, and so forth.

The reliability of the estimate of changes in quantities obtained by the

foregoing procedure depends primarily on the composition and internal stability of the value series in relation to the particiilar price index employed.

Changes in value may reflect shifts between the various products included in

the value aggregate and not a change in total quantity or price. If such

shifts are important, serious errors may arise from using a price index

incorporating weights that remain fixed over relatively long periods of time.

In addition, the weights used in the price index may not necessarily be related to the relative importance of the products in the value aggregate even

in the base period. To the extent that they differ, the estimate of quantity

differs from the actual quantity. However, if the relative importance of the

products in the value aggregate remains relatively stable over time, changes

in the derived quantities should approximate changes in actual quantities. A

somewhat similar consideration applies if the prices in the two series differ.

In some instances, the value aggregate may be divided into several parts.

Given appropriate price indexes, estimates of changes in quantity may be obtained for each component. If combination of the breakdowns from a quantitative standpoint is feasible, that is, if the commodities to be combined can be

expressed in common quantitative terms, summing the estimates for the sub-#

groups will probably give a better estimate of changes in total quantity than

deriving the total estimate directly. In any event, examination of the

results at less aggregative levels, where possible, should assist the analyst

in evaluating the total estimate.

26/ This section was prepared by Martin S. Simon and Carroll Downey,

Agricultural Economic Statisticians, Agricultural Marketing Service.

27/ Or, conversely, estimating changes in price from value data through

division by a quantity series.
- 37 -

Use of Arithmetic or Logarithmic Variables and Other

Transformations that Relate to Functional Form

Linear (arithmetic) and logarithmic equations are the principal f\mctional forms used in economic analysis. Linear equations give results which,

when translated into total value-supply curves, make more economic sense at

the extremes than do the results obtained from logarithmic equations. This is

true because the total value drops to zero when supply is zero and when supplies are veiy large. In most cases, however, no data are available for these

extreme values; therefore we have little interest in them. In many analyses

that have been run within the Agricultural Marketing Service based on undeflated data, logarithmic equations appear to fit the data better than do

arithmetic equations. When projections are made from a pre-World War II

period into a post-World War II period this is particularly evident. Whether

this would hold if the analyses were based on deflated data is not known.

Logarithmic equations have the mechanical advantage of yielding curves that

have a constant elasticity, but this is not a valid criterion for deciding on

their use.

I^rom a statistical viewpoint, logai'ithmic equations shg^Xd. be iised when

(IJL^the relationships between the variables^ axe^beli^^

rather than addi€Iveri^yj^£rilât are believed to be mpre stable ija percentagre than in Q'^soJLEË£^'££îB^ S^A, ¿llTthe^un are belieyed

"^QL^-^^^Q^^ uniform over the "range of^^ti^^Tnïegende^ expressed

in^piercentage rather than absolut'^^^^^ some exTent, These"*items are

different^as^c1Er''or"^''t15e''sam^^^^ two conditions are more likely

to hold for analyses based on undeflated data than for those based on deflated

data, although they might hold in either instance.

Ths use of logarithmic equations to handle relationships that are believed to be multiplicative can be illustrated by the following example.

Assume that we use a logarithmic relationship of the following form:

log X = log a + b log Y + c log Z. (62)

When this is translated into natural nimibers, the equation becomes:

X = aYV. (62.1)

Thus, whenever we use an equation in which all of the variables are converted

to logarithms, we implicitly assume that the relationship among the variables

when expressed in natural numbers is of the form shown in equation (62.1).

Frequently, in economic problems, a careful consideration of the variables

involved in a particular equation will give strong reasons for expecting the

relationships to be either additive or multiplicative. Particularly in those

cases where the underlying relationships are believed to be additive, use of

logarithmic curves to yield the mechanical convenience of constant elasticity
- 38 -

should be avoided. 28/ Methods for computing the elasticity for each point

on a curve are discussed in the section "beginning on page T8,

In certain analyses^ the effects of some variables on the dependent variable are additive, whereas the effects of other variables are multiplicative.

A study of factors that affect prices of corn from June to September by Foote

(26, pp. 30-^1) is of this nature. Four components of supply were included in

the initial formulation; these should be treated as additive variables, but

the nature of the variables was such as to require that their respective

weights be determined by regression analysis. However, the combined supply

and the two demand shifters included in the analysis were believed to affect

the dependent variable simultaneously, in a multiplicative way. A method by

which the several regression coefficients can be estimated through an iterative approach in analyses of this sort is discussed in the section beginning

on page 122. In other studies, the variables in each of several demand equations may be related to utilization in a multiplicative way but the several

utilizations may be involved in an additive identity. Such is the case with

the economic model for dairy products described on page 12. This situation

can be handled by using a semi-logarithmic statistical equation; that is, by

expressing utilization in actual nimibers, and the other variables in logarithms. Relations that have multiplicative aspects can be obtained, of

coTirse, in ways other than by use of logarithms, as by the use of crossproduct terms as additional variables. An equation of this form might be the

following:

X = ai + bi2Y + bi3Z + b^^YZ. (63)

1

If Y and Z are each predetermined variables, the variable YZ is computed for

each observation, designated as an additional variable, and the entire

analysis is treated as a linear regression with three independent variables.

In this equation, the effect of Y and Z as such each enters the equation in an

additive way but their "joint" or multiplicative effect is brought in as an

additional variable. If either Y or Z is endogenous within a system of equations, then the method described in the section beginning on page 71 for

handling the multiplicative term is preferred.

28/ Wold and Jureen (106^ pp. 258-259) point out, "the demand for food has

an income elasticity that presents a clear tendency to decrease as income increases. Such tendencies must always be kept in mind when dealing with demand

elasticities that are estimated on the hypothesis of constancy. ... Constant

elasticities should in general be interpreted as average values, and in principle they will be valid only for the range of income covered by the data

employed. Constant elasticities should, accordingly, not be given without

indicating the range to which they refer, and it is necessary in any case to

indicate the average income level in the sample." A similar situation probably exists, in some instances, when constant elasticity curves are used to

measure the elasticity of demand with respect to price from time series data.
- 39 -

Sometimes economic relations are of such character as to require curves

of a very different sort* In a study of smoking habits based on survey data

by Sackrin (82, pp. 2-6), the quantity of cigarettes consumed was believed to

be related t~age and income, but the age relationship was of such a nature

that a peak was reached at middle age. A'satisfactory equation was obtained

by using a relationship of the following form, where Q represents consumption,

D represents income, A represents age, and A represents the age at which

smoking is believed to reach a peak:

Q = ^1 + ^12^ + "^ISCA - A) + b-Li^(A - A)2. (64)

When h-^o is zero or positive and b]_4 is negative, the desired inverted parabola on age is obtained.

In general, the type of functional form to be used should be decided

before the analysis is run. Except in a few special cases, usually involving

thousands of observations, the form of the equation cannot be determined from

the data. [See Foote (28, pp. 1-2).]

Problems Related to the Use of Time as a Variable

Time frequently is introduced as a variable into an analysis as a measure

of sources of corrnTñüoüs Ty'stenlETic variation for which no data are available,

if s'^^""¥oïïrcës'^are believed to be important before the analysis is run and lí

thejfcime.. effect is believed to be linear, or curvilinear to only a moderate

degree, then a'time^variât)ïé probably should Ibe" incîu^^ the initial analysis. If its partial regression coefficient fails "to differ significantiy'from

zero, it may be omitted. However, if the time effect is strongly curvilinear

—for example, U-shaped—then a nonsignificant coefficient may be obtained

which is, in a sense, misleading. Thus, if while formulating the model the

research worker is uncertain as to the importance of unmeasured factors that

change systematically over time, or of the nature of their effect, time probably should be omitted from the initial analysis, but the imexplained

residuals should be plotted against time to determine whether they exhibit a

nonrandom pattern. If they do, an attempt should be made to discover an

economic cause for the pattern.

Sometimes an additiprial_v§,i:j,able can be, found that is related to the unexplained residbügi^IliPh^^^^^^

tfîe'TesJjduals graph!caíly"2a¿Snst_t^ . Unle§js. the..<new^vajcÍB.ble

is correlated To a considerable degree with'the"^

iñ jgHe "n^^ analy'ïïl^^^^-^e^avap^ a good ijidication of the

partial correlation_on the new variable. If the new'y^^^ to be a

promising one,'"lîSe ajJí^SIiTlf^SSSSYIH^^^ "^^

Q^BtglTT'^the^ true multiple and partial Veli.tionships. If no such variable can

bé^'found, but an explanaTioT'oF'^'tñ'e^^Tíme^ of technological or

institutional developments is available, then a time variable should be introduced into the analysis through the use of an appropriate graphic or mathematical relationship.
- 4o -

Sometimes the imexplained residuals deviate from a random pattern for

only part of the period. If a satisfactoiy economic reason can be foxmd for

the deviation, the analysis can be adjusted by use of charts that indicate the

partial relationships between the dependent variable and each of the independent variables. (Methods for constructing such charts are given in the section

beginning on page 205 and an example of their use is shown in the section beginning on page 17^.) The following steps are used: (l) The unexplained

residuals for the years involved are plotted against time, and a free-hand

trend is drawn through them that*coincides with the assumed effect of the

explanatory cause. (2) Deviations from this trend are substituted for the

original deviations on each of the partial charts. (3) If a change in the

slope or general shape of the relation is suggested, and this appears to be in

line with expectations based on economic theory and knowledge about the item,

a graphic adjustment in the relation is made, (k) An appropriate adjustment

is made in the various mathematical coefficients obtained from the original

analysis.

Handling of time in logarithmic equations.—In analyses for which the

other variables are converted to logarithms, time (t) frequently is used

either in a converted or an unconverted form. When it is converted, the first

year should be assigned the nimiber 1, aç the logarithm of zero is undefined.

The following equations illustrate the effect of these alternative treatments:

(1) Use of converted data:

log Y = log a + b log X + c log t (65)

Y = axH^. (65-1)

Here c normally is close to zero and may be positive or negative. As t increases, we raise a progressively larger nimiber to a certain power.

(2) Use of \mconverted data:

log Y = log a + b log X + (log c) t {66)

Y = aX^c^. (66.1)

Here c (the antilog of log c) normally is close to 1 and always is positive,

but log c may be positive or negative and noiTnally is close to zero. As t

increases, we raise c (a constant) to a progressively larger power. Results

of the two approaches are shown in table 2 and in figure k.

In effect, these alternative approaches give (l) a power function in t

and (2) an exponential function in t. The power function increases or decreases more rapidly for small values of t than does the exponential fimction,

but it soon becomes nearly flat if c is less than unity. The exponential

fiinction never declines below zero, and hence for negative time trends becomes

flatter as t becomes very large; but when it is used for an increasing time

trend, it increases more rapidly as t increases. The typical time trend is
- lu -

ALTERNATIVE POWER AND

EXPONENTIAL FUNCTIONS OF TIME

Section A. Power functions obtained when t is

converted to logarithms**

Dependent

variable

When c is greater than unity

-•—When c equals unity

When c is positive but

less than unity

When c is negative

Time - t

Section B. Exponential functions obtained when

t is not converted***

when c is greater than

unity

When c is less than unity

Time - t

* For specified values of coefficient c as shown in text equations.

** This function is undefined for t = 0 and has little relevance for t <1.

*** With this function, c cannot be negative.

U. S. DEPARTMENT OF AGRICULTURE NEC. 4431-57(8) AGRICULTURAL MARKETING SERVICE

Figure ii-.—When analyses are run in terms of logarithms and time is included as

a variable, time may be used in a logarithmic or non-logarithmic form. Conversion of time to logarithms gives a greater variety of possibilities, as

shown in section A, and is recommended in most cases.
. 1^2

one that increases rapidly during a period of grovth and then tends to flatten

out. Thus the power function, which requires that t he converted to logarithms, appears more logical for most prohlems. The vide variety of alternatives permitted hy its use is a further advantage.

Data in table 2 illustrate an important aspect of the use of time as a

variable. The coefficients have been chosen so that the two curves are

similar for t in the neighborhood of 1 to 15 at the lower and higher values of

c respectively. But for extrapolations beyond that, the two curves differ

markedly. Extrapolation always is dangerous—with time as a variable it is

particularly so. And it is most dangerous when a polynomial involving t is

used, or when an exponential curve is involved for which the c coefficient is

greater than 1.

è

Table 2.--Results of using time converted to logarithms or not converted in

analyses for which the other variables are converted to logarithms

Contribution of the time trend to the estimated

value of the dependent variable i(ihen t is—

t ;

Converted to logarithms Not converted to logarithms

and c equals— and c eqv&ls—

-0.1 0.1 0.98 : 1.02

0 " 1.00 1.00

1 ! 1.00 1.00 .98 1.02

2 • 93 1.07 .96 1.04

5 : .85 1.17 .90 1.10

10 : .79 1.26 .82 1.22

15 •• : .76 1.31 .7^^ 1.35

20 ! .7^^ 1.35 .67 l.k9

30 : .71 1.41 .55 1.81

ko : .69 1.1^5 M 2.21

50 : .68 1.1^8 .36 2.69

75 : .65 1.51^ .22 k.k2

100 : .63 1.59 .13 7.25

When t is not converted, we can get the value of c by obtaining the antilogarithm of the coefficient attached to t in equation (66). 29/ Each additional year has the effect of raising this coefficient to one higher power.

29/ Mechanically, this can be done most easily by getting 100c by taking the

antilogarithm of the coefficient plus 2.
- k3 -

Thus^ in the cases used as examples in table 2, we can say that time in effect

either raises or lowers the. value of Y by roughly 2 percent each year. When

the logarithm of t is used, however, no such statement can be made because the

percentage effect of a 1-unit increase in t becomes less as t becomes larger.

This fact has led some analysts to use t in an unconverted form, but they may

have failed to consider the effect of this on the general shape of the curve.

Time trends in relation to first difference analyses.—If first differences are used, the constant value in the equation represents the linear

effect of time as, if different from zero, it implies that some change in the

dependent variable would occur from the preceding year even if there were no

changes in the independent variables. In non-first difference analyses, the

constant term has little economic meaning and hence a standard error for it is

seldom obtained. When working with first differences, however, the analyst in

general will wish to determine whether this coefficient differs significantly

from zero. A formula for the standard error of a function for single-equation

analyses based on any number of variables is given in Friedman and Foote (^0,

pp. 17-19), and in certain statistical textbooks. The standard error of the

constant term can be obtained from this formula by carrying out the computations when all the independent variables are set eqxxal to zero. This test may

be biased if the residuals from the analysis are serially correlated. However,

as discussed on page 30, first differences in general are used only when it is

believed that the residuals will not be serially correlated.

When all variables in a first difference analysis are converted to logarithms and the constant term differs significantly from zero, we can find the

percentage effect of time in each year most easily by taking the antilogarithm

of the constant term plus 2. For example, if this tenn in logarithms was

0.015, the antilogarithm of 2.015 is 103-5^ so we could say that the time

trend alone tended to increase the dependent variable by 3.5 percent per year.

If this term in logarithms was -0.015, the antilogarithm of I.985 is 96.6, so

we could say that the time trend alone tended to decrease the dependent variable by 3*^ percent per year. In such analyses, time has an effect similar to

that obtained from the exponential analyses for studies based on actual data,

discussed on page ^0.

Other effects of time.—Time also may affect an analysis in several other

ways. Ways of measuring possible changes in basic structural relationships

over time are discussed in the section entitled "Years to be Included" (see

page 21). If the coefficients as such have changed in a systematic way over

time, this can be measured by the method discussed in the section beginning on

page 110. In some studies, certain variables may enter into the equation with

a time lag. Such variables can be incorporated easily into the equation if

the nature of the lag can be assumed in advance. Use of the theory of distributed lags at times is helpfixl, particularly if we wish to measure differences

between long-run and short-run elasticities of demand or supply. Methods of

analysis that appear useful in this area are discussed in the section beginning on page HI. Other ways of incorporating lagged variables into the

equations are discussed in the section beginning on page II6.
- kk .

Deciding Whether Certain Variables are Endogenous

or Predetermined

As discussed in a later section, if we wish to estimate certain structxiral coefficients, such as elasticities, the recommended method of fitting

individual equations depends on the number of endogenous variables contained

in them and the degree of identification (see page6l). Hence we must decide

whether each variable is to "be classified as endogenous or predetermined. If

we are concerned only with equations designed for forecasting a single variable, the method of least squares may be preferred and this distinction is less

important than when structural coefficients are to be estimated. Here we use^

as the dependent variable that item for which a forecast is desired and as

independent variables all relevant items that are expected to be known at the

time of forecast (see page 128). A distinction between endogenous and pre- '

determined variables must be made in formulating a system of equations

designed for forecasting or analytical purposes, since we must have one equation for each endogenous variable in the system.

An endogenous variableJ,g_fcâchMcally d^^^^ correjjd^ed

with theimexplained rejid^l^^ whijchjt^^^

A"preaeteiM^fl?^^ is indep§jaàikLa:£^the^

structural equation in which it agg^a^^ Predeteimniad-^ysçriables are generaîï^jdeJj^^^T^^ jDr those detemined out^^^'^e

pa?ti'Piilajf^-- eja.Qni;)mi.£^^c^r^u!^ consideratiäi, iSäTTaased"M endogenous variables•

At times, the nature of the economic model itself determines whether a

particular variable is assumed to be endogenous or predetermined, as when we

decide to take the price of feed as given, in a study relating to the supplydemand structure for eggs, but to let it be endogenous in an analysis of the

feed-livestock economy. In making this choice the research analyst is guided

by two conflicting aims: (l) To allow for as many interdependent relationships

as possible and (2) to keep his models simple enough for them to be fitted

statistically without undue effort. Common sense, supplemented by the tests

and considerations outlined here, is perhaps the best means for bringing about

a reconciliation. Three types of variables for which the classification frequently is not clear are those relating to (l) production, (2) consumption,

and (3) the general economy, such as disposable income or prices of all

commodities.

Production.—For many industrial products, production is determined to a

large degree by current price in relation to certain cost factors. Unexplained

residuals that affect the demand equation may well also affect the supply

equation, so that such variables probably should be treated as endogenous in

a study of the supply or demand structure. In such cases, one or more supply

equations are included in the complete economic model. For some livestock

products produced on farms this is true, but to a lesser extent than for industrial products. Insofar as production is affected by changes in feeding

rates in response to changes in c"urrent coimnodity-feed price ratios or by

culling or within-year replacement of producing animals, it may tend to be
. 45 -

endogenous, and supply equations may "be needed in the economic model• To the

extent that production is affected by numbers of animals on hand at the start

of the year or that were produced as a result of decisions made before the

start of the year, it is independent of the imexplained residuals in the demand equation and hence should be classified as predetermined. For many

crops, production is determined chiefly by weather conditions during the growing season and by economic factors before the- start of the marketing (and

frequently the planting) season. Here production is almost completely a predetermined variable although, as noted on page 11, if only part of the crop is

harvested for economic reasons and we refer to that part as production, then

to that extent production may tend to be partially endogenous.

A detailed analysis for hogs given by Fox (33, pp. 28-31) is useful in

illustrating the sort of considerations that enter into a decision as to

whether production should be classified as a predetermined variable or may be

at least partially endogenous. We reproduce his discussion in full in the

paragraphs that follow.

"Production of pork obviously is a direct function of the

number of hogs slaughtered, their average weight, and the percentage yield of pork per hog. Year-to-year variations in

production of pork result mainly from changes in the n\imber

of hogs slaughtered.

"The nwñber slaughtered in any given year is determined

mainly by the number of sows bred in the preceding year. For

example, hogs marketed from September to March were born 6 to

9 months previously, from sows bred 10 to 13 months previously.

About June 22, when the size of the spring pig crop is known,

a forecast can be made of the number of hogs that will be

slaughtered from September to March. Similarly, about December 22, when the size of the fall pig crop is known, the

approximate number of hogs that will be slaughtered from April

to August of the following year can be forecast.

"The average age at which hogs are marketed can be varied

by a few weeks according to how much they are forced during

raising and feeding, and by perhaps a week or two according to

the exact time chosen for marketing. For example, economic

influences current toward the end of a marketing season may

determine whether more spring pigs than usirnl will be carried

over into the period for marketing fall pigs, or whether more .

fall pigs will be marketed early, along with spring pigs.

Variations in average marketing dates are directly related to

the average weight per hog slaughtered, as late marketings

mean heavy weights, early marketings light weights. Variations

in the number of gilts saved for breeding mean opposite variations in the number slaughtered currently. These factors

influence production of pork relatively little in most years.
^ k6 ^

"The nature of available official data means that calendaryear estimates of pork consi^mption must be used in deriving a

consumer demand equation. This unit splits the marketing season

for spring pigs. However, the logical basis for considering

that calendar-year production of pork is predetermined, or nearly

so, rests on the 10 to 13 months required for gestation'and feeding to market weight, plus a decision-making interval before

actual breeding.

"The relevant statistical question in this connection is.

What proportion of the variation in calendar-year production of

pork is associated with factors known or determined before

January 1 and with noneconomic variables operating during the

current year? For this purpose, production of pork can be

considered to be determined by some or all of the following

variables:

Spring pig crop, previous year;

Fall pig crop, previous year;

Breeding intentions for current spring pig crop—that is,

niomber of sows to farrow (reported in previous December) —

multiplied by actual number of pigs saved per litter, which

depends mainly on natural conditions, including weather, at

farrowing;

Supply of corn, previous year;

Hog-corn price ratio, preceding September-December;

Production of corn, current year; and

Short-term expectations regarding price trends which could

affect age and weights at marketing.

"The first three variables accoimted for more than 93 percent of the variation in production of pork during 1924-^1-1 • The

report of breeding intentions reflects the influence of other

variables such as supplies of feed grain and relative prices of

hogs and corn, current and anticipated . Supplies of feed on

January 1 relative to numbers of livestock also affect production of pork because of their influence on average slaughter

weight and yield of pork per hog. The current yearns production

of feed grains, which depends primarily on weather, could be

introduced as an additional factor which may influence the

weights at which hogs are marketed during the latter part of

the calendar year and the number of gilts saved for breeding

purposes after January 1.

"From this analysis it appears that in 192^-^1 variations

in calendar-year production of pork were about 95 percent predetermined. The explanation of production is not significantly

increased by including the current price of hogs or pork."
-A7 -

In a similar analysis for beef. Fox concludes that about 85 percent of

the variation in production can be explained by variables measured or existing

at the beginning of the calendar year or before, supplemented by noneconomic

factors that operate during the calendar year. Under such circumstances, it

can be presumed that production is correlated only slightly with the imexplained residuals in the demand equation.

Eather than base conclusions on a test of this sort, the analyst may prefer to follow the path chosen by Gerra (see page Ik) and design a structural

analysis to test the extent to which production is in part endogenous. The

amount of work required by the Gerra approach, which necessitates the simultaneous fitting of a number of equations, versus the Fox approach, which permits

the use of a single least sqimres analysis to estimate the coefficients in the

demand equation, provided production (and consimption) is found to be nearly

predetermined, is 10 to 20 times as great in a model comparable in size to

that used by Gerra. Therefore, in circumstances of this sort, a system of

equations should be used only when it is important to deteimine the extent to

which production is endogenous. It is conceivable that a study like the one

Fox made could show that production is only about 10 percent endogenous and

yet coefficients from a structural analysis that differ from zero by a statistically significant amount could be obtained on the variables that relate

to endogenous supply.

Cons\amption.—If production is considered to be endogenous, then consxamption also nonnally is considered endogenous, but if production is found to be

nearly enough predetermined to so treat it in the analysis, then a separate

decision must be made with respect to consumption. Here Fox (33^ PP» 12-13)

again outlines some relevant considerations. His comments in full follow:

"Suppose the supply of a given commodity entering the marketing system is not affected by the current market price.

Suppose further that the marketing system passes on this supply

in a routine way, so that, except for normal wastes and losses

in the marketing process, the supply that reaches consumers is

exactly equal to that marketed by farmers. In this case, consumption is not determined by prices during the marketing

period; it can be used as a predetermined variable.

"If consximption is not exactly equal to fann supply, because

of change in stocks or because of existence of foreign trade,

cases that are more complicated will arise. Theoretically, the

existence of imports soiggests one or more supply curves for producers or dealers in the countries from which imports are obtained;

the existence of exports denotes the presence in a complete model

of the demand cxxrves of one or more foreign coxmtries. If domestic

stocks change significantly, in addition to the demand cxirve for

final consumption, a demand curve for storage holdings exists.

Thus, in a strict sense, it is clear that if foreign trade or

changes in stocks are important, a multiple-equation model is

required.
~ 48 -

"However, suppose the changes in stocks and in net trade are

both small relative to observed changes in consimption and that

domestic consumption, accumulation of stocks, and net exports

move in the same direction in response to changes in supply. In

such cases, changes in domestic consumption can be estimated with

considerable accuracy on the basis of changes in supply. If supply is predetermined, consumption also can probably be treated as

predetermined imder such conditions."

Fox (33^ P- 31) then applies these considerations to pork. He says,

"Exports and changes in stocks of pork and pork products

(excluding lard) normally are small. In terms of year-to-year

changes, 93 percent of the variation in consumption of pork

during the calendar-years 1922-^1 was associated with variation

in the quantity of pork produced. When both variables were

expressed in millions of po\mds dressed weight, the regression

equation was as follows:

Q = 50.2 + 0.75 S (67)

(0.05)

where Q is consimiption, S is production, and the nxmiber in

parentheses is the standard error of the regression coefficient.

Thus, a l-million-pound change in production normally was associated with an 0.75-inillion-pound change in consumption.

"As 95 percent of the variation in production of pork

during the interwar period was apparently predetermined, it

appears that at least 88 percent (O.95 times O.93 times 100)

of the variation in consumption of pork was predetermined.

Alternatively, consumption of pork could have been expressed

directly as a function of the variables used to explain production of pork. When this was done, 90 percent of the variation in consumption of pork was associated with the known

p:çedeteimLned factors affecting production. In this instance,

the bias that may result from treating consumption as a predetermined variable and using the single-equation approach is

probably small."

To increase our knowledge of the effects of changes in production on the

several alternative outlets, a set of first difference analyses can be run,

with production as the independent variable and with each of the components of

the corresponding disposition as dependent variables in a series of simple regression analyses. Year-to-year changes in production are exactly equal to

the sum of year-to-year changes in (l) domestic consumption, (2) net foreign

trauie, and (3) the net change in inventories. These utilization groups could

be fvirther subdivided if the data permitted and the problem required it. The

technique consists of calculating the simple least squares regression of each

distribution category separately upon the production variable, using first
. 1^9 -

differences of actual data in each case. When this is done, the sum of the

slopes ("b" values) of these regression equations equals 1.0 and the sum of

the constant terms ("a" values) equals 0. 30/ Such a set of analyses indicates the change in domestic consumption, net exports, and stocks, respectively, that would he associated on the average with a given change in

production.

If more than one important domestic outlet is available for the commodity, then separate equations within an equation system are needed to measure

the structural coefficients that relate to demand in each outlet, and the consumption of each shoiad he considered as endogenous. For some crops, however,

varieties grown for processing and those grown for fresh use differ. In such

cases, each variety can he considered as a separate commodity, and consumption

may he considered as predetermined, provided the other considerations that

have heen noted permit such an assxmption.

As noted previously, if two or more competing commodities are involved,

then a system of equations is required to obtain their respective direct- and

cross-elasticities of demand even though the constmiption of each is thought to

be essentially a predetermined variable. However, the nature of the system

differs considerably, depending on whether all of the variables relating to

consxmiption can be considered as predetermined or whether some of them are

considered a^ endogenous. The formulation and statistical fitting of such

systems is discussed in detail beginning on page 87.

Variables that relate to the general economy;—In arriving at a decision

on whether to classify variables that relate to the general economy as endogenous or predetermined, the following kinds of considerations are involved.

If we deal with a perishable item, then factors that affect its consumption

but are not included in our equation may be correlated only slightly, if at

all, with the variables that relate to the entire economy; in such cases, we

probably can assiame that variables that relate to the general economy are

basically predetermined. On the other hand, if we deal with a commodity that

can be stored easily by consumers and that frequently is stored by them, then

factors that affect the level of current demand (for storage and consumption)

but are not included in our equation, are likely to affect the general economy

to some extent. In such case a correlation may prevail between the variables

that relate to the general economy and the \mexplained disturbances in the

demand eqtxation.

Consider, for example, the effect of a disturbance like the Korean conflict on purchases by consumers of sugar. Purchases might well increase by

more than normally would be expected due to the associated increase in consumer incomes. Thus the xmexplained residual for a period immediately following the disturbance probably would be positive and hence partially correlated

30/ This approach was developed by Fox and first published in Foote and Fox

(29, p. 8) . A proof that the s\mi of the slopes equals 1 and the sum of the

constant terms equals 0 is given in Foote and Fox, pp. 79~8l.
- 50 -

with the level of cons-umer income, which also would tend to increase as a result of the Korean conflict. In this instance, we might merely omit one yearof data from the analysis, hut if disturbances of this sort occurred frequently, and a variable explaining them could not be introduced into the

analysis, then income should be treated as endogenous.

If data were available to correct for changes in stocks held by consimiers, so that the demand equation reflected only the demand for immediate

consumption, then the situation would be similar to that for a perishable

commodity. In some cases, use of the year-to-year change in income in addition to the level of income may eliminate the correlation between the unexplained residuals and the general variables. Use of distributed lags, as

described in the section beginning on page 111, also wotild tend to eliminate

this correlation. In any of these cases, the general variables can be taken

as predeteimined.

A similar situation prevails if we consider the supply-demand structure

for livestock or livestock products. Farmers essentially sell milk or eggs as

fast as they are produced; here general economic variables probably can be

considered as predetermined. In relation to livestock that can be held in

inventory, a change in general business conditions is likely to affect the

nature of the relationship. If this effect is not allowed for specifically in

the model, it shows up in the unexplained residuals and a correlation is generated between these residuals and the change in business conditions. Here

factors that measure these conditions should be considered as endogenous.

Harberger 31/ clarifies and extends these views. He says, "any price set

by the competition of many types of buyers and sellers may frequently be

viewed as exogenous when the purpose is to analyze the behavior of only one or

a few classes of bi:orers and sellers, and (consumer) income may often be

treated as exogenous when we wish to study the demand for products we believe

to be insensitive to the general state of expectations in the economy." As he

later points out, not only are durable and semidurable goods likely to be

sensitive to the "general state of expectations," but so also are conpodities

that are considered by the consumer as true luxuries, since the user would

tend to "hold off purchases of these if he looked forward to a period of

financial stringency and insecurity."

These comments suggest that the treatment of disposable income and certain variables relating to wage rates or other costs as predetermined variables in the models that are discussed in the section beginning on page 10

probably is satisfactory.

Some authors have argued that the relevant consideration is the extent to

which consumer income, say, is affected by changes in price or consumption of

the given commodity. They have shown that for even the most important

31/ Harberger, Arnold C. On the Estimation of Economic Parameters. Cowles

Commission Discussion Paper: Economics No. 2088, 1953> P» 31* [Processed.]
- 51 -

individual farm products, such as beef, pork, or fluid milk, the retail value

is equivalent only to 2 or 3 percent of disposable income. Hence these

authors have assumed that disposable income can be treated as a predetermined

variable in statistical analyses of demand for farm products, either singly or

in moderately large groups. As each of the items cited is a commodity which,

at least until recently, has not been stored in quantity by consumers,* these

authors come to the same conclusion as that given in the previous paragraph,

but for a different reason. Their reasoning suggests that disposable income

also co\ild be considered as a predetermined variable when dealing with durable

or semidurable products, so long as in value these products represent only a

small part of total income. But we conclude that special characteristics of

the commodity itself may tend to require that disposable income, and similar

variables, should be considered as endogenous, unless the equations are modified in such a way as to eliminate the correlation between these variables and

the unexplained residuals from the analysis.

Choice of the Dependent Variable in Least Squares Analyses

Designed to Estimate Structural Coefficients

When equations are fitted statistically by a simultaneous equations

approach, we in effect relate a linear combination of the endogenous variables

to the predetermined variables in the equation; hence no choice is required as

to which variable to consider as dependent. If only one endogenous variable

is involved in the equation, estimates of the structural coefficients that are

statistically consistent can be obtained when the equation is fitted by the

method of least squares, provided we take as dependent the single endogenous

variable (see page 57)- Pox (33) has emphasized, as previously noted, that

for many agricultural commodities aU„pJLJhe variables. in.„a.^^^^

except price can be considered as predetermined, .Hence, he argued, /the ,correct,way ,to, esti^a^^^^^^ iß to use prij?f\^as

'thË„dependent v^^^ ,th.e .equation to show

consumption as^ a function of price in deriving the elasticity of Téî]^dr''^^This

is a sound proœïïïïr^'-whHr^t^é l)asic assm^^ hold, thätTs,''when'^^^^^^

tion, consimer income, and other variables in the demand equation can be taken

as predetermined and there are no close substitutes or complements. 32/

For reasons discussed in the section beginning on page 6?, however, we

may at times wish to use the method of least squares to estimate structural

coefficients when more than one endogenous variable is included in the equation. Here we must make a choice as to which variable to treat as dependent.

Hildreth and Jarrett (ifg, p. 71) say with respect to this situation, "Intui-

^ÎI£^£9MîaH.^ij5B?.^^^^^ suggest that least-squares bias.might be^minimTzear by

|re£ting.,..âS«J^^ ^o^^currènt endogenous variables that are mosF

^"^££5^1^^ i^l^^û^^a by^'prèdrèTéSrnea v

32/ Fox apparently was unaware of the latter requirement, as was the author

of this handbook at the time when Foote and Fox (29) published. Its importance is emphasized by Meinken, Rojko, and King (727.
- 52 -

"being estimated« ..•" This is directly analogous to the arguments given by

Fox when he reasons that production (or consumption) can be treated as a predetermined variable if it were chiefly determined by variables which themselves were predetermined and did not enter into the demand equation*

Brindow (10^ p. 17) notes that application of this criterion suggests

that price should be treated as dependent in demand equations but that utilization should be treated as dependent in supply equations for particular outlets. He then says, "The grounds for making a choice are so xincertain,

however, that solutions were obtained with both quantity and price dependent.

... In two cases (the equations for supply of fresh apples and demand for

canning apples) coefficients obtained by least squares with quantity dependent

were much closer to" those obtained by a simultaneous equations approach "than

were coefficients inferred from solutions with price dependent. In the other

two cases, differences were small." He concludes that if the coefficients

dei^i?d,neiO!X»^3. ''are accepted as 'c'fiteria,

leasjt^sqjxares- results obtained wLth^^quantity dependent were generally better

tEan those obtained with prince dapendent."

Further empirical research appears to be needed adequately to test the

approach suggested on intuitive grounds by Hildreth and Jarrett if further

comparisons are to be made of res-ults obtained when a least squares versus a

simultaneous equations approach is used to estimate structural coefficients

for equations that contain more than one endogenous variable. Such empirical

research might well include the measurement of least squares relations when

alternative variables are treated as dependent.

Particularly when working with price level relations, we may find that

the unexplained residuals are extremely small, so that the choice of the dependent variable in a least_squares fit is essentially arbitrary. If the

correlation coefficient is nearly equal to 1, it makes little difference which

variable is chosen as dependent. When the relationships are no closer than

for most of the items shown in table 1, however, a simultaneous equations fit

may be preferred if we have an interest in the structural coefficients. Such

equations always contain two or more endogenous variables, and the HildrethJarrett criterion as applied to least sq-uares analyses is of no value in

choosing the.one to treat as dependent.

As shown by the experimental results summarized in the section beginning

on page 128, if we are interested only in equations designed for forecasting

a single variable, then a least squares analysis \mder certain circimistances

may result in smaller errors of estimate than are obtained from equations

fitted by methods that allow for the simultaneity implied by the model. ,If

this is the case, we use as dej^endent that variable for which a forecast is

desired'./, '^""^ ^''^' " ^'"""" ' \ "' "
. - 53 -

CONSIDERATIONS IH CHOOSING A STATISTICAL PROCEDURE FOR FITTING THE El^UATIONS

WHEN WE WISH TO ESTIMATE THE STRUCTURAL COEFFICIENTS 33/

Nearly all ec[uations relating to demand before 195O were fitted by the

method of least squares. In 19^3^ Haavelmo (U5) emphasized that entire

systems of equations had to be considered as a xmlt in order to understand and

measure quantitatively many econoiaic relationships; staff members of tne

Cowles Commission for Research in Economics spent a considerable part of the

next 10 years in developing statistical methods to provide such meastirements.

As a result of their work; some analysts concluded that the method of least

squares was completely outmoded. Others, however, feel that methods that

apply to simultaneous equations are so complex and computationally expensive

that they should be avoided whenever possible.

In the paragraphs that follow we start with some elementary economic concepts and then proceed step by step in such a way as to show precisely when

and why simultaneous equations techniques are needed. In later sections, computational aspects are discussed.

Some Economic Considerations

In 1927, Working (107) discussed what now is called the identification

problem in his classic paper, "What Do Statistical 'Demand Curves' Show?" 3^/

In this paper. Working pointed out that when a research worker begins a demand

study, he is confronted with a set of dots like that shown in section A of

figure 5- He knows that each can be thought of as the intersection of a demand and a supply curve, as in section B, but, without further information,

neither curve can be determined from the data. Working then noted that if the

demand curve has shifted over time but the supply cui*ve has remained relatively stable, as in section C, the dots trace out a supply curve; conversely,

if the supply curve has shifted but the demand curve has remained stable, as

in section D, the dots trace out a demand cu3rve. If shifts for each curve

have taken place, as in section E, the dots trace out what may look like a

structural demand or supply curve, but the slope will be too flat or too

steep.

In many analyses of .the demand for agricultural products, factors that

cause the demand curve to shift over time are included as separate variables

in a miatiple regression equation. In effect, we are then able to derive from

our estimating equation an average demand curve. This is indicated in a rough

way in section F. As discussed on pages kk-^hS, in some analyses we cem assume

that the quantity supplied is essentially unaffected by cxirrent price. When

price is plotted on the vertical scale, the supply curve in such a case is a

vertical line, and year-to-year shifts in the supply curve trace out a demand

33/ Part of the material in the first three subsections of this section is

adapted from that in the article by Foote (27).

3^/ A similar line of reasoning is followed by Koopmans (61).
- 5^^-

SUPPLY-DEMAND RELATIONSHIPS

SECTION A SECTION B

• • •

SECTION C

SECTION E

S3

SECTION 6

C---S3

U. S. DEPARTMENT OF AGRICULTURE

FREQUENCY

30

20

10

Truepi^

Limited

Informotion

method

SECTION H

iiA_li

Q

^Least

squares

method

L.

1^-

-• .46 .48 .50 .52 .64 +•

ESTIMATE OF ß I

NEG. 1695-^55 (6) AGRICULTURAL MARKETING SERVICE

Figure 5.-The relationships shown here bear upon the cónditions^üñaei-^íhich

Sasticities of demand can he measured by using a single-equation analysis.

They are discussed in detail in the text.
- 55 -

curve, just as they did in section D. Under these circumstances we may be

able to obtain valid estimates of the elasticity of demand by use of a least

squares multiple regression analysis for vhich price is the dependent variable

and supply and some demand shifters are used as independent variables. This

point was noted by Working (107, p. 223)^ emphasized by Ezekiel in a paper

published in I928 (19), and reconsidered in 1953 "by Fox (33) in the light of

modern simultaneous equations theory. For a nuniber of agricultural products,

this set of circumstances permits us to estimate elasticities of demand with

respect to price by use of single equation methods. Two points, however,

should be kept in mind: (l) Price must be used as the dependent variable in

order to obtain elasticity estimates that are statistically consistent—to use

the least sqxmres technique, the supply curve for a given marketing year must

be a vertical line. (2) An algebraic transformation must be made after the

equation has been fitted to derive the appropriate coefficient of elasticity;

the definition as normally used in economic literature is in terms ofthe percentage. change in .quantli^^^^^^^^^^ witli, a given percentage change in pripe.

Other circumstances imder which least squares equations can be used to derive

coefficients of elasticity are discussed in the section beginning on page 67•

What happens if we have a supply curve that is not a vertical line? If

we consider any single point, as in section G, we have no way of knowing on

which demand and supply curve of a whole family of curves it lies. The basic

problem of indeterminateness is similar to that in which shifts in the demand

and supply cuirves take place. What is needed is some hypothesis, adequately

tested and proven to be sound, as to the nature of the joint relationships

between supply and demand. We should then be able to UQtangle the two and to

obtain a reliable estimate of the slope of each curve. This is essentially

what is involved in the simultaneous equations approach.

Suppose, however, that the analyst has no interest in the true demand and

supply curves but only wants a method that will assist him in studying probable future trends in price$. Working had some suggestions on this point, too.

He said, "It does not follow from the foregoing analysis that, when conditions

are such that shifts of the supply and demand curves are correlated, an

attempt to construct a demand cuarve will give a result that will be useless.

Even though shifts of the supply and demand curves are correlated, a curve

which is fitted to the points of intersection will be usefiil for purposes of

price forecasting, provided no new factors are Introduced which did not affect

the price d.uring the period of study. Thus, so long as the shifts of the supply and demand curves remain correlated in the same way, and so long as they

shift through approximately the same range, the curve of regression of price

upon quantity can be used as a means of estimating price from quantity" (107,

p. 227). The experiment described in the section beginning on page 128 tends

to verify Working's conclusions, although this analysis deals with a different

aspect of simultaneity. However, this experiment suggests that more accurate

forecasts can be obtained by including as independent variables in the forecasting equation all of the variables in the system that are expected to be

known at the time of forecast. Other aspects of this experiment that relate

to forecasting are s-ummarized in the section beginning on page 1^1.
- 56 -

Öiir experimental model suggests that^ even when no change in structure

takes place, least squares equations under certain circumstances give poor

estimates of the structural coefficients. Under other circxmstances, least

squares estimates of the structural coefficients are nearly identical with

those obtained hy a method that allows for the simultaneity implied by the

model, and are close to the true coefficients used to generate the data. The

relevant circimistances are discussed in the section "beginning on page 67.

Effects of certain changes in structure on the forecasting merits of the two

sets of equations are discussed in the section beginning on page 139.

Marschak (66) gives an interesting example of the importance of changes

in stiTUcture on the need for using a complete system of equations. He considers the old problem of taxation of a monopoly. He points out, "Knowledge

is useful if it helps to make the best decisions." He considers, among other

things, the kinds of knowledge that are useful to guide the firm in its choice

of the most profitable output level. If the tax rate has not changed in the

past and is not expected to change, the firm can fit an empirical curve to

observed data on output and profits and iimnediately derive the point of maximum revenue. If the tax rate has not changed in the past but is expected to

change in the future, the firm coiild, if it so desired, vary its output and

profits under the new tax structure and derive a new empirical relation. But

this takes time, and substantial losses might occur during the experimental

period. If the firm had taken the trouble to derive the structural demand and

net revenue curves, it could immediately determine its most profitable output

under the new tax structure. If the tax rate had varied during the initial

period, an empirical regression of net revenue on output and the tax rate

could have been fitted and xised to find the most profitable output \mder the

new tax structure. In the example cited, however, this relation is a quadratic one and, particularly if based on a small sample, the analyst might

not be aware of this, whereas if the structural relations had been obtained,

this fact woiald be revealed by algebraic manipulation.

In many real-life situations, changes in structure are frequent. Hence

Marschak concludes: "... a theory may appear "unnecessary for policy decisions

until a certain structural change is expected or intended. It becomes necessary then. Since it is difficult to specify in advance what structural

chaniges may be visualized later, it is almost certain that a broad analysis of

economic structure, later to be filled out in detail according to needs, is

not a wasted effort" (66, p. 26).

This argument in no way invalidates the lose of a single equation to estimate elasticities of demand in those cases for which the equation contains

only a single endogenous variable and no substitute or complementary products

are involved. In such cases, we may obtain estimates of the structural parameters that can be used in the same way as any other statistically-valid

estimates. Instead, Marschak is arguing that only rarely should the economic

analyst be satisfied with a purely empirical fit if he can obtain structural

relationships with some additional work.
- 57 -

Some Statistical Considerations

We now turn to some statistical considerations that have a hearing on the

extent to which we can use least squares to estimate the coefficients in a set

of structural equations. In the relationships illustrated in figure 5, we

assiomed^ more or less implicitly, that the points lie exactly on the demand or

supply curve. In actiaal statistical analyses this is never true, since some

variables that cause the curves to shift always are omitted and the precise

shape of the curves to he fitted are not known. Thus we normally deal with

stochastic rather than fiinctional relations. j^.,.AÍuSto£j^^

is one that includes a set ofjj^e^qpi^ájasa^ or error terms whose

direction and magnitude are usually not known exactly for any particular set

ofc^^^^^mm^r^^^Më' t)éhavíor on tne average mèr^ ^repeated samples^^n

To take a concrete example, let us consider a single equation from a

structural set:

Y = a + hiZi + h2Z2 + u. (68)

Here Y is a variable for which an estimate is desired., the Z*s are two variables which are known to affect Y, and u is an error term. We also have an

interest in estimating a and the b^s in a way that will meet certain desirable

statistical criteria. We assiome that for a number of periods we know the

value of Y and the Z^s. We do not know the value of u but can estimate it in

a rough way for any given period as the difference between the value of Y computed from, the equation and its actual value.

We know that estimates of the regression coefficients will differ for

different sets of observations. We would like to estimate them in such a way

that the average value for a large number of periods or samples equals, or

nearly equals, the value that would be obtained from a similar calculation

based on the combined evidence of all possible samples. Estimates for which

these two values are exactly equal are known statistically as unbiased estimates. ¿5/ ^^ also would like the variation of the estimates about their

average or true value to be as small as possible since, under this circumstance, we would have more confidence in any single estimate than if the

variation about the average is large. Estimating procedures that give the

smallest possible variance within a given set of estimating procedures are

known as best estimates. Despite their name, such estimates possess no more

desirable properties than many alternative estimates. Although the choice of

35/ As Mood (74, p. 1^9) points out, "it is obviously of some advantage to

construct an unbiased estimator, but this is not a very crucial requirement.

If the mean of an estimator differs but little from the parameter value relative to the standard deviation of the estimator, the estimator may be quite

satisfactory." Unfortimately, it frequently is difficult to ascertain the

size of the difference to be expected when we work with estimators that are

not known to be unbiased.
- 58 -

terminology is unfortxmate, the name has become common in statistical literature. In certain circxamstances, we may be unable to obtain best unbiased

estimates but able to obtain estimates that are efficient and consistent, A

consistent estimate is one that is unbiased when we work with all possible

data; it may or may not be biased in small samples. 36/ In actual practice,

of course, we never have all possible data, but estimation procedures that

give consistent estimates presxmiably are better even with small samples than

are those that are known to be biased even with an infinitely large sample.

Efficient estimates are similar to "best" estimates, except that they are

known to give the smallest possible variance only when we work with all possible data. 37/

If we use the method of least squares to estimate the coefficients in

equation (68), we obtain best unbiased estimates if the u's and Z's meet

certain rather rigid specifications. These specifications are given by the

well-known Markoff theorem. Some of the specifications that relate to the u*s

are difficult to state precisely in nonmathematical tenns. Essentially,- they

require that the u's follow some (not necessarily a normal) probability distribution, that their average or expected value be zero, that their variance

be finite and independent of the particular values of the Z's (that is, that

they be homoscedastic) and, finally, that they shall be serially independent.

When working with economic data, we usually assume that these specifications

hold, but we may test at least the one regarding serial independence of the

residuals after we have run the analysis. As discussed on pages 30-32 and 37-

38, in some cases we transform the data to make the variance of u less dependent upon the Z's or to reduce the serial correlation in the u's.

An additional Markoff specification regarding the Z*s is easily stated

but economists frequently disregard it. To be certain that the least squares

approach gives best xxnbiased estimates, each Z must be a set of known nimibers,

in contrast to a random variable. When attempting to obtain elasticities of

demand, this specification holds only in rare instailces. One case is that for

which prices are arbitrarily set at certain levels, as in a retail store experiment, and the quantities boxight by consxmiers at these prices are -recorded.

We know of only three experiments that have been conducted in this way—those

36/ The following comments from Mood (7^, p. 150) may help to clarify the

meaning of the teiTns unbiased and consistent. Tor consistent estimates, "the

estimate becomes near the true parameter value with probability approaching

one as the sample size increases without limit. ... A consistent estimator is

obviously xanbiased in the limit (that is, as the sample size approaches infinity), but for finite sample sizes it may be biased, thoiigh in such a way that

the bias approaches zero as n becomes large. An unbiased estimator may or may

not be consistent depending on whether or not its distribution becomes concentrated near its mean as the sample size increases."

37/ These definitions have been adopted by mathematical statisticians in

part because they simplify certain mathematical proofs and theorems. They are

not necessarily considered ideal definitions from the standpoint of applied

users of statistical methods.
- 59 -

by Berry^ et al (9)^ Godwin (h^) and Jasper (52). The least squares method

was developed for use in connection with experiments in the physical sciences,

where the independent variables frequently are sets of known ntmibers, or for

the study of relationships between variables, such as heights of fathers and

heights of sons, where no "structural" coefficients are involved.

Econometricians have shown that the least squares approach gives estimates of the structural coefficients that are statistically efficient and

consistent—^but not necessarily best and unbiased--provided the u's meet

approximately the same requirements as for the previous case, if the Z^s are

predetermined variables as defined on page 58. A simplified proof of this is

given by Klein (58, pp. 8O-85), though a knowledge of calciilus and the principles of probability given in chapter 2 of his book are required to follow

his development.

We now can reconsider an example cited earlier. In section F we showed a

diagrammatic representation of a situation for which the least squares method

could be used to estimate the slope of the deiijand curve. We now know that

this estimate will be statistically efficient and consistent only if the quantity consumed and the demand shifters each can be classified as an exogenous

or lagged endogenous variable and the other statistical requirements that have

been noted, such as lack of serial correlation in the residuals, are met. As

noted previously, Fox (33) has argued that the least squares method can be

used to estimate the coefficients in the structural demand equations for a

considerable number of agricultural products, including meat, poultry and

eggs, feed grains, and several fresh fruits and vegetables. 38/ Under the

assumptions of Fox, market price is used as the dependent variable and the

independent variables are production (which is assixmed to be highly correlated

with consumption) and some relevant demand shifters.

Another situation under which we can use the method of least squares to

estimate elasticities of demand is that for which data are available on purchases or consimiption of individual consumers, as prices that confront consumers are determined chiefly by factors other than those that affect their

purchases. In this case, consumption is taken as the dependent variable and

retail prices of the various items, family income, and perhaps other household

characteristics, are taken as independent variables.

As noted previously, least squares equations may yield satisfactory forecasts--and, at times, better forecasts than any other method of fit, depending

on the particular circumstances that apply (see page l4l).

One further problem needs to be mentioned before leaving the statistical

aspects of this subject. In all cases given previously, we have assumed implicitly that the variables are known without error. Any analyst who has been

38/ Research currently underway within the Agricultural Marketing Service is

designed to measure the extent to which production is a predetermined variable

for certain livestock products.
- 6o -

connected with the compilation of data from original sources knows that errors

of one sort or another always creep in. These can result from memory bias on

the part of respondents^ inability to find all of the people in a complete

census, errors of sampling, and a host of other reasons beyond the control of

the most careful investigator. Whenever we work with economic series, nonnegligible errors in the data are known to exist. A reasonably complete consideration of this subject at this point would break the continuity of our

discussion. This topic is covered in the section beginning on page 143-

Some Econometric Considerations

Discussion in the preceding section suggests, and econometricians have

shown, that we obtain estimates that are statistically biased if we use the

least squares approach to estimate the coefficients in an equation that contains current values of two or more endogenous variables, where an endogenous

variable is defined as one that is correlated with the unexplained residual.

The mathematical nature of the bias has been shown by a number of authors in a

supposedly popular way, 39/ but none of these explanations is completely satisfactory for a nonmathematician. We do, however, have some experimental

evidence of the kind of statistical bias that results when the method of least

squares is applied in such cases. Methods which have been developed to estimate the coefficients in equations that contain two or more endogenous variables are known to give results that are statistically consistent, but methods

are not now available that are known to be statistically unbiased. Thus, we

cannot say for sure what happens when we work with the small samples that

usually are involved in economic research.

An experiment was designed by Wagner {96) to measure the kind of statistical bias that arises when we apply these methods to small samples. As a

byproduct of this experiment, we have some concrete evidence of the kind of

bias that may arise when we use the method of least squares instead.

In this experiment, a simple 3-equation model was formulated with known

coefficients. Variables generated by the model were obtained and random error

terms added to them. Two thousand observations were obtained in this way, and

they were then divided into 100 samples of 20 observations each. The first

equation contains two endogenous variables. Coefficients were obtained for

this equation for each sample by the limited information approach, which is

discussed on page 63, and by the method of least squares. Since there were

100 samples, 100 separate estimates of the single regression coefficient involved were obtained by each approach. Frequency distributions of these

estimates are shown in section H of figure 5, together with the true value of

the coefficient. Each method gives estimates that are statistically biased,

since the average for the 100 samples differs from the true value, but the 3

highest frequencies for the limited information approach are grouped about the

39/ See, for example, Bennion (8), Bronfenbrenner (12), and Meyer and Miller

(lIT- ~
- 61 .

true value, whereas the 3 highest frequencies for the least squares approach

each are to the right of the true value. The average bias of the least

squares estimates is almost three times as large as that for the limited in-

' formation approach. For a second version of this model the two frequency

distributions are more nearly similar, but the average bias in the least

squares estimates still is almost double that for the limited information

estimates. In this study, carried out at Stanford University, a large scale

electronic computer was used. Further experiments of similar character, with

different sorts of models, currently are underway at Yale University.

The experimental study by the Agricultural Marketing Service, referred to

previously and discussed in the section beginning on page 128, also gives evidence of the degree of statistical bias that may result when the least squares

approach is used to estimate the coefficients in equations that contain more

than a single endogenous variable. As discussed in the section beginning on

page 67, it is sometimes possible to estimate in advance the degree of bias

that is likely to exist.

In discussing methods that are used to estimate the structural coefficients in equations that involve more than one endogenous variable, it is

convenient to introduce a mathematical concept that deals with the degree of ^

identification. We saw earlier that it is sometimes impossible to estimate"

the coefficients in certain structural equations with the kind of statistical

data available (see page 55). Such equations are said to lack identifiability

or to be underidentified. In this connection, Dxirbin (17, p. 35I+) says, "It

is easy to get formal identification by adding extra variables to the model,

but if the effect of these variables on the system is small, then for small

samples the standard errors of the coefficients will be so large that the

situation is almost as bad as if the model were not identified. Unfortunately,

nothing can be done to remedy this." Fortunately, most equations within

models that relate to the demand and price structure for agricultural commodities are identifiable and, in the discussion that follows, we deal only with

-identifiable equations. Such equations may be Just identified or overidentified. In general, the degree of identification relates to individual equations in a system, not to the entire system.

By algebraic manipulation, we always can write down the equations in a

complete system so that the number of equations equals the number of endogenous variables. WejttLen^^ar^^ of these ñB ^n equations „in^.a^wtopyn

ggdogenous variables, and*we~can always solve the equations so that each

endogenous ^^^v^^^ as a fujiction. qf .ajà, of tl>e pre^eteji^^

Z^riaMe£ll0^jsy^^ reduced form equations. 'Since each

^^^ coefficients m these ecmarta^ns^^^a^^ statistically effiçieat«aM..consistent can be obtained by use of the method of least sguares. However, as

discussed m the section beginning on page 63, we may at times wish to obtain

these coefficients instead by an algebraic derivation from structural coefficients that have been estimated from statistical data.
- 62 .

Just identified equations «—If each equation in the system is just identified^ there always exists a -unique transformation by which we can go from

the coefficients in the reduced form equations to the coefficients in the

structiiral equations. Suppose we have the following structural demand and

supply equations^ where q is production or consxmiption, p is price, d is consumer income, w represents important weather factors that affect supply, and

each variable is expressed in terms of deviations from its respective mean:

p = b3_iq -f ^12^ (demand relation) (69)

q = t)2iP + 't>22^* Tí' (supply relation) (70)

A ru;Le„ of tliH!Jb^^tn determine the^^degree^f_ident^^^^ jj^tes^,jtjaat each..

equation.iâ.justT3enïîîlea^^ niAer^'pf\varia^ in the system minus the

nuäEer of variables in each ecjuation equals ^ the number of endogenous variables

iff'"'tîTé' system minus one; more exact rules depend on the rank of certain ma^-=^

trices. As there are k variables in the system and 3 variables in each equatiOTL^and we have two endogenous variables in the system, each equation is just

identified.

If we substitute the right hand side of equation (70) for q in equation

(69) and the right hand side of equation (69) for p in equation (70) and simplify terms, we obtain the following:

P =

bllb22 ^ ^ .^12 ^ (^^)

l-biib2i l-^ll"b21

1^22 ^12^^21 . f^r,\

q = -T-T:—^— V + •=—T:—t:— d. (72; ^ l-biib2i 1-^11^21

Since the denominator of each term on the right of the equality sign is identical, we can ignore these denominators for the moment. If we divide the

coefficient of w in equation (71) "by the coefficient of w in equation (72), w^^^^^

obtain an estimate of bn- If ^^ divide the 'coefficient of d in equation-.(72)

by the coefficient of d in equation (71)^ we obtain an estimate of b2i. Given

an estimate of b-^i and '^21? we can estimate b2^2 '^^om the coefficient of d in

equation (71) and b22 i'^om "the coefficient of w in equation (72). This gives

the four coefficients needed for our structural equations. Estimates that are

\miquely equivalent are obtained by any alternative algebraic manipulation.

Since the b*s are known to be statistically consistent estimates, provided the

necessary criteria with respect to the u's are met (see page 58) and the predetermined variables are known without error, the estimates of the structural

coefficients obtained in this way are statistically consistent.

Computationally, we may wish to estimate the coefficients in another way,

but the answers obtained are identical with those that would be gotten by an

algebraic manipulation of the regression coefficients from the reduced form

equations.
- 63 -

Overidentified eq.uati ons.--Let us now consider an equation that is overidentified. Suppose that equation (70) on supply contains a second predetermined variable^ z, that represents lagged values of prices. We now have 5

variables in the system. The supply equation still is just identified^ since

5-^ equals the nxomber of endogenous variables in the system minus one. However, equation (6s) on demand is overidentified, since 5-3 is greater than

2-1. The new reduced form equations can be obtained by the saiae general

approach as used previously, but the result now looks like this:

^11^22 ^ ^11^23 ^ _, ^12 , /^.x

1-^11^21 1-^11^21 1-^11^21

^22 ^23 ^12"^21 . fryUS

q = T-T: ^ ^ + -, . ^. Z + TT-rr—T-— d. (7^)

^ 1-^11^21 1-^11^21 1"-T^11^>21

With this set of equations, b]_2. could be estimated either by dividing the coefficient of w in equation (73) "by the coefficient of w in equation (7^) or^^

dividing the coefficient of z in equation (73) "by the coefficient of,z in

equation (7^). Different answers are obtained from the two estimates. It is

in this way that overidentified equations differ from just identified ones;

for overidentified equations, we have an oversiofficiency of information and no

direct way to decide which answer to use. In fact, neither answer obtained by

the use of reduced form equations is statistically consistent.

It would be possible to estimate the several coefficients in the two

structural equations directly by use of a maximum likelihood approach. Maximum

likelihood estimates are known to be statistically consistent and efficient.

They are used widely in statistical work because the necessary equations

always can be derived by performing certain mathematical operations that involve the maximization of the so-called likelihood fimction. The general

approach is the same as for any maximization process by use of calculus, and

it is not difficult. For complex systems of equations, however, the mathematics involved in solving the resulting equations is generally complex. That

part of Klein (58) referred to on page 59 involved the derivation of maximum

likelihood estimates. Methods for obtaining maximum likelihood estimates

based on a simiü.taneous solution for all of tHe structural equations are discussed by Klein (¿8) and Chernoff and Divinsky (13) and are called fullinformation maximum likelihood estimates but, to quote Klein, the computations

involved in general are "formidable." Hence this method is seldom used.

Another method, developed by staff members of the Cowles Commission for

Research in Economics, is called the single-equation limited-'information

maximxmoi-likelihood method. In this approach, equations are fitted one at a

time and less information is utilized than in the full-information approach.

In essence, a least squares fit is obtained between a linear combination of

the endogenous variables in the equation on the predetennined variables in the

equation, subject to the condition that the predetermined variables in the
^ 6k -

system but not in the equation have a minimum effect on the combination of endogenous variables* Since, if the model is properly formulated, the latter

variables should have no effect on the endogenous variables in the particxilar

equation, this is a reasonable procedure. Kiis method is known to give estimates of the coefficients that are statistically consistent and as efficient

as any other method that utilizes the same amo\mt of information, provided the

usual criteria with respect to the u*s and errors in the data are met. A

compromise between this and the full-information method is one called the

limited-information subsystem method, in which selected groups of equations

are fitted as a unit. Most of the systems of simultaneous eqxaations that have

been fitted and that involve overidentified equations have been based on the

single-equation limited information approach. This is the approach that generally is meant when we refer to the use of the limited information method.

Alternative Methods

Three alternative methods of fitting equations have been proposed, each

of which, if properly used, will yield estimates of the structural coefficients that are statistically consistent, even though more than one endogenous

variable is contained in the equation. These methods, however, are not

necessarily efficient in a statistical sense. The methods are (l) the recursive approach, which has been advocated particiolarly by Herman Wold, (2) the

method of instriamental variables, and (3) a method proposed by H. Theil.

Recursive systems.—Estimates of the structural coefficients that are

statistically consistent are obtained from the recursive approach only when

the system of equations has a special form; if it has this form, it is known

as a recursive system. A recursive system is made up in the following way:

(1) At least one equation contains only a single endogenous variable. As we

have pointed out, consistent estimates of the coefficients in such equations

can be obtained by fitting them directly by least squares, provided the endogenous variable is treated as dependent. In some systems, several equations

contain only a single endogenous variable; each of these is fitted directly by

least squares, using the endogenous variable as dependent. (2) At least one

other equation must contain only one endogenous variable in addition to those

contained in the first set. Consistent estimates of the coefficients in 'these

equations can be obtained if they are fitted directly by least squares,

provided calculated values of the endogenous variables included in the equations referred to in item (l) are substituted for actual values before making

the computations and the single new endogenous variable is treated as dependent. 4o/ (3) The recursive system as a whole must be of such a nature that by

to/ Reasons for using calculated values rather than actual values are the

following: (1) The unexplained residuals in the several equations within an

equation system are assumed to be correlated one with another and, by definition, an endogenous variable in a particular equation is assumed to be correlated with the unexplained residuals in that equation. (2) Calculated values

for a given variable in a given equation are known to be uncorrelated with the
. 65 -

successive steps each of the equations can be transformed into one that contains only a single endogenous variable other than those which have been

treated as dependent in prior analyses. Wold believes that many economic

systems are of this nature^ whereas certain other econometricians believe that

systems of simultaneous relations of a nonrecursive type are more typical.

These points of view are interestingly expressed in Wold (10^4-) and the discussion which follows his paper.

Instrumental variables.--The method of instrumental variables is described by Klein (50^ pp. 122-125). He says^ "Choose as many exogenous variables in the system as there are imknown parameter coefficients of endogenous

variables in the given equation. Multiply each variable in the equation by

one of the exogenous variables, and sum the equation over all sample observations. Repeat this procedure for each exogenous variable selected from the

group outside the given equation and for each exogenous variable in the

equation. This will provide as many linear equations in the ijinknown parameters as there are unknown parameters. Solve for the estimates of the

parameters.'' The consistency property of the estimates remains if lagged

endogenous variables as well as exogenous variables are used.

Klein goes on to say, "Obviously, the objection to this technique is that

the set of instrumental variables is not unique" except when each of the equations is just identified, in which case the method of reduced forms is equally

easy. "In general ... the estimates obtained for the parameters will depend

upon the particular set of instrumental variables selected, and this arbitrariness lessens the attractiveness of the method. Some principles can be indicated for the selection of the instrumental set from the total number of

exogenous variables in the system, and these principles may help to reduce the

degree of arbitrariness involved. One should avoid choosing a set of instrumental variables which is highly intercorrelated and should try to choose that

set which shows the greatest correlation with the endogenous variables in the

given equation. By the same reasoning, one should choose instrumental variables which are the least correlated with the predetermined variables already

present in the equation being studied. ... The use of the method of instrumental variables would seem to be desirable when quick resxats are wanted in

exploratory studies. It should be especially useful in screening multiple

possibilities among a group of admissible hypotheses."

Theil^s two-rounds estimates.—Descriptions of his method given by Theil

(9Q> 9lFâre not likely to be "understood by most readers, but Klein (59) discusses the method in terms of the matrices used in connection with liSted

information estimates. The discussion of Theil»s method given here is adapted

from that in Klein.

unexplained residuals in that equation because the residuals are ignored in

the computations. (3) Hence, calculated values for an endogenous variable obtained from one equation are uncorrelated with the unexplained residuals in

another equation within the sajne system, and the calculated series becomes in

effect a predetermined variable.
^ 66 ^

In o*btaining the coefficients in an individual equation that contains

more than one endogenous variable^ we first write the equation with one endogenous variable expressed as a function of the other variables in the equation. We then take each of the predetermined variables in the system in turn

and perform computations similar to those used in the method of instrumental

variables. If the initial equation is overidentified, this process yields

more equations than there are coefficients to be estimated. To make use of

all of the information available^ Theil runs a least squares relationship

between the variable on the left of the equality sign in each of these equations and the several variables on the right of the equality sign^ with the

value in each equation treated as an observation. This procedure gives estimates of the coefficients on the corresponding variables to the right of the

equality sign in the initial equation. These estimates have statistical

properties that are equivalent to those of coefficients estimated by the

single-equation limited-information method. Klein (59^ p. 1^9) proves that

"Theil*s method of two-rounds estimation is the same thing as the method

of instrumental variables, using a particular linear combination of instrumental variables and leaving no room for arbitrary selection.'' 4l/

The exact procedure can be clarified by a concrete example. Suppose we

have W observations for a system of equations that contains 3 endogenous and k

predetermined variables. Assume that one of the equations in the system contains the following variables, where the y^s refer to endogenous and the z's

to predetermined variables, each expressed as deviations from their respective

means :

^1 = ^11^2 -^ ^12^3 + "^13^1' ('75)

This equation is overidentified, since the number of variables in the system

minus the ntmiber of variables in the equation is greater than the number of

endogenous variables in the system less one. If we wish to estimate the coefficients in this equation by the method proposed by Theil, the following steps

are used:

(l) Obtain the following equations by multiplying each observation for

equation (75) by the indicated predetermined variable and simiming over the N

observations:

Xz]_yi = b^i LZ1J2 + ^12 ^^1^3 + "^13 ^'^1 (T^)

Lz^-^ = b^-L Lz2y2 + ^12 ^'^2^'i + ^13 rz2Zi (77)

Xzoy-j_ = '^•\2_^'^'^2 "*" ^12-^^3^3 "^ "^2^3X2221 (78)

Xz^y-L = '^iiLz\^Y2 + "^12^2:4^3 + ^lZ^'^k'^1^ (''^9)

kl/ This point was proved independently by Basmann (7).
. 67 -

(2) Fit the following equation by least squares^ using as observations

the corresponding four values from equations (76) through (79):

^^1^1 = "^ii^^i^a **" ^12 ^^1^3 '^ ^13 -^^i^i- (^ = 1.2,3,^) (80)

The estimates of h-^^^ ^12^ ^^^ ^13 ^^^^ "^^^ least squares fit of equation (80)

are the desired ones for the corresponding coefficients in equation (75)•

Radner and Bobkoski 42/ compare the time required to use this approach

vith that of the limited information method for two examples. For one example^

Theil's method required 58 percent as long and for the other example, 92 percent. They feel that in neither example does the fxoll advantage of Theil's

method become apparent.

The chief disadvantage of Theil's method, in relation to the method of

limited information, is that different res\ilts are obtained depending on which

variable is written to the left of the equality sign.

Reasons Why a Least Squares Fit May Be Satisfactory Even

Though the Estimates of the Structural Coefficients

are Subject to Some Statistical Bias

Before Christ (l^) had published an article, our staff had decided to

abandon the practice of publishing structural coefficients obtained by fitting

both by least squares and by a simultaneous equations method, such as that of

limited information. Instead, we had expected to publish only the results obtained by the latter approach. However, Christ (jA, pp. 397-398) says, "...

the least-squares method (of estimating the coefficients) in a system of

(struct\iral) equations can be likened to a shotgun that scatters its shot

fairly close together, but not centered on the bullseye. ... the limitedinformation method can be likened to a shotgun that scatters its shot less

close together than the least-squares shotgun does, and not centered right on

the bullseye either, but becoming better centered and approaching perfect

centering as the sample size approached infinity. Thus the question of which

method to use for any finite sajnple size is still open, for we do not know how

to tell whether the bias of the limited-information method at a given sample

size is smaller than that of the least-squares method by enough to compensate

for its bigger variance." Christ recommends a continuation of the practice of

publishing coefficients obtained by both methods in the hope that empirical

evidence eventually will provide a guide to the orle to be preferred under specified circumstances. We concur in his recommendation.

Wold and Faxer (105) indicate why in certain instances results of the two

approaches are so nearly similar. Suppose the only error which the analyst

makes in specifying the nature of the statistical model is one of assuming that

fr2/ Radner, R., and Bobkoski, F. Sme Recent Work of H. Theil on Estimation

in Systems of Simultaneous Equations. Cowles Commission Discussion Paper:

Statistics No. 385- 195^- 10pp. [Processed.]
- 68 -

a certain variable is predetermined vhen in fact it is endogenous. This is

equivalent to assuming that an independent variable in a least squares analysis is uncorrelated with the imexplained residual when in fact a non-zero

correlation exists. In such a case. Wold and Faxer show that the degree of

specification error, as contrasted with the sampling error which applies in

any case, in a 2-variable analysis is eq\aal to the degree of correlation

between the independent variable and the unexplained residual times the imexplained variance as a ratio to the total variance in the dependent variable.

Although the exact formiila is slightly more complex, a similar situation holds

when more than two variables are involved in the analysis. Thus, if the

analysis explains a substantial part of the variation in the dependent variable, the degree of specification error, must of necessity be relatively small.

This explains why the least squares approach frequently is satisfactory to

estimate structural coefficients when used for price level relations even

though each of the price variables clearly is endogenous. Also, if the degree

of correlation between the assimied predetermined but actually endogenous variable and the xmexplained residual is small, the degree of Specification error

also must be relatively small. This justifies the argument advanced by Fox

(33> P- 8) when he said, "... there are certain cases, particularly in

analysis of agricultural prices, in which simultaneity is of limited importance. In such cases it is doubtful whether the elaborate procedures of the

Cowles Commission will improve or even change the results of the single-equation approach within the limits of sampling error."

Results from an experiment.--A Monte Carlo experiment was designed by us

to determine the relative merits of equations fitted by (l) least squares and

(2) limited information for use in forecasting under specified conditions.

This experiment is described in detail in the section beginning on page

Some of the results obtained from it have a direct bearing on the question

being considered in this section, namely the probable size of the statistical

bias in estimating the coefficients in an equation by least squares versus

limited information.

The model used in this experiment is similar to that for wheat described

in the section beginning on page 11, except that it involves fewer equations

and variables. The initial model was formulated in such a way as to bring

about a squared correlation coefficient (or coefficient of detennination) of

about 0.3 Among the \mexplained residuals in the three stochastic equations.

We anticipated that this degree of correlation would induce a fairly high correlation between the endogenous variable to be treated as independent in the

least sqixares fit and the -unexplained residuals in each of the three equations. However, the latter correlation coefficient squared in no case

exceeded O.05. As the multiple coefficient of determination for each of these

structural equations exceeded O.9, differences between the coefficients estimated by (1) least squares and (2) limited information were negligible. This

confirms empirically the theorem of Wold and Faxer.

The model was reform\xlated to bring about a squared correlation coefficient of more than O.9 between the xmexplained residuals in the three equations and to reduce the multiple coefficient of determination for the
^ 69 ^

structural equations moderately. These changes were designed to assure that

different coefficients would be obtained by the two approaches. This change in

the model resulted in increasing the squared correlation coefficients between

the endogenous variable written on the right of the equality sign and the unexplained residuals in each structural equation to 0.7 or more. Based on the

theorem of Wold and Faxer, a substantial degree of statistical bias would be

expected if the coefficients in these equations were estimated by the method

of least squares and, in fact, the coefficients estimated by least squares did

differ considerably from the structural coefficients and by more than would be

expected based on their standard errors (see page I3I). Results when these

and related equations were used for forecasting are discussed in the section

beginning on page 128.

This experiment suggests that a high degree of correlation between a so--

called endogenous variable and the unexplained residuals in a particular

structural equation will exist only when a high degree of correlation exists

among the unexplained residuals in the several equations within the system.

Thus, unless the latter correlations are high, coefficients estimated by least

squares may be nearly the same as those obtained by a method that allows for

the simultaneity implied by the system, such as limited information. If this

is true, the computational simplicity of the method of least squares suggests

that it could be used to provide approximate estimates of the structural coefficients, except when a high degree of correlation among the unexplained residuals in the several equations is anticipated.

Results from empirical research.--This hypothesis, if subsequently verified, would explain why some systems of equations given coefficients that are

nearly identical when fitted by least squares versus limited information, as

was true for the 6-equation model for wheat used by Meinken (71), and why

other systems give quite different answers by the two methods~as was true for

the 9-equation dairy models used by Rojko (81). The four utilization equations

in the model for wheat relate to food, feed, export, and storage, respectively,

and negligible correlations between the unexplained residuals in the several

equations would be anticipated. Factors that affect the consumption of the

several dairy products but are not included in the equations are more homogeneous, so that a fairly high degree of correlation would be expected among

the unexplained residuals in the several equations. Systems of equations for

certain other commodities have been fitted by these alternative approaches,

and results obtained also are consistent with those that would be anticipated

from an application of this hypothesis. Further mathematical, experimental,

or empirical evidence will be required for complete vqirification.

M EXAMPLE THAT ILLUSTRATES SOME FITTING. PROCEDURES ^3/

Beginning on page ik, we discuss an 11-equation model of the egg economy

At an earlier time, this model contained 9 equations, but the economic basis

for it is similar to that for the 11-equation model. In this section, we

43/ This section is based chiefly on material prepared by Martin J. Gerra

agricultural economic statistician. Agricultural Marketing Service. A knowledge of the notation and procedures given in Friedman and Foote (ko) is

assumed. ^—^
- 70 -

discuss the fitting procedure for the 9-equation model, giving only such economic aspects as are required in connection with the fitting process. Methods

for fitting systems of equations by the limited information approach, or a

slight modification of it for equations that are just identified, are given in

detail in Friedman and Foote (^). The discussion included^here supplements

that given hy these authors, chiefly in two ways: (l) More detail is shown

with respect to obtaining linear approximations for non-linear variables and

(2) steps that can be run efficiently on certain electronic computers are

delineated. The latter, of course, depend on the nature of the data processing equipment available and the computational programs that have been written.

Equations and Variables in the Statistical Model

The following variables are taken as endogenous in this model: QE^ PR ^°-*

p¿, Lpi, Cp, LQ, Pp and Pp, S». Their exact economic meaning need not be specified. The remaining variables are assumed to be predetennined.

Structural form. —The following structxH'al equations are involved in a

system representing the supply and demand for eggs. The nine equations shown

correspond to the nine endogenous variables in the system. Therefore, this is

a complete system.

Qg T

-g— = ai + ti2 PR + cii -^ + cj^2^ + ci3 Pc + 014 Pß +

CI5 Pt + G16 Po + ^1 (Öl)

QE = (QA^P) - A (82)

Lp = Lj + Cy - M - Lc (83)

P'

Cp = al^ + b42 -¿- + \% {^)

Le = a5 +t52-^ + U5 (85)

Pp = a5 + 1362 PR + C61 W + \i6 (8^)

QF S' . . I

PR = aj + 1372 —^ + 1373 ^- + C71 -^ + C72 Pj« + C73 Pc +

07!^ P¿ + C75 Pt + C76 P¿ + U7 (87)

Pp = as > H2 PR + C81 W + U8 (88)

-|: = a9 + b92 ^ç^^IJ,) -»-CgiF-hUg (89)
^ - 71 -

In this notation, the a's represent the constant tenorin each equation,

the b*s and c*s represent structural coefficients that apply to the endogenous

and predetermined variables, respectively, and the u's represent random error

terms. No coefficients or random error terms are involved in equations (82)

and (83) because they are identities; they need not be fitted by statistical

means•

Linearized form.—Because the equations in the egg model are stated in

linear terms, the endogenous variables in the model used in nonlinear coinbinations (for example. —1-, QA • LF> ——r Q^cL so forth) were transformed into

linear approximations by use of formiilas given by Klein (58, pp. 120-121). The

linear approximations are then substituted for the original variables. Combinations of variables that are assxmed to be entirely predetermined, however,

are treated as a single composite without linearization as, within the analysis, they are assumed to be given.

The formulas given by Klein are:

XY = YX + XY - XY

^/:^2y X/Y = X/Y + X/Y - (X/Y"^) Y

(90)

(91)

where X and Y are the means of X and Y, respectively. If either the product

or the quotient is mtiltiplied by a constant, then each term in the transformation is multiplied by the constant.

Rewriting the structural-equations in linearized form, we obtain:

-=— +

H

1 ^E I

QE --^- H = ai + bi2PR + cii -^ + C12PM + C13PC +

H H

CI4PB + Ci5Pt + C16P0 + %

QE = (QA • %- - QA • L?) + (^F • "^A) - A

Lp = Lj + Cp - M - Le

Cp = ai^ + b42

^G ^G

^F

p.2 PG + Ul^

(81.1)

(82.1)

(83)

(84.1)

+ b 52

f-F 1

-— + -— PT!. -

p_2

Pp = ag + bggPj^ + C53_W + U5

'G

•G + Uc (85.1)

(86)
P¿ = ay + lDj2

'13

- 72 -

QF I 1

_ H H

QF -

QF

[s- ^ 1

_H H

S' - S' H1

12 J

+ C71 -i- + CY2PM +

CY3PC + CY^Pg + CyjPt + C-jS^o + "7

Pp = as + bgaPp + C81W + U8

(87.1)

(88)

H

_1_

H 12

H + b92

QF QF

(QE + A) (QE + A)2

(Qp + A) +

(QE + A)

QF + c91F + uc (89.1)

Fitting the Linearized Model

In fitting the model of the egg economy, equations (82,1) and (83) are

excluded because they have no statistical coefficients. However, in using the

model for analytical purposes or forecasting, these equations are used together with the seven equations that involve statistical coefficients. The

seven equations each contain more than one endogenous variable and are each

overidentified. Therefore, they are fitted by the limited information method.

The analysis is based on first difference values for the years 1931-^1

and 19^6-5^. Conditions during these years are assumed to have been sufficiently homogeneous to permit their inclusion in a single analysis. The war

years 19^2-^5 are excluded because of abnormal circumstances that influenced

supply and demand in that period. First differences are used because of the

several trend factors that have affected supply and demand.

Variables used in the Mzz and Mzy matrices.—The Mzz matrix usually contains moments for al~the predetermined variables in the system. In some

cases, however, the number of predetermined variables in the system exceeds

the number of available observations and modifications must be made. To prevent the obtaining of indeterminate results, we must have at least one more

year of data than there are predetermined variables in the Mzz matrix.
- 73 -

In the model of the egg economy, there are 21 predetermined variables and

18 observations for each variable; at least k predetermined variables must be

omitted. For reasons given in subsequent paragraphs, 6 variables are excluded

from the matrix of predetermined variables M^;^^ leaving 15 variables in this

matrix.

Moments for the 5 predetermined variables—Pj^, PQ, Pg, F^ and PQ—that

relate to prices of competing or complementary goods during the first half of

the year, are excluded from the matrix of predetermined variables for the entire system, because these variables are highly correlated with their respective annual averages—Pj^, PQ, Pg, P-j;^, and P^^* Moments for these variables,

however, are included in certain other matrices when equation (87.I) is fitted

individually. With the method of limited information, moments for some predetermined variables can be dropped from the matrix of predetermined variables

for the entire system if this appears desirable, provided sufficient predetermined variables are used to provide identification, although the estimates of

the coefficients tend to be less efficient in a statistical sense than those

in which moments for all the predetermined variables in the system are used.

Moments for the predetermined variable M, which relates to mortality, are

omitted from the Mgz matrix because reliable data are not available, kk/ Experience in fitting models of this sort indicates that improved results are

obtained when variables that poorly represent the economic factor that should

be in the structural model are omitted from the M^z matrix. Moments for such

variables should be used when fitting the particular eqixation in which they

are found, but, in this case, the omitted variable is in an identity, hence no

statistical fitting is involved.

The linearized values of each of the predetermined variables used in combination with an endogenous variable in the structural equations are used only

once in computing moments for the Mg^ matrix. For example, Ql-^/Ey when transformed into linearized form, is:

^E 1 ^ %

+ -^QE --=^H.

H • H "^ 12

Similar last terms are involved for Q]^/H and SVH- AS the respective last

teims in each are perfectly correlated, only one, namely - (Qg/l^) H, is used

in computing moments for the Mgz matrix. As an alternative, moments for H as

such might have been used.

In computing moments for the endogenous variables, however, the entire

linearized value of the combination of endogenous and predetermined variables

is used in place of the original variable. For example, for Q^/H, the following linearized form is included in computing the moments for the M^y matrix:

^4/ Mortality may be, in fact, an endogenous variable but to so treat it

would have required an additional equation to keep the model complete. If it

depends on variables that are included in the Mzz matrix, then no loss in

efficiency is engendered by omitting it from this matrix.
-1^

Table 3.—Variables used in obtaining moments for specified matrices and equatic

Variables used 1nM^z matrix

- "^^ H = ^1

I/H = Z2

^i = ^3

Pc = Z4

PB = 25

Pt = Z6

PO = 27

LpQA == Z8

A = Z9

Lj = ZlO

p; ,.

P:2^ü = ^11

V ^° ^ '^'

QE

F = zu,

W = Z15

Mzy matrix

^E 1 , QE

H - H^

Cp = y3

yi

p;

-—- + PT? -

i^F f

^Tg % = yii.

PG PG PG'

% 1 ^F

% PG PG

PF = y.T

PR = ys

S^ 1 . S

T^T^ -^^^ = ^10

QF

PF = yii

QF

(QE + A) (QE + A)2

1

(QE +A) +

(QE + A)

QF = yi2

Predetermined variables excluded from Mgz matrix but used in individual equations

PM = Z16

Pc = ziY

Pt = Z19
and for S'/E, we include;

- 75 -

H H H"^

S^ +4^ S^ - 4^ H.

H H H^

This is done because a single structiiral coefficient applies to the entire

combination.

If "both the nxjmerator and the denominator are endogenous variables, then

no terms from the linearized variable are involved in the M^z matrix. As in

the previous case, the entire linearized variable is treated as a unit in obtaining moments for the matrices that involve endogenous variables.

The next step in the fitting process is to assign numbered y^s and z*s to

all of the variables, making use of the linearized forms. The letter z is

used if the variable is predetermined and the letter y if it is endogenous. In

table 3, the variables are designated as being included in either the M^z

matrix or the Mzy matrix, or as being excluded from the Mzz matrix but used

in fitting the individual equations. The y's included in the identities are

omitted from the Mzy matrix as these equations need not be statistically

fitted.

Variables used in fitting the individual equations.—Listed in table k

are the variables used in fitting each of the seven equations—they are designated by their structural symbols. The endogenous variables are classified

under Y* and the predetermined variables under Z*^. The corresponding y*s and

z's are indicated in parentheses.

Considerable care must be used in computing the linearized values of the

variables, particularly when first differences are used, to assure that enough

decimals are carried to make certain that the final differences shall contain

several significant digits. The linearized values in all cases are obtained

before the first differences are computed.

In deriving the reduced form forecasting equations after the model has

been fitted, the term in the linearized value containing the endogenous variable is treated as an unknown value while the remaining terms in the linearized

value are treated as known values. For example, equation (8U.1), after values

have been obtained for a]^ and '^^,2^ ^^ rewritten as:

PG PG PG'
PF

If first differences are used in fitting the models the term b]^2 ^T ^^ '^'^^

forecasting equation equals zero "because the first difference of a constant is

zero.

Table if*--Variables used in fitting specified eqiiations

Equation Y^ Z^

(81) : QE/H. PR (yi, 72)

(Ok) '. Cp, Pp/P¿ (yj, yi^)

(85) :' Lc. PF/PG (^5^ yó)

(86) : PF. PR (y?, ya)

(87) '. P¿. QF/H. SVH

: (y8> y9> yio)

(88) ': PF, P¿ (yii. 78)

(89) : S/H, Q¿/(QE + A)

: (yio. yi2)

W (Z15)

F (zii^)

yH; p^. Pc^ %^ ^t^ -^0

(Z2P Z3^ Z^^ Z^^ Zg^ Zy)

W (Z15)

l/H, PM, P¿, P¿, Pt, P¿

(^2^ 216^ ZiY> Z18. 219, Z20)

For certain economic variables that normally change slowly from month to

month, an annual average is used in conjunction with other more volatile variables that relate to the first half of the year. For example, annual disposable income was used in the January-J\me equation (87.l). This is done to

simplify the fitting process.

Computations on an Electronic Computer

This section outlines the procedure for estimating on an electronic computer (SEAG) 45/ the structural coefficients and their standard errors for

this system of simultaneous,equations. In using the limited information

approach, we estimate the coefficients of one equation at a time, but two

preliminary computations are made which are applicable to all equations in

the system. These are (l) computing adjusted augmented moments for all the

45/ The National Bureau of Standards, Washington, D. C, designated the computer in use when these computations were made as Standards' Electronic

Automatic Computer, or by the sobriquet SEAC. The entire fitting process

could have been programmed for SEAC However, costs of programming were so
- 77 -

variables in the system; and (2) obtaining the matrix product Myg M^z Mgy

which^ when the entire operation is done on desk calculators, is obtained by

performing a forward Doolittle solution, using all the variables listed above

as being included in the M^z Q'^^^ ^zy iiaatrices. 46/

Computing adjusted augmented moments >—As the first step in computing the

adjusted augmented moments, the first differences and means for the 27 variables in M^z Q'^d Mgy/ 9'^^ ^ check simi, are obtained on a desk calculator. In

setting up the worksheet for this step, the Z values shoiild be listed first in

the sequence in which they occur in the system, followed by the Y values, also

listed in sequential order, with the check sum at the end. The uncorrected

sums of squares and cross-products then are obtained on SEAC by use of an A'A

routine. V// The tincorrected sums of squares and cross-products are corrected,

augmented, and adjusted, in the usual manner kQ/ on a desk calculator.

Because the predetermined variables zig, zjjj, ... zgo Q-^re excluded from

the matrix of predetermined variables (see page 73)> it is necessary to carry

out an additional computation to obtain their adjusted augmented moments.

These predetermined variables (zi6^ ••• ^2o) ^^® ^^ ^^ used in fitting equation (87.1), and consequently the corrected sums of squares and cross products

of zi5, ... Z20^ the endogenous variables appearing in equation (87.1), namely

yd> y^f ^^^ yiO^ ^^^ the predetermined variable zg are obtained directly on

SEAC and then are augmented and adjusted in the usual way on desk calculators.

large that, instead, certain operations were done on SEAC and others, on desk

calculators. Slightly different procedures probably woixld be involved in

using a different electronic computer, but the basic principles illustrated

here would be applicable.

k6/ If this operation is done on desk calculators, only that part of the

product matrix actually needed is computed. See Friedman and Foote (i+O,

P- 67). —

47/ A program for obtaining corrected svias of squares and cross-products is

available on SEAC, but the machine does not have sufficient internal memory to

handle a problem of this magnitude. Uncorrected values can be obtained by a

matrix multiplication process for which the machine has a larger capacity. A

program to obtain adjusted augmented moments for up to 28 variables has been

written by the Agricultural Marketing Service for the IBM 650.

kS/ See Friedman and Foote (ko, pp. 3-7)- Particularly in a problem of this

size, it is desirable to combine tables 1, 2, and the upper part of table 3,

as shown in their publication. The k^ should be entered as a third row in the

first section of table 1. Computations shown in the second and third row of

each section other than the first of their table 1 can be retained in the machine and the difference recorded directly by use of negative miatiplication.

Two additional rows then should be added to each such section of table 1, the

first of which contains the k^kj and the second, the adjusted augmented moments. When this is done, the adjusted augmented moments are entered directly

in the first row of each section other than the first in table 3 and the first

section of table 3 is omitted. This procedure has been adopted as a standard

one for any problem in our central computing unit.
- 78 -

OlDtainlng Mzy Mzz My2>--In this step the matrices M^z and Mzy are set up.

The Mzz ^matrix includes the adjusted augmented moments of the predetermined

variables Z]_^ Z2^ .•. z^^^ and is^ therefore^ a 15 x 15 symmetric matrix. The

Mzy matrix includes the adjusted augmented moments of the z's (z^, Z2i ..•

z^^) on the y's in the seven equations to he fitted by limited information

(yi^ 72> ••• yi2)> and is a 15 x 12 matrix. The matrix product is obtained

directly on SEAC^ using standard routines for matrix inversion and multiplication. The latter can be carried out by making use of an A'BA routine^ where A

is the transpose of Mzy and B is the inverse of MzzAdditional steps in the limited information fit.--Values for the seven

submatrices My*z Mzz Mzy* are obtained directly from the complete matrix

Myz Mil Mzy.

The remaining steps to obtain the structural coefficients and their

standard errors in a limited information fit for each of the seven equations

follow the procedure outlined by Friedman and Foote {ko). With two exceptions, these computations are performed on desk calculators. Because equations

(81.1) and (87.1) contain 6 predetennined variables each, it would be a laborious process to compute W^^z"^, P% and My*z*P' on a desk calculator. These

values for equations (81.I) and (87.l) are,, therefore, obtained on SEAC, using

standard matrix routines. 49/

SPECIAL ECONOMETRIC TOPICS

In this section we consider points that would have led us too far astray

from our main theme had we taken them up e9.rlier.

Elasticity of Demand 50/

Most research analysts are familiar with the term, '>l,airiièiffi|t^^^ ."

which was popularized, if not invented, by Alfred Marshall. Elasticity'^^de-^

mand is the ratio of the percentage change in consumption of a,caDMQdity to""

^m..3,ñm£¿miM BOZfi^sijAagel^ka^^^^^^^^^^^^ notation it Tu

written as *^'

49/ By noting the matrix operations that are involved in getting Myx-z^I'S i*

will be seen that

Myîfz-î^P' = My^z-^z^z^z^**

If a matrix routine is available on the computer to get A'BA, then this triple

product can be obtained directly once M^iz^ has been obtained. However, if

this is done,

P' = Mz^z-^z-i^

must be obtained as a separate operation, using an ordinary AB routine.

^0/ Some of the material in this section is adapted from that in Waugh (100,

pp. 56-57) and Foote and Fox (29, pp. 36-38). Similar material is given in

many textbooks on economics.
- T9 -

where dq/dp is the derivative of the demand curve at the point (p, q.) when

quantity is plotted on the vertical scale. The magnitude of dq/dp can be

measured empirically by finding the change in quantity associated with a small

change in price at the particular point on the demand curve.

Many economists have trouble with coefficients of elasticity. Figure 6

shows how this coefficient can be measured graphically. The curved line on

the diagram represents an assumed demand curve for eggs. The scales for consimiption and prices would not need to be shown. They are unimportant^ because

the coefficient of elasticity is invariant to changes in scale provided that

the axes start at the origin. Suppose we want the coefficient of elasticity

at the point (p=a^ Q.=c). We draw the indicated straight line tangent to the

demand curve at that point. The elasticity in question is -a/b, as shown in

the following paragraph. For this example^ this equals -35-5 divided by 6k.^

based on the scales shown. In terms of small squares on the grid^ this equals

-17.75 divided by 32.25. Either computation indicates an elasticity of -0.55*

This piece of graphics comes from Marshall (67^ pp. 102-103)- It derives

from the definition of elasticity given in equation (92). In this charts

dq/dp = - (c+d)/(a+b) and, by similar triangles, (c+d)/(a+b) = c/b. Also

p = a and q = c. So

-t---i = T--|-=-A- (53)

Some economists have found the concept of elasticity so difficult that

they have used "arc elasticity," or the "average elasticity of a curve." If

the graphic approach to elasticity is used, there is little need for such concepts. The elasticity coefficient shown here is identical to the mathematical

point elasticity and is easy to compute.

For most demand curves, including linear ones, the elasticity differs at

every point on the curve. If a linear relation is given when both quantity

and price are expressed in logarithms, then the elasticity is the ssime at

every point on the curve and equals the slope of the curve. A proof of this

can be derived as follows. For such a curve,

log q = a + b log p. {^k)

Applying standard principles of differentiation from calculus, we obtain

q dp p ^^^^

or

= ^- (95.1)

A comparison with equation (92) shows that, for this curve, b = "n .
ELASTIC

Demand For

CONSUMPTION (NO. PER CAPITA)

r Î 1

:iTY

Eggs

-aM

Í

s^ 4

QAn ^^ <w - . oUU "'N^^^

W^

C - ^^H^

9 5i

Add i ^"^^5^ ' N>> ^

4UU i^ jr '^S. ^""^^^JIJÍ^ "c " X "

^^ ^^ ^'•1

^¡^

^*s '^"'s^

^ SI

"^S,

^^

1 ^w n a- -3- b *"s^

0 20 40 60 80 100

RETAIL PRICE (< PER DOZ)-p

U.S. DEPARTMENT OF AGRICULTURE NEG. 1335-55(1) AGRICULTURAL MARKETING SERVICE

00

o

Figure 6.—The curved line in this diagram represents an assumed daoaand curve for eggs. A graphic method

for measuring the elasticity of demand at a particular point on any demand curve is illustrated.
« 81 -

The cross elasticity..Qt..de»^.âj..JfeCto^^§^J^^ change In _

cQ]g^p±iD2X.^Q£..Qne...^ associated with a 1 percentcM^*lx:"the

^riœ^^ £Qm«iai31t>y>. if the direct eia¥ticïly aS^^^^H^o^T^^^agWü-.

i^^^'Sav^ the same sign, so that an increase in price of the second commodity

resiilts in a decrease in consumption of the first commodity, we say that they

are complementary products. If the direct and cross elasticities are of opposite sign, so that an increase in price of the second commodity results in an

increase in consumption of the first commodity, we say that they are competing

products. If the direct and cross elasticities are of the same magnitude when

the respective quantities and prices are expressed in comparable terms, we say

that the two commodities are perfect complements or substitutes. If the cross

elasticity equals zero, or nearly so, we assume that the two products are independent in demand. Special techniques for studying relations among complementary and substitute commodities are given in the section beginning on page

8T.

If the elasticity, when defined as a positive number, is less than -unity,

so that a given percentage change in price results in a smaller percentage

change in consiomption,. .we ,„say.„^i¿te±..AeMaad is ineiastie.^ For such commodities,

as supply increases the total value tends to decrease. Demand for most farm

products, at least at the local market level, tends to be inelastic. ^|.f,....ih,e ^^^

elasticity is larger than unity, so that a given percentage change in price

resultsTrîS''a'larger percentage change in consumption, we say that, demand is

ela^±iß^. For such commodities, as supply increases the total value tends to

increase. If the elasticity equals unity, then the total value remains unchanged regardless of the quantity,

Henry L. Moore, a pioneer in the field of price analysis, was interested

in the reclprgcal_of demand elasticity, whi^^^Jie^^^^^^^e^^J^'^g^^ "

Price flexibility is the ratio"^óf^ a percent in the price of the commodity to the associated percentage change in its cons\miption. But the term

is also applied to the ratio of percentage change in price to the associated

percentage change in whatever supply variable is used in the analysis (consijuarption, production, or total supply) • Obviously, there is no logical reason

why the percentage relationship between price and production or price and supply should be the exact reciprocal of the ratio between changes in consimiption

and changes in price.

Reasons for different elasticities of demahçL.—The immediate pbject of a

statistical demand analysis is the measurement of relationships raiher than an

explanation of the particular values obtained. But the analyst feelè "under

pressure to rationalize the numerical resuilts on either a commonsense or a

theoretical basis. It is often said, for example, that the demand for an item

that takes up an almost infinitesimal fraction of total income is likely to be

highly inelastic. But this is not the relevant factor that tends to make for

inelasticity; rather it is that the commodity has no close substitutes. A

commodity such as potatoes or onions, on this score, tends to have a less

elastic demand than a commodity such as beef, pork, or chicken which has

several fairly close substitutes. Whether two commodities are effective economic substitutes depends on consimaer attitudes toward them.
- 82 .

Problems involved in interpretation.—Many possible complications arise

in interpreting coefficients that are intended as elasticities of demand. For

example^ moderate changes in relative prices might not cause measureable

changes in relative consumption of two commodities. But sharp increases in

the price of one or a continued wide price differential might lead to a substantial and possibly cumulative or irreversible shift from this commodity to

the other. Although the immediate cause was physical shortage of butter

rather than its high price, the change in relative consumption and in consumer

attitudes toward butter and margarine during World War II is one of the most

dramatic cases on record.

Some economists run an analysis in which price is made a function of the

P^o^^^.'ÖSLPX^ÄHBEÜU^^ They refer -^gg-^EETîeclpÎS

nB^snmrè^man one outlet (including storage), the ^asticity of demand in any

one of these outlets need not equal the reciprocal of "price flexibility with

respect to production." Changes in commercial stocks and in net exports of

meat, for example, tend to cushion the effects of a change in production of

meat upon its retail price. The reciprocal of the price-production relationship suggests a xmit "elasticity of demand" for meat. But the regression of

consumption of meat upon its retail price gives an.elasticity of demand of

around -0.6. Year-to-year changes in consumption normally are highly correlated with changes in production, but during 1921-41 they were only 70 percent

as large. Hence the elasticity of consiomer demand is only 70 percent as large

as the reciprocal of the price-production flexibility.

An elasticity of. demand for "meat to store" and one for exports of meat

from this countr;^^ could be calciliated. An elasticity of supply for imports of

meat (perhaps a separate one for each major country from which meat is

imported) also would be involved. Ordinarily, elasticity^of demand means the

^^?5'feè5Î"^y of* ^oïïÊs;tic cons\miption wrtE"';'J^pecr^ is likely thaï"

the^^las-ti-cj^tie^^

than the elasticity of demand for consumption. "'" '""*""''^^-.

The three major categories of utilization for wheat have very different

demand curves and elasticities. For example, the price elasticity of demand

for wheat for domestic food use is less than -0.1 (see table 5). The demand

for wheat as a livestock feed is inelastic as long as the price of wheat is

considerably above the price of feed grains. But if the price of wheat falls

to or slightly below the price of corn on a pound-for-poxmd basis, the use of

wheat for feed is likely to increase tremendously. In other words, the demand

for wheat in the range of (say) 20 cents a bushel below the price of corn to 5

or 10 cents above is highly elastic/ The elasticity of demand for exports of

wheat has varied in the last 30 years. In the late 1920»s, when exports of

wheat from this country amounted to about a sixth of total wor3,d exports and

exports to Europe amounted to 5 to 7 percent of European production, the elasticity of demand for our wheat exports may have been substantially greater

than one. Dollar rationing as such—that is, the setting aside by other countries of a specific dollar amount to be spent for our wheat—would imply a
- 83 -

unit elasticity of demand with respect to price. An elasticity of demand for

wheat for export iinder conditions of extensive import and exchange regiilations

may have little meaning.

The implication of this is that research workers should indicate specifically the particular utilization, or set of utilizations, to which a given

coefficient of elasticity refers. The ratio of a percentage change in total

utilization of wheat to a percentage change in the price of wheat sho\ild he a

weighted average of the elasticity in each utilization group.

A further problem in intearpretation is that, except for double logarithmic curves, the elasticity differs at every point on the curve and, in a statistical analysis, also differs, depending on the particular values assumed by

other variables in the analysis. This problem frequently is ignored, in a

sense, by computing the elasticities when all variables are at their average

values. But, in comparing results from one analysis with those from another,

this practice is imdesirable because the averages depend on the particular

years on which the analysis is based. More reliable comparisons could be obtained from elasticities derived for a uniform year or period of years. In

such computations, use should be made of calculated values for the dependentvariable rather than actual values.

Table 5, adapted from Meihken (21, p. ^3)> shows the effect on several

elasticity and price flexibility coefficients of using values of the variables

for specified time periods. The analysis was based on data for 1921-29 and

1931-38, and results of the elasticity computations are shown based on averages for these years and also for I93I and 1953, respectively, to contrast

levels in years of depression and prosperity. Publication of tables of this

sort to illxistrate the range in elasticities that are suggested by an analysis

lander specified conditions appears desirable.

The value of statistical regression coefficients depends upon the extent

to which they improve our ability to act intelligently and appropriately in

specific situations. An application that is frequently encoxontered is the

question of whether compensatory payments or purchase and diversion programs

would be less costly in supporting the price of perishable commodities. So

far as cost to the public treasury is concerned, the answer to this question

turns largely on the elasticity of demand for the commodity. If the elasticity

of demand is unity, this tends to make the costs to Government of the two

methods of price support identical. If demand is highly inelastic, purchase

and diversion is less expensive to the Government; if demand is more than unit

elastic, compensatory payments presumably are less expensive to Government as

well as more satisfactory to consumers. The reasons are discussed in detail

by Pox (32). The accioracy of statistical regression coefficients and their

validity in the particular context—time, place, and duration of time with

which a projected program is concerned—bear upon the qiaality of administrative actions and the overall effectiveness of Government programs.
. 84 -

Table 5.—Wheat: Coefficients obtained by a simultaneous-equations approach

expressed in percentage terms based on values of the variables for specified periods

Determined at values for years

beginning July—

Coefficient ] Average

1921-29 '

and

1931-38

i 1931 i 1953

Demand elasticity: :

Use for food with respect to Kansas

: -0*04 -0*02 -0.11

Use for net export with respect to :

spread between world and Kansas City '

: .18 ,23 .39

Use for feed with respect to spread

between price of wheat and com

Use for storage with respect to deflated Kansas City price of wheat •.•.

Income elasticity of use for food

Price flexibility (world price) with

respect to—

Wnvld sunrjlv .••••••••••••••••••••••»••

\ -AO -.02 -1.75

i -2.73 --75 -^-76

: .20 .16 .76

I -l,i|-3 -2.48 -.7^^

Wm^l ñ TiT»"! OP 1 PVPT .......««•••••«••••«• Î 1.78 l.SP l.tó

Adapted from Meinken (71^ P* ^3)*

Problems of Aggregation 31/

Most applied economists in the supply^ demand, and prices area are interested chiefly in the behavior of market aggregates. This interest stems from

their desire: (l) to forecast such economic variables as prices, production,

and utilization, and (2) to analyze probable effects on these variables of

changes in institutional factors such as price supports, freight rates,

tariffs, import quotas, and so forth.

51/ This section is based chiefly on an impublished paper by Richard J.

Foote and Marc Nerlove, Agriciiltural Marketing Service, entitled "Why Applied

Research Workers are Interested in Problems of Aggregation." These problems

are discussed in detail by Theil (¿g) and, more briefly, Allen (2> pp. 69^-

72ÍÍ-).
.85^

As is well known^ however, economic theory with respect to supply and

deiaand is couched largely in terms of the individual. More important, we have

a great mass of data, some aggregative, some not. Hypotheses consistent with

most or all of our data may or may not "be useful for forecasting or for

assessing the effects of alternative policies; but there is surely some presumption that such hypotheses should he better for both purposes than those

consistent with only a small part of the available data. Thus, applied economists are interested in a body of theory that, in many specific situations,

will help to develop hypotheses, both qualitative and quantitative, consistent

with available data that relate both to aggregates and individuals, or relatively small groups, and in methods by which these hypotheses can be tested

against available data.

Althoxigh much remains to be done in this area, some progress has been

made. In this section, we mention briefly some research studies that have

been published. Our discussion can be shortened by using the terms "microtheory," "microvariables," "aggregates," and "macrotheoiy." Microtheoary is

that body of economic theory that is concerned with the behavior of individuals, either consxmiers or production or marketing firms. Microvariables are

those that relate to such individuals. Aggregates, as their name implies, are

totals or index numbers of the microvariables. Macrotheory deals with economic relationships that should hold between aggregative variables.

The consistency approach.—Consider the traditional trichotomy: microtheory, aggregates, macrotheory. The consistency approach may be characterized

as follows: Given any two of the foregoing, determine the third in such a way

that it is consistent with the other two. Thus the consistency approach has

three variants: (1) Given the microtheory and certain properties of the macrotheory, construct aggregates of the microvariables consistent with the microtheory and the given properties of the macrotheory. This approach was taken

by Klein (¿6). (2) Given the microtheory and certain aggregates of the microvariables, find a macrotheory which holds between the aggregates and which is

consistent with the microtheory. This approach was taken by May (68, 69).

(3) Given a macrotheory which holds between certain aggregates, find which

microtheories are consistent with the macrotheory and under what assiamptions

we have consistency. This approach was taken by Klein (57) in a second study

in this area.

Theil (^, p. 5) gives a concrete example that relates to studies of this

sort. He says, "Suppose, for instance, that a microtheory tells us that each

family's consumption is an increasing function of its income. Suppose also

that all family incomes move up and down simultaneously. Then total consumption must increase whenever total personal income does, so that a macrotheory

in terms of these aggregates which tells us that total consumption is a decreasing function of total income is certainly not consistent with the

microtheory^"
- 86 .

The analogy approach,—Holte ¿2/ and Theil (^) are included in vhat we

call the analogy approach.

Theil's method may be characterized as follows: Suppose we are given a

microtheory and some simple aggregates of the microvariahles. We assume a

macromodel, analogous to the micromodel, \diich relates the aggregates. Now we

estimate the macromodel statistically and ask what the meaning of the estimated parameters is in terms of the parameters of the microtheory. In

particular we ask how the estimated parameters differ from simple combinations

of the microparameters^ such as s\ms or averages. Allen (2, p. 709) simimarizes some of Theil's results as follows:

"In the simple case where individual demands for a commodity are aggregated into a single relation between total demand and aggregate income, the

results can be expressed and interpreted:

"(i) an exact total demand relation is obtained by aggregating incomes

with weights proportional to individual marginal propensities to consume (or

individual income elasticities);

"(ii) with simple aggregation, the total demand relation is determined

statistically and its parameters generally depend on the movement of incomes

over the time period considered;

"(iii) with simple aggregation, again, the only situation in which there

is never contradiction in prediction between individual and total demand relations is when all individual marginal propensities to consume are equal;

"(iv) the only system of fixed weighting in aggregate income which leads

to no contradiction in prediction is that where weights are proportional to

individual marginal propensities to consxme, as in (i)."

Holte's approach is similar to Theil's in aim, although it differs in

method. In describing his approach. Holte ¿3/ writes "Another approach is to

specify a mlcromodel, postulate a macromodel and state that the macroparameters are such simple functions of the mlcroparameters as implicitly assumed

by many economists. We will then generally get a macromodel which is wrong

(if the micromodel is assumed to be true). By investigating what determines

'the degree of wrongness' in the macromodel we may perhaps obtain knowledge

which makes it possible to describe some types of economic situations in which

the analogy approach is a sound one, and other types in which it is dangerous . "

The formal approach.—The "formal" approach, given that title for lack of

a better one, is characterized as follows: Given a microtheory and some distributional assumptions concerning some or all of the relevant variables which

$2/ Holte, Fritz C. A New Approach to the Aggregation Problem. Cowles

Foundation Discussion Paper No. 21. I956. 26 pp. (Processed.)

12/ Ibid., p. 2.
- 87 -

serve to differentiate the individual \mits, derive both the appropriate aggregates and the macrotheory* Althoiigh this approach is an old one, we

mention only a few recent examples: (l) Tohin (22) used the assiimption of a

random distribution of tastes to derive aggregate demand fxinctions from individtial demand ftmctions vinder rationing, (2) Farrell (21, 22) followed

Tobin's lead and applied the results to combine cross-section data on incomes

and ownership of automobiles with time series data on income, prices, and

stocks of automobiles of various ages» While Farrell's eiapirical results are

not precise, his approach shows clearly the relation between problems of

aggregation and the construction of hypotheses consistent with different typesj

of data* (3) In an interesting paper, Houthakker (¿l) showed how to aggregate

the technological possibilities of individual producing imits into a production function for a group of units such as an industry. Houthakker showed

that, if we assimie a linear programming model for each producing unit and a

Pareto distribution of fixed resources among producing units, then we can

derive a Cobb-Douglas production fvinction for the industry as a whole.

Houthakker's approach is especially interesting, because it points out a way

in which technological and engineering data may be combined with time-series

or cross-section data on inputs and outputs. Its major defect is that it

deals only with firms and industries producing a single product, (k) Friedman

(hi) also used the "formal*' approach. He divides consxmrption and income into

two components, permanent and transitory. He shows that the relation between

consumption and income in a cross-section depends on certain characteristics

of the joint distribution of permanent and transitory canaponents of consumption and income among individuals and on the macro- and micro-relations

between the permanent components. Similarly the relation between consumption

and income over time depends on certain characteristics of the joint distribution among time periods. Although Friedman does not explicitly investigate

the direct combination of cross-section and time-series data, his approach

lends itself to such investigations.

Each of these approaches to problems of aggregation undoubtedly has something to offer the applied research worker, although the "formal" approach may

prove the most fruitful of the three in generating hypotheses that are consistent with several types of data. Further research by économetricians

appears to be needed before practical techniques for applied research workers

become available. Futtire developments in this area should be watched closely.

Equation Systems for Competing Commodities

Meinken, Rojko, and King (72, pp. 711-712) point out, "Since the days of

Walras, at least, economists have recognized the importance of substitution in

demand. Theoretically, the consumption of a commodity depends not only upon

its own price, but also upon prices of all other commodities. The Walrasian

formulation of the problem is too unwieldy for purposes of statistical measurement. Many statisticians,, however, have measured the interrelationships

among the demands for two or three commodities—neglecting the minor effects
- 88 -

of other coimnodities and services. Thus, they have meas"ured competition between beef and pork, without worrying too much about the undoubted fact that

beef consimiption may be slightly affected by the price of gasoline, shoes, and

movie tickets.

"Most of the statistical work done on this problem has been based either

upon demand and cross elasticities, or upon the relation of cons\miption ratios

to price ratios. Recently Wa\agh (^) has derived a partial indifference surface from market data. ... Since the demand functions for two commodities,

the relation of the consimption ratios to the price ratios, and the indifference function all involve the same variables—prices, quantities, and income—

and provide measures that are designed to indicate the competitive relationship between two commodities, it appears, on an intuitive basis at least, that

it would be possible by appropriate mathematical transformations to go

directly from any given approach to the other two. We shall, in fact, show

that it is possible to go directly from the demand functions to the ratios, or

to the indifference surface, but it is not possible to go the other way."

These authors next consider in detail three approaches that have been

commonly used to study relationships among competing products. They show that

one of these typically has been used in an incorrect way and that a second

frequently yields misleading conclusions.

They say, "Demand theory conventionally specifies that, for an individ\ial

consumer, the quantity of beef or pork consumed depends upon prices of beef,

pork, and all other commodities, the individual constmier's income, and factors

that reflect changes in tastes and preferences." If time series data that

relate to purchases by individual consumers are available, a least squares

regression of consumption of either product on their respective prices and

other relevant variables provides coefficients that can be used to derive

estimates of the direct and cross elasticities that are statistically consistent.

These authors then say, "Market demand, which is the summation of these

individual demands, may be defined as follows:

Qb = f(Pb. Pp. Y. ^i) (96)

Qp - f(Pb, Pp, Y, ug) (97)

where the Q*s represent the aggregate consimiption of beef (Q-^) and pork (Qp),

the P*s represent market prices of beef (P^^) and pork (Pp); Y represents

aggregate consimier income; and the u's represent random disturbances that •

affect consumption of beef and pork. As no separate allowance is made for

substitute commodities such as other meats and fish, the u*s also include the

effect of changes in the price or supply of these. Other meats and fish are

believed to be relatively unimportant in affecting the quantity of beef and

pork consumed.
- 89 -

"If time series data on prices^ qioantities and incomes are given^ the

method used to estimate the coefficients in these demand relations depends on

assumptions that are made regarding the type of functional relation that generates the observed data. For beef and pork, as for many other agricultural

commodities, production suid consumption in any given period are essentially

predetermined, that is, the supply curve is completely or almost completely

inelastic. ... When consumption of two competing agricxiltural commodities can

be assimed for all practical purposes to be independent of prices in the current period, the procedure normally has been to estimate the coefficients for

the relations that express the price of each good as a function of the two

qixantities and income. This procedxare gives (statistically) biased results

when applied to two or more competing commodities because ... prices for each

commodity are simultaneously determined by the interaction of demand factors

and the supply of each. Thus a given combination of production of heef and

pork results in a unique set of market prices that is simultaneously determined. To obtain estimates of the elasticities of demand that are statistically consistent, the parameters in the structural demand equations (96) and

(97) must be estimated by a statistical method that allows for this simultaneity. Equations of the sort discussed here always are just identified.

Hence, the reduced-form method of fitting simultaneous equations can be used

to estimate the coefficients" (72, pp. 713-715)• This method is discussed in

detail in the section beginning at the bottom of this page.

With respect to, the third method, these authors (72, p. 717, 726, 732-

733) say, "Various research workers have derived statistical estimates of the

elasticity of substitution (Eg) hy a short-cut method of relating price ratios

and consumption ratios. ... Two conclusions with respect to the elasticity of

substitution can be drawn. First, an empirical estimate of the true Eg can be

obtained by relating price ratios and consuiiiption ratios only under very restrictive conditions, ... When price ratios and quantity ratios are related, a

poor fit may be indicated for direct substitutes and a good fit for independent commodities. Moreover, the regression coefficient will not necessarily

tell us whether goods are competing, independent, or complementary, let alone

the »ease of substitution.» Second, it is demonstrated that even knowing that

two goods are competing, the nimierical value of Eg [O - 00] tells us little

about the *ease of substitution* or the degree of competitiveness between the

goods, its designed puirpose. This follows since the measure Eg is a combination of direct price elasticities and cross price elasticities and income

elasticities." Hence they conclude "that the statistical estimation of Eg is

of dubious value, and its estimation by the price-ratio consumption-ratio

method, with no other check as to the nature of the demand interrelationship,

is meaningless."

Estimation of direct and cross elasticities from market data when consxmiption of each commodit;^ can be taken, as a predetermined variable.—In this

section, we consider methods that can be used to estimate direct and cross

elasticities of demand, as commonly defined in economic literature, for three

commodities that are believed to be competing, when consumption of each is

believed to be a predetermined variable (see page ^7). The use of three
- 90 -

commodities, rather than two as done frequently in the past, illustrates certain aspects of the method, namely, that of providing an analytical framework

which readily can be extended to more competing commodities. To simplify the

notation and equations, we assxmie that all variables that depend in part on

(l) the size of the population are expressed in per capita terms and (2) the

general price level are deflated. Consumer income is considered as a predetermined variable and consumption of each item is assimied to be a linear

fianction of retail prices of the several competing items and consumer income.

Most of these assumptions must be made implicitly in order for eqtaatioias (96)

and (97) to be consistent with the general principles developed so far in this

handbook.

We thus imply that our model consists of the following structural equations. The subscripts a, b, and c are used respectively for the three commodities. The variables are assimied to be expressed as deviations from their

respective means so that the constant terms can be omitted.

qa = "bllPa + T^12Pb + ^13Pc + ^ikY (98)

q-b = b2iPa + l:>22Pb + ^23Pc + '^2ky (99)

ÇLc = "bslPa + ^32Pb + T^33Pc + ^Sl^y* (lOO)

It is now clear why such equations are often just identified. To be just

identified, the number of variables in the system minus the number of variables in each equation must equal the n\miber of endogenous variables in the ^

^yj?i:?5_^iîH^, one. Stated another way, the nimiber of variables omitted from

each equation must equal the number of endogenous variables less one. If we

have n commodities and hence n endogenous prices, we always omit consuniption

(a predetermined variable) of n-1 commodities from each equation; hence each

equation is just identified. This condition holds only if each predetermined

variable (any number) in the system, but excluding the consumption variables,

appears in each equation. For example, equations (98) to (lOO) remain just

identified if we allow for the nonlinear per capita and deflated variables by

the method described in the section beginning on page 71, as we add two predetermined variables to the system but incorporate each into each equation, so

that the counting rule is unaffected. /

However, if any variable besides prices of the n commodities is ass\amed

to be endogenous, as for example consxmier income, the system of equations as

given is incomplete as there are more endogenous variables than equations. If

consxmier income is the additional endogenous variable, balance may be restored

by adding an income equation which includes special factors that explain income behavior and that can be treated as predetermined variables. The demand

equations then are overidentified unless only one special factor is used to

explain income, and the income equation always is overidentified. In the

remainder of this discussion, we consider all variables other than price to

be predetermined.
- 91 "

As discussed in the section beginning on page 62> in the method of reduced forms each endogenous variable is expressed as a function of all of the

predetermined variables in the system. These equations are known as reduced

form equations; estimates of the coefficients that are statistically consistent and efficient can be obtained if they are fitted directly by least squares

as each contains only a single endogenous variable. In this instance, the

three reduced form equations are as follows:

Pa = Bll^a + Bigqb + Bißqc + Bi^y (lOl)

Pb = B2iqa + B22qb + B23qc + B2i^y (l02)

Pc ^ ^Sl^a "^ ^32% + ^33^0 + S34y- (l03)

By the same kind of algebraic manipulations as shown on page 62 the

relations between the coefficients in the reduced form and the structtiral

equations can be derived. Such relations are given by Meinken, Rojko, and

King (72, p. 73^) for a system of equations for two commodities. They point

out that the reciprocal of the "price flexibility" (see page 8l ) for beef

equals the direct price elasticity for beef only if the cross elasticities of

beef on pork and pork on beef each are zero. They say further, "The algebraic

relation of the coefficients indicates clearly that the reciprocal of the

cross price flexibility does not give the cross elasticity of demand. The

same holds for the income coefficient."

Algebraic relationships of this kind become complex when we deal with

more than two commodities, but the relationships can be stated simply by use

of a matrix notation. It is suggested that readers who are •unacquainted with

this notation skip the next three paragraphs. Meinken, Rojko, and King introduce time as a variable into each of their equations, and in the example given

here we do likewise so as to include two predeteimined variables in each

equation.

In order to use a matrix notation, it is convenient to write our equations making use of somewhat different symbols. The reduced form equations

with which we deal in this section are as follows:

Pi = ^11^1 + ^12^2 + ^13^3 + Ciiy + Ci2t (l04)

P2 = ^21^1 * ^22^2 "*" ^23^3 + ^21^ " " ^22* (l05)

P3 "" ^31^1 "* ^32^2 "* /'^33^3 "*" ^31^ + C22t. (106)

In connection with systems of equations, the predetermined variables frequently are designated by the letter z. Here we wish to treat the predetermined variables that relate to consumption in one way and the other predetermined variables in another way. So we include y and t in a vector (z), the

three quantities in a vector (Q), and the three prices in a vector (P). Primes
92

are used on the vectors in equations (llO) and (llO.l) to indicate that they

are to be written as column vectors. We also work with a loaatrix (B) of the

Bij coefficients and a matrix (C) of the Cij coefficients.

Each of the reduced form equations is fitted by least squares^ using the

single endogenous variable as dependent. After obtaining estimates of the B±¿

and Cij coefficients^ we wish to transform these algebraically into the structtiral coefficients—the b^j and Cij—that are used in the following structural

equations :

q^ = bi3_p;L + ^12P2 + ^13^3 + ^11^ "*" ^12^

^2 = "'^2lPl + 'b22P2 + ^23P3 + ^21^ + ^22"^

^3 " "''Sl^l *• ^32^2 + ^33^3 "* ^3iy *" ^32*-

(107)

(108)

(109)

Equations (l04) to (l06), treated as a unit, can be written in the following matrix notation:

P' = BQ' + CZ'. (110)

If we multiply each term of equation (llO) by B~^ and rearrange the resultant

terms, we obtain:

Q» = B-ip, « B-lcz^ (llO.l)

In this form, we have a matrix equation that is equivalent to our structural

equations, where the bij coefficients are found in matrix B""l and the cij coefficients in the product matrix - B^^C. By carrying out the matrix operations indicated, we obtain the structural coefficients. The matrix equation

applies directly to any number of variables.

We now summarize the above for readers who are not acquainted with matrix

notation although we shall, in fact, make use of such notation. Derivation of

the structural coefficients involves the following steps:

(1) We fit the reduced form equations (104) to (l06) by the method of

least squares. In doing so, advantage should be taken of the fact that the

same independent or predetermined variables are used in each equation [see

Friedman and Foote (kO, pp. 69-76)].

(2) We take the B coefficients obtained in this way and write them in the

following form:

"Bii B12 B13

B21 B22 B23

B31 B32 B33 ^
- 93 -

This is in effect a matrix and must be inverted according to standard rules

for a nonsymmetric matrix. Any mathematician and most statisticians know how

to do this, and programs for inverting matrices are available for any electronic computer. ¿4/ The answer is in the same fona as this, but different

numbers appear in the cells.

(3) The b coefficients for the structural equations (lO?) to (109) are

found in the following locations within the inverse of the matrix of B coefficients :

^11 TD12 1=13

b2i b22 ^23

.^31 1332 b33

(h) We form a matrix of the C coefficients obtained by least squares of

the following form:

Cii CigC21 C22

C31 C32

and premultiply it by the inverse of the matrix of the B coefficients. This

again is an operation that can be performed by any mathematician and most

statisticians or can be done electronically. 35/ Upon completion of this

operation, a new set of nxombers are formed, within which the c coefficients

for the structural equations are foimd at the following locations:

^11 ^12

C21 C22

f31 ^32^

By making use of these simple principles and calling upon the aid of

mathematicians to perforai what for them are simple mechanical operations, we

can go from the coefficients of the reduced form equations to the coefficients

of the structural equations with greater ease than by carrying out algebraic

manipulations of the sort described on page 62 even for two competing commodities, and we can handle as many commodities as are involved in the analysis.

5^/ An outline for carrying out such computations on desk calculators is

given by Friedman and Foote (1+0, pp. 98-IOO), and elsewhere.

¿5/-Matrix multiplication on desk calculators is described by Friedman and

Foote (40, pp. 23-26), and elsewhere.
- 9k -

Unfortxinately, it is difficxxlt to obtain standard errors for the structural

coefficients even for two competing commodities, although methods to derive

these standard errors are available. 36/

Estimation of direct and cross elasticities from market data vhen

consimiption of one or more commodities is at least partially endogenous > —When

we drop the assumption that consumption is a predetermined variable, we face a

variety of equation systems. However, as some of the equations normally are

overidentified, the aspects of simultaneity that cause complications when we

fit by the method of reduced forms are taken care of automatically, so that

the derivation of direct and cross elasticities from such systems frequently

is a simple matter. This, point can be illustrated by reference to two equation systems referred to previously.

In the model for dairy products shown on page 13^ three equations involve

cross elasticities, namely equation (l4) for butter, (15) '^OT cheese, and (17)

for margarine. In the economic model, two endogenous variables are shown on

the left of the semicolon in each case, namely the quantity of the product in

question and the price of either that commodity or its competitor. In writing

down the equations in the statistical model, however, the quantity of the

product normally is written to the left of the equality sign in systems of

this type, and the direct and cross elasticities can be obtained directly from

the coefficients shown. Standard errors for the structural coefficients are

given directly by the limited information method. A similar situation prevails with respect to the model for asparagus shown on page l6« Althoiigh fo\ir

endogenous variables are shown to the left of the semicolon, only a single

endogenous quantity is involved and this normally is shown on the left of the

equality sign in the statistical model. Again, the direct and cross elasticities can be obtained directly from the structural coefficients when the equations are written in this way.

The author, in cooperation with 01man Hee, Anthony S. Rojko, and Will

Simmons of the Agricultural Marketing Service, recently developed a model for

potatoes during late winter and spring that illustrates the kind,of algebraic

manipulations that are required when consumption is at least partially endogenous and when the struct\iral equations contain more than a single quantity

variable. As a unique set of manipulations in general is required for each

such model, only the general approach is outlined here. In this model, we

consider the demand for old and new potatoes separately in each of two periods

—late winter (January-April) and spring (May-Jvme). Price relationships

between new crop potatoes in the two periods also are measured. The supply of

old crop potatoes to be consimied during both periods is determined by stocks

on January 1, but the quantity used in each period is determined simultaneously by the economic variables that enter into the model. Production of new

56/ The general approach is described by Klein (¿8, pp. 258-259), but

certain partial derivatives must be obtained which for these equations axe

algebraically complex.
- 95 -

crop potatoes within each period is assumed to he a predetermined variahle.

Demand for new potatoes in period II is assimaed to depend in part on the quantity of new potatoes availahle in period I*

The following variables enter into the economic model. As in previous

models discussed in this section, we assume that quantitative variables are

expressed in per capita terms and that variables affected by the price level

are deflated. As prices at retail are not available in the detail required, a

marketing margin variable is included in each of the demand equations (see

pages 214.-26).

PQJ - Local market price of old potatoes in period I

POTJ" Local market price of old potatoes in period II

PNJ - Local market price of new potatoes in period I

PUjj- Local market price of new potatoes in period II

Sj - Stocks of old potatoes used for consumption in period I

Sji - Stocks of old potatoes used for consuinption in period II

S « Stocks of old potatoes on hand January 1

Ql T Production of new potatoes in period I ] In each case, most

> of these are used

Qll « Production of new potatoes in period II j directly for food

Yi - Disposable consimier income in period I

Yji - Disposable consimier income in period II

M - Factors that affect marketing costsThe first 6 variables are assimied to be endogenous.

The following equations are involved in the economic model:

Sj, PQJ, PNJ ; Yl^ M (Demand for old potatoes in period I) (ill)

^Oi^ %I ^ Ql> Yl^ M (Demand for new potatoes in period l) (ll2)

Sll> ^Oii^ %ii i ^11^ M (Demand for old potatoes in period II) (ll3)

^Oii^ PNII ; Qll> ^Iß Yllí M (Demand for new potatoes in period II) (ll^)
- 96 -

Pjf-, Pu-^ j Ql, Qll (Normal relation between prices of new

potatoes in periods I and II) (ll5)

Sj + Su = S. (identity with respect to stocks) (ll6)

Equation (llH) contains more than one variable that relates to quantity,

whereas, to determine direct and cross elasticities, as normally defined in

economic literat\ire, we need a single quantitative variable as a fimction of

the several prices and other variables used in the model. An equation in the

needed form can be obtained by the following algebraic operation:

Substitute equation (ll5) into equation (11^) to eliminate Qj and get

Qii = f(Poii> %!. PNII. YII. M). (117)

Any person who has a good grasp of college algebra can make substitutions of

this sort.

Standard errors of the coefficients involved in the modified equation ce.n

be determined by the method given by Klein (j^, pp. 258-259)- The amount of

work involved in using this method depends on the nature of the algebraic

transformations. In some cases, the task is complex; for this eq\aation, the

task is easy, given a knowledge of calculus.

A partial indifference surface.—In this section, we present a method developed by Waugh (99) to derive a partial indifference surface from market

data, although the discussion follows the summary of his approach given by

Meinken, Rojko, and King (72, pp. 726-732). They say, "The partial indifference function used by Waugh is

Z = A [ll [^ :SJj Q^l+C^-f) + Qpl-('^-s) (118)

where Z represents the elevation of a siorface. For each value of Z, there

exists a contoior line (partial indifference curve) that represents substitution possibilities of beef and pork for which consumers are indifferent. In

contrast to a production function, in which Z can be taken as a direct measure

of output, the Z in (ll8) is not intended to give a direct measure of the

level of utility. Instead, Z may be considered a monotonie [or steadily increasing or decreasing] fimction of utility from which it is possible to

obtain contours (indifference curves) for each value of Z."

If Pj» equals the ratio of beef prices to pork prices at retail, this

partial indifference function implies that

Pr = Aft^í^-^^Qpí^"^) 2L/ (119)

57/ The reasoning involved is given by Melnken, Rojko, and King (^2, p. 727)

and is svanmarized here. If Z is a single valued, monotonie function of utility, say Z = 0 (U), the marginal rate of substitution (or the slope of a
- 97 -

In logarithinic form, this eq.\ials

log Pr = log A + (b - f) log Qb + (c - g) log Qp. (ll9-l)

Estimates of the coefficients in eqviation (ll8) can be ohtained algebraically

from the coefficients in eqxiation (ll9) or (ll9*l) which, in turn, can be

derived from an equation, fitted by least squares, of the form

log Pr = log Ve + (b - f) log Qb + (c - g) log Qp + (d - h) log Y, (120)

which includes income as a variable. For any given level of income, if we

substitute the value of Y in eqxmtion (120) and transform to the original nonlogarithmic variables, we obtain an equation like (II9), from which we can

obtain the coefficients in equation (II8). In so doing, the value (d - h) log

Y must be combined with log ^¡^ to get log A from which the value of A is

derived•

Waugh's analysis implies that for each level of money income, there is a

definite indifference surface for beef and pork. This means that the constant

(A) in equation (II9) may vary with income. This also is true for the constant in equation (II8). In a sense, equation (II8) is that of a threedimensional surface, and may be thought of as a cross section of a fourdimensional surface, holding income constant at some specified level.

Meinken, Rojko, and King then apply this method, with some modification,

to their data on Canadian beef and pork consumption and prices, and state,

"Figure 7 shows, for each year, the contour line (partial curve) associated

with the quantities of beef and pork consumed had per capita real income been

contour line) is

BZ y BZ ^ B[0 (U)] / B[0(U)]

BQb ' BQp BQ^ / BQp

where B [ 0 (U)]/ BQb eqimls the marginal utility of beef and B[ 0 (u)3/B Qp

equals the marginal utility of pork. In competitive equilibrium, the marginal

rate of substitution between beef and pork must equal the ratio of their

prices or

BZ , BZ^

BQb / BQ^Ip - ^^

But the slope of the contours of Z in equation (II8) at any point is

which also equals the price ratio, that is.
. 98 -

$6if8.53 in that year» ... Figure 7 also shows the adjusted price ratio (P^)

associated with the quantities of beef and pork consumed during that year had

per capita real income remained constant at $648.53» ... If conditions of

competitive equilibrium are met and if the contour function represents the

true substitution possibilities between beef and pork, we would expect the

adjusted price ratio to be tangent to the contour line for the particiaar

quantities of beef and pork consumed for that year. 58/ Inspection of figure

7 tends to substantiate this formulation." A chart of this sort also is presented in the original article by Waugh (^).

Measuring Relationships Among Complementary Products $9/

Some commodities, like edible fats and oils and sugar, are used almost

exclusively as ingredients in combinations of foods in which they often

account for a relatively small part of the total cost. At any given time,

consumption of such commodities tends to be related to cons\mg)tion of other

foods by a set of technical coefficients—nuniber of teaspoonfuls of sugar per

cup of coffee, ounces of butter-plus-margarine per po\md of bread, ounces of

salad oil per imit of salad vegetables, and so forth. If the demand for

sugar-using foods increases with consumer income, the demand for sugar also

will appear to increase with income, even though no consumer actively values

sugar for its own sake.

Consider the following two demand equations for sugar:

qg = ai + biPs + ciy (l2l)

and

m

qg = a2 + bgPs + cgy + d D Viq.i^ (l22)

i=l

^8/ The shape of the indifference curve depends on the degree of substitution between a pair of commodities and becomes a straight line for goods that

are perfect substitutes. The shape of the empirical indifference curve presented here depends directly on a particular combination of elasticities of

demand for beef and pork. As implied in the discussion of the elasticity of

substitution, results obtained from a method of estimation that is based on

some combination of direct and cross price elasticities do not provide sufficient information (l) to conclude that a pair of commodities are substitutes

and (2) to deteimLne the precise degree of substitution when they are known to

be substitutes. Thus, the empirical indifference curve given here also is

subject to these limitations. We know that beef and pork are substitutes, but

the relative flatness of the curve does not tell us the precise degree of

substitution between these commodities.

59/ This material was developed initially by Fox and was first published by

Foote and Fox (29, p. 18). A brief discussion along similar lines is given by

Fox (33, PP- lé^7).
- 99 -

PARTIAL INDIFFERENCE SURFACE

FOR BEEF AND PORK*

PORK (LB.)^

20 25 35 40

BEEF (LB.)

ABASED ON PER CAPITA CONSUMPTION AND ADJUSTED PRICE RATIOS. CAHADA, 1928-41, ?94«-5? ANO 1953

U. S. DEPARTMENT OF AGRICULTURE NEC. 3092.56j('2) AGRICULTURAL MARKETING SERVICE

Figui-e ?•—In the method developed by Waugh (99), the extent to which the adjusted price ratio approximates the tangent to the contour curve for the

particular quantities of beef and pork consumed for that year provides confirmation of the belief that these contour curves represent true indifference

surfaces for a given level of consumer income.
- 100 -

in which the last term of equation (l22) is an index nxjimber of consxanrption of

sxigar-using foods weighted "by the quantity of sugar (WJL) customarily used with

a unit of each food. Given a certain consumption level for sxigar-using foods,

hoth price and income elasticities of demand for sugar are "believed to be

small. Hence, in general, b2 would he considerably smaller than bi and C2

woiild be considerably smaller thaja c^.

Other examples of completing goods may be fo\md in rayon and cotton in

the very short rvn (a few weeks or months). If the proportion of rayon in a

rayon-cotton blend is inflexible, an increased price for cotton, passed

through into the finished product, may curtail consumption of both cotton and

rayon. During a longer period and for the textile market as a whole, the

competitive relation between rayon and cotton would predominate. Completing

relationships also may exist among different types of tobacco used in a

standard blend.

Derived Demand Equations, Partially Reduced Form Equations, and

the Estimation of Demand at Different Market Levels 6o/

In this section we consider a commodity that is generally sold by producers to intermediaries, who may be processors, wholesalers, retailers, and

the like. We think of a market as divided into three sectors or gro\aps of

individuals: (l) consumers, (2) the processing and marketing group, and

(3) producers. For given values of other relevant variables, we may think of

the quantity produced, the quantity consumed, the price paid to producers, and

the price paid by consumers (that is, the retail price) as being determined by

four relations: (l) the producers* supply relation, (2) the marketing group's

demand relation, (3) the marketing group's supply relation, and (h) the conSTjmers* demand relation. Let

QQ = quantity consumed

Qp = quantity produced

Py = retail price

P^ = price paid to producers

D = disposable income

Zx = other factors affecting consimier demand

60/ This section was prepared by Marc Nerlove, agricultural economic statistician. Agricultural Marketing Service. It is based in part on Hildreth and

Jarrett (^2^ PP* 107-112) and in part on Nerlove, Marc: The Predictive Test as

a Tool for Research: The Demand for Meat in the United States, M. A. thesis,

Johns Hopkins University) 1955^ especially pp. 5-18.
- 101 -

Z2 == other factors affecting the marketing group's

supply and demand

Zo = other factors affecting producer supply

The four relations can be written in the following fianctional form, where

the Greek letter indicates that we have an eqxaation of some sort that involves

the variables shown in the parentheses on the left of the equality sign:

^ (QC> Pr> ^p ^i) = 0 (consumer demand) (123)

S(QC^ Pr> ^w^ ^2) = 0 (marketing group's supply) (12^)

a(Qp, Pr^ ^w^ ^2) == 0* (marketing group's demand) 61/ (l25)

M.(Qp, P^, Z3) =0. (producer supply) (l26)

In this model, we neglect problems raised when a large part of the quantity produced is exported or stored or when a large part of the quantity consumed is imported or comes at times from stocks• If the commodity is highly

perishable, or if storage is sufficiently expensive, we may assime that the

quantity sold to consixmers is the same as that sold to the marketing group. In

this case Qe = Qp> when the two variables are measured in comparable units.

If equations (l23) and (125) have an appropriate form—for example,

linear or linear in the logarithms of the variables—the marketing group's

supply and demand relation may be represented by a single equation, which can

be called the marketing group's behavior relation:

^(Qc> Pr> ^w^ Z2) = 0. (marketing group's behavior) 62/ (12?)

Equations (123), (126), and (127) constitute a simple model; more complicated forms are likely to be encountered in applied work. But the model shown

is useful to illustrate several fundamental points. Equation (l27) is a reduced form eqioation, although it is not, strictly speaking, a derived demand

or supply equation.

Derived demand and partially reduced form equations.—If equations (123)

and (127) are of an appropriate form, we can eliminate the retail price from

each. This follows from the equilibrium condition that the qimntity supplied

61/ a is a demand function for a factor of production, namely, the raw

material; actually our system contains demand functions for other factors, but

we neglect them here.

62/ This equation corresponds to equations (57) or (60) on pages ^ and 25,

respectively. This equation is interpreted as a supply function for marketing

services in subsequent paragraphs. It can be obtained from equations (l23-

126) in a variety of ways; in each, exactly the same equation is given.
- 102 -

by the marketing group eqiaals the quantity demanded by the consumer group. If

this is done, we obtain a relationship among the variables Qe> Pw^ D, Zi, and

ZQ which might be called a derived demand relation:

Ö (Qc, P^, D, Zi, Z2) =0- (derived demand relation) 6¿/ (128)

Alternatively, under the conditions specified, we can eliminate the price paid

to producers« In this way, we obtain a relationship among the variables Qc

Pp, Z2> and Z3:

^ (Qci Pr> 2;2, Z3) = 0. (partially reduced form supply) 6k/ (129)

An important aspect of eq-uations (l28), and (129) is that each is a partially reduced form equation derived from a structural eq\mtion by the elimination of a price through the use of an equilibrium condition.

Hildreth and Jarrett (j¿2, p. I08) make the following interesting point

concerning partially reduced form equations: "Equations obtained by simultaneously eliminating one or more eqioations and one or more endogenous variables

from á model have been called partially reduced form equations in various

discussions. In a certain fundamental sense, all equations we are likely to

deal with may be regarded as partially reduced form relations. It is always

possible to imagine a more fundamental explanation of the phenomena that we

observe, involving more equations and more endogenous variables. If the mo^del

we use is a reasonable one, it should, in principle^ be possible to derive it,

either exactly or approximately, from the more fundamental model by successive

elimination of variables."

Thus, additional equations and equilibrixmi conditions might be used to

eliminate more variables than P^. or P^ from the system defined by the equations (123-126). For example, if Z2 and Z3 are prices of inputs or other

outputs of the marketing or producer group, we coiild obtain the demand or

supply schedules for the factors or products to which these prices apply from

a knowledge of the appropriate production fxmctions. With these equations and

the appropriate supply equations for factors or demand eqiaations for products,

we could, by using the equations already in the model, eliminate Z2 and

Z3. 6¿/

63/ This equation corresponds to equations (58) or (61.2) on pages 2^^ and

26, respectively.

6k/ This type of relationship was used by Nerlove, op. cit., in his study of

the demand for meat.

65/ Conceptually, this procedure is analogous to that used in showing the

relationship between the Walrasian general eq\iilibriim framework and the

Marshallian partial equilibrium framework.

The basis for the statement made on page 2k ^ that M cannot be eliminated

from equations (5?) or (58) unless further eqxaations are specified, follows

from the discussion in the text.
- 103 -

If data are available on all the variables, we may estiioate the parameters in equations (l23)^ (l26), and (12?) jointly. If, however, retail

prices or prices paid to producers are not available, we must use a model

which includes a partially reduced form eqxiation; that is, we must use the

model consisting of equations (l23) and (129) if prices paid to producers are

not available, or the model consisting of equations (126) and (128) if retail

prices are not available. Unless additional equations and equilibrium conditions are specified, the variables Z2 and/or Z3 cannot be eliminated from the

system; hence, the procedure of estimating the elasticity of demand at the

producer level of the market by using an equation which does not contain Z2,

such as

0' (Qc> Pw. D. Zl) = 0 (130)

is incorrect, unless we assxmie that no variables other than Q^, P^ and Py

enter the marketing group's behavior relation.

Thus, if retail prices are not available, we cannot simply substitute

prices at some other level of the market; additional variables relating to the

behavior of the group of individuals between the final cons\mer and the level

of the market to which the price variable refers should, almost always, be included in the demand equation. For example, if wholesale rather than retail

prices are used, variables that relate to the behavior of retailers should be

Included in the demand fvmction, since the latter is really a partially reduced form equation or a derived demand equation and not a consumer demand

equation,, If the coefficients of these additional variables do not differ

from zero by a statistically significant amount, they may be dropped from the

analysis on the assumption that they do not affect the behavior of any group

of individuals standing between the final consumer and the level of the market

at which demand is measxared.

Elasticities of demand and supply from partially reduced form equations

in relation to those from structural equations.—Since, in many instances, we

may be forced to use a partially reduced form equation in our models rather

than estimating equations (123), (126), and (12?) Jointly, the relation between the elasticities of demand or supply derived from the partially reduced

forms, equations (l28) or (129), and those that would be derived from the

original structural equations, (l23) or (12?), is of some interest. Hildreth

and Jarrett (^, pp. IO8-II2) obtain such a relation when a derived demand

equation is estimated rather than a true consumer demand equation. Nerlove

(op. cit.) obtained such a relation when a partially reduced form supply equation is estimated rather than a true producer supply eq-uation. Here we state

their results without proof. Beginning on page 106, a simple proof of one of

the Hildreth-Jarrett results is given.

Let

E^ p = elasticity of demand at retail with respect

to price
l^çD = elasticity of demand at retail with respect

to disposable income

E(^^p^ = elasticity of demand vith respect to the price

paid producers derived from estimates of the

coefficients in equation (l28), the derived

demand relation

E^¿D = elasticity of demand with respect to disposable

income derived from estimates of the coefficients in equation (l28), the derived demand

relation

All these elasticities are defined so as to be positive. What is the relation

between EQ^P^, and ^¿p^ and between BQ^D and íQ¿DÍ

Let Ep^^ be the elasticity of the retail price with respect to the price

paid producers. This elsusticity could, if data were available, be derived

from estimates of the coefficients in equation (l27). It might be called the

"elasticity of price transmission." Hildreth and Jarrett show that

1Q P • Ep p

^cPv = 1 - A . i'^ > (131)

where A is the change in retail price associated with a imit change in the

quantity consumed in relation (l27)> that is,

and B is the change in the quantity demanded by consimiers associated with a

unit change in retail price, that is,

The minus sign is introduced on the right because E^^p-, is defined as a posi-

/BQc\ tive quantity whereas H^p" c ^^> o^ coxjrse, typically negative. Similarly,

Hildreth and Jarrett show that

^cD , .^

where A and B are defined as before.
- 105 -

Two important groups of relations follow almost immediately frcan eq.-uations (131) and (13^):

(1) If the quantity passing through the marketing sector does not influence the retail price, so that A = 0, the income elasticity of derived demand

equals the income elasticity of consumer demand at retail, ^j Beginning on

page lOT, however, a discussion is given which suggests that, in general, A is

a positive nimiber different from zero. Since B normally is negative, it then

follows from eq\iation (13^^) that the income elasticity of derived demand tends

to he less than the income elasticity of consumer demand at retail. Thus, in

general.

Hence, if retail prices are imavailable, the best we can do is to set a lower

limit to the income elasticity of consiimer demand at retail from the coefficients in the partially reduced form equation.

(2) Prom eq.uation (131) we see that, even if A = 0, the price elasticity

of derived demand differs from the price elasticity of consumer demand at

retail by an amount depending on the elasticity of price transmission• Unless

we can specify whether Ep^P^ is greater than, less than, or equal to one, we

cannot say whether ^^^ is a lower or upper limit to B^^p^. For a large

class of cases, however, we expect the elasticity of price transmission to be

less than or equal to one. This class includes such cases as (a) a constant

dollar marketing margin, (b) a constant percentage marketing margin, and (c) a

marketing margin that is any increasing monotonie function of the quantity

passing through the marketing system. 67/ If Ep^^p^ is less than or equal to

one, it follows from equation (131) that

^cPw^ ^cPr- (136)

Thus, in á wide variety of cases, the price elasticity of derived demand is a

lower limit to the price elasticity of consiomer demand at retail.

We next state Nerlove's result for the relation between the elasticity of

supply derived from the partially reduced fonn supply function and the time

producer supply elasticity. 68/

(>(>l The reader should remember that equation (127) shows essentially the

relation be.tween retail and wholesale prices. If A = 0, then the marketing

margin is xmaffected by the quantity moving through the marketing sector.

67/ Proofs of these statements are given in the Appendix, pages 203-205.

38/ This result is given and proved by Nerlove (op. cit., pp. 8-IO). The

proof there follows the saone general approach as that used by Hildreth and

Jarrett (^^ PP- IO8-III).
- 106 -

Let

EQçP^ = elasticity of producer supply with respect to the

price paid to producers, that is, as derived from

estimates of the parameters of equation (l26)

EQ^P^ = elasticity of supply at retail, that is, as derived

from estimates of the parameters of equ&tion (l29),

the partially reduced form supply relation

Both BQCPW ^* ^c^r "tend, of course, to "be positive. Nerlove shows that

rt ^^c w r^w

^cPr = 1 - A C ' (137)

where C is the effect of a unit change in the price paid to producers on the

qtiantity supplied, that is,

/ ^^c \ ~ Qc

'= l"W7f^ ,=^cPw--^- (138)

The result Which follows from equation (l37) is not as clear cut as the

results which follow from equations (l3l) ^nd (l3^). If the quantity passing

through the marketing system has no effect on the retail price, A = 0. In

this case, by the arguments given ahove, the elasticity of supply at retail

tends to he less than the elasticity of supply at the producer level, since

Ep^^ tends to be less than imity. On the other hand, since C is positive, if

the qtiantity passing through the marketing system does affect the retail price

strongly, 1 - A • C may be much less than one^ and tend to offset the effect of

the elasticity of price transmission. It may thus happen that the elasticity

of supply at retail exceeds the elasticity of supply at the producer level.

For commodities like eggs, meat, or milk, there may be both a significant producer supply response to current price and a significant effect of the quantity passing through the marketing sector on the marketing margin. In these

cases, the elasticity of supply at retail may well exceed the elasticity of

supply at the producer level. In such instances, results from the use of

least squares procedxires to estimate the consimier demand function at retail

may be more seriously biased, in a statistical sense, than consideration of

the elasticity of producer supply would lead one to expect.

Proof that the price elasticity of derived demand is typically less than

the price elasticity of demand at retail.--A simple proof of equation (ISFJ

may be obtained from a particular interpretation of the marketing group's

behavior relation—equation (12T). It is primarily because of an interest in

this interpretation that we give the proof here.
- 107 -

Equation (l27) states that there is a relation "between the quantity passing through the marketing sector^ the retail price, the price paid to producers, and other variables related to the marketing sector. In order to

write this relation, we assume that the quantity supplied "by producers is

approximately the same as the quantity used hy consumers. The marketing group

may be viewed as adding certain services to the commodity under consideration

as it passes through the marketing sector which, in total, are roughly proportional to the quantity passing through. Consequently, equation (12?) may be

interpreted as a particular kind of supply fxmction, namely, the supply

function for marketing services.

A supply function for a commodity or service usually contains the price

of that commodity or service, but equation (12?) contains two prices, the

retail price and the price paid to producers, and neither is the price of marketing services. If the function r) is of the appropriate form, we can remedy

this situation. 62/ Let the difference between the retail price and the price

paid to producers be Pg, so that

Pg = Pp - Pw . (139)

Ps is the dollar amoiont per "unit of the commodity passing through the marketing system which goes to individuals in the marketing group. If the services

which they supply are roughly proportional to the quantity passing through, Pg

may be interpreted as the price of marketing services; hence, if r) is of an

appropriate form, we may derive the following equation from equations (12?)

and (139) and this can be interpreted as the supply of marketing services

relation:

^ (QC^ ^B> ^2) "= 0* (supply ot marketing services) (l^O)

From equation (l^O) we see that the supply of marketing services has a welldefined price elasticity

^A - (-1-) ^ -ë- • t^^^)

Under conditions of competition in the marketing sector, S^^Pg tends to be

positive.

We are now ready to show that, in general, the price elasticity of consxomer demand at retail is greater than the price elasticity of derived demand.

If we differentiate the identity (l39) with respect to Q^ ^^^ multiply both

sides of the result by Qc/^r^ ^^ have

3Ps Qc 3Pr Qc 3Pw Qc

BQc Pr 3Qc Pr BQc Pr

69/ Whatever the form of r] ^ this always is possible if we allow approximations by a Taylor series.
- 108

With scane manipvilation, we Imve

Recalling that elasticities of demand are defined to be positive, we have,

from equation (l^3),

BQePs ^r EJ^ePr ^c^w ^r ' ^^^^

By use of equation (l39)> equation (iMi.) can be rewritten

^cPs V'^l " ^c^ ^ ^c^ Í • ^^^^^

Adding and subtracting ---1— • JÏ frc»a the right hand side of equation (l^5)

^c^r ^^ and performing some algebraic manipulation, we find

^cPw ^cPr

(11^6)

Since the retail price always exceeds the price paid to producers, and since

E^^^Pg and Eft^P^^ are defined so as to be positive, we see that the right hand

side of equation (ikS) is positive* Thus

^cPw ^cP;

(Ihl)

c*r

But equation (1^7) implies that

^c^w ~ ^c^r ' ^^36)

that is, that the price elasticity of derived demand is a lower limit to the

price elasticity of consumer demand at retail, which is what we set out to

prove.

In-passing, we shoiald point out that it is incorrect to include the marketing'margin, Pg, in any partially reduced form equation derived from equations ( 123), (1^6), and (l27). In particular Pg should not be used in place

of the variable Zg in the derived demand equation, since the identity (l39)

must be included in any system containing equation (l4o); when it is, it

becomes clear that Pg is not an independent variable, but a derived one.

Problems of estimation.—^We now turn to some problems of estimation.

These are: (l) If we know the price elasticity of consxmer deznand at retail,

how can we estimate the price elasticity of derived demand? (2) If we know

the price elasticity of derived demand, how can we estimate the elasticity of
- 109 -

demand at retail? unless we can assume that the quantity passing through the

marketing system has no effect on the marketing margin and unless we know the

elasticity of price transmission^ it is clear that the second question cannot

be answered.

If the system containing equations (l23)^ (l26)^ and (l27) has been estimated jointly, 70/ then we have estimates of EQ^Pr^ ^PrPc^ ^> ^^^ ^' consequently, the elasticity of derived demand, that is, the price elasticity of

demand at the producer level of the market, may he estimated by applying

equation (131). If the quantity passing through the marketing system does not

affect the marketing margin, A = 0 and the price elasticity of derived demand

equals the price elasticity of consumer demand at retail times the elasticity

of price transmission. 71/ As indicated, the elasticity of derived demand is

less than the elasticity of demand at retail, because Ep^.p^ typically is less

than one.

In order to derive the price elasticity of consumer demand at retail from

a derived demand function, we must also estimate the marketing group's behavior relation or the supply of marketing services relation. Since retail

prices are typically not available when we estimate a derived demand relation

directly from the data, it usually is not possible to estimate the elasticity

of price transmission by estimating the marketing behavior relation. If,

however, we can obtain an elasticity of price transmission on the basis of

outside knowledge or incomplete data, we can obtain the price elasticity of

cons\mier demand at retail by simply dividing the price elasticity of derived

demand by the elasticity of price transmission. This procedure rests, of

course, on the assimiption that the quantity passing through the marketing

sector does not affect the marketing margin.

Variables to be included in the marketing group's behavior relation.—In

general we should include in the marketing group's behavior relation (1) the

quantity consumed, (2) either (a) the retail price and the price paid to producers or (b) the difference between these prices and (3) other variables,

referred to collectively as Z2, that affect the i^arketing margin.

Interesting problems arise when we consider the additional variables

represented by Zg. Price series for many factors of production used in the

marketing sector are rarely available; furthermore the marketing sectors for

many commodities are characterized by rapid technological change. Wage rates

in industries connected with the marketing sectors for many farm commodities,

however, are frequently available. One way to allow for the marketing margin

70/ If the quantity supplied does not depend on cvcrvent price, equation

(l2o) may be omitted and quantity taken as independent.

71/ The procedure is in fact the one suggested by Fox in Foote and Fox (29^

p. 40). Illustrative computations were given in the publication cited, but we

have not reproduced these results here because the price elasticities of consumer demand at retail, from which the derived demand elasticities were

computed, were obtained from analyses based on undeflated data.
- lio -

to depend on (l) the amoimt of commodity flowing through the marketing sector,

(2) the level of technology in the marketing sector, aaid (3) the one factor

price available (namely, wage rates in the marketing industry) is to assimie

that the marketing margin is siinply a function of unit labor cost in marketing. 72/

Allowing for Systematic Shifts in the

Regression Coefficients Over Time

Foytik (¿2), in a study of the demand for plimis, uses an interesting

technique to determine whether the regression coefficients on the several

variables in the demand equation change in a systematic way as the season

progresses. He states that two alternative methods might be used. In the

first, data for each week are considered as a separate set of observations,

and a demand curve for each week is derived. The regression coefficients

obtained are listed in sequence. Unless they seem to follow a systematic

pattern over time, the differences between weeks are assumed to be statistically nonsignificant. The second approach, which was the one used by Foytik,

considers all the weekly observations as an entirety, and tests whether

systematic changes in the regression coefficients are statistically significant. This is done by use of the following equation, where i = W = week of

season, P = price, Q = quantity sold, and D = consumer income:

Pi = a + (b + b'W)Qi + (c + c'W)Di + (d + d*W)Qi.i + (e + e»W)W. {lk&)

If the coefficients on the cross-product terms b', c', d', and e* differ from

zero by a statistically significant amount, the coefficients within the basic

demand equation can be assumed to change in a systematic way over time. Eq\mtions of this type restrict the changes in the net regression coefficients to

a well-defined, smooth pattern, although the use of second and higher degree

terms of W in the parentheses wo\ild permit the rates at which the coefficients

change over the season to themselves increase or decrease gradually. Foytik

(39^ P» ^^5) states, ''It should be pointed out that the more complex the

function is made, by the use of such additional terms, the more difficult it

is to make a suitable economic justification even though the statistical fit

is improved."

Foytik fitted an equation of this type to 270 observations covering

weekly prices and sales at the New York auction market for 1928-48, excluding

the years during World War II. His final equation retained all of the coefficients shown in equation (l48) except b' and d*. Thus his study suggests

that, for plums in the New York market, the relationships between price and

(1) current and (2) lagged quantity remain essentially unchanged throughout

the season, but the relationship between price and cons-umer income changes in

a systematic way, with changes in income having a less important effect on

price as the season progresses. This result appears reasonable.

72/ This approach was taken by Gerra in the model referred to on page ik and

by Nerlove, op. cit.
- Ill -

Use of Distributed Lags in Demand Analysis 73/

larving Fisher (24) was the first to use and discuss the concept of a distributed lag. Althotagh the idea that one economic variable depends on another

variable lagged in time is an old one, the use of lagged variables in empirical research has been restricted primarily to areas other than demand

analysis. In this section we discuss briefly some of the analytical problems

involved in attempting to introduce variables with a distributed lag into a

demand study and then illustrate the general approach with an application to

measuring the long-run demand for automobiles.

In economics, a cause often produces its effect only after a lapse of

time. For example, a drop in the price of potatoes in the fall cannot affect

potato acreage until the following spring, nor can it decrease potato production until the following fall. The lapse of time between a cause and its

effect is called a lag. The lag may be a specific time, say three months, or

one year. But in many cases, the effects of an economic cause are spread over

many months, or even many years. In such cases, we have a distributed lag.

Distributed lags with respect to variables that affect consumption may

arise for the following reasons:

(1) Psychological. Under this category we include forces of habit and

assiïtnptions on the part of the consimier that changes may be only temporary.

(2) Technological. These include factors such as, in a general case,

lack of knowledge about possible substitutes or, in a specific case, the inability to increase greatly the use of frozen foods without first acquiring

adequate freezer storage space.

(3) Institutional. This category includes (a) situations in which certain contractual items of expenditure or savings may need to be adjusted

before shifts can be made in consumption patterns, and (b) situations resulting from the fact that some markets, particularly for durable goods, Are

imperfect in an economic sense.

Analytical approaches.—One way to measure the degree of lag with respect

to a particular variable is to find by statistical analj^sis that distribution

of lag which maximizes the effect of the causal factor. Empirical analyses

based on this approach have been run which (l) make no assumption as to the

form of the distribution of lag, (2) are designed to estimate certain characteristics of an assimed general form for the distribution, or (3) derive and

statistically fit a model based on thé fimdamental cause of the distributed

lag but which yields a specific distribution of lag only incidentally.

One way to formulate models for generating distributed lags is to assume

that the lags arise chiefly because of technological and institutional

73/ Material in this section is condensed from Nerlove (7¿, 76).
- 112 -

rigidities. The traditional distinction between long- and short-run elasticities rests largely on causes of this sort. Although ohservahle points each

lie on a short-run curve^ the coefficients of the long-run cxirve can be estimated if the model is properly formulated. Relations between long- and

short-inin demand curves depend chiefly on the path that observed consumption

would follow if it moved directly toward its long-nm equilibri"um level

following an initial change in .a causal variable. The shape and form of such

paths is determined by the type of institutional and technological rigidities

that exist.

Another way to forarulate models that generate distributed lags is to

assume that technological and institutional rigidities are absent but that

uncertainty about the future exists and that habit is a powerful force. Under

conditions of uncertainty, a change in price or income may be thought of by

consumers as divided into two components—one permanent and one transitory.

The permanent part changes expectations in all relevant future years, whereas

the transitory part changes expectations in only some future years, or perhaps

in no future year. The average level about which future prices or incomes are

expected to fluctuate is called the "expected normal;" this level is affected

only by that part of the initial change in a causal factor that is considered

permanent. If forces of habit are strong, the effects of a change in current

price or income on consumer behavior are slight compared with the effects of a

change in the expected normal. Hence, models need to be formulated so as to

emphasize the effects of changes in the expected normal.

Hicks {k8, p. 205) defines "the elasticity of a particular person's expectations of the price of a commodity x as the ratio of the proportional rise

in expected future prices of x to the proportional rise in its current price."

The elasticity has been found by empirical research to normally range between

0 and 1. Many factors affect the elasticity of expectations; some affect all

commodities equally, whereas others affect different commodities differently.

Among the latter is the typical variance of prices of the commodity. For any

given commodity the elasticity of expectations need not be stable over time,

but models are simpler if we assimie that it is stable. Frequently an eq-uation

that contains one or more distributed lags can be reduced by algebraic manipulation to'an equation that does not contain such lags. The reduced equation

then can be fitted statistically and the results used to obtain certain

characteristics of the lag. A difficulty with this approach is that the

number of variables added in the reduced equation is greater than or equal to

the square of the number of variables with distributed lags that enter the

initial equation. Thus the method is not feasible when the initial equation

contains more than two or three variables with distributed lags.

In working with commodities that substitute for or complement each other,

we generally are interested in a system of equations rather than a single

equation. Each eqtiation frequently contains values of the same variables.

The BTUltiple equation method of reduction of a demand equation which

involves distributed lags takes advantage of the existence of all the interrelated equations. By way of contrast with the single equation method of reduction described in the preceding paragraph, if each original demand equation
- 113 -

contains n expected normal prices and expected normal income, each reduced

demand equation in the milltiple set includes, in addition to n + 1 cxirrent

values of prices and income, n + 1 lagged values of the qioantities demanded

and aggregate consijmption of all commodities. Thus, whereas the single equation method leads to one reduced demand equation containing (n + l) + (n + 1)2

independent variables, the multiple equation method leads to n reduced demand

equations each containing 2(n + l) independent variables. For statistical

purposes, the latter are clearly preferable when n exceeds 1 or 2. Some

systeins of demand eqxmtions are of such a natxxre as to be "separable" in a

certain mathematical sense. An example is a system that contains several

demand equations and a consumption function; here we may (l) reduce the whole

system, including the consumption function or (2) solve the consumption function for expected income, substitute this into the demand equations, and then

reduce only the demand eqiiations. This is the sense in which the term

"separable" is used.

Equations that contain distributed lags that are due only to technological or institutional rigidities caji be reduced easily. Reduction becomes

more complex when the lags are due to uncertainty about the futiire, but a considerable degree of simplification may be obtained by the muJLtiple equation

method of reduction. When the lags are due both to uncertainty and rigidities,

in general, no simplification of reduction is possible. If, however, the

distributions of lag are of a special form, simple reduction is possible.

Statistical estimation of the coefficients in demand equations that contain distributed lags can be done theoretically in two ways: (l) by dealing

directly with the equation that involves the distributed lags or (2) by using

one or more reduced equations. Maximimi likelihood procedures under the first

approach are available. This approach requires a large nimiber of repeated

steps; for this reason, it is called the "iterative" method of estimation. In

order to use the iterative method, the unexplained residuals in each equation

that contains a distributed lag must be normally and independently distributed. If the lags arise solely because of technological and institutional

rigidities, this will be tarue only under special conditions. If the lags are

caused by uncertainty, problems of serial correlation in the residuals do not

arise but the iterative procedure is computationally feasible only in the

simplest cases. Thus the iterative approach can be used only under special

circumstances.

If a distributed lag is due only to rigidities of a teclinological or

institutional nature, the coefficients can be estimated easily by using a

single reduced equation fitted by least squares. Estimation becomes more

complicated.if the lags are caused by uncertainty about the future. If three

or more variables with distributed lags are involved and we use the singleequation method of reduction, the number of variables in the reduced equation

becomes so large as to make statistical fitting virtually impossible with time

series of normal length. Complications from serial correlation of the residuals also enter. Only in the simplest of cases sho\ild the noniterative method

be based on a single reduced equation. In using the m\iltiple reduced equation

approach, we must assume that the distributions of lag for the same variable
in each of the eq-uatlons is the same. If the multiple eqiiation method can be

used^ it is computationally much simpler than the single equation method.

In analyzing the demand for commodities in general, if anticipated distributed lags result from uncertainty about the future, ve should specify a

system of equations before proceeding with any estimation. If the distributed

lags are believed to result only from technological or institutional rigidities, only a single demand eqioation for the particular commodity need be

specified.

An empirical example.—The example described here relates to the measurement of the long-mn demand for automobiles. The same general approach,

however, may prove useful for other consumer durables or near-durables, such

as certain items of clothing and other textile products, household appliances,

and so forth. Only enough detail with respect to the specific variables used

is given to indicate the economic basis for the equations in the model.

In this system of equations, we assimie that the quantity of automobiles

demanded by consimiers can be represented by the total stock of cars adjusted

in some manner for age, make, and model. As cars are known to depreciate

rapidly the first few years and more slowly thereafter, we can adjust for the

age factor approximately by applying a constant percentage rate of depreciation. Use of such a depreciation rate permits us to derive the total stock of

automobiles, adjusted for age, from an index of past purchases of new cars

which takes into account differences due to make and model. Let s^ be the

stock of automobiles during period t, d be the percentage rate of depreciation

and x-t be new car purchases during period t. Then we may write

St = xt + (l-d)xt-i + (l-d)^t-2 + ..• (1^9)

= xt + (l-d)st-i. (1^9.1)

A specific value for st is not needed in fitting the model, but a measxire of

xt is required.

The demand for the total stock of automobiles depends on the usual variables that enter into a demand eqxxation, namely relative price, consvmier

income, and technological factors such as miles of highways, degree of urbanization, and so forth. However, consxmiers may be -unable or unwilling to adjust

stocks immediately to the equilibriimi level suggested by current income and

prices. For example, consimiers may prefer to pay off certain installment

debts before purchasing a new car, or they may believe that a current change

in income is not pennanent and hence defer purchases of expensive items until

they feel that it is permanent. Each of such factors tends to cause a lag in

adjustment of purchases to current economic stimuli.

Since these economic stimuli are constantly changing, consimiers never

reach an equilibrium position—they only move toward it. Thus we cannot expect a direct relation between observed stocks of cars, or new purchases, and

current values of prices or consumer income. Instead, purchases of new cars
- 115 -

are related to current and prior values of the economic stimuli. As we show

in the paragraphs that follow^ we can deduce the long-run equilibriimi relations from observed changes in new car purchases, price, income, and other

variables by making certain assxmptions as to how consumers adjust toward an

equilibri-um position.

Let s^ be the long-run equilibriimi stock of cars desired by consumers, pt

be current relative price, y-^ be current real income, and z-t be other variables that affect long-run demand. Then

St = a + bipt + b2yt + 'b3zt. (l50)

It seems reasonable to suppose that the closer consumers are to equilibriutn, the less incentive they have to overcome the costs and frictions of

adjustment. When current stocks are far out of line with equilibrium stocks,

on the other hand, consumers have strong incentives to make necessary adjustments. Thus it seems reasonable to assimie that the rate of adjustment is

proportional to the amount of imbalance or that

St - st-i = ß (st - st-i). (151)

We now use these basic equations to derive an equation that can be used

for statistical estimation. This is done as follows:

(1) Rewrite equation (151) to get

St = ßst + (l-ß)st-i. (151.1)

(2) Lag equation (151.I) by one period to get

st-1 = ßst-1 + (l-ß)st-2- (151.2)

(3) Multiply equation (15I.2) by (l - d) to get

(1 - d)st.i = ß (1 - ^)H-1 + (1 -ß)(l - d)st-.2. (152)

(k) Subtract equation (152) from equation (151.I) to get

St - (1 - d)st.i = ß [si - (1 - d)sti] + (1 -ß )[st.i - (1 - d)st.2]- (153)

(5) From equation (1U9.I)

^t = St - (1 - d)st«i (1^9.2)

xt-i = st«i - (1 - d)st^2- (1^^9.3)

(6) By substitution in equation (153) we get

xt = ß[si - (1 - d)sti] + (1 -ß)xt.i. (153.1)
. 116 .

(T) Using equation (150) to substitute for s^ and st-i, we get

xt = ß [a + bipt + b2yt + t3Zt " (l-d)(a + ^il9t-l + ^gyt-l +

^3Zt-.l)l + (1 -ß)xt.l (153.2)

= A + BiPt + BgPt.i + Bgyt + Bi^yt.i + ^5^ + Boz^-i + t^Xt-i- (153-3)

If we fit equation (153.3) ty least squares, we can estimate the coefficients in equations (1^1-9.1), (150), and (151) as follows:

ß = 1 - By (154)

^>1 = Bi/ß (155)

b2 = B3/ß (156)

b3 - Bj/ß (157)

a = A/(ß + d - 1). (158)

But

d = (B2 + bi)/bi (159)

= (Bi^ + b2)/b2 (160)

= (B6 + t3)/b3. (161)

The estimate of d in equation (153*3) is overdetermined, the alternative

answers for d obtained from equations (l59) - (161) are not the same. Herlove

suggests using that B^ which relates to the variable that is measiired most

accurately by available data.

Results of an analysis of this sort for 1922-41 and 1948-53 are given in

Nerlove (75> pp. 62-63). By relating new car piurchases to current and lagged

price, current and lagged consumer income, and lagged new car p\irchases, he

obtains a multiple coefficient of determination of 0.9, a value for ß of 0.7

and, when d is estimated from the coefficient on lagged income, a value of

0.5. The long-run elasticity on relative price is estimated to be about

unity, and on real income, about 4.

Measuring Cyclical Factors that Affect Demand

Sometimes the residuals from a demand analysis follow a pattern that suggests a cyclical shift in the demand relations. Three examples are cited in

this section.

Fats and oils.—In a study of factors that affect wholesale prices of

fats and oils used in food products, Armore (¿, pp. 56-58) states.
- 117 -

"When the final analysis was completed, the residuals

appeared to be correlated with changes in the general price

level and with changes in prices of fats and oils. As discussed in the hody of this bulletin, accumulation and reduction of inventories by members of the fats and oils trade

during periods of chsûiging prices would be expected to have

such an effect. To throw more light on this point, year-toyear changes in the dependent variable were added as a fourth

independent variable. The multiple coefficient of determination was raised from O.92 to O.96, so that the unexplained

variation was reduced from 8 to ^ percent, and a statistically significant partial coefficient of determination of O.52

was obtained for the new variable. The addition of this

variable has little longer-term forecasting value but it does

confirm the importance of allowing for the effects of changes

in inventories in making short-term forecasts."

Coffee.—In a similar study of factors that affect prices of coffee, Hopp

and Foote {30, pp. i1.33-.l1.3U) write,

"When the tmexplained residuals were plotted against

time as a check on the degree of serial correlation, it was

found that they followed a definite cyclical pattern. ...

For coffee, however, use of the change in price as an additional independent variable had practically no effect on

the regression equation or the residuals. ... Further study

of the \inexplained residuals ... indicated that in general

they are positive when prices are rising or remain relatively

high and they are negative when prices are declining or remain relatively low. If these are designated as inflationary

and deflationary periods, respectively, 18 out of 20 residimls

for the inflationary years were positive and 25 out of 34

residuals for the deflationary years were negative. On the

average, prices were 21 percent higher during inflationary

periods than woxild have been expected from the regression

equation and 11 percent lower during deflationary periods.

... When the computed prices were adjusted in this way, the

percentage of variation explained by the analysis was increased from 70 to 8k percent. This improvement by stratification of the variation into these 2 classes was highly

significant statistically.

"The economic explanation of this appears to be similar

to that for fats and oils. When supplies are declining,

efforts are made to maintain coffee inventories, and prices

tend to be higher than would be expected from the level of

supply in relation to current consumption.. When supplies

are increasing, inventories can be reduced; hence prices

tend to be lower than would be expected based on relative

supplies.
- 118 -

"The only notable exception to this cyclical effect

vas during 1931-3^• In this deflationary period, prices

declined, but they did not fall as low as woxild have been

expected based on the original regression equation. In

the extreme situation that existed at that time, when

values fell exceptionally low as supply reached an alltime peak in relation to consxmiption, some market resistance apparently developed to further price declines."

Allowance for cyclic position significantly improved forecasts made from

this analysis for the years 19^9-53•

Mill demand for cotton.—Lowenstein and Simon (6¿, pp. 106-110), in a

study of factors that affect mill consumption of cotton, use a different

approach* They write,

"In any given period mill consimiption of cotton may be

out of balance with consumer pxirchases of cotton products

because of changes in inventories at various levels of fabrication and distribution. For example, when inventories of

cotton products are being built up at any level of marketing,

the increments represent an increase in demand for cotton

fabrics, and hence for cotton, over and above current consimrption.

"Of importance among factors that affect inventories are

changes in sales or expectations thereof. Merchants may try

to keep inventories in a fixed—or relatively fixed—ratio

to their rate of sales. If consumer purchases of cotton goods

decline, merchants would try, other factors being the same,

to adjust inventories to a level commensurate with the changed

conditions of demand. Current demand would be satisfied

temporarily from stocks of goods produced in the past. Orders

for cotton products thus would tend to decline by an amount

greater than that of the decrease in retail sales. As the

decline in demand spreads to preceding stages of distribution

and manufactiire, it would grow in intensity to the extent

that inventories on these levels also are reduced proportionately. Ultimately, the magnified reduction in consimer demand

is reflected back to the mill level. A result of the adjustments in inventories along the line is a rate of cotton

consumption less than that indicated by the decrease in consumer demand. Although inventory changes may bear some

relationship to changes in consumer income, the latter measure

could not be expected to account fully for the effect of the

former on mill consumption of cotton.

"Other factors that affect demand for goods for inventory, given the marketing and technological structure of the

industry, include fears of shortages, expectations concerning
- 119 -

price changes, and other facets of the economic outlook*

Prices themselves are not immune to inventory changes. For

example, the decision to acquire additional stocks, perhaps

in line with a rise in consumer purchases, would add to the

upward pressure on prices for textiles. The price rise, in

t\im, induces, for speculative and precautionaiy reasons,

further increases in stocks and an inventory-price-inflation

spiral may develop. Thus actions taken with respect to

stocks may affect and be affected "by prices.

"In addition to changes in inventory, the size of the

total inventory in the cotton textile system is to be noted,

obviously, the larger this inventory, the deeper will be the

effect of, and the longer the adjustment to, a decline in

consvmier demand« Conversely, if demand were to increase

suddenly and sharply, with inventories overly low, an

industry-wide speculative movement could be generated because of the prevailing tight supply condition. Stocks of

cotton goods thus affect demand schedules for cloth and

hence mill demand for cotton.

"A change in demand for cotton goods is translated at

the mill level' into a change in volume of new business both

for immediate and future delivery. The reaction of output

to a change in demand, however, is usually not instantaneous.

It takes time for production to adjust to a new level of

sales. Influential among reasons for the relatively slow

response of production are the momentum of the manufacturing

process, uncertainty concerning the lasting nature of the

change, the time it takes to obtain additional materials or

to cancel orders, and cost and time considerations relating

to removing shifts and shutting down^ looms or to adding

shifts and starting up idle equipment. Initially, jnill

stocks of cotton textiles would tend to bear the brunt of

a fehange in demand, probably varying inversely to it. Theoretically, adjustment in output, when it comes, woxild account

for the involuntary change in stocks plus any tendency for

mill inventories of textiles to be broxjght into line with the

new level of demand. Hence cotton cons-umption would be expected to reflect both the lag in response of output to a

change in demand and the resulting adjustment in level of

stocks.

"Adjustment in production is often carried too far;

that is, output is foimd to be forthcoming at a rate too

high or too low when compared with the level of demand.

Hence it may more than compensate for the earlier change

in stocks. If output were maintained at a level above

that of demand, textile stocks would tend to accimulate.

But if output were cut back and maintained below the level
-120 -

of demand, stocks would tend to decline. If, concurrently,

demand were to shift in the opposite direction the iidbalance could be magnified. Ultimately production woixld have

to "be adjusted and, if carried too far again, coiild affect

stocks similarly "but in an opposite direction.

"Prices of cotton textiles, actual or expected, have

been omitted from the preceding discussion but actually

they are interwoven into the dynamics of the industiy. The

apparent tendency of price to respond almost immediately

to changes in the demand for cotton textiles is prima facie

evidence of the lagged output response. If the change in

demand is sudden, following a period of stock accimoiulation

or liquidation, the effect on prices can be extreme until

the necessary adjustment is made in stocks and output. If

the change in price initiates a further change in inventoiy

demand and possibly a price-inventory spiral up the line,

the shift in demand would be greater and the supply adjustment required by the industry magnified. When prices are

thought to be f\illy discounted and production c\irtailment

is proceeding apace, the desire to cover forward at low

prices or the incentive to acqiiire stocks in anticipation

of higher prices may initiate a buying wave. At this

point risks connected with stock acquirement may be low

compared with those associated with the continued deferment of needs. The low textile prices also may lead to

an increase in retail sales with all its back ramifications

on the demand for textiles. On the other side, expectations of lower prices may cause a general falling off in

demand.

"Largely as a consequence of these varied forces and

their interrelationships, mill product stocks tend to

change in a cyclical fashion, frequently being out of line

with demand. The tendency for output to be kept at relatively high levels in the short run despite unfavorable

economic conditions apparently is characteristic of the

cotton textile industry. Clearly the tendency on the part

of the industry not to respond readily or properly to

changes in demand can affect the timing and extent of mill

consumption of cottor^ considerably and, if possible, should

be accounted for in ijhe mathematical formulation of mill

demand for cotton.

"Recently the American Cotton Maniifacturers Institute,

Inc.—henceforth designated as the Institute—made available to the United States Department of Agricultxire for

research purposes data on production, stocks, and unfilled

orders of cotton broad woven goods in physical units at

the mill level. ... Stock and \mfilled orders data represent
- 121 -

the mills' position as of the end of a reporting period,

generally at or.near the end of a calendar month. Data on

production cover the intervening period; hence they coincide approximately with a calendar month. ...

"Because mills participate in the Institute's statistical program on a voluntary basis, the percentage of the

industry covered by the reports tends to vary. ... Because

of ... the lack of strict comparability over time, it was

foxmd advisable to use the data in a ratio form to adjust

roiaghly for changes in the reporting sample.

"The ratio of 331ÍII stocks of cotton cloth to unfilled

orders was computed as of the end of each month for the fxall

period covered by the data. ... The ratio reflects the degree of imbalance between stocks, output, and demand at the

mill level. When the ratio is relatively high, unless an

increase in demand is forthcoming, a downward adjustment in

output to reduce stocks in indicated. Conversely, a relatively low ratio suggests the likelihood of a higher output

rate in the near future. The ratio indicates also the

cyclical character of changes in inventories of mill products .

"Obviously some inventory is necessary if a business

is to fixnction properly and efficiently. The amount of

inventory not considered excessive may vary directly with

the volume of business, so that a relatively constant ratio

between the two is sought. Whether mill stocks of cotton

cloth are too high or too low at a given time probably

depends more on the amount of business expected in the

near future—reasonably approximated by the level of unfilled orders—than on past volume.

"Some 'normal' ratio of stocks to unfilled orders thus

may be postulated about which the actual ratio would fluctuate and toward which it would tend. Departure from

normal—indicative of imbalance in the industry—would be

expected to lead to changes in mill consumption of cotton.

For want of information, it was decided to use the average

of the ratios as noimal.

"The residuals from Analysis II in logarithms were

fovind to be fairly closely correlated with actual deviations from normal of the stock-unfilled order ratio for

cotton cloth. The best result—a coefficient of correlation of -0.85—was obtained when the new variable led the

residuals series by 5 months. This lead is consistent

with the 3 to 6 months' lead that would have been expected

from a priori considerations.
- 122 -

"To throw more light on the importance of the effect of

imbalance in mill inventories of cotton cloth on mill consumption of cotton, the deviations of the stock-imfilled

order ratio from normal were added as a fifth independent

variable in Analysis III. As the new variable is believed

to affect mill consumption of cotton in an additive fashion,

actual deviations from nonnal were used. The other variables ... were kept in logarithms when the analysis was run.

Analysis III is based on 1927-32, 1935-^0, and 1948-52, the

only full years for which data on stocks and unfilled orders

are available when a lead of 5 months is employed. The results ... are surprisingly good. ... All of the coefficients

are statistically significant. ... Changes in actual deviations from normal of the stock-^unfilled order ratio, on the

average, account for a larger percentage of the variation in

mill consumption of cotton than does the price of cotton or

the consumption of synthetic fibers, after allowing for the

effects of the other independent variables. Changes in

disposable personal income and changes in the ratio of the

current to the preceding year's income, in that order, are

still more important in this respect."

Whenever the residuals from an analysis, plotted against time, appear to

follow a pattern, an attempt should be made to ascertain whether cyclical

effects could be anticipated for the industry being studied and, if so, appropriate allowance should be made. At times, the analyst may know in advance

that such cyclical effects may be important and, if so, a special variable

like that used by Lowenstein and Simon might be incoirorated into the analysis, or checks like those used by (l) Armore (¿) and (2) Hopp and Foote (¿),

respectively, might be made.

Obtaining Additive Weights for the Conrponents of Variables to

be Used in a Logarithmic Analysis ?¥/

The following method can be used whenever additive weights are desired

for the components of one or more variables to be used in a logarithmic analysis. The method has been applied mainly to problems that involve the determination of proper weights for several components of a supply variable. Two

research papers have contributed to the statistical problem considered here,

Foote (2¿) demonstrated that a successive approximation method similar to the

one outlined here will converge to the least-sq\iares value. Ezekiel (20, pp.

390-ÍI-03) proposed methods along these lines for studying joint relationships

while holding other variables constant.

Steps in the analysis.—The following steps are used when weights for

components of a supply variable are desired and price is used as the dependent

variables :

Jk/ Material in this section is adapted from Foote (26, pp. 3^1-41).
- 123 -

1. The components of the supply factor are combined in some simple way.

As a first step in the example for livestock feed^ they were converted from

bushels to tons and added together. The composite is used as one independent

variable in a logarithmic analysis.

2. Values of the dependent variable from this analysis are adjusted for

the effects of the independent variables other than supply^ and the antilogarithms obtained. In the case of first-difference analyses, some additional

calculations are involved. The calculated value plus 2 has for its antilogarithm a figure in terms of percentage of the preceding year. This should be

applied to the actual price in the preceding year to get an estimated price.

The difference between this estimate and the actual price in the preceding

year is the adjusted dependent variable that should be used as the dependent

variable in the linear first-difference analysis discussed in step 3.

3* The adjusted dependent variable is used as the dependent variable in

a linear analysis in which the separate components of the supply factor are ,

used as independent variables. The partial regression coefficients from this

analysis are proportionate to the weights that should be applied to the individual supply components.

k. The following method may be used to test whether these partial regression coefficients differ significantly from each other: (a) Conrpute the

simple linear correlation between the dependent variable used in step 3 and

the supply factor based on constant weights for the components. In the case

considered here, the latter is the same as the supply variable used in step 1.

Compute the imexplained s\m of sq,uares_for this analysis. If XQ is the dependent variable, this equals £{XQ - Xo)^ (l - 3:^^01 ) • i'^) Compute the

iinexplained simi of squares for the linear multiple correlation analysis in

step 3- This equals X(Xo - Xo)^ (l - K^Q^I ... p). (c) Take the difference

between these two sums of squares and divide it by P-1. This represents the

variance owing to the differences between the regression coefficients in the

linear analysis, (d) Divide the sum of squares obtained in step b by N-P-1.

This represents the error or remainder variance, (e) Compute the ratio

between the variance in step c and the variance in step d, and compare this

with the tabular values in an F table, using P-1 and N-P-1 degrees of freedom.

If the ratio obtained is larger than the tabxalar value at the 5-percent point,

the differences between the regression coefficients are statistically significant. If this is approximately true, the weights determined in step 3

should be used, provided they appear logical.

A method of successive approximation to refine the various coefficients

is next considered. This is used only if the F-ratio obtained in step he is

equal to or larger than that at, say, a 10-percent probability level. If the

regression coefficients do not differ significantly, the simple weighting procedure originally used in step 1 is satisfactory, and that analysis becomes

the final one.
- 121^ -

5* obtain a new supply composite by multiplying each component by its

partial regression coefficient from the linear analysis or by weights that are

proportionate to these coefficients. In the case of first-difference analyses, the actual values are multiplied by the weights, the total obtained, and

the first-difference logarithms are derived from this total.

6. Use this composite with the other variables originally used in step 1

to obtain a second approximation for the logarithmic analysis.

7. If the second approximations for the partial regression coefficients

for the variables other than supply differ from the first approximations obtained in step 1, get a new series for the dependent variable adjusted for the

effects of these nonsupply factors and obtain the antilogarithms of these.

8. Use the variable obtained in step 7 as the dependent variable in a

linear analysis, with the same components of supply as independent variables

as in step 3«

9. If the regression coefficients obtained in step 8 differ from those

obtained in step 3, repeat steps 5, 6, 7, and 8 until the results converge to

stable values. The final results yield a set of weights to be applied to the

components of supply to give a composite factor of supply that will maximize

the multiple correlation in the logarithmic analysis, for which price is the

dependent variable and supply is one of several independent variables.

Application to an analysis relating to com.--For the series of analyses

discussed here, the basic analysis is a logarithmic one, using first differences, for which the dependent variable is the Jime-to-September average price

received by farmers for corn. The independent variables are: (l) A composite

feed-concentrate-supply variable, for which the weights are determined from a

separate linear analysis; (2) a coDiposite imits of livestock production variable, for which the weights were partially determined from a separate linear

analysis; and (3) average June-to-September index numbers of prices received

by farmers for livestock and livestock products.

The dependent variable for the linear analysis involving the separate

components of the supply factor was determined by adjusting year-to-year

changes in the price of corn for the effects of year-to-year changes in imits

of livestock production and prices of livestock, using the highest-order

partial regression coefficients from the logarithmic analysis. Likewise, the

dependent variable for the linear analysis involving the separate components

of the units-of-livestock-production variable was determined by adjusting

year-to-year changes in the price of corn for the effects of year-to-year

changes in supply of feed and prices of livestock, using the highest-order

partial regression coefficients from the logarithmic analysis. Results for

each succeeding approximation were used to improve the results for the following analysis.
- 125 -

In an iterative analysis of this t^^pe^ values for certain coefficients

inay be assiimed at the start, as the results of successive iterations converge

toward the statistically most likely or "true" values.- During the process of

analysis, certain coefficients may be obtained which are inconsistent with a

priori expectations. Particularly if the standard errors of these coefficients are large, consistent coefficients may be assumed or the variable may

be omitted from the analysis, in which case a regression coefficient of zero

is assuimed. Table 6 lists the coefficients obtained or assxmied for succeeding

approximations for each of the three analyses involved in the over-all analysis of factors that affect the price of com from June to September.

Results obtained from the second approximation for the analysis dealing

with the components of the supply of feed concentrates were tested to determine whether the regression coefficients differed significantly from each

other. An F of 3*30 "was obtained, compared with a 5-percent probability value

of 3»24. As the individual coefficients in general seemed reasonable, we decided to use them. As the coefficient for concentrates other than com, oats,

and barley fed (bolf..i23) "wras positive and had such a high standard error, this

factor was omitted. Year-to-year changes in this item are extremely small.

In the first analysis, data on livestock production units for the Julyto-September quarter only were used. In the second approximation, data for

the July to September quarter were weighted by 2 and for the October-toDecember quarter by 1. The series on prices of corn was then adjusted for the

effects of supplies of feed concentrates and prices of livestock. The adjusted

price was used as the dependent variable in a linear analysis in which the two

livestock production series were used as separate independent variables. These

two variables explained only 7 percent of the residual variation in prices of

corn. This is consistent with the highest-order partial coefficient of determination between prices of com and production of livestock from the logarithmic analysis, which was only 0.05 for this approximation. The analysis indicated, however, that, if anything, October-to-Deceniber livestock production

units (for which the sign of the regression coefficient was consistent with a

priori expectations) were more important than July to September livestock

production units (for which the regression coefficient had the "wrong" sign).

In subsequent logarithmic analyses, these two series were given equal weights.

In the basic analysis of the factors that affect June-to-September prices

of corn, the regression coefficient for the units-of-livestock-production

factor (bo2.i3) had the wrong sign on the first approximation. This occasionally occurs in analyses of this type and may merely indicate the need for

additional approximations. In this instance, the dependent variable, in terms

of first-difference logarithms, for the second approximation for the linear

analysis involving the components of the supply of feed concentrates was obtained by use of the formiaa X0.3 = XQ - bo3.i (X3 - X3). In other words, the

computations for this approximation were based on results that would have been

obtained had the Xg factor been omitted from the analysis.
- 126 -

Table 6.—Corn: Successive approximations for the three sets of multiple correlation analyses used in connection with the hasic analysis of factors

that affect June to September prices

AMLYSIS DEALING WITH FACTORS THAT AFFECT THE PRICE OF CORN, JUISE TO SEPTEMBER

Coefficient

R^O.123

^01.23 t SbQ3^,23 •*

^02.13 t SbQ2,i3 **

1303.12 t Sbo3^12 ••

Approximation

0.634 0.738

-.82 + 0.36 -1.65 + 0.39

-1.06 + .5U .23 + .69

1.21 + .29 1.10 + .2lt

0.770

-1.70 + 0.3^^

.76 + .69

.87 + .23

0.758

-1.59 t 0.34

.70 + .70

.81* + .23

ANALYSIS DEALING WITH COMPONENTS OF THE SUPPLY OF FEED CONCENTRATES

R^0.123l^ ••

^01.231* Í s^oi.234

^02.13i^ t ^02.13i*

b03.12lf Í Sbo3^i24

t>04.123 t ^oi^.123

1/1

1/1

1/1

1/1

0.530 0.587

.2.69 t 0.69 -2.88

-.68 + M -.78

-.85 + .87 -.61

1.93 Í 3.80 1/0

0.656

-3.19

-.81

-.5i*

1/0

ANALYSIS DEALING WIŒH COiPONENTS OF PRODUCTION OF LIVESTOCK

R^0.12

^»01.2 •

^02.1

— 0.071 ---

1/1 1/2 1/2/1 1/1

1/0 1/1 1/2/1 1/1

1/. Assumed values.

2/ Calcillated values for this analysis were as follows:

l>01.2 t Sboi.2, "l-^^ t 1.62; ho2.1 t sbo2.1, ^-^^ - ^-^^^
- 127 -

In general, the multiple correlation coefficient for the basic analysis

shoxild "be the same or higher for each succeeding approximation, and the regression coefficients should change in an orderly way. In the analyses discussed here, this takes place throvigh the third approximation but does not

hold for the fourth. This is believed to reflect rounding errors. The weights

for the components of the feed-concentrate-supply factor were carried to two

decimals only, which is equivalent to two significant figures for the smaller

coefficients. Thus, there is no sound way of choosing between the third and

the fourth approximations. The fourth was chosen mainly because the weights

for the supply factor continue to change in an orderly way through that

approximation.

Conrparison of results for corn, oats, and barley. —Meinken (70) ran

similar analyses for oats and barley for July to October when marketings are

heaviest for these crops. In each case, the price received by farmers was the

dependent variable and a first-difference logarithmic analysis was used covering the years 1922 to 19^1. In these analyses, as for corn, the three

Table 7«—Corn, oats, and barley: Relative weights by which specified components of the supply of feed grains should be xnultiplied to obtain a

composite supply factor for use in analyses of factors that affect their

respective prices l/

Item Corn Barley

Months covered by analysis'•••••••••••: June-Sept.

Weight obtained for specified :

component of supply:

Stock of com, July 1:

Weight : Ö.6h

Standard error 2/ :' .11

Prospective new crop of corn:

Weight • : .16

Standard error 2/ : .06

New-crop supply of— :

Oats: :

Weight : ¿/.lO

Standard error 2/ .. • : .13

Barley: :

Weight : ¿/.lO

Standard error 2/ : .13

Sum of weights : 1.00

July-Oct. J\ily-0c0.19

.11

0.11

.08

,2k

.06

.Ik

.Ok

.5T

.17

.20

.18

0 .55

.k2

1.00 1.00

1/ When all components are expressed in tons.

2/ Based on standard errors of respective regression coefficients.

3/ Components for oats and barley were combined in the analysis for corn.
- 128 -

variables—(l) a composite of supplies of old- and new-crop com and new-crop

oats and barley, (2) production of livestock from July to December, and

(3) prices received by farmers for livestock and livestock products for the

months included in the analysis—were used as independent factors. Weights

for the components of supply were determined by a linear first-difference

analysis throxagh the method discussed in this section.

Table 7 (see page 127) compares the weights obtained from the analyses

for oats and barley with those obtained for corn. In each case, the c\arrent

supply of the commodity as such carries the largest weight, and the weights

for other components are consistent with a priori expectations. The regression

coefficient for barley in the analysis for oats had the wrong sign but, as it

did not differ significantly from zero, its value was assumed to be zero in

these computations. Results for the analyses for oats and barley confirm

those obtained for com in indicating that this general approach is useful in

analyzing problems of this type.

An Experiment to Test the Relative Merits of Least Squares

and Limited Information Coefficients for Forecasting

Under Specified Conditions

As discussed in the section beginning on page 57, if an equation within a

system of equations co^itains two or more variables that are endogenous and

this equation is fitted statistically by the method of least sqioares, the estimates of the structural coefficients obtained will tend to be biased in the

sense defined on page 57* Under the circumstances described beginning on page

67 p the degree of bias may be small, but in general more accurate estimates

of the structural coefficients can be obtained by methods that allow for any

simultaneity that may exist within a system of equations than by a direct fit

by the method of least squares.

But in practical matters of business and politics, the economist is not

ordinarily asked to compute a structiiral coefficient. Rather, he is asked to

estimate the expected value of one or more variables. These estimates—or

forecasts—frequently are obtained by using coefficients estimated by statistical means and known or assximed values for certain related variables. A

farmer may want an estimate of the expected price of apples in late winter.

An economic analyst may provide such an estimate based on given data on apple

production and consiamer income.

With respect to problems of this type, there seems to be less agreement

about the relative merits of least squares and of structural analysis. Some

econometricians and many applied research workers believe that the best unbiased forecasts are obtained from equations fitted directly by least squares.

Others apparently believe that more accurate forecasts can be made from equations derived by limited information. Klein (¿8, p. 253) implies that better

estimates are obtained from "reduced fona" equations derived from a structural

approach because these take into account certain restrictions within the

model.
- 129 -

To attempt to clarify questions with respect to the best method when we

are chiefly interested in forecasts for a single variable, the author, in

cooperation with Frederick V. Waugh, recently made a Monte Carlo analysis of

some constructed data. We believe the results throw some light upon the

proper uses of both the least-squares regressions and the structural equations. Possibly these resxalts would have been anticipated by économetri cians

who have specialized in this area. Each of us must admit that we were surprised by some of the results. We realize also that this one test will not

settle forever all arguments about the relative merits of these two approaches

in economic research. But we sincerely hope that these results will help to

bring about more \mderstanding than now exists.

The model.—Other writers, such as Bennion (8), Bronfenbrenner (l2), and

Marschak (66), have used either theoretical or Monte Carlo models in an

attempt to shed light on the relative merits of these alternative approaches.

In general, none of these models are appropriate for the particular tests that

we had in mind. The one used by Bennion, for example, yields exact values for

any one variable given exact values of the other two. Hence it cannot be used

to test the relative merits of alternative methods when we wish to predict one

of the endogenous variables given a knowledge of other variables in the model.

Our model is a simplified version of the one used for wheat by Meinken

(71) (see page ll). In economic terms, we have three price-determined utilizations (YI, Y2} ^3) that add to a fixed supply (Zij.), and price (Yif) must be

at a level to equate demand with supply. Each utilization is a function of

(1) price and (2) a demand shifter that is unrelated to this system of equations. The disturbances in the three demand equations were coniputed so as to

be correlated. To add variety, we specified that the coefficients on (l) price

and (2) the demand shifter for eqimtion (2) are identical. Coefficients on

price were chosen so that one was greater than unity, one equaled unity, and

one was close to zero. The following equations were used:

Yi = -0.1 Yl^ + Zi + ui (162)

Yg = -(Y4 - Zg) + U2 (163)

Y3 = -4 Y4 + Z3 + U3 (164)

Yi + Y2 + Y3 = Zk (165)

u-L = è(5v + wi) (166)

ug = 2(5v + wg) (167)

U3 = lf(5v + W3). (168)

The Z's were drawn from a rectangular distribution, making use of random

digits published in the Journal of the American Statistical Association. Each

column of five digits was divided into two 2-digit columns and a 1-digit

column that was ignored. All nimibers of 40 or less in the 2-digit columns
- 130 -

were recorded until sufficient numbers were obtained for the initial analyses.

For those analyses involving forecasts when the Z's vary over twice the range

used in the period of fit, all numbers of 80 or less were recorded. When the

Z's fall outside the initial range, numbers of 60 to 8o were recorded.

The v's and w*s were obtained in the following way^ making use of the

same tables of random digits: (l) Each column of 5 digits was treated as a

unit. (2) If the last digit was even (or zero), the nvimber was recorded as

positive; if the last digit was odd, the number was recorded as negative.

Numbers of 49998 or above were ignored. (3) Use was made of a table in Ostle

(19, PP* 442-443) entitled "Areas of the Standard Normal Curve." A decimal

point was mentally placed at the left of the first digit and the interval that

included it in the right hand coliamn of each section of the table was used to

select the assigned number. For example, the first random nimiber was 48190.

The relevant portion of the table shows

2.09 -48169

2.10 .48214

Hence the first recorded number is 2.095. It is positive because the last

digit in. the 5-digit nxunber is even. Similar procedures were used to obtain

the remaining observations. Thus the v's and w's, and therefore the u's, are

approximately normally distributed with mean zero.

The u's and Y's were then calculated from the given values of the Z's,

v's, and w's. To avoid possible small sample bias, 100 observations were used

in fitting the structixral equations by, respectively, least squares and

limited information. The results are shown in the tabulation on page 131.

If the first and third equations and the first formulation of the second

equation are used, as in the original model, then the coefficients estimated

by limited information are considerably closer to the coefficients used to

generate the data than are those estimated by least squares and each coefficient is within two standard errors of the "true" value. All of the least

squares coefficients, on the other hand, differ from the coefficients used to

generate the data by more than three times their respective standard errors,

although the standard errors estimated by least squares are, in each case,

moderately to materially smaller than those estimated by limited information.

It is possible, however, that the expected value, in a statistical sense, of

the coefficients estimated by least squares differs from the coefficients used

to generate the data and represents instead an average of a large number of

equations that best forecast the Y^ (i = 1, 2, 3) given YL. and the respective

Z's.

Neither method of fit gives good results for the second formulation of

the second equation. The limited information coefficients differ from the

"true" value by more than do the coefficie„its estimated by least squares, but

they are each within two standard errors of the true value whereas the least

squares estimates differ from the coefficients used to generate the data by

materially more than twice their smaller standard errors.
. 131 -

The following tabulation shows the resvilts obtained by fitting, respectively,

by limited information and least sqxiares:

First eqijation:

Used to generate data Y^ = 0 - 0.1 Yij. + 1 ^1 "•" ^1

Limited information fit l/ Y^^ = l.k - 0.l6 Yu + O.963 Zn

(0.11) (0.028)

Least squares fit l/ Yj^ « -1.5 + O.19 Y4 + O.95Ô Z^

(0.02) (0.013)

o P 2

Related coefficients R'^^ = O.98 ^kUi * ^*^^ ^Z u * ^'^^

Second eqiiation:

First formulation:

Used to generate data Yg = 0 - 1 (Y4-Z2) -•• Ug

Limited information fit l/ .. Y2 = -O.5 - l.l4 (Yi^-Z2) + U2

(0.13)

Least squares fit 1/ Yp = 6.6 - O.53 (YJ.-ZO)

(0.08) ^ "^

Related coefficients r^ = O.31 r?y .g; )ii^ =" 0*26

Second fonnulation:

Used to generate'data Y2 = 0 - 1 Y4 + 1 Z2 + U2

Limited information fit 1/ .. Yp = 6.0 - 2.5 Yi, + 1.^ Zo

(1.6) ^ (0.4) ^

Least squares fit l/ Yp = -5-5 + 0.2? YL + O.78 Zp

(0.07) (0.05)

Related coefficients R^ « 0.79 rv „ =0.73 rf = 0.004

14U2 Z2U2

Third equation:

Used to generate data Y3 = 0 - 4 Y4 + 1 Z3 + Uo

Limited information fit 1/ .... Yo = 6.6 - k.k Yu + O.93 Zo

(1.1) ^ (0.27) "^

Least squares fit l/ Y3 = -11.0 - 1.^5 Y¡^ + O.56 Zo

(0.12) (0.09)

Related coefficients R^ « 0.62 r| = 0.77 r| ^ = O.OO3

1/ Numbers in parentheses are the standard errors of the respective coefficients.
- 132 -

Certain coefficients of multiple or simple determination are shovn belov

each eq\aation. The first is the coefficient of multiple determination estimated hy least squares, except for the first formulation of the second equation, where the simple coefficient is relevant. The second shows the simple

coefficient hetween the endogenous variable treated as independent in the

method of least squares and the unexplained residual in that equation. The

third shows thé simple coefficient between the predetermined variable and the

unexplained resid\ial in each equation. The latter is asstmied by the model to

be negligible and in no case exceeds 0.02.

Forecasts from the price-utilization equations.—In this experiment, we

first assume that the économetrician is asked to forecast—or estimate—^Yi,

Y2> and Y3, using equations like (162), (163), and (l64). He does not know

the parameters in these eq\aations, but must estimate them from sample data. We

assume that he knows the value of Y^ and the Z's in the forecast period from

an independent source. For purposes of estimation, he may fit three leastsq-uares regression equations, using Yi, Y2> and Y3, respectively, as dependent

variables and Yl^ and the respective Z's as independent variables. 75/ Or he

may use the limited information method to obtain estimates of the structural

coefficients. Here the several Y's—or simultaneously determined variables—

are treated as endogenous and the Z's, as predetermined. A second approach

for predicting the first three Y's from the Z's, with or without a knowledge

of Yi|., is discussed in the next section.

Four types of test samples were used, each involving paired samples of 25

observations: (l) When the Z's were drawn over the same range as for the observations used in the initial fit and (a) varied from observation to observation or (b) were fixed over the 25 observations, (2) were drawn over twice the

range, and (3) were drawn completely outside the range of the Z's used in the

initial fit. The latter have some relevance in applied work, as for example

when an analysis for a period prior to World War II is used for forecasting in

a period following World War II. In each case, the range in the v's and w's

was the same as for the initial observations.

By making use of the alternative estimates of the coefficients in the

price-utilization equations, as shown in the tabulation on page 131^ and known

values for Yi|. and the several Z's, estimates of the first three Y's were obtained for each of the 8 test samples. Differences between the actioal and

coarputed utilizations for each equation were obtained. For the observations

used in the initial fit, these differences add to zero. In a test sample,

however, they would not necessarily add to zero and, for some methods and some

samples, the arithmetic sum differed considerably from zero. If the symbol d

75/ As discussed in the section beginning on page 51, there are several

schools of thought concerning the correct method to choose the variable to be

treated as dependent in a least-squares analysis. Here we choose as the dependent variable the one that is to be estimated and as independent variables

those whose values are given at the time of forecast.
- 133 -

is used to represent these differences, estimates of the mean squared error

sho\ild he computed hy the formula:

This is used rather than the usual formula for the square of the standard

deviation which is

,2 2 Xd^ - dXd

^cL " N - 1

(170)

because the latter does not take into account errors caused by a faulty estimate of the level of the regression curve.

A comparison of the mean squared errors for the h samples for which the

Z's were drawn over the same range as in the period of fit indicates that in

every case they are smaller for the estimates based on coefficients obtained

by the method of least squares than for those obtained by limited information

even though these coefficients differ materially from those used in constructing the model. (See table 8.) Ratios of the alternative pairs of mean

sqxaared errors range from l.k to l4.6. For the two samples for which the Z's

were drawn over twice the range, the mean sqxmred errors based on coefficients

obtained from the equations fitted by least sqiiares are smaller for h comparisons and larger for 2 comparisons. For the two samples in which the Z's lie

outside the initial range, the mean squared errors based on least squares are

smaller for 2 comparisons and larger for k comparisons. In each of the lastnamed comparisons, however, the ratio of the larger to the smaller mean

squared error ranges only between 1.1 and 2.1. Variations between saanples

within any pair of samples in general are greater when the Z*s are drawn

outside the initial range.

Mean squared errors are smaller in most cases for the least sqimres coefficients than for those coniputed by limited infonmtion, even though the

latter are closer to the coefficients used to generate the data, because the

\mexplained residuals in the structural relations are correlated with the

values of YI4.. The structural coefficients do not allow for this correlation,

whereas the least sqtiares estimates include both the structural relation

between Yi (i = 1, 2, 3) and Yk and also the relation between Ylj. and the

respective u^. This is the reason both for (l) the difference between the

least squares estimates of the coefficients and the structural coefficients

and (2) the better forecasts of Y^ given by the least squares equations

given a knowledge of Yk*

ïhis point is so fundamental that it seems desirable to illustrate it

graphically. The samples for which the Z's are fixed are ideal for this purpose, because the scatter between the respective Yi (i = 1, 2, 3) and Yl^ are
- 13^^ -

Table 8.--Difference between act\aal and computed utilizations when price is

given and estimates are made from the price-utilization equations: Mean

squared error for alternative methods and samples

When the Z's are

drawn over—

Sample

First Second

Least

squares

Limited

information

Least

squares

Limited

information

The same range as in the :

period of fit and— :

They vaiy: :

Yl : 3.3

Y2 : 95.^

Y3 .•.•...•.... : 12lf.3

Are fixed: :

Yi ...•: 3.0

Y2 •..• : 70.5

Y3 •.•.•••- : 99-5

Twice the range: :

Yi ..• : 7-7

Y2 • : 121.2

Y3 • : 755-3

Outside the range: :

Yi : Ik.k

Y2 : 221.5

Y3 .....: 931.8

8.5

134.5

549.7

2.1

85.4

209.4

9.4

170.4

675.9

9.8 .8 11.1

150.3

603.0

107.3

94.4

173.3

703.0

9.9

99.4

435.8

7.9

165.3

515.7

10.8

208.3

622.0

13.1

189.7

498.2

13.8

339.3

924.2

24.7

158.2

952.2

in effect partial relationships for the fixed Z. Figure 8 shows the scatter

"between Y^ and Y4 for the first of these samples. Similar results ,are given

for Y2 and Y3. Because a positive correlation exists "between Yi|. and ui, the

line of best fit differs in slope from the structioral relation between these

two variables. The least squares eqiiation based on the initial sample of 100

observations is not a perfect fit for this sample of 25 observations^ but it

is much closer to a fit than is either the limited information estimate or the

structural relation. We believe that this is the point that Bennion (8) had

in mind, but this seems a clearer example.

Forecasts based on all of the predetermined variables.—An alternative

method of estimating the Y's, which can be used without a knowledge of Y4, is

to make use of the reduced form equations that involve the four given or predetermined variables in the model. To make this test, the following equations

first were fitted by least sqiaares, using the respective Y*s as dependent

variables and the several Z*s as independent variables. 76/

76/ The numbers beneath the coefficients in each case are the respective

s tandard errors•
- 135 -

RELATION BETWEEN Yl AND Y4 IN

A FORECAST SAMPLE WHEN Zi = 12

Yl

30.0

25.0

20.0

15.0

10.0

5.0

Least squares fit

to initial sample

Limited information

fit to initial sample

~ !

6 »3

—']^Z^>?3=^3?f2*i3"9 '^ Structural Structural relation relation

•14 23* •5o»12 ^*"**"^'Zr""*"""'----

lO» •• 17 ^"^--•^

•15 25 ^ —^

5.0 Í0.0 15.0 20.0 25.0 30.0 35.0

Y4

U. S. DEPARTMENT OF AGRICULTURE NEC. 4627-57 (10) AGRICULTURAL MARKETING SERVICE

Figure 8. ^A regression equation obtained by least squares more nearly fits the

data that show the relationship between two endogenous variables than does the

true structural relation because the endogenous variable that is treated as

independent in the least squares equation is correlated with the unexplained

residuals in the structural equation.
. 136 -

Yi = 1.3 + 0,95 Zi - 0.01 Z2 - 0.05 Zo . 0.00 Z4 R2 = 0.97

(0.02) (0.02) (0.02) (0.02)

Yp = 3-5 - 0.27 Z-, + 0.81 Zp - 0.2if Zo + 0.16 Zk R2 = 0.87

(0.04) (O.Oil) (0.04) ^ (0.04)

Y3 = -5.0 - 0.68 Zi - 0.80 Z2 + 0.28 Zo + 0.85 Zk R2 = 0.85

(0.06) (0.05) (0.06) "^ (0.06)

Yk = 4.2 + 0.10 Zn + 0.24 Zp + 0.l4 Zo - 0.27 Zk R^ = 0.27

(0.06) " (0.06) (0.06) ^ (0.06)

The last equation shows price as a function of production and the deinand

shifters in the respective demand equations. The other equations normally

would not be used except in connection with a system of equations.

Next, the following equation for Yl^. was computed by algebra from the

coefficients obtained by the limited information approach, using the method

described in Friedman and Foote (4o, p. 84):

Y4 = -1.33 + 0.17 Z3_ + 0.20 Zg + 0.16 Z3 - 0.18 Zi,..

Once a value for Y4 is obtained, this value and the known values of the Z's

can be substituted in the first and third equations and the first formulation

of the second equation shown in the tabiilation on page 131, using the equations obtained by the limited information fit, to obtain computed values for

the remaining Y*s. Results of this approach differ from those obtained from

the reduced form equations, as fitted by least squares, because each of the

equations in this system is overidentified.

As before, differences between the calculated and actual values for the

four Y's were obtained, and related mean squared errors were computed for each

of the first six samples used in the preceding test. The results are shown in

table 9.

To justify the elaborate fitting procedures involved in the limited information approach so far as forecasting is concerned, better forecasts shoixld

be given by the algebraically-derived limited information equations than by

the reduced form eqiiations fitted by least squares. In 17 out of 24 comparisons, the mean squared error is less for the limited information than the

least squares forecasts, but the average difference differs from zero by a

statistically significant amoiint only if we use a probability level of about

20 percent. 77/ Except for greater variability of results in the latter case.

77/ In Cowles Fo^undation Discussion Paper No. 11, dated April 20, 1956, entitled "The Efficiency of Estimation in Econometric Models," L. R. Klein

emphasizes that, "While the squared discrepancies between predicted and actual

values of endogenous variables in the model v/ill be smaller over the sample

period if calciilated from (unrestricted) least squares estimates of the
- 137 -

no consistent differences in these relationships were fo\md when the Z's were

drawn over (l) the same range as in the period of fit and (2) twice the range.

Table 9.—^Difference between actual and coarputed endogenous variables when

estimates are made from all of the predetermined variables in the system:

Mean sqioared error for alternative methods and samples

When the Z's are

drawn over—

The sajne range as in the :

period of fit and—

They vary: :

Yi : h.k

Y2 ..-.•... : 18.1

Y3 : 38.4

Yk : ^7.^

Are fixed: :

Yi : ^.8

Y2 ....•: 2k.k

Y3 : 1^8.9

Yk : 53.0

Twice the range: :

Yi • .: 5.5

Y2 ••....•...: 26.7

Y3 ...: 55-1

Yk •.- : 58.0

Sample

»e ; First ; Second

• Least ; Limited

\ squares . information

' Least * Limited

[ squares [ information

3.8 lt.6 3.9

18.9 23.2 25.9

36.6 k6.0 45.7

kk,0 55.3 51^.2

h.3 5.3 4.8

22.6 21.0 23.3

114.3 1^3.8 1^5.7

kB.6 57.1 57.0

5.8 5.3 5.2

22.9 25.2 50.9

29.1 50.6 70.5

33.0 59.0 53.1

Comparison of the respective mean squared errors shown in tables 8 and 9

gave results that at first thought were surprising but^ on further examination, appear reasonable. In the computations shown in table 8 we assxmie that

price (YU) is known and then estimate each of the utilizations based on this

price and the exogenous variable in the respective structural equation. In

table 9, we ignore this available information with respect to price but make

reduced form equations than if calculated from any set of estimates of structural parameters using a priori restrictions," the algebraically-derived

estimates will tend to give better predictions outside the sample. A proof is

given of the proposition that "the more one uses valid information in the form

^^ £ P^^io^i restrictions imposed on the system, the more efficient are the

estimates."
- 138 -

use of all of the Z's in the system. Mean squared errors for the limited information estimates in table 9 are in every case smaller than the corresponding mean squared errors in table 8 and in many cases are materially smaller.

Moreover^ they also are smaller than the corresponding least squares estimate

in table 8 in each comparison except for Y^ when the Z*s are drawn over the

same range as in the period of fit. Again the differences are substantial in

most instances. Mean squared errors for the least squares estimates in

table 9 are smaller than the corresponding values in table 8 in the same set

of circumstances. The implication of these results is that even if somehow we

knew or could estimate Yl^. in advance of making a specific forecast^ we would

do better to forecast the several utilizations from all of the predetermined

variables in the structural system than by making use of the estimate of the

endogenous variable Yi|. and the specific Z occurring in equations (162), (l^S)^

or {16k) in either the limited information or the least squares estimate of

the structural equation for the particular utilization. These resiilts have

important implications with respect to the kinds of variables that should be

included in eqiiations designed basically for forecasting.

Forecasts based on all of the predetermined variables and the known

price.—\ThB,t happens if we make use of the known price in addition to all of

the predetermined variables in the system? At least for the initial sample^

we know that our least squares estimates can be no worse than if we ignore

this information. But, on first thought, we might not expect much improvement

since this price itself is based on the several predetermined variables in the

model. As degrees of freedom, as well as computational time, are always a

problem when we work with multiple regression analyses, it seemed worthwhile

to find out how much we gain by including in our estimating equation such endogenous variables as are expected to be known at the time of forecast.

This test was conducted in two stages: (l) We determined the extent to

which the multiple coefficient of deteimination for the initial sample was

increased when the least squares estimating equations for Y^, Y2; and Y3,

respectively, include Y¡^ in addition to the four Z*s in the model. The results are shown in table 10^ In every case the multiple coefficient of determination is increased substantially by the inclusion of Yl}.. (2) For the first

and third pairs of test samples, we determined the accuracy of forecasts based

on the least squares regression equations when Yij. is included. The results

are shown in table 11. In every case the mean squared errors are smaller than

by any of the other methods that were used. Thus it is evident that if we

know the value of one of the endogenous variables at the time of forecast, we

should fit a least squares equation that includes this as one of the independent variables and use this eqtiation as a forecasting mechanism.

When we work with the algebraically-derived equations, we have no way of

making use of an advance knowledge of Y¡^ except by using this price directly

in the structural equations which, as we have seen, results in poor forecasts.

At least for this model, the slightly greater accuracy of prediction which the

algebraically-derived reduced form equations have over the least squares
- 139 -

reduced form equations disappears completely if we compare the limited-information results as shown in tatle 9 with those obtained from the least squares

equations when Yi^ is included as an independent variable as shown in table 11.

Table 10.—Initial sample: Multiple coefficients of determination for least

squares equations that include all of the predetermined variables

Dependent variable

When price (Yi|.) is—

Yi i Y2 i Y3

Excluded ••••••••• •••••••••' 0.97 0.87 0.85

Tncluded ...•«•••••••*•••••*•••• .997 .98 .98

Table 11,—Difference between actual and computed endogenous variables when

estimates are made from Yl¡. and all of the predetermined variables in the

system: Mean squared error for alternative samples when xise is made of

least squares regressions

Sample

When the Z*s are

drawn over—

First Second

Yl , i ^2 i ^3 Yl ! ^2 ; Y3

The same range as in

the period of fit .... :

Twice the rsmge .......

0.5

, .4

5.06

5.39

3.35

5.39

0.1

.3

2.56

5.12

5.31

Forecasts imder a changed structure.—For the two samples in which the

Z's vary from observation to observation but were drawn over the same range as

for the observations used in the initial fit^ the model was recomputed by

omitting the second equation. In a real situation this would be representative of a commodity with three major utilizations in the period of fit but

only two in the period of forecast. The res\ilts, which are shown in table 12,

differ only slightly from those obtained for the basic model. The least

squares estimates are far superior to those from the limited information

approach when use is made of the price-utilization equations. However, use of

the forecasting formulas derived by algebra from the limited information
- lií-0 -

coefficients when the second eqxmtion is omitted restated in errors that are

smaller or about the sajne for Y^ but much smaller for Y3 than either set obtained by use of the price-utilization equations*

We first thought that no use could be made of the reduced form equations

fitted by least squares under this sort of a change in structure. Subsequently^ we decided to try these equations when ZQ is omitted. An adjustment

was, of course, required in the constetnt term. Results are shown in the table

when Yi^ is (l) excluded and (2) included. Particularly when Yl^ is included,

the mean squared errors are considerably smaller than when estimates are made

from the structural equations. Best estimates for Y3, however, are obtained

from the algebraically-derived reduced form equations. This may result from

the particular form of the system of equations, since the error in Y3 can be

no greater than that for Yi since these two endogenous variables must add to a

fixed sum. For this phase of the experiment, a model that involved four

utilizations would have been preferable.

Table 12.—Difference between actual and computed endogenous variables when Yg

is omitted from the structure: Mean squared e,rror for alternative methods

when the Z's are drawn from the same range as in the period of fit

Sample

Method :

First Second

Yi : Y3 :' Yk 1 Yi ; Y3 ': Yi,

When Ylj. is given and use is made :

of price-utilization eq.uations :

fitted ty— •

Least snnares ..•••••••..••••••.. : 4.5 181.9 — 3.9 316.8

Limited information •••••.••.••••! Q.k 5l)-4.2 — 8.8 6U3.3

Estimated from Z^, Z3 and Z^.

using equations obtained by—

Least squares when Yl|. is—

: k.l 166.4 kk.k 5.6 155.3 34.5

TncTuded «.....•.•..••..••...•• : 2.3 88.8 — 2.3 139.0

Algebraic derivation from the

limited information coefficients

when the second equation is

n^m^ i'A'.&ñ fTr»n the model «.«••.... • li.C) U-P ^^.? k.i U.l ^8.Q

•

Other tests that might be made.—^The tests discussed here answer most of

our questions with respect to how to forecast from a price-utilization model

when price is at its equilibriimi level. Additional tests would be desirable
. 141.

to determine (l) how to forecast several interrelated utilizations, either

singly or in combination, when price is at a level considerably different from

its eguilihrium value, as is apt to be the case under Governmental regulation

or support, and (2) forecasting techniques when supply is simultaneously determined with demand.

Conclusions.--1» Table 13 sximmarizes the basic conclusions with respect

to the relative merits of the several methods of estimation when applied to

samples outside the initial period of fit. Results are compared for five alternative methods of estimating the respective utilizations and for two

alternative methods of estimating price. For three of the methods of estimating the utilizations, price is assumed to be known exactly from an outside

source. In comparing some of the alternative methods, either (a) no consistent superiority was in evidence or (b) the results, although slightly superior

for one method or the other, were not sufficiently different to be of practical significance. In such cases, an identical ranking number is assigned to

each method.

In forecasting price, the two methods, each based on all of the predetermined variables in the system, are closely similar, but the algebraicallyderived limited information coefficients have a slight advantage, particularly

when the Z's are from twice the initial range.

With respect to forecasts of the utilizations, the least squares estimates derived from all of the Z^s and the known Yi|. give the best forecasts

except when Y2 is omitted from the structure; in the last named case, the

algebraically-derived limited information estimates based on all the Z's are

best, but this may result from a particular characteristic of this model.

Poorest forecasts are obtained when use is made of the structural equations

and the known Y^j in this phase of the experiment, the coefficients obtained

by least squares give better forecasts than those obtained by limited information even though the latter are closer to the true structural values. This

results because the imexplained residuals are correlated with Yl^..

2. The importance of classifying variables into (a) endogenous and

(b) predetermined—which is equivalent to specifying whether they are or are

not correlated with the unexplained residuals in the stinactural equations—is

upheld when we are interested in estimating the structural* coefficients, although (as noted on page 67 ) the statistical bias that results if we use the

method of least squares rather than a simultaneous equations approach to estimate struct\nral coefficients tends to be small unless the several imexplained

residimls are themselves highly correlated. In connection with forecasts,

however, this distinction is not important. We should use as independent all

variables that are expected to be known at the time of forecast. As noted in

item 1, the forecast eqtmtion, if fitted by least squares, should include all

predetermined variables in the system plus any endogenous variables that are

expected to be known.
3» A-ll of these conclusions apply when price is at its equililDriijm

level. Further research is needed to determine how forecasts should be made

when price is at some other level, as it might he under government controls»

Tahle 13.—^Approximate ranking of alternative methods of forecast in order of

minimum mean squared error: Specified conditions l/

WITH RESPECT TO THE THREE UTILIZATIONS

Method

With no change in

structure when the Z's

are from the—

Same range

as in the

period

of fit

With a change in

structure due to

the omission of

¥2 when the Z*s

are from the

same range as

in the period

of fit

When yi|. is given and use is made

of the price-utilization

equations fitted hy—

Least squares

Limited information

Estimated from all predetermined

variables in the system using

eqixations obtained by—

Least squares when Ylj. is—

Excluded •

Included •

Algebraic derivation from the

limited infoniiation

coefficients

3

h

3

3

2

1

2

1

k

5

3

2

WITH RESPECT TO PRICE

Estimated from all predetei^ned

variables in the system using

equations obtained by--

Least squares

Algebraic derivation from the

limited information

coefficients

1/ Identical rankings indicate inconclusive or nearly identical results. The

best method is identified by the nxmiber 1,
- Ilf3 -

Allowance for Errors lu the Data

In a statistical analysis^ we have no way of distinguishing between that

part of the unexplained residual that is due to errors in the data and that

part that is due to omitted variables. 78/ For this reason, methods have been

developed to handle equations for which the variables are (l) subject to error

but in which the true variables are assumed to be related in a functional way

—called "error" models—or (2) known without error but assumed to be related

in a stochastic way—called "shock" models, but practically no progress has

been made in developing methods for handling stochastic relationships when the

variables are subject to error—a "shock-error" model* The latter is, of

course, the situation which prevails in any empirical analysis based on time

series data/ All of the statistical methods discussed so far are based on an

asstmiption that the independent or predetermined variables, at least, are

known without error*

Adjusting the sums of squares for assxmied errors —^Fox suggested a way to

modify the method of least squares to correct for errors in the data, provided

we are willing to make an informed guess about the average magnitude of the

errors. 79/ Such guesses or estimates can be made by carefully reading how

the series was compiled, or preferably, by talking to the person in charge of

the compilation. For most series, we have a fairly good idea as to whether

the error is in terms of, say, a few percent, 10 to 20 percent, or something

higher. If we wish, we can experiment with various assumptions about the

level of error, compare the effect on our coefficients, and then arrive at a

rather good idea as to the effect which likely errors may have on their magnitude

The nature of this correction can be shown easily in the following way.

When we express variables in terms of deviations from their respective mesuis,

the least squares regression coefficient between two variables is given by:

Suppose that y and x are each subject to error and that the magnitude of the

error is given by d and e, respectively. The simple regression coefficient

then equals:

Let us now expand the numerator and denominator. We obtain:

78/ Effects of the omitted variables frequently are referred to as "shocks"

or"'^random shocks" in econometric literature.

79/ This approach and some examples based on it first was published in Foote

and Fox (2^, pp. 29-35).
X(y+<i)(x+e) = Zyx + Lye + Xdx + Xde (173)

X(x+e)2 ^ ¿;^2 ^ 2£^^ ^ 2;e2. (174)

If d and e are random and independent^ and we have a large sample, any

svanmation tena that involves one or hoth of them as a cross-product equals

approximately zero. Summation terms that involve their squares, however, do

not equal zero. Applying this principle, when the variables are subject to

error, we can write the value of the regression coefficient in the following

way:

This gives a biased estimate of by^ because the denominator is too large. But

if we reduce the sum of squares for the independent variable by an amount proportional to the percentage error, then we obtain an estimate of the regression coefficient that is approximately unbiased provided the only source of

bias is that due to errors in the data. Such a correction can be made easily

if we know the level of error approximately. The reader will note that the

bias is independent of any error in y, as the only term involving y is a

cross-product.

One way to obtain the simple coefficient of determination, or the square

of the correlation coefficient, is from the following formula:

Althoxigh an algebraic derivation comparable to that used in obtaining equation

(175) involves too much detail to give here, it is intuitively obvious and

easily can be shown that if we reduce the sum of squares for both the dependent and the independent variables by an amount proportional to their respective percentage errors, we obtain an estimate of the coefficient of determination that is approximately unbiased provided the only source of bias is that

due to errors in the data.

In working with multiple regression and correlation coefficients, similar

reasoning suggests the desirability of reducing the sxmis of squares for all

variables by amounts proportional to their respective errors. Insofar as this

general approach is correct, we see no reason why the same sort of correction

could not be applied to the sums of sqiaares that are involved in equations to

be fitted by the simultaneous equations approach. Some empirical evidence

that the whole approach may be in error is presented immediately after the

following example which shows one way to use this method of allowing for estimated errors in the data as an analytical tool.

In analyses of the demand and price structure for lettuce based on single

equations, Sh\iffett (84, pp. 21-28) obtained coefficients of multiple determination ranging between O.O6 and O.19. He cites evidence to support the
- 1^5 -

hypothesis that the poor results may be caused at least in part by (l) simultaneity between demand and supply relationships, since part of the crop frequently is unharvested for economic reasons and (2) errors in the data. As

partially unharvested acreage and some totally unhaarvested quantities are not

reported in the statistics on lettuce, he felt that the simultaneous relations

could not be fitted with available data. By use of the technique siiggested by

Fox, however, he did arrive at estimates of possible effects of errors in the

data.

He says: "Some of the unexplained price variation for lettuce may have

occurred because of the errors in measurement of the variables involved.

According to estimates of the former Bureau of Agricultural Economics (9^),-.

between 25 and 50 percent of the total supply of lettuce was not included in

commercial supplies during the years for which the statistical analyses were

fitted. Noncommercial production may be correlated with commercial production

to some extent, but the home-garden production would tend to remain relatively

more stable from one year to another than commercial production, as production

for home use woiild not be expected to respond much to prices in previous

years. Some error also is involved in the price estimates because the point

'\rtthin the marketing system to which the prices apply varies from State to

State. As relative production varies from year to year, this factor would

cause some distortion in the series relating to season average prices for the

country as a whole" (84, p. 29).

He considers an analysis of factors that affect the season average price

of lettuce which uses as independent variables annual production and disposable income per capita. All variables were expressed as first differences of

logarithms and the equation was fitted for the years 1921-^1. A coefficient

of multiple determination of 0.08 was obtained when none of the variables were

adjusted for possible error, and the absolute value of the net regression coefficient on production was 0.4. An analysis based on the same variables but

assuming random measurement errors of 25 percent for production and 10 percent

for season average price increased the coefficient of multiple determination

to 0.12 and the absolute value of the regression coefficient on production to

0.7* Computations based on a 50-percent random error component in production

eind 25 percent in disposable income raised the coefficient of multiple determination to 0.4l and the regression coefficient to 2.3-

Shuffett (84, p. 32) concludes, "These computations indicate that an extremely high level of error must be assiamed in order to raise the coefficient

of multiple determination to a statistically significant level. They also

indicate that, although the coefficients for the unadjusted analysis may be

biased toward zero to a considerable extent, an extremely high level of error

must be assimied to change the conclusion that demand at the farm level is

elastic. Satisfactory analyses designed to measure the elasticity of demand

for lettuce probably would require both an allowance for a moderate level of

error in the data, as in the first adjusted analysis illustrated here, and an

allowance for the simultaneous determination of quantities moving into marketing channels and price as discussed in the preceding section."
. 11+6 -

A Monte Carlo study of the effect of errors of observation,—Ladd (63)

presents results from an experimental study of an overidentified 2-equation

model for which the only basic assiimption (see page 58) which is violated is

that the variables are knovn without error. Thirty samples of 30 observations

each were constructed and fitted by both the method of least sqioares and that

of single equation limited information• In one sample, all of the variables

were taken without error, whereas in a set of 30 samples, normally distributed

random errors of a"magnitude such as to give an average relative error variance ranging from O.O8 to O.265 for each of the endogenous and predetermined

variables were used. The errors were computed in such a way as to be distributed serially independently and independently of each other, of the true

variables, and of the random shocks. Comparisons were made between the

results obtained from the single sample for which^ the variables were known

without error and an average for the 30 samples for which errors were taken

in the variables.

Although the results of any experimental study of this sort may reflect

in part the particular model used in the test, the findings are helpful in

appraising the potential usefulness of methods like that proposed by Fox. The

following simmiary of this study is based essentially on that given by Ladd

(63, pp. 294-295):

(1) The presence of errors of observation imparts little statistical bias

to either the least squares or the limited information coefficients, but does

increase the standard errors of these coefficients. (The first finding

directly contradicts the assimiption made by Fox.)

(2) The distribution of the limited information estimates approaches the

normal distribution quite rapidly.

(3) A standard t-test (see page I78) rejects the null hypothesis that the

two methods of estimation give the same average values on five of the six coefficients; on two because of their sizeable least squares bias, on one

because of the different impact of the random shocks on the two methods, and

on two because of the differential effects of errors of observation on the two

methods.

(k) The least squares method can be applied directly to a structural

equation having endogenous variables on the right of the eqioality sign if the

covariance between each explanatory endogenous variable and the random shock

is small. (This confirms the viewpoint presented on page 67.) Even in cases

in which this covariance is quite high, a coefficient may be obtained whose

least squares bias is negligible, but we have no way of knowing in advance

whether this will be the case.

(5) The estimated limited information standard errors and variances

understate the reliability of the corresponding coefficients because of an
inherent characteristic of the method^ the presence of errors of observation

in the data^ or a combination of these two factors.

(6) In an equation with moderate or high covariance between an explanatory endogenous variable and the random shock, the least squares standard

errors overestimate the true standard errors. (This contradicts the findings

from the experimental model discussed beginning on page 128.)

Comments of Kendall and Wold with respect to this problem.—In part because of the conflict in conclusions with respect to the effect of errors in

the data on the coefficients as assumed by Fox and as found by experiment by

Ladd, we conclude this section by quotations from two mathematical statisticians who have investigated this area rather carefully.

In a detailed discussion of structural, regression, and functional relationships, Kendall (¿5, pp. 20-24) includes comments with respect to a number

of 2-variable caaes in which one or both of the variables is subject to error.

If X* and y' are the observed variables that are known to be subject to error,

Kendall mentions that Lindley {6k) has shown, among other things, that the

regression of y' on x' is not linear even if that of y on x (the true variables) is linear unless certain conditions hold. In particTilar, if the distribution of X is normal, only normality in v (the error in x) will preserve the

linearity of y' on x'. Kendall then derives a regression coefficient similar

to that given in equation (l75) hut comments that even if we assume that the

distribution of v (the error in x) is known with known variance, the problem

of testing it for significance is fonnidable. He concludes his discussion of

this topic as follows: "The errors of observation impair our estimators,

vitiate our tests of significance and even bend our regression lines unless

we are prepared to postulate normality in the variâtes, which is asking rather

a lot, particularly in economic work. But they are not quite as devastating

as they look. A slight departure from linearity will sometimes allow the

ordinary theory to be used as an approximation. None the less, it would

appear that much more effort is needed to reduce errors of observation than

has sometimes been supposed" (¿5, pp. 23-2^).

As a summary to a similar discussion of the effects of errors of observation. Wold (104, p. hh) concludes as follows: Corrections for errors in the

explanatory variables "have been worked out on the assumptions that the errors

are independent of the error-free variables and that the error variances are

known a priori. ... The disadvantage in such correction methods is that in

practice we have little or no information about the observational errors,

neither of their presence nor of their distributional properties. Hence, if

it is felt that the data contain observational errors that are not negligible

(negligible relative to the influences upon the effect variable that are due

to unspecified causal factors), it is dangerous to employ the data for more

than a tentative orientation."
- 11^8 -

Effects of Serial Correlation in the Error Teims 8o/

In this section we siimmarize recent work on the statistical effects of

possible nonindependence in the true error terms of an economic relationship.

This detailed summary is included in this handbook for two purposes: (l) The

original literature is highly mathematical and is not simimarized well in any

other place and (2) actually little is known in a general way about the

effects of serial correlation in the error terms. By spelling out in detail

what is known^ the reader can see by default how much is not known in this

area.

Nonindependence in the error terms, according to Cochrane and Orcutt (l$>

pp. 36-38), usually takes the form of positive serial correlation of the error

terms for the following reasons: "(l) Systematic errors may arise from a

faulty choice of the form of relationship assumed to exist between economic

variables. Since the economic variables are positively autocorrelated, 81/

then, in general, errors of this type will be positively autocorrelated. Further

the shortness of most available time series makes the statistical results

meaningless if very complicated relationships are adopted, so that errors of

this type are inevitable.

"(2) Error terms may arise owing to the omission of variables, both economic and noneconomic, from the analysis. Important variables may be omitted

either because they are not available or because their importance is not

realized. Furthermore, because of the brevity of available time series, it is

also frequently necessary to neglect variables which individually have but

small influence. Nevertheless, it is evident that the total influence of a

number of such variables may be very substantial and highly positively autocorrelated. ...

*'(3) The series of data used may not measure exactly what is required for

the particular analysis. Insofar as the discrepancy is one of coverage, it

seems reasonable to believe that the error term involved will have much the

same autoregressive properties as economic series in general. Insofar as the

discrepancy is more nearly what might perhaps be called pure error of observation, it would appear more difficult to say anything about whether or not it

is autocorrelated. 82/ ..."

80/ This section was prepared by Marc Nerlove, agricultural economic statisti'cian. Agricultural Marketing Service.

81/ Evidence for the tendency of economic variables to be positively autocorrelated is presented in Orcutt (77)• Economic variables are, of course,

not necessarily positively autocorrelated.

82/ Because of the way in which agricultiiral series are constructed, it

seems likely that such errors in the series as occur will tend to be positively autocorrelated. For a description of the estimating procedures used to

construct agricultural series see U. S. Dept. Agr. Mis. Pub. 703 (2^); and for

a discussion of some of their statistical properties, see Nerlove, Marc,
- Iit9 -

Thus^ we not only expect nonindependence in the error terms^ but we expect this nonindependence to take the form of positive autocorrelation, In

most of the following discussion, however, we do not restrict ourselves to

positive autocorrelation but consider temporal nonindependence more generally.

In order to understand some of the recent work on the statistical effects of

nonindependence, we introduce the notion of a stochastic process.

Stochastic processes 83/*—^A stochastic process is simply a generalization of the notion of a random variable. 84/ As is well-known, the probability that a random variable x is less than a particular value u defines the

cumulative distribution function of the random variable, F(u). This fimction

has the following properties:

0 ^ F(u) < 1

(177)

lim F(u) =

VL ^ 00

lim F(U) =

U -i" - 00

f Ï 0. du

The idea of a random variable may be generalized easily to that of a

random vector. In this case the cumulative distribution function has an many

arguments as there are components of the random vector. If, for example, the

random vector has n components, its cumulative distribution function has n

arguments.

In principle, the notion of a random vector and a cumulative distribution

function are valid irrespective of the nature of the measurements or of the

dimensionality. Specifically, they apply if the random vector is a time

series of any finite number of observations. Since the observations of a time

series are, in principle, unlimited in number, an extension of the notions of

an n-dimensional random vector and cumulative distribution function is required. We are thus led to consider a random vector of the form

Estimates of the Elasticities of Supply of Corn, Cotton, and Wheat, Ph.D.

thesis, Johns Hopkins University, 1956, pp. IOI-I38. Statistical problems

resulting from errors in the variables are discussed briefly in the preceding

section.

83/ This section is based largely on Wold (l02, pp. 1-8, 31-66) and Wold and

Jureen (106, pp. 1^9-168). Readers who are unacquainted with calculus and

matrix algebra may prefer to skip to the concluding paragraphs of this section, which begin on page 169.

8h/ Cramer (16, pp. 57-58) gives a clear but nonrigorous definition of a

random variable.
- 150 -

X = (.•., xt-i, xt, xt+i, ...) (178)

where t runs thro-ugh all the integers from -00 to +00. The variables x^ are

all random, "but they differ from random variables in general in that they are

ordered. The random vector x—of infinite dimensionality—is called a

stochastic process; it expresses the ordered development (usually, although not

necessarily, throvigh time) of a series of events suhject to random influences.

Any particular sequence, x, is«called a realization of the process. For

example, the error terms of an economic relationship are a realization of a

stochastic process.

Since any m components of the vector x form a random vector of finite

dimensions, we have a joint cumxilative distribution fxmction for the m random

variable components which we loay write as

F(t, m; ui, ..., Um). (lT9)

The particular vector to which equation (l79) refers is

(^t+l> ^t+2> •••> xt+Bi)* (180)

Thus, for a stochastic process, we have a whole set of cumulative distribution

functions, rather than a single function as we have when dealing with a random

variable or a random vector. This set of distribution functions, is generated,

for a fixed positive integer m, by letting t range over the integers from - 00

to 00. These distribution fxmctions must be consistent in the sense that

lim F(t, n; ui, ..., u^) = F(t, m; u^, ... u^^), (181)

%rf.l> . . ., Uj^ -> 00

where m < n and - 00 < t < 00 . A set of distribution fxmctions which satisfy

equation (181) may be said to determine the probability density of a particular realization of a stochastic process. 83/ The notion of a stochastic

process may be extended to cover that of a stochastic vector process. In this

case each component of x is interpreted as a vector.

In our discussion of the effects of nonindependence of the error teims in

an economic relationship, it is useful to distinguish between two fundamental

types of stochastic processes: stationary processes and evolutionary

processes. These two classes are mut\mlly exclusive and exhaust the totality

of all stochastic processes. A process is called stationary if and only if

the set of distribution functions (F), generated from equation (l79) "^y letting t range over all the integers from - 00 to 00, satisfies the following

relation

F(t, m; ui, ..., Um) = F(t+v, m; ui, ..., u^) (182)

85/ See Wold and Jioreen (106, p. 153)
- 151 -

for all m > 0 and all t and all 11^ (i=l, ...^ m) between -00 and 00. The

definition of stationarity contained in equation (182) amounts to treating

time^ or the ordering variable t^ in a purely relativistic sense. The probability laws assimed to rule a stationary stochastic process "... depend on

time in such a way that if we replacé time as measured from a fixed time point

by a time variable measured from another time point, the probability laws will

remain the same. In other words, if the development in a time series [or

stochastic process] is known up to a certain time point, say t, the probability laws ruling the continued development will depend only on the behavior of

the time series up to the time point t, not on the actual value of t." Wold

(102, p. U). All stochastic processes which are not stationary are

evolutionary in the sense that the properties of their sets of distribution

functions do depend on the partic\ilar point from which we measure time.

For a stationary stochastic process we define the mean and variance as

follows

/OO

u du F(t, Ij u)

-00

/OO

(u-m)2 d^F(t, 1; u).

> (183)

The correlation of x^ with xt+n or xt^n i'^'^^ "two are equal if the process is

stationary) is called the nth order autocorrelation coefficient and is defined

as

Pn = -9— • (18^)

The set of autocorrelation coefficients generated by letting n range over the

integers from - 00 to 00 is called the correlogram of the stationary process.

The fact that equation (182) holds for a stationary process is crucial in

the above definitions, for this equation clearly implies that the mean, variance, and correlogram of the process are independent of the ordering variable

t. If this were not the case, m , a ^, and i p^} - oo<n<<=« } all would be

functions of time, and therefore not useful as characteristics of the process.

Although stationarity of a process implies that it has a mean, variance, and

correlogram independent of time, the converse is not generally true. If a

process has-a mean, variance, and correlogram independent of time, we say that

it is stationary to the second order. Stationarity implies stationarity to

the second order, but a process stationary to the second order may be evolutionary in the sense that it does not satisfy equation (182).
- 152 -

In most applications of the theory of stochastic processes to economic

relationships, the error term is taken to he a stationary process. Intuitively, this implies that the economic structiare imderlying the relationship

has been in its present form for some time past, so that the random influences

at work have had sufficient time to settle down to an orderly existence. Not

only, however, is it usually difficult to find periods of sufficient length

over which we may assume no change in structtire within the particular set of

relationships under consideration, hut it is even more difficult to find a

period hefore which no structural changes have recently occurred. Be this as

it may, stationary processes are the most susceptible of precise formulation,

and no very useful results have yet been obtained without the assxmiption of

stationarity.

In subsequent discussion it is usefxol to distinguish between two fundamental types of stationary stochastic processes: (l) a purely random process,

and (2) a process of moving summation. A stochastic process

X = (..., x^.i, x^, x^^i, ...) (178)

is called purely random if, for all t and n, its distribution functions satisfy the following condition:

n

F(t, n; Un, ..., Uj^) = TT F(t, 1; u^). (185)

i=l

Thus, the error terms of a purely random process are assumed to be completely

independent of one another. For economic series, this assumption is not

likely to be justified. For any purely random process the correlogram reduces to

PO = 1^ ^1 = ^2 =•••== 0» (^^^)

Let e = (..., et-i, e-^, ©t+l^ •••) ^^ ^ purely random process with zero

expectation. For all t between - 00and 00, we may form the sum

z^ = ao e^ + a-L ^t'-l "*" •••^ (I8T)

where the a's are constants, independent of t^ such that ^ s-i is finite (thus

insuring that zt is a finite n\miber).

The stochastic process defined by

z = (..., zt-i, Zt, zt+i, ..•) (188)

is clearly stationaiy, for the set of distributions {F} generated by conibining equations (185) and (187) satisfies the condition stated by equation

(182), provided, of course, that the a's of equation (187) do not depend on

time. For the expectation and the variance of z we have

\
- 153 -

mz = Ezt = O

(189)

2

where u g is the variance of the purely remdom process e. The correlogram of

the process z is given hy

f H ^+i

Pn= ¿:^2 (190)

t^i

for n ranging over the integers from - oo to oo. The process z is called A

process of moving sxinmaation.

The process of moving s\3ma3ation has two special cases which are of interest: (l) the process of moving averages, and (2) the autoregressive

process. The latter plays a major role in current discussions of the statistical effects of serial correlation. The process of moving averages is

defined "by equation (18?) when the siammation is taken over a finite nuiaber of

terms only. Thus, if

H = ^H + ^l^t-l + ••• + s-hH-h (191)

the process z is one of moving averages. The expectation of the process and

its variance are given "by equations {189) except that the si:immations in these

formulae are only over a finite number of tenns. The correlogram is given by

equation (19O) except that /^n = 0 for n > h. The name is unfortunate, as

this is not an average in the usual sense, "but it has become established.

An autoregress ive process is a sort of inverse of a process of moving

simimation. Suppose that for all t between -00 and 00 we have

^t + ^l^t-l + • • • + T^h^t-h = ^U (192)

where the process e is a purely random process. Equation (192) is an h-order

difference equation in zt, and may be solved for zt as a fimction of the

series e^ when t lies between - 00 and 00. The solution is of the form

z^ = et + a^e^^^i + a2et-2 + • • • > (l93)

where the a's are functions of the b's in equation (192). The process z defined by equation (l93) is a special case of the process of moving simimation

in which OQ = 1. The process defined by an equation such as (192) or (l93) is

called an h-order autoregressive process, where h is the number of past values

of zt in equation (I92); thus,

H + ^l^t-l = ^t (19^)
- 15^ -

is a first-order autoregressive process • In order that z-^, as given by eq.xaation (193)^ be a finite mjiiiber, the'a's and hence the h's of equation (192)

must satisfy certain conditions. We need not, however, discuss these here.

For further details see Wold and Jureen (l06> pp. I62-I63)•

A coDimonly discussed stochastic process is the Markov process. A stochastic process x is a Markov process if

E(xt I xt-i, xt-.2^ •••> xt-n) = E(xt I xt-i) (195)

for all n > 2. Thus an autoregressive process is a Markov process if and only

if it is first-order (that is, h=l). For an autoregressive process vhich is

Markov, the correlogram has a particularly simple fo3rm:

^n = (-^l)"" (^ = 0, 1, 2, ...). (196)

Ordinary least squares in matrix form 86/.—In a subsequent section, we

take up Aitken's method of generalized least squares, a method developed to

meet complications introduced when the error terms of an economic relationship

cannot be assumed independent. Since Aitken's method cannot conveniently be

discussed except in matrix form, we first discuss ordinary least squares in

matrix form.

Consider the vectors

y =

X1 =

ß =

Jt.

^i2

Xit/

\

(i=l, ..., n)

86/ This section is based on Kempthorne ($^, pp. 3^^-39)
- 155 -

and

e = > (197)

and the matrix

= [x^^ X2> •.., xj. (198)

We suppose that, at any time t, a relationship holds hetweeh y^, the Xj^^'s,

and e^:

yt =f ßi ^it + n- (199)

The last relationship can be more conveniently written in matrix form as

y = Xß+e. (199.1)

In the theory of ordinary least squares, we generally assime that the matrix X

is given, that is, is nonstochastic, and that the error terms e have the same

distribution, zero mean, and finite variance. If X is nonstochastic, the distribution of e^ cannot involve any of the x^^.

The error terms, or the vector e, may be thought of as part of the realization of a stochastic process. If the process is stationary, the probability

laws holding for the particular part represented by e are characteristic of

the entire process. In the ordinary theory of least squares, e is assumed to

come from a purely random stationary stochastic process. Its correlogram must

be that given by equation (186), so that its variance-covariance matrix, Eee*,

may be written

(j 2 (200)

where o^ ±3 the variance of the process and I is an identity matrix. Under

oiir assumptions we also have that

Ee = 0. (201)

The true error terms should be carefully distinguished from the unexplained residuals which might be calculated from equation (l99) given y and X

and an estimate of ß . Let b be an estimate of ß and u be the vector of

calculated residuals, then by definition we have

y = Xß + u. (202)
- 156 -

The least squares estimate of ß ^ say h, is defined as that estimate^ given y

and X^ which minimizes the s\m of squared residuals^ u'u. From equation (202)

we see that u'u^^may he written

u^u = (y - Xb)V(y - Xb) (203)

= (y' - T:)'XO(y - Xb) (203.1)

= y'y - 2h'X'y + b'X'Xb. (203*2)

Eqxaation (203.2) may be differentiated with respect to b^^ b2^ ...^ b^. These

derivatives^ when set eqiml to zero, give n equations which may be written as

- 2X'y + 2(X»X)b = 0 (204)

or

X'Xb = X'y. 87/ (204.1)

Equation (204.l) may be solved for the vector b, giving

b = (X'X)-l X'y , (205)

and the resulting estimates minimize the sum of the squared residuals, that

is, they are least squares estimates of ß .

An estimate of ß is called linear if it is a linear function of the

vector y, say Ay. The estimates b are clearly linear since we may set

A = (X'X)'"-^ X'. Under the assumptions, the estimate b is also an unbiased

estimate of ß . To see this, we substitute equation (l99»l) in equation (205)

and take the expected value:

E b = E(X'X)"1 X'y

= E(X»X)-1 X'(Xß + e)

= E[ß + (X'X)"1 X'e]

= ß + E(X'X)"-1 X'e . (206)

Since, by assumption, X is nonstochastic and Ee = 0, the final term in equation (206) vanishes; hence, Eb = ß , that is, b is an unbiased estimate of ß .

87/ The reader can verify that equation (20U.I) gives the usual normal

equations and that the process of deriving them by differentiation of equation

(203.2) is exactly equivalent to the usual procedure.
- 157 -

The variance-covariance matrix of b is given by

E(b - ß )(b - ß )' = E[(X'X)-1 X»ee'X(X'X)""l]

= (X'X)""lX' E(eeOx(X'X)"l

= cj2 (X'X)""^ (207)

by equation (200)« The diagonal elements of cj ^ (x»X)""^ are the variances of

the estimates b and are clearly positive.

We may now ask the question as to whether the estimates b have minimimi

variance as compared with other linear unbiased estimates^ that is, whether or

not they are statistically efficient as compared with other linear unbiased

estimates. Consider a linear estimate of ß , By. If the estimate By is to

be unbiased we must have

E(By) = E(BXß + Be) = ß . (208)

Since B and X are nonstochastic and Ee = 0, equation (208) implies that

BX = I. (209)

V7e may arbitrarily write the matrix B as

B = [(X^X)"1 X'+ C]. (210)

By equation (209) we have

CX = 0, (211)

where 0 is the zero matrix. Hence,

b = [(X'X)-l X' + C]y (212)

is an arbitrary linear imbiased estimate of ß subject to equation (21l).

Substituting equation (l99*l) i^i equation (212) and proceeding as before, we

see that the variance-covariance matrix of b is

E(b - ß )(b . ß )» - E { [(X'X)""^ X» + C]ee» [X(X»X)"^ + C»]

= a 2 [(x^X)"""" + CC'i, (213)

since Eee' = cr "^ I and CX = 0. Since each diagonal element of CC is simply

the sum of sq\mres of the elements in the corresponding row of C, they must be

positive. Hence, unless the matrix C is the zero matrix, the variances of the

estimates % are greater than those of b. For this reason the least squares
158

estimates. ID, are called ""best" linear -unbiased estimates. In passing, we

note that the assiamption that Eee' = cr^I, that is, that the error terms are

part of the realization of a pxorely random stationary stochastic process, was

essential in the derivation of the fact that the least sqixares estimates are

efficient. This assiamption was not necessary in the derivation of the fact

that b is an \mhiased estimate of ß .

Wold (103) also has shown that, \mder the assumptions, the least squares

estimates, b, are consistent estimates of ß . This proof likewise does not

depend on the asstamption that Eee' = o ^I.

Effects of nonindependence on ordinary least squares estimates•—When e

is not assiamed to form part of the realization of a purely random stationary

stochastic process, we no longer can assume that Eee' = cr ^j. hence, it no

longer follows that the ordinary least squares estimates are "best" in the

sense of being most efficient. Watson {^D discusses the loss of efficiency

that restilts when Eee* is incorrectly assumed to be equal to ^^I* He does

this quite generally under the assumption that the e form part of the realization of a stationary process or a process stationary to the second order. In

a second article, Watson and Hannan (¿8) apply the results of Watson (^) to

the special cases in which e forms part of the realization of an autoregressive or a moving average type of stationaiy process. Gurland {kh) investigates the loss of efficiency in least squares analysis when the error tenns

form part of an evolutionary stochastic process. Wold (103) discusses least

squares regression with autocorrelated variables and error terms, but his results are more properly described when we come to methods for dealing with

nonindependence.

Suppose that the error vector, e, forms part of a realization of a stationary stochastic process* As we have seen, there exists for such a process

a well-defined mean, variance, and correlogram. The correlogram for such a

process may be written

p^ = p^^ for -^00 < n< 00 , (21^1-)

since the correlation of the t*^ component with the t + nth component and the

correlation of the t*^ component with the t - nth component are the same if

the process is stationary. Thus

Eee' = ß , (215)

where

'1 Pi P 2 • • • P •

Ü (216)

Pn ^n-1 ^n-2
159 -

and a ^ is the variance of the process. The matrix ß is not only symmetric

hut the elements are eqxml along any northwest-southeast diagonal. 88/ If the

process ass-umed is not at least stationaiy to the second order^ the correlogram of the process is in general undefined 89/ and Q does not possess the

desirahle properties just mentioned.

The form of the matrix Q may be illustrated for several types of stationary stochastic processes:

(1) For a purely random process,

Û = a %. (217)

(2) For a process of moving summation, reference to equations (187) and

(190) indicates that

i HH+1

1

Q = a

i ^i^i+2

HH+1

(218)

(3) The same expression for ß holds vhen the process is one of moving

averages except that the summations are now over a finite number of terms.

(k) The general autoregressive process also falls into the same class as

the process of moving summation, since an autoregressive process of finite

order may be transformed into a process of moving sxmmiation of infinite order;

hence, Q may also be expressed as in equation (2l8).

(5) If the autoregressive process is Markov, ve have by equation (196):

ß = CT

bi -b^^

'^1^ (-bi)^-l(-bi)^-2

(219)

88/ Such a matrix is lmo\m as a Laurent matrix; see Wold and Jureen (106,

P- 156).

89/ Note, however, that it is defined if the process is evolutionary but

stationary to the second order.
- l60 -

The particularly simple form of Q in equation (219) has led to the frequent

assumption that the error process is autoregressive and Markov.

If the variance-covariance matrix of the error term is actually ß hut we

take it to be cr ^i in order to justify least squares^ what will he the result?

As indicated in the preceding section^ our least squares estimates^ h^ of ß

are still unbiased and consistent; they are not^ however^ statistically efficient nor will the ordinary t- and F-tests apply (see pages IT8-I83).

Let us consider the relationship to be estimated as being given by equation (199.1). Our least squares estimates of ß are given by

b = (X'X)-l X»y. (205)

We would take the variance-covariance matrix of the b to be

V(b) = CT2 (X^X)-I, (220)

whereas in fact it is not. Substituting Eee' = ß in equation (20?), we find

that actually

v(b) = (x»x)~i X'ñx(x»x)"-"^. (221)

Our estimate of the error variance would be

s2 =TÍr [(y - Xb)' (y -Xb)]. (222)

o—n

This estimate is "biased as:

E(s2) = -r- E[(e-X(X'X)-l x'e)' (e-X(X'X)-l X'e)]

"C—n

^ ^ E[e'e - e»X(X'X)"^ X'e], (223) t-n

where the expression in the brackets in equation (223) is obtained from equation (222) by substituting for y and b from equations (l99-l) and (205). ^l^e

trace of a matrix is the sum of its diagonal elements; if Eee* = ß , Ee*e =

trß, where tr ß means the trace of the matrix ß . By some manipulation it

may also be shown that

E[e' X(X»X)^^ X'e] = tr[X' Q X(X'X)-^], (22^)

so that

E(s^) = T^- [tr ß - tr (X'ß X(X»X)-1)]. (225)

"t-n

Q is a t X t matrix; if Q. = 'cf^I, tr ñ = «^ ^t. X' Q X(X'X)-1 is an n X n
- l6i -

matrix: if ß = a %^ tr (X' ß X(X'X)""^) = a^n; hence^ equation (225) shows

that s^ is unbiased only if ß = a%.

Watson (£7^ p. 329) is a"ble to set limits on the maximum and minimum bias

in the estimate s2. These limits depend on the characteristic roots of the

matrix Q . 90/ Watson is able to derive these limits only for the case in

which the independent variables are uncorrelated and form an orthonormal set,

that is, only when

x'x = I. (226)

The limits in this case are as follows:

Max. bias = cr2 (-the mean of the t-n greatest characteristic

roots of ß - the least root of ß )

Min. bias = C3"^ (the mean of the t-n least characteristic

roots of ß - the greatest root of ß ). (227)

It is clear that the upper limit is always positive and the lower limit always

negative. Iñien ß = cj%^ both limits are zero. The presence of nonindependent error terms is commonly believed to make the least squares estimate of the

variance of the coefficients deceptively small; equation (227) shows that this

may not always be the case. In practice, the limits in equation (227) are not

very useful for, although we can always transform the independent variables in

such a way that they are uncorrelated, that is, so that X'X is diagonal, we

cannot always transform them in such a way that X'X = I, as required if they

are to be orthonormal.

In the preceding section we showed that the least squares estimates are

statistically efficient as compared with any other linear unbiased estimates

provided the error terms e form part of the realization of a purely random

process. When the error terms do not do so, the ordinary least squares estimates in general are not efficient. Our definition of the efficiency of one

estimate relative to another, as given on page 208, is not rigorous; we now

need a somewhat sharper concept. The joint efficiency of an estimate b*^ relative to an estimate b, where b*^ and b are both vectors, is- defined as the

ratio of the determinant of the variance-covariance matrix of b* to the

detenninant of the variance-covariance matrix of b, that is.

Eff(b) =

Vfb

"vtïï

b^) (228)

90/ The characteristic roots of a matrix A may be found by solving the polynomial ) A-X I I =0 for X. If the matrix A is nXn, the equation | A-X I |

= 0 is a polynomial of degree n. It therefore has n possible solutions for

X j if complex solutions are allowed. See Klein (¿8, pp. SUO-S^fl).
- 162 -

where "b-í^ is an alternative estiniate to ID against which we measiire the efficiency of "b. 91/ If "t>*. is the most efficient linear estimate^ then the ratio

given in equation (228) becomes closer to one as the linear estimate h becomes

more efficient as an estimate of ß . Hence^ provided we can find the most

efficient linear estimate^ the ratio (228) can be used to measure the loss in

efficiency if b is used instead of b*.

Aitken (l) has shown that when the variance-covariance matrix of the

error terms is Q ^ the most efficient linear estimates, b^, have a variancecovariance matrix

V(b*) = (X' ß ""^X)"^. ¿2/ (229)

Consequently by equations (22l) and (229) ^^ have

I (x'x)"i X' ñ x(x'x)~i I

= ÍJL2L1 (2-^Q)

Ix'fi x| Ix'fi -1x1 ' ^ ^

since the determinant of an inverse of a matrix is the reciprocal of its determinant and the determinant of the product of two square matrices is the

product of their determinants. Watson (913 PP* 330-331) shows that the ratio

(230) is indeed less than one, so that the least squares estimates, b, are

statistically inefficient. Watson also shows that this ratio tends to one as

Q tends to a 2i and that it has, in general, a definite lower limit which

depends on Q, and on X. His proof does not require the assumption that X*X

= I. In a second article, Watson and Hannan (98) show that the loss of efficiency may be very great, that is, the lower limit to the ratio (230) may be

close to zero, if strong serial correlation of the error terms is present due

to the fact that they form part of a realization of an autoregressive or

moving average process.

Since the estimate of the error variance, s^, is biased, the ordinary tand F-tests do not apply to the least squares estimates if tlae error terms do

not form part of the realization of a purely random process. Watson (97^ PP»

331-3^0) shows that the attributed level of significance at which the ordinai^r

t- and F-tests are carried out is not the true level of significance; he is

able, however, to set limits to the true level of significance. These limits

depend on the variance-covariance matrix of the error terms and are difficult

to compute except for the simplest cases. The limits converge to the usual

levels of significance asñ^ a^i.

91/ See Watson (^7^ P- 330) and Gurland {kh, p. 220).

92/ More is said about this in the following section. See equation (237)•
- 163 -

Gurland {kh) investigates the case in which the error term forms part of

the realization of an evolutionary stochastic process but which the process

would be autoregressive and Markov given sufficient time. Gurland assimies

that the origin of the process is not sufficiently far back in time so that

the initial conditions may be neglected. He shows that additional loss of

efficiency can result from not specifying the initial conditions of the

process. Thus, even if we use the methods suggested in the following section,

which are based on the assxmiption of stationarity, large losses in efficiency

may result because the processes are not stationary but evolutionary. As indicated earlier, evolutionary processes appear more appropriate in economics

than stationary ones, and the reader should bear this in mind when reading the

following section.

Aitken's generalized least squares and other methods for dealing with

nonindependence.--Aitken (l), Cochrane and Orcutt (1$), Klein (58, pp. 86-89),

and Wold (103) all have suggested methods for coping with the problems raised

by nonindependence. The methods of Cochrane and Orcutt and Klein each are

derived from Aitken^s method of generalized least squares; Wold*s method is

not directly derived from Aitken's method. All methods rest on an assumption

of stationarity or stationarity to the second order. All methods except

Aitken*s involve considerable simplification with regard to the stochastic

process underlying the error terms.

Beginning on page 160, we discuss the situation in which Eee' = Q but in

which we assvime that Eee' = ^^1^ that is, a special form for ß . We indicate

that (a) a considerable loss of statistical efficiency may result from such

incorrect specification, and (b) in such a situation the ordinary t- and Ftests of the regression coefficients do not apply. Aitken's generalized least

squares aims at both statistical efficiency and appropriate t- and F-tests; it

solves the general problem by transforming the variables y and X between which

the regression is taken. Instead of taking the regression as stated in equation (202), Aitken recommends transforming all the yariables by premiiltiplication by a matrix H,

Hy = HXb + Hu, (231)

where H is such that

Hß H* = I (2:32)

where we set a ^ equal to one without loss of generality. By equation (199.I)

we have the corresponding transformation of the true equation

Hy - HXß + He. (233)

The variance-covariance matrix of the transformed error terms is

E Hee»H' = H(Eee')H» = H Û H» =1 (234)
- l6k -

by equation (232)^ the definition of H. The transformation procedure, therefore, brings us back to the case in which ordinary least squares is appropriate and efficient, albeit the variables are now Hy and HX rather than y and X.

If we substitute Hy for y and HX for X in equation (205), we obtain the least

squares estimates appropriate to the case in which Eee* = ß ,

b* - (X'H'HX)""^ X»H«Hy

= (X» ß -Ix)-! X' Q -^. (235)

The variance-covariance matrix of b**^ may be found as follows: Substituting

equation (233) for Hy in equation (235) we have

b* = (X'H'HX)-l X'H'HX ß + (X'H»HX)-1 X»H»He

= ß + (X» Q "1X)-1X' Q -le. (236)

Hence, Eb*^ = ß and

E(b* - ß )(b^ • ß )^ = V(b*)

= (X» ß •"lx)"lx»ß -lee' ß-^X (X» ß "^X)"!

= (X» ß -lx)-l. 93/ (237)

The residual variance, s^, is estimated by

s^ = T^TIK ^^^^ " ^^*^' i^y • HXb*)], (238)

where t* is the number of rows in Hy and HX. Note that because of the transformation by premultiplication by H, the vectors Hy and HX do not have the

same number of rows as y and H; they have, in fact, fewer rows for a finite

sample size. It can be shown that E(s^) = a2, so that the estimate s^ is

unbiased. The ordinaiy t- and F-tests apply to the coefficients b*.

In order to use Aitken*s generalized least squares, we must be able to

specify ß exactly. It is naturally never possible to do so, although reasonable assimptions can sometimes be made. Watson's analysis, discussed in the

preceding section, may be applied to the sitimtion in which ß is specified

incorrectly; similar conclusions result. Hence, incorrect guesses about ß

may be costly in terms of the statistical efficiency of the analysis and the

accuracy of t- and F-tests.

Aitken's method may be illustrated by an example: Suppose that the stochastic process from which e is derived is autoregressive and Markov, so that

93/ Hote that the fact that both ß and X* ß ""^X are symmetric is used to

derive equation (237). ß is symmetric only on the assvimption of stationarity.
- 165 -

(239)

where Ht is purely random and hence independently distributed. By equation

(219) the matrix Q may be written

... (-bi)°'

Q =

1 ^! A

^î 1 ^î

..J ^f 1 (21*0)

^A ""^2. j • • • • •

where we again asstmie that a^ = 1. If b^ is known or assumed, it is possible

to specify the laatrix H iised to transform the variables, transform them, and

estimate by ordinary least squares. When ß is as in equation (24o), the

matrix H is a t-1 x t matrix of the form

H =

^1 1 0 ... 0

°\

0 t>l 1 ... 0 o\

0 0 ^1 • • • 0 0

• . . . .

iit/ (241)

Equation (2i+l) may be derived quite simply: Consider one of the equations

(199),

yt = -Ç ßi ^it + Sf (199)

^/ Actually

Q

-1 _

1+bn

1+bi

0

whereas, in this case.
- 166 .

Lag equation (l99) one period, multiply the result by "bi, and add the result

to equation (l99)• We get

yt + ^iyt-1 = f ßi (Xit + ^1 Xit-l) + et + hi et-i. (242)

By equation (239)

^t + ^1 ^t-l = r)t> (239)

where r\ ^ forms part of the realization of a purely random process. When e is

not autoregressive and Markov, it is difficult to specify the transfonnation

matrix H; it is always possible, however, to use equations (235) and (237) to

estimate b* and V(b*), respectively. This latter procedure, actually, is

simpler than transforming the variables, even in the case where the error

process is autoregressive and Markov.

Klein (^, pp. 86-89) discusses a simple generalization of Aitken^s

method for a special case. Klein supposes that the error process is autoregressive and Markov but that the autoregressive coefficient, say bo, is

unknown. Let H be defined by equation (2ifl), except that we substitute bg for

bi in this equation. We may substitute Hy for y and HX for X in equation

(203.2) to get

u'u = y'H'Hy - 2b'X'H'Hy + b'X'H'Hb, (2^1-3)

where u'u is the residual s\mi of squares. In theory we can minimize this residual sum of squares with respect to any number of parameters, that is, the

b's and whatever other parameters enter H'H = Q -1; in practice, however, only

the case which Klein has chosen is practical. In Klein's case, only one additional parameter, bg, is present. Even this case presents computational

difficulties of a high order; Klein shows that the solution of the n + 1 equations, obtained by minimizing u*u with respect to b and bo, requires the

solution of a high order polynomial. The order of the polynomial varies with

bi 0 .

l+b-L^ bi .

H'H =1 0 bi l+bi^.

0

The discrepancy is only l-b^^^ in the upper left hand corner. The discrepancy

is due, of course, to the fact that we have only a finite number of observations, t, and, therefore, that the matrix H cannot be square. It can be shown

that this discrepancy does not affect the validity of Aitken's method [see

Stone (87, p. 289)].
. 167 -

n, the n"U33íber of independent variables. For example, when n=2, the polynomial

is of order 9; when n=3, it is of order 13; and when n=4, it is of order 17.

The reader is referred to Klein (58, pp. 86-89) for further details.

Cochrane and Orcutt (l¿) propose an iterative procedure based on Aitken's

method. Their method can be applied only when the error process is of a relatively simple type; for example, first- or second-order autoregressive. The

suggested procedure, in their words, is as follows: "First estimate the

desired regression coefficients by ordinary least squares and ohtain the resulting series of residuals. Then estimate from those residuals by least

squares the autoregressive parameters of a one or two lag difference equation.

Use the autoregressive parameters to make an autoregressive transformation of

the observed series aimed at randomizing the error term [that is, to determine

the matrix H], and reestimate the desired regression coefficients. 93/ P^t

these revised estimates back in the original equation, obtain the resulting

series of residuals and estimate their autoregressive parameters. Use these

to make a new autoregressive transformation of the original series [sicI] 96/

and so on until estiioates of the desired regression coefficients are obtained

which are consistent with estimates of the autoregressive parameters of the

residuals in the sense that no further adjustments are necessary."

Wold (103) shows that the autoregressive parameters estimated from the

calculated residuals of a least squares regression are consistent estimates of

the true autoregressive parameters. In general, however, by the nature of

least squares, we expect the estimated autoregressive parameters to be biased

towards zero. Consequently, the Cochrane-Orcutt method may stop short of the

desired goal. Watson and Hannan (S§) show that even slight misspecification

of ñ may result in serious loss of statistical efficiency and significance

levels for t- and F-tests that are greatly out of line with the true significance levels. Nevertheless, the Cochrane-Orcutt method is probably a good

one, ,in first approximation.

Wold (103) 97/ proposes a method, not derived directly from Aitken's

method, which aims only at correct t- and F-tests for the ordinary least

squares estimates and not for statistical efficiency as well. Wold discusses

the case in which there is autocorrelation both in the independent variables

25/ An alternative to Cochrane*s and Orcutt*s procedure at this point is to

use the autoregressive parameters, obtained from the series of calculated residuals, to obtain an estimate of the matrix Q "1. The latter can then be

inserted in equations (235) Q^d (237) in order to obtain b* and V(b*). The

estimates so obtained are slightly different from those of Cochrane and Orcutt

because of the finite sample size.

96/ The author of this section interprets this sentence to mean that the

second-round autoregressive parameters should be used to transform the series

already transformed by using the autoregressive parameters estimated in the

first round, and so on. As it stands the sentence does not make much sense.

97/ See also Wold and Jureen (106).
- 168 -

and in the error term, that is, when both the independent variables and the

error term form part of the realization of an autoregressive process. Wold

first proves that the least squares estimates are unbiased and consistent

estimates of ß . Unbiasedness, as we have already seen, is a consequence of

the fact that the proof of the unbiasedness of ordinary least squares estimates does not depend on any assimptions about the error variance-covariance

matrix, Unbiasedness and consistency are highly desirable properties, and

efficiency may well be disregarded, which is exactly what Wold does. Wold

makes use of equation (22l). When the error variance-covariance matrix is Q ,

but we obtain the ordinary least squares estimates, b, their variancecovariance matrix is given by

V(b) = (X'X)"lx» Q X(X'X)""1. (221)

Wold shows that the least sqixares estimate, s^, of the error variance, a^, is

statistically consistent. 98/ Consequently, we may replace a^ by s^ in equation (221). Wold also shows that the least squares estimates of the autoregressive parameters, obtained from the calciliated residual terms, are consistent estimates of the true autoregressive parameters. We may, therefore,

estimate the matrix Q . Replacing ß in equation (22l) by its estimate we

obtain consistent, but not necessarily -unbiased, standard errors for the least

squares estimates. For small samples, these estimates may be severely biased,

probably toward zero. Wold points out that the standard errors so obtained

are usually, but not always, greater than the ordinary standard errors. When

estimating the correlogram of the error term, simplifying assumptions usually

have to be made; typically one cannot handle the assiomptions of an autoregressive process of more than first or second order.

In the siinple case in which there is only one independent variable,

formula (22l), with the appropriate substitutions, results, after some manipulation, in

var b-L = -^ ^ I ^ ^ ^-^ , {2kk)

where s is the estimated residual variance, s? is the variance of the independent variable xj^, (l, P]_, P2f •••) is the estimated correlogram of the

error process, and (l, r^ -', ri^^ .••) in the estimated correlogram of the

independent variable. 99/

98/ This follows from equation (223), hut the proof is involved.

99/ For the simple case presented in eqviation (2U4), the standard error of

bj reduces to the ordinary least squares standard error when ri^^ = rl-^^ =

rll) = ... =0, that is, when the independent variable is not serially correlated. Thus, for nonstochastic independent variables, the ordinary least
- l69 -

Conclusions.--We have seen that nonlndependence leads^ in least squares

procedures^ to loss of statistical efficiency and incorrect t- and F-tests; it

does not lead to statistical bias or inconsistency. Various approximate procedures have been developed for regaining some of the lost efficiency and

deriving correct t- and F-tests of the regression coefficients.' These procedures are based on the assumption of stationarity, which, as we have indicated, is not particularly appropriate in the economic realm. Moreover,

although the procedures possess large-sample validity, it is doubtful that

they are good approximate procedures for dealing with the short series which

are typical in econoiaics. Nevertheless, if severe serial correlation is

present, the methods discussed in this section probably are better than

nothing.

We have not discussed the effects of nonlndependence in the context of

simultaneous eqimtions. Little is known about this subject. We close with

two remarks: (l) When a system is just identified, our entire discussion

applies to the least sq-uares estimates of the reduced form parameters. We do

not know the effects of nonlndependence on the limited information estimates

of the parameters in an overidentified equation. (2) When lagged endogenous

variables are included as predetermined variables, the situation is much worse

than that described above. The least squares estimates of the reduced form

parameters not only are inefficient in a statistical sense but they also are

biased and inconsistent if the reduced form error terms are not independent.

Spatial E(^uilibrium Analyses 100/

The analyses and techniques discussed so far deal chiefly with studies

designed to obtain quantitative measurements of the demand and price structure

for a given market area, where this area frequently is the United States as a

whole or, at times, the entire world. Spatial equilibrium analyses instead

are concerned with studying interrelationships between regions or countries.

They deal with geographical price equilibriums under stated conditions or

changes in conditions and flows of commodities between several areas.

This type of model is valuable because it is operational and the computations are manageable. Information obtained from solving spatial price equilibrixmi models permit one to make predictions as to the direction and magnitude in which the variables of the system will change when some change is made

sqiiares standard errors are consistent but not unbiased estimates of the true

standard errors. This result, however, holds only asymptotically for large

samples. Watson (^^ pp. 24-26) shows that, for the simple case of one independent variable, this result holds even for small samples. In dealing with

economic time series, however, nonserially correlated independent variables

are rare indeed.

^Watson, G. S. Serial Correlation in Regression Analysis. Ph.D. thesis.

North Carolina State College, 1951.

100/ Most of the material in this section is adapted from Judge' (¿3).
- 170 -

in the data of the problem^ for example^ the effect of a rise in iinit transport costs "between regions i and J* By utilizing the resxilts of these

equilibrium models^ certain insights into the changing character of a particular industry can be obtained.

The conceptual framework used to explain the mechanism which generates

the level and location of consumption, the market prices and the geographical

flows among regions is a perfectly competitive market in space, form, and

time/ The necessary condition for a perfectly competitive market that is

relevant in this model is a xmiform price which differs only by transportation

and handling costs•

All firms are assumed to have the objective of maximizing net profits and

thus to choose that market which, deducting transportation charges, will yield

the largest possible profit. The supply sources and markets within each

region are ass\mied to be at a single point. Transportation cost per unit

between each possible pair of regions is assimied to be independent of volume

and direction. Commodities are assumed to be homogenous, that is, an item

produced in one region is the same as that produced in another. The time

period considered usually is one year and the quantity available by regions,

population, and disposable income are assumed predetermined variables within

this time period. The model employed normally is static, that is, it does not

allow for accimiulation or depletion of inventories. Therefore, all demands

must be met from current production. It is fxirther assumed that the market

demand schedules for each region are known; they may or may not be identical.

Given the predetermined geographical supplies and a unit cost of shipping

from source i to market j, the problem is to maximize profit (minimize transportation costs) to each source subject to the side conditions given by the

market demand curve in each region. In addition, this maximization is also

subject to the following conditions involving shipments: (l) all of the supplies from each source must be sold, (2) no negative amounts can be shipped,

and (3) shipments are made only to markets where per imit transportation costs

do not exceed price. The first assumption can be dropped if this appears desirable, as when purchases by an individual firm or Governmental agency from a

number of suppliers are being studied. It usually is retained when studies

for an entire industry are made.

Our problem involves the simultaneous maximization of profits by a n\mo¡ber

of entrepreneurs, but it can be transformed into a single maximization

problem. We may visualize a low calibre "mastermind" who is in control of all

sources of supply and tries to maximize his total profits but does not realize

his decisions affect prices. In that case he will direct his manager at every

supply source to ship to those markets and only those markets yielding the

greatest per unit profit. The optimum set of flows can be defined as the set

chosen by a monopolistic firm which encompasses the entire industiy and wishes

to minimize its cost of meeting given demands. This problem is formulated in

the framework of linear programming; it can be shown, however, that the resulting minimimi cost set of flows is the one that would be determined under conditions of a perfect market.
. - 171 -

A solution to this problem exists provided the quantities to he disposed

of hy the various so\irces are not so large that they cannot be sold at prices

covering the transportation costs. The final solution must of course satisfy

the conditions of the regional supply and market demand relationships and must

be consistent with the decision riiles given previously regarding shipments.

The solution obtained is unique except for the case when two or more sources

find two markets equally profitable. In this case more than one optimum shipment plan exists.

The solution yields a price for each source and market^ quantities that

will be shipped over each path^ and thus the net sxirplus or deficit for each

region. Once the model has been fitted, it can be used to measiire the effect

on prices and shipments of (l) a change in transportation costs, (2) a change

or shift in demand within one or more regions, or (3) changes in production in

total or by areas. If the model is an international one, it can be used to

measure the iinpact on prices and foreign trade in the several countries of

changes in tariffs, export subsidies, quotas, and the like. Simple examples

of this sort are described by Fox (¿5) •

Methods of analysis are discussed by Judge (¿3) and in the references

cited by him. Applied analyses have been made by Judge for eggs and by Fox

(3^) and Fox and Taeuber (38) for livestock and feed grains. A number of

studies that relate to industrial commodities, either for individual firms or

entire industries, also have been published.

STATISTICAL CONSIDERATIONS IN INTERPRETING THE COMPLETED ANALYSIS 101/

Considering the Signs and Magnitudes of the

Statistical Coefficients

The first check that normally is made on a regression or structural

analysis is to determine whether the signs and general magnitude of the coefficients are consistent with expectations. Price-quantity coefficients, for

example, always should be negative and income coefficients for most commodities should be positive. At times, advance information is available that

strongly siiggests whether certain coefficients should be elastic or inelastic

and the nature of the signs that should hold for certain cross-elasticities.

At other times, a purpose of the analysis is to ascertain the basic nature of

these coefficients and, in such circxmistances, these criteria cannot be used

for checking purposes. "Wrong" signs or magnitudes from statistical analyses

may occasionally lead to a revamping of the underlying theory but they are

more likely to indicate the need for a different statistical approach in the

analysis.

101/ Part of this material was originally published by Foote and Fox (22,

pp. 25-29, 35-36) and part is adapted from Foote (28). Other sources are indicated by footnote.
- 172 -

As noted on page ^3^ the constant value in a first-difference analysis

should be consistent in sign and approxiiuate ¡Magnitude with expectations based

on any lonmeasured trends that are believed to affect the dependent variable,

provided this coefficient differs from zero by a statistically significant

amoimt.

Considering the Unexplained Residxmls

In the 1920's and the early 1930's, major emphasis in regression analysis

frequently was placed on obtaining a high multiple correlation coefficient.

Dozens of analyses often were run, each differing slightly from the others by

the use of (l) slightly different forms of variables of major importance,

(2) alternative variables, each of which were assumed to have only a minor

effect on the dependent variable, or (3) alternative methods of manipulating

the data. If rxm by the graphic approach, several successive approximations

would be obtained for each variant. Multiple correlation coefficients of 0.99

or above were a common occurrence; yet frequently these analyses had little

value for forecasting for even a year or two. The trouble lay in the fact

that the analyses had been manipulated to such an extent that they gave a

false sense of security—an almost perfect fit was obtained for the particular

sample but this sample, which seldom contained more than 20 to 30 observations, in a sense no longer represented the entire universe.

Now it is expected that part of the variation in the dependent variable

will remain unexplained. Factors that cause unexplained residuals can be

divided into three types:

1. Those due to errors in the data. (See page 1^3 for methods that can

be used to estimate the importance of this factor.)

2. Those due to the omission of certain variables. This may be because

the analyst fails to think of them, because no data are available, or because

in most years they are so minor as not to be worth including in the study.

This is the kind of random error which normally is assxmied in a least-squares

analysis and it is the type of error allowed for in simultaneous-equation

"shock" models.

3. Those resulting from the \ise of wrong types of curves> incorrect

lags, and similar factors. Charts that indicate the degree of partial correlation help to measure the relative importance of the nature of the chosen

curves in causing residuals (see page 17^).

Residuals for years not included in the analysis frequently are larger

than those for the years included. The increase in unexplained variation may

be caused by the following factors: (l) Extrapolation beyond the range of data

included in the analysis. Some tests designed to measure the importance of

this are discussed in the section beginning on page l84. (2) A change in the

basic structure of the relationships may have occurred. This might reflect a

change in the nature of the curves or the increased importance of some omitted
- 173 -

factor. The charts described in the section beginning on page 17^ help to

indicate whether the nature of the curves has changed. (3) The regression

curves in the analysis tend to adjust so far as possible to compensate for the

types of errors discussed in the preceding paragraphs. Deviations in the

later years reflect not only the true variability but also the extent to which

the regressions were warped from their true shape to keep the deviations for

the years included in the analysis as small as possible. This factor was particiilarly important in connection with some of the analyses run by the graphic

method in the early 1930*s.

If analyses are carefully formulated^ making use of the basic principles

given in this handboolc^ and manipulations purely for the purpose of raising

the multiple correlation coefficient or increasing the degree of fit in some

short test period are avoided, the results have a good chance of applying in

future periods with as high a degree of accuracy as found for the years included in the analysis. Of course if a basic change in structure has

occurred, an adjustment to allow for this is required (see page 21).

Testing for Serial Correlation in tíie Residuals

Durbin and Watson (l8) developed a method by which the imexplained residuals from an equation fitted by least squares can be tested to see if

successive values are correlated. Use of the limits shown in their table must

be regarded as approximate when this test is applied to residuals from equations fitted by the limited information approach or to equations fitted by

least squares that contain a lagged endogenous variable. But no exact test is

available for such residuals.

In using this test, we compute the following statistic:

N

^ (dt - dt.i)^

d« = ^^^ It , ^ (245)

t=l ^

where d^ is the unexplained residual for observation t. If gaps occur in the

data, as when certain war years are omitted, the nuniber of observations that

enter into the numerator are reduced by one for each gap, since these observations consist of first differences. To retain internal consistency in the

formula, the number of observations that enter into the denominator also

should be reduced by one for each such gap. This can be done by omitting the

^ 2

d+ that immediately follows the gap in obtaining X d^.

t=l

If the dependent variable in the analysis is converted to logarithms, so

that the residuals are in these terms, the Durbin-Watson test should be

applied to the residuals in logarithms.
- 17^ -

Friedman and Foote (4o^ pp. 77-78) discuss the use of this test and reproduce the required table from the original Durhin-Watson article. The test

as described by them applies when positive or negative serial correlation is

believed equally likely. Anderson (3^ p. Il8) points out that "In most cases,

the experimenter desires a test of the null hypothesis against the alternative

of positive correlation.'' (Reasons are given beginning on page l48 of this

handbook.) In the notation \A^ed by Friedman and Foote, \mder these circxmistances we expect a small value of d' when the null hypothesis is false, and

the following testing procedure should be used: If the computed value of d' is

less than the tabulated value for dL, the null hypothesis is rejected. If d'

is greater than dy, the null hypothesis is assumed to be true. If d' lies

between dL and d^j, the test is inconclusive. When use is made of this onetailed test, the level of significance is half that of the two-tailed test

described by Friedman and Foote.

Testing for Honlinearity in an Equation

If the equation of a straight line is fitted to data that are from a nonlinear relationship, adjacent deviations tend to be of the same sign, indicating that the residtaals from the regression line are not distributed randomly.

A meas\ire of the relationship between adjacent deviations can be obtained by

modifying the statistic developed by Durbin and Watson (l8) to test for serial

correlation. The modification consists of letting t represent obseirvations

arranged in order of magnitude for a specified independent variable. If more

than one independent variable is used in the analysis, a separate test must be

made for each one. The test is precisely the same as that described in the

preceding section. If d' or 4-d* is less than dj^, we assume that the relationship may be curvilinear. If both d' and ^-d* are greater than dy, we

assume that there is no-departure from linearity. If neither of the computed

values is less than d^, but one of them lies between dL and dy^ the test is

inconclusive.

Charts that Indicate the Degree of Partial Correlation

A method is described in the Appendix (page 205) which permits the construction of charts that indicate approximately the degree of partial correlation between the dependent variable in a least sqimres analysis and each of

the independent variables. An excellent example of the use of such charts to

determine, after running a study initially, whether the correct form of the

relationships was assumed is an analysis by Armore and Burtis (6), made in

1950, of factors that affect consvmrption of fats and oils other than butter.

The analysis first was win in linear terms, following which such a set of

charts was developed. The chart in the upper left hand corner of figure 9

shows, for example, the relation between consumption and price after adjusting

the dependent variable for the effects of the other variables in the analysis. 102/ Linear relations give a fairly good fit for the years prior to

102/ As shown by Foote (2^), the simple regression between such variables

eq\aals the partial regression but the simple correlation equals the part
- 175 -

CONSUMPTION OF FATS AND OILS OTHER THAN BUTTER

Alternative Statistical Analyses

The residuals sugo**t a curvilinear relationship between X4 and Xi

. X,''32.234*.0093b-¥.139X3-^.401X4 < yttart 192240)

'301 "29 -26 •30 -32 '34 -Se^^^l

'' V" I J^ • '4*0

By using logarithms of Xa and X4, the residual variance is reduced ^ ^ ^

I I I I II

X¡' " -i7.036-.0;4X2 +3Í.7/3 Zog X3-t-6.691 log X4 (years 1922-40)

•34 \ •3sl / '29

•33'•40 ^30/ ? \^¡7í

23 '•26

•28 -29^«;

'36 V39/<',l

2»i%-26

1922 •"X»'23 _

'•3I>J'24" •34rf'^35

'30

fZ'23 _

•^.y-'lo-'h

Addition of 1947 and 1948 does not materially alter the regression relationships

1 \ \

_ X,'» -;4.060-.079X2 •t-30.528 log X3 + 6.463 log X4 (yarn 1922-40. 47-48 )

I— • •, -1 23 I >*37 -35 -•--

3, ^y^» 26

1922 VC?"

34X35^

-/:

11111II11 40

Addition of 1949-56 does not alter the relations when allowance is made for on appropriate timfi_t£ftnd

60 80 100 120 140 Xj 75 100 125 150 175 200 X, 1925 1930 1935 1940 1945 1950 1955 X4

Xt o^¡uil»é,for n»t inflvncmi of X» and X4 Xt adjuil^d for not Influoncoi of Xt and X4 Xi adiii$lodior n«! inflw*iic«t of Xi and Xi

Xj'Domastic disappoarance per person ( pouada ) X3'Induatrial production per peraon ( index: 1935 - 39 '100 )

JifaF Wholesale price, deflated ( index: 1935-39 '100 ) Jf^« Year ( 1922-1 )

*ÍN FIRST TWO SECTIONS, REFERS TO YEARS NOT USED TO DETERMHE THE REGRESSIOM EQUATIONS. IH LAST SECTION,

REPRESENTS DEVIATIONS FROH ADJUSTED TREND.

U.S. DEPARTMENT OF AGRICULTURE NEC. 4341-57(7) AGRICULTURAL MARKETING SERVICE

Figure 9.—^The first three rows of small charts illustrate how this analysis

was modified at the initial fit to take account of ideas obtained from the

charts. The last row of small charts indicates further adjustments when it

was "brought up-to-date through 1956.
- 176 -

World War II on which the analysis is based^ although a definite curvilinear

relation is suggested for "time" even for these years• Addition of dots that

relate to 19^7 sind 19^8, the only post-World War II data available at that

time^ suggested the need for a curvilinear relation for both X3 and Xij.. Use

of a semi-logarithmic relation for these variables (l) improved the fit materially for the years included in the analysis, (2) gave fairly good forecasts

for the postwar years, and (3) resulted in a sign on the variable that relates

to price that is consistent with economic expectations. Subsequent inclusion

of data for 19^7 Q^nd 19^8 resulted in a slightly steeper regression on price

and a moderate improvement in fit for these years.

Although some of the variables used in this analysis no longer are available, other variables that are designed to measiire the same economic phenomena

can be obtained. As a matter of interest, this analysis was brought to date

through 1956. The resxilts are shown in the bottom section of figure 9,. Technological developments in the fats and oils industry suggested that a U-shaped

time trend might be indicated if this analysis were brought to date, but the

sharp downturn suggested by the statistical analysis was not foreseen. It is

generally known, however, that the demand for fats and oils in this country,

even when butter is excluded, has dropped sharply due to increased use of

synthetic products, particularly in soaps and cleaners, paints and varnishes,

and emulsifiers. This downtrend may well have beguji about 1950^ s-s new plants

for use of synthetics that had been planned immediately following the end of

World War II came into production. Thus the dashed line in the small chart in

the lower right hand section of figure 9 inay well represent a true measure of

the effect of technological developments on the domestic demand for fats and

oils during the period 1950-56.

This study f\irnishes an excellent example of how charts of this sort can

be used to revise an analysis graphically. It would be difficult, if not impossible, to obtain a mathematical function that would reproduce the time

trend shown for the up-to-date analysis. The graphic ciirve, however, is all

that is needed. Deviations from the dashed line in this chart can be plotted

aroxmd the original regression curves shown for the other variables to determine whether the new data suggest a change in these relations. These deviations are plotted as circles in the two relevant charts. In neither case is

a change in relation suggested, even though the new observations with respect

to industrial production are mostly outside the range of the original observations. Thus, our study suggests that the basic relationships between

consumption, prices, and industrial production that held during 1922-48 also

hold during I949-56 after making allowance for trends that resulted from

technological developments. Other uses for such charts are described on page

172.

correlation. However, unless the degree of intercorrelation among the several

independent variables is large, the part correlation nearly eqioals the partial

correlation.
- 177 -

The Use and Interpretation of Tests of Significance

Yule and Kendall (l08, p. ^37) state, "It cannot be over-enrphasized that

estimates from small samples are of little value in indicating the true value

of the parameter which is estimated. ... Nevertheless, circumstances sometimes

drive us to base inferences ... on scanty data. In such cases we can rarely,

if ever, make any confident attempt at locating the value of a parameter

within seirviceahly narrow limits. For this reason we are usually concerned,

in the theory of small samples, not with estimating the act"ual value of a

parameter, hut in ascertaining whether observed values can have arisen by

sampling fluctuations from some value given in advance." Tests of significance as commonly used are designed to measure whether the observed value

differs significantly from zero. This is referred to as "the null hypothesis."

In most cases, a test could equally well be made as to whether the observed

value differs significantly from some other value.

Tests of significance may be made under any of the following conditions:

(l) With no previous knowledge; or (2) applied to a factor that is believed to

be unimportant. In these cases, a nonsignificant value for a regression or

correlation coefficient would indicate that the factor should be omitted from

the analysis. (3) Applied to a factor which for theoretical reasons is believed to be important. A nonsignificant result in this instance does not

indicate that the factor is not important. It does indicate a need for further evidence to prove beyond reasonable doubt that the factor is important.

Therefore, such a factor might be left in an analysis, particularly if its

effect is believed to be small. If repeated samples of 20 obseirvations were

drawn from a population for which the true partial correlation was 0.3^ nonsignificant results, based on a 5-P^3^cent probability standard, would be

obtained about three-fourths of the time. Such a factor would not greatly

affect the calculated value in most years, but if interest were centered on

the structural nature of the relationship, the analyst might wish to include

it tentatively pending further evidence.

Requirements for tests to be valid.—^Yule and Kendall (l08, pp. 437-^38)

state, "Our results will be strictly true only for the normal universe. ...

It appears that, provided the divergence of the parent from normality is not

too great, the results which are given below as true for normal universes are

true to a large extent for other universes. ... If there is any good reason

to suspect that the parent is markedly skew, e.g. U- or J-shaped, the methods

... cannot be applied with any confidence." Nonparametric tests which are

independent of the nature of the distribution from which the sample was drawn

are available in certain cases [see Siegel (8^)]. But in general, the requirement of normality is not very restrictive.

As discussed in the sections beginning on page 1^3 and l48, respectively,

if tests of significance as applied to correlation measurements are to be

valid, the independent or predetermined variables must be known without error

and the Tonexplained residuals must be randomly distributed. Certain procedures that can be used if these conditions are not met are discussed in those'

sections.
- 178 -

Same common tests.—1. Student's t is defined as the difference between

any particular value assumed to be the true value in the universe for the statistical measure examined and the value determined from the sample divided by

the standard error of this coefficient. In common practice, the assumed value

is taken as zero, so that t is given as the sample value divided by its standard error. In such cases, the n\ill hypothesis is tested. If we do not know

whether a regression or correlation coefficient should be positive or negative, then we use the conventional or 2-tailed test. If economic theory leads

us to expect one sign or the other, then we should use a 1-tailed test, for

which the level of significance is half that of the 2-tailed test. The t-test

can be used equally well to test for significant deviations from any other

assumed value, such as a slope of -1 for a price-quantity regression coefficient. The t-test may be applied with reasonable confidence to sample values

that depart somewhat from noniial in their distribution, but it should not be

used, of course, where other more appropriate tests are available.

2. For samples of 20 or 30 observations, the standard error of a correlation coefficient cannot be estimated with much reliability if the correlation in the universe is high, whether positive or negative. Therefore, when

testing hypotheses other than the null hypothesis. Fisher's z transformation

should be used. See Ferber (23> pp. 381-386). Snedecor (86, pp. 113, 286)

provides convenient tables for testing whether simple, partial, and multiple

correlations differ significantly from zero.

3- Chi-square is used to determine whether a series of frequencies

differ significantly from a theoretical or expected set of frequencies. Chisquare is defined as

X2=i: i^, (246)

in which X is the observed and m is the expected frequency. A somewhat different use of chi-square is described beginning on page l84.

k. The F-test is used to determine whether the assembly of data in

various classes defined by certain restrictions exhibit on the average greater

differences between members of different classes than between members of the

same class. If they do, it is concluded that the restrictions result in significant separation of the data into at least two classes; that is, that the

means of at least two classes differ. Fox: example, the F-test was used to

learn whether the weights differed significantly for the several supply components in the analysis for corn discussed on page 122.

The F-test also can be \ised in connection with a problem in which the

value of certain regression coefficients are assumed in advance. The following steps are used in such cases:

a. Minimize the siom of squares, making no assumption about the

coefficients. The degrees of freedom attached to this sum of squares

equals the number of observations minus the total number of coefficients .
- 179 -

"b. Minimize the sinn of squares after assigning values to such

coefficients as are to he tested.

c. Find the difference between the first and the second sums

of squares. The degrees of freedom attached to this difference

equals the number of coefficients to which a value was assigned.

d. Obtain the two mean squares by dividing the sum of squares

obtained in steps 1 and 3 hy the appropriate number of degrees of

freedom.

e. F equals the ratio of the two mean squares. If it differs

significantly from zero, the sample indicates that the assigned

values should not be used. If it does not differ significantly

from zero, we have no reason to doubt that the sample came from a

population for which the true values were equal to the assigned

values and the assigned values can be used.

5. A test for overidentifying restrictions in overidentified equations

within a system of equations, developed by Anderson and Rubin (4, p. 56), is

described in Friedman and Foote (4o, pp. 79-8l) and elsewhere.

Application of the F-test to Some Problems in. Regression Analysis

A test to determine whether one or more regression coefficients differ

from an assiamed value.—Suppose we have an equation of the following type:

C = a + biP + b2Y, (247)

in which C equals consiomption, P equals price, Y equals income, and all variables are expressed in logarithms. We wish to test whether bi can be assxmed

to'equal -1, that Is, that the price elasticity is equal to unity.

The usual least-sqiiares regression analysis is run and the following

value computed: (X C^ - CZC) (l-R^). The degrees of freedom attached to this

sum of squares is N - 3, where N is the nxamber of years included in the analysis. The following series next is obtained: C = C + P. This is the value

that C would have in each year if bi were equal to -1 and Y and the unexplained residual were at their given levels. A simple correlation between C*

and Y then is computed and the value {LC^ - C'XC*) (l-r^) obtained. Obtain

the difference between these two sums of squares. The degrees of freedom

attached to this sum of squares equals 1, as a value was assigned to 1 parameter. F is computed as described in steps h and 5 above and looked up in an

F-table, using N - 3 and 1 degrees of freedom.

In this case. Student's t could be used instead. The t-value would be

obtained as follows:
- 180 -

bl - (-1)

and lcx>kecL up in the usiial table, using N - 3 degrees of freedom. But if

values were to be ass\amed for more than one of the parameters and a single

test for all of the assigned values simultaneously were desired, the F-test

would be needed.

A test to ascertain whether the corresponding regression coefficients in

three or more multiple regression equations all are equal.—This example and

the following one are taken from Meinken (70, pp. 100-102). The test was

applied to analyses of factors that affect the price of com, oats, and barley, respectively, in summer. In each analysis, a composite supply factor

based on the iterative approach discussed beginning on page 122 was developed

and used as an independent variable. Two additional independent variables

were used in each analysis—livestock production and prices of livestock and

livestock products. The hypothesis to be tested was that each of the three

independent variables affects prices of com., oats, and barley similarly. The

following eq\aations more definitely indicate the nature of the hypothesis:

> (249)

X¿ «a' + b¿i^23^Í + ^¿2.13X2 + ^03.12^3

XQ = a + ^01.23X1 + ^02.13^2 + ^03.12^3

ÄQ = a + ^01.23^1 + ^02.13-^2 "* ^^03.12^3

where

XQ is the price of com during June to September,

XQ is the price of oats during July to October,

XQ is the price of barley during July to October,

X-|_, X-j^ , and X^ are the respective weighted supply variables,

Xg is the production of livestock during July to December, and

X3 and X3 are the average prices of livestock and products for the months

to which the respective XQ'S apply.

The hypothesis to be tested is that the following relationships hold

simultaneously:

It . t fI

^01.23 " ^01.23 " ''^01.23 " ^01.23

t II III

D02.13 " ''02.13 " ''oa.is " ^02.13

^03.12 = '''03.12 = '''03.12 = '''03.12

^02.13 " '''Ó2.13 = '•'¿¿lis = '''02.13 / (250).

t I I
- l8l -

The following steps are involved in making the test:

1. For each of the three analyses^ commute the unexplained SIM of

squares. In each case, this equals X (XQ - XQ)^ (l - î^^O.123)* ^^^ "^^^ ^®"'

suits. The degrees of freedom attached to this sum of squares is N-P, where

N is the total n\mber of observations in the several analyses and P is the

number of restrictions, that is the number of coefficients involved in the 3

equations in (2ÍÍ-9). In this case, each analysis was based on 20 years. Hence,

N = 60, and P = 12 (9 regression coefficients plus 3 constant terras).

2. Combine the respective s"ums of squares and cross-products, after

correcting for the respective means, for the three analyses. In the notation

used above

Xio^ =Xx¿2+i:x¿'2 +i;x¿'*2^ etc. (251)

Rerun the analysis, using these totals in obtaining the regression coefficients. Compute the unexplained sum of squares for the combined analysis,

which equals X XQ^ (l - R2Q^J^23)» The degrees of freedom attached to this sum

of sq"uares is N-6, the 6 representing the three regression coefficients from

the combined analysis plus the 3 constant terms for the separate equations.

3. Subtract the result in step 1 from that in step 2 and divide by the

difference between the respective degrees of freedom. In this case, the difference in degrees of freedom is 6 [N-6 - (N-12)]. This represents the mean

sqxaare res\ilting from differences among the regression coefficients in the

three analyses.

k. Divide the final result in step 1 by the degrees of freedom attached

to it, namely N-12, in this case. This represents the error or remainder mean

square.

5. Compute the ratio between the mean square in step 3 and the mean

square in step 4, and compare this with the tabular values in an P table,

using 6 and N-P degrees of freedom. If the ratio obtained is larger than the

tabular value at the 5-P^^cent point, the differences between the regression

coefficients from the respective equations are statistically significant and

the hypothesis is rejected. In this case, the ratio was smaller than the

tabular value, as would have been expected, and the hypothesis was accepted.

Had we wished to include in (250) the hypothesis that a'=a*'=a*''=

ä, the same test could be used, except that deviations from a common mean

would have been used in step 2 and the degrees of freedom attached to it

would be N-^, the k representing the three regression coefficients from the

combined analysis plus the one constant term. The divisor in step 3 then

would be 8 [N-^ - (N-12)], and appropriate adjustments would be required in

step 5«
- 182 -

A test to ascertain whether a particular set of regression coefficients

in three or more regression equations are egual.—This test was applied to

analyses of factors that affect the ratio of prices of oats^ barley^ and sorgh-um grains', respectively, to prices of corn. In each analysis, the respective supply ratio was used as an independent variable. The hypothesis to be

tested was that the regression coefficients for the supply ratio were

identical for the three analyses. The test was complicated somewhat by the

fact that two of the regression equations involved only a single independent

variable, whereas the third involved three. The three equations can be

written as follows:

AQ = a + boi Xi

Xo = a + boi X-L > (252)

„ » t f I t t t Î t I t t

XQ = a + ^01.23^1 "^^02.13^2 "* .^03.12^3^

where

XQ is the ratio between the price of oats and the price of corn,

XQ is the ratio between the prices of barley and that of corn,

XQ is the ratio between the price of sorghum grains and that of corn.

Xi, Xi , and Xi are the ratios between the supply of the respective

items and of corn, and Xg and X3 are related variables.

The hypothesis to be tested is that the following relationship holds:

^¿1 = m = ^oi!23 = *'^01.23- (253)

The steps involved are the same as those given above, except that the

sums of squares and cross-products involving Xg and Xo in step 2 are based

only on the variables from the third equation in (252), and P = 8 (5 regression coefficients plus 3 constant terms). The divisor in step 3 equals

2 [W-6 - (N-8)], and in step 4 equals N-8. Appropriate adjustments are made

in step 5. In this case, the ratio between the mean sqi^are in step 3 and

that in step h was larger than the tabular value at the 5-percent point, as

would have been expected, and the hypothesis was rejected.

A test to ascertain whether the composite effect of several variables in

an analysis is statistically significant.—This exaiirpie is based on an unpublished study made by the Stanford Research Institute of the effects of an

advertising and promotion program for a certain commodity. A regression

analysis was run using eight separate variables that related to promotion, in

addition to price, consumer income, and temperature. Wrong signs and coefficients that did not differ from zero by a statistically significant amount

were obtained for a number of the variables relating to promotion. Hence, a

test was desired of whether the eight variables considered as a group had a
- l83 -

significant effect. This test was r\m in the following way. The initial

analysis contained 12 variables in all. The study was rerun, omitting the 8

promotional variables. The square of the multiple correlation coefficient

from each of these studies was used as an indicator of the proporticÄi of the

initial variation in the dependent variable explained. These valuesrare shown

in the first column of table ik; the appropriate degrees of freedom are placed

in the second col"umn. The increment explained by the additional eight variables is obtained by subtraction and, naturally, eight degrees of freedom are

used to obtain this increment. The unexplained variation for the 12-variáble

analysis is obtained by taking 1 minus the explained variation. An estimate

of the mean squares is obtained in each case by dividing the figure in the

first column by the degrees of freedom in the second column. The P-ratio is

obtained by taking the ratio of the two numbers shown in the last column. An

F-ratio of 2.69 is indicated, compared with a 1-percent critical value of

2.72. Thus, we woiad expect to get an F-ratio of this magnitude only slightly

more than 1 percent of the time if, in fact, the 8 additional variables had no

more effect on the mialtiple correlation coefficient than would occur if they

were completely unrelated to the dependent variable. Thus, the 8 variables

collectively have a statistically-significant effect on the dependent variable, even though it was impossible to obtain statistically significant res\ilts

for each of them separately.

Table l4.—Analysis of variance of the added effect of 8 variables that relate

to promotional activities

Type of variance

Proportion

of initial

variation

iil

Estimate

of mean

square

Explained by-- :

12 variables : 0.^98

k variables : .38O

Increment by the additional 8 variables.: .118

Unexplained by 12 variables .....: . 502

92

100

92

0.0148

.0055

VALIDITY OF AN ESTIMATE FROM A MULTIPLE REGRESSION EQUATION

Conditions Required for Valid Forecasts 103/

Under what conditions does the following equation give a valid estimate

of X-L?

103/ This section is based on material contributed by Frederick V. Waugh,

Director^ Agricultiiral Economics Division^ Agriciiltxiral Marketing Searvice, and

published by Foote and Fox (22, pp. 35-36).
f

- 184 -

Xi = a + 1)12.3 X2 +^13.2 X3. (254)

1. There must be a significant "scatter" "between Xg and X3 - that is,

r23 must differ significantly from 1. This is not a matter of sampling error.

A correlation of 0.99 inay differ significantly from 1 so far as sampling is

concerned. It is solely a matter of errors of observation. A statistician

should know enough about his data to judge whether the observed scatter is

larger than could happen as a result of errors in X2 and X3. If not, the

scatter is nonsignificant, and the equation is worthless.

2. Ustaally the equation is invalid in the case of extrapolation beyond

observed values of X2 and X3. This can be tested by drawing a simple scatter

diagram for X2 and X3. If they are highly correlated, the observations lie

within, a narrow ellipse. The values of X^ associated with combinations of X2

and X3 which lie within this ellipse can be estimated from the equation.

Unless we are willing to extrapolate, we cannot estimate the value of X^ associated with any combination of X2 and X3 lying outside the ellipse. A chisquare test described in subsequent paragraphs can be used to measure the

degree of extrapolation involved in equations having more than 2 independent

variables.

3. The error of a forecast of X^ is composed of two parts: First, the

standard error of estimate; ajid second the error associated with the regression plane,

Xi - a - bi2.3 Xg - bi3.2 X3 = 0. (251^.1)

The first of these errors is constant; the second varies with the particular

values of X2 and X3. The error of the regression plane is least at the center

of the ellipse mentioned above. We could draw a series of ellipses around

this center; as we moved away from the center, each successive ellipse would

connect conibinations of X2 and X3 for which the error of the regression plane

would be equal. And each successive ellipse would indicate a larger error.

The standard error of a forecast as described by Ezekiel (20, pp. 3^2-3^5)

allows for both types of errors. It applies exactly only for a set of independent variables identical to that included in the original analysis. However, it may be assimied to hold approximately for other values that lie within

this range.

A Chi-sqioare Test to Estimate the Degree of Extrapolation

in a Multivariate Scatter 10^/

Each value of the several independent variables used in a multiple regression analysis may lie within the range included in the analysis, but when

considered together, groups of variables may lie outside the range of the

original set.

10V The test here described was developed by Wa\igh and Been (lOl); the

description of its computation and use is based in part on Armore and Bxirtis

(6, pp. 7, 9)-
X

- l85 -

In this event, the new group is an extrapolation from the original

scatter just as much as if one of the values lay outside the range of the

original set for that variable. Hidden extrapolation of this kind becomes

difficult to detect graphically when there are three independent variables

and practically impossible to discover with more than three.

The N observations of the n independent variables used in a regression

analysis may be represented by K points scattered in n dimensional space. The

pattern and degree of concentration of this scatter depend on the structure of

intercorrelation among the independent variables as well as the variances of

the variables. Waugh and Been suggest that for any number of independent

variables, a chi-square can be calculated for each combination of observations

to indicate its position with respect to the grouping tendency of the whole

set of obsei^ed combinations, as defined by the pattern and degree of concentration of the observed scatter. When the values of all independent variables

are at their means, chi-square equals zero. As the values depart from their

means, chi-square increases. However, chi-square also depends on the structure of intercorrelation among the independent variables in such a way that it

indicates the position of any given combination of values of the independent

variables with respect to the grouping tendency of the whole set of observed

combinations.

A value for chi-square can be obtained as a preliminary step in the computations of the standard error of a fxanction or, if the latter has been

computed, may be obtained from it. The following fonmilas show the relations:

.2 2 (1 4- X^) .

X2=-!2l.Sii^.l. (255.1)

^1.23^+5

Methods for computing X in connection with equation (255) are obvious, given

the formula for the standard error of a fimction [see, for example, Friedman

and Foote (4o, pp. 17-19)].

The theoretical probability of each chi-square can be foxmd in a chisquare table, which is given in many statistical textbooks. This indicates

the probability of occurrence of the given combination, or one farther from

the grouping tendency, in sampling from a universe implied by the scatter of

the base-period data. A separate chi-square is computed for each observation.

If the computed value for observations outside the initial sample is larger

than any of those for the period on which the analysis is based, then the observation must be considered an extrapolation. The higher the chi-square the

greater the extrapolation.
- 186 -

USE OF SYSTEMS OP EQUATIONS FOR ANALYTICAL PURPOSES

In using systems of eq\xations for forecasting or analytical purposes^ the

standard procedure is to insert values for the predetermined variables within

the system and to simultaneously estimate the several endogenous variables.

Methods for doing this are discussed by Friedman and Foote (ko, pp, ÖI-85) and

elsewhere. Other types of eqij^tions that may be useful in forecasting are

discussed in the section beginning on page 128. In this section, we describe

instead (l) methods by which coefficients from a variety of sources can be

combined within a structural system to make specified analyses and (2) procedures by which a system can be modified to allow for structural changes that

are believed to have taken place.

Assigning Values to Specified Parameters 10$/

The material that follows presents a method by which a rough quantitative

measurement can be made of how a price support program for farm products can

contribute, directly or indirectly, to stability in the remainder of the

economy. Figure 10 shows some of the major lines of influence along which the

effects of a price support program may be transmitted to other parts of the

economy. The coefficient beside each arrow represents the estimated percentage change in the variable to which the arrow points which is associated with

a 1-percent change in the variable from which the arrow leads. Most of these

"path coefficients" are based upon known factors, such as the weights of particular components of official index numbers, or the coefficients of statistical demand fxmctions. Others seem reasonable, but could be checked by

empirical analysis. One coefficients assumes a "multiplier" of 2, and one

operates with a time lag* Values of the three coefficients marked with

asterisks are pure assumptions.

As indicated in figxire 10, the farm price support program during a period

of recession woxild have three immediate effects: It (l) raises the average

level of prices received by farmers; (2) reduces farm output, at least after

the first 12 months of recession; and (3) reduces the commercial utilization

of farm products.

Suppose, for example, that the direct effect of a price support program

is to increase farm prices by 10 percent. If marketing margins remain constant, this will increase retail food prices about 5 percent. As retail food

prices carry a weight of 30 percent in the consimier price index, that index

will rise 1.5 percent.

The "influence" of the consumer price index upon wage rates is based on

pure assumption. This index figures in some important wage contracts, and is

widely used as a talking point in wage disputes. The coefficient in figure 10

implies that a I.5 percent increase in the consumer price index would have the

10¿/ Material in this section is adapted from Fox (^, pp. 334.337).
LINES OF INFLUENCE OF PRESENT FARM PRICE SUPPORT

PROGRAM ON VARIOUS ECONOMIC MAGNITUDES'"

Ê FARM PRICE \

1 SUPPORT Hl^i^

> PRICES

PAID BY

FARMERS

4^ 0^5

RETAIL

PRICES,

NONFOODS

f^ ^^ t

FARM

PRICES

0.5 RETAIL

FOOD

PRICES

"•' \

CONSUMER

PRICE

INDEX

0.S*

m PKU(9KAI¥I

/

A

N

^

Sa

FARM

OUTPUT

COMMERCIAL

UTILIZATION ^^ ^ «1

0.3 DISPOSABLE

PERSONAL

INCOME

WAGE

OF FARM RATES

PRODUCTS

^» «• «

! '»N. ^^y^ho

.I8t

0.7

}

\

NET, PICKUP

INCCC

STOCKS

(PERCENT OF

FARM OUTPUT)

-^-

y^

GROSS

NATIONAL 4

PRODUCT 1 '—"

\^ .09

"

u.

o NUMERS BESIDE ARROWS AKE UULTIPUERS APPLICABLE TO PERCENTAGE CHANGES IH VARIABLE FROM

miCH ARRO^ LEADS. THOSE láARKED * ARE ASSUMED VliTHOUT EMPIRICAL CHECK.

A OPERATES WTH TIHÍE LAG OF ONE YEAR t ASSUIÁES 'MULTIPLIER' OF 2.0

S. DEPARTMENT OF AGRICULTURE NEC. 620-54(4) AGRICULTURAL MARKETING SERVICE

Figure 10.—Coefficients from a variety of sources can be inserted into a structural system and the results used for analytical pxirposes. Coefficients shovn in this chart are used to measure effects of

a price support program for farm products on the general economy during an economic recession.
- 188 -^

effect of maintaining wage rates 1 percent higher than they would otherwise

have been. The influences of wage rates upon gross national product and upon

the retail prices of nonfood products also rest on assxraption. The latter

coefficient assumes that wages constitute ahout 50 percent of value added in

manufacturing and distributing processes^ and that most nonfann prices are

administered in such a way that these direct wage costs are covered, even

during a recession.

Figure 10 implies that the initial or direct increase in the consumer

price index generates a further increase in the same index. An increase in

the cons\mier price index increases wage rates, which increase nonfood prices,

which enter the consumer price index with a weight of 70 percent. Hence, the

total effect upon the consumer price index of an increase in farm prices consists of the direct influence plus this "feed-back" effect.

An initial increase of 10 percent in prices received by farmers leads

directly, through prices of purchased livestock, feed, and food products, to

something like a 1.5 percent increase in the index of prices paid by farmers.

There is also an indirect effect operating through the consumer price index,

wage rates, and retail prices of nonfood products. This effect is only about

a fourth as large as the direct one.

The dotted arrow running from "prices paid" to "farm prices" reflects the

use of the prices paid index as a basis for setting price supports. As such

supports are announced in advance of the planting season, this coefficient

operates with a time lag of one year. Under the present price support program, the direct influence of a 1 percent increase in the prices paid index

would apply to products accoimting for only ^5 percent of cash farm income;

hence the direct effect on the average level of all faxm prices would be only

0.45 percent. The coefficient of O.67 shown in figure 10 allows for the influence of price support levels for feed grains upon the unsupported prices of

meat animals, poultry and eggs.

Figure 10 also shows three chains of influence of price supports upon

disposable personal income, particularly that of nonfarm people. An increase

in disposable income raises prices of those farm products which are not supported and whose market supplies at any given time are fixed; it also increases commercial utilization (but not prices) of farm products which are in

surplus at their applicable support prices. The direct effect of the farm

price support program in raising the disposable income of farm operators is

not adequately allowed for in figure 10.

The net increase in Commodity Credit Corporation stocks as a result of

the price support program represents an injection of money from outside the

private economy. It is equivalent to a purchase of goods by the Federal

Government, with, in the practical situation, no simultaneous increase in

government revenue. In figure 10 we apply a multiplier of 2 to the anniml

rate of increase in CCC price support stocks.
- 189 -

The model in figure 10 contains a nmiber of implicit "path-multipliers."

If we follow the arrow from farm prices through the consimier price index and

hack through disposable personal income to farm prices, we find that the initial increase in farm prices generates a secondary increase 4.2 percent as

large as the first one. The second increase woxild generate a third order increase, and so on. Using a well-known formula for the sum of a power series,

the final effect of a 1-percent increase in farm prices along this path would

be equal to l/(l-0.042), or 1.044 percent.

Some minor additional effects of the same sort are found if we consider

the secondary "loops" centering around the cons\mier price index and disposable

personal income. The chain of influences from farm prices through prices paid

by farmers and back again involves a similar power series, but with a time lag

of 1 year between the change in prices paid and the next-order change in farm

prices. The effects of the price support program upon farm output (by means

of acreage restrictions) and upon the net increase in CGC price support stocks

may be additive to the initial effect of the program upon farm prices and may

not involve power series multipliers.

In a stationary equilibrium, eliminating the effect of the one time lag

in the system, it appears that if the existing price support program initially

or directly increased farm prices by 10 percent it would result in a final

level of farm prices about 12.3 percent higher than they would have been in

the absence of a price support program. This figure depends, of co\irse, on

the coefficients in figiire 10, and would be altered if some of these coefficients were revised.

Modifying Structural Systems 106/

Two sets of statistical analyses are basic for the studies reported in

this section, and these are supplemented by certain other analyses. The first

set is an equation that shows the effect of certain factors on the price of

corn from November through May, when marketings are heaviest. The other set

is a system of 6 equations that shows the sim\iltaneous effect of ik given

variables on domestic and world prices for wheat and on domestic utilization

for food, feed, export, and storage of wheat for the July to Jxme marketing

year. The supplemental analyses include studies of (l) normal seasonal variation in prices and (2) relationships among prices at local and specified terminal markets. These are mentioned in later sections.

The analysis for corn is described by Foote (26, pp. 5-12). The following variables were used:

XQ - price per bushel received by farmers for corn, average for

November to May, cents.

Material in this section is adapted from Foote and Weingarten (^).
- 190 -

X-^ - total supply of feed concentrates for the year beginning in

October^ million tons.

X2 - grain-consinning animal units fed on farms dxiring the year beginning in October^ millions.

Xo - price received by farmers for livestock and livestock products^

index nimibers (l910-lU=100), average for November to May.

The following regression equation applies:

Log X¿ = -0.95 - 1-82 log Xi + 1.71 log X2 + 1.36 log X3. (256)

For any given year^ if expected values for X2 and X3 are inserted, this equation can be written in the following way:

Log X¿ = log Al - 1.82 log Xj^, (256.1)

where log A^ = -0.95 + 1-71 log X2 + I.36 log Xo for that year. In the rest

of this paper, the form shown by equation (256.I) is used. The reader should

remember, however, that the applicable value for log A^ must be obtained from

equation (256).

The system of equations for wheat is described in detail by Meinken (71,

pp. 36-50). Because of space limitations, a list of all variables taken as

given for this system of equations cannot be included here. Included among

these given variables is the price of com, but, as is shown later, the system

can be modified to include com prices among the variables that are simultaneously determined within the system.

Variables that are assumed to be determined simviltaneously within the

original system of equations for wheat include the following:

P^ - wholesale price per bushel of wheat at Liverpool, England, converted to United States currency, cents.

P¿ - wholesale price per bushel of No. 2 Hard Winter wheat at Kansas

City, cents.

Cf - domestic use of wheat for feed, million bushels.

Ce - domestic net exports of wheat and flour on a wheat equivalent

basis, million bushels.

Cg - domestic end-of-year stocks of wheat, million bushels.

^h ~ domestic use of wheat and wheat products for food by civilians

on a wheat equivalent basis, million bushels.
- 191 -

All variables relate to a marketing year b.eginning in July, Cg is

assumed to apply to stocks held in commercial hands. When a price-support

program is in effect, end-of-year stocks under loan or held hy the Commodity

Credit Corporation are computed as a residual.

P^ is assumed to depend directly on certain given variables; hence its

value in any year can be obtained by a direct solution of a single equation

similar to equation (256) for corn. It then can be treated as though it were

given. The values of the given variables and the calculated value of P^ for

any year can be substituted in each equation. By making computations similar

to those used in obtaining log A^, new constant tenns can be obtained for each

equation. The eqxmtions then can be written conveniently in the following

form. These equations bear the same relation to the original equations as

equation (256.l) does to (256).

Ch + Cf + Ce + Cg = A2 (257)

Ch + o.ooi5LPd = LA3 (258)

Cf + 2.5Pd = A4 (259)

Ce + 7.8Pd = A5 (260)

Cß + i^ll(Pd/ld) = A6 (261)

Two given variables are involved in these equations. They are (l) L, the

total population eating out of civilian supplies, in millions, and (2) I^,

wholesale prices of all commodities in this country as computed by the Bureau

of Labor Statistics (l926=100). They cannot be included in the modified constants because they appear as a multiplier or divisor, respectively, of P^.

By subtracting the last k equations from equation (257) and solving the

resulting equation for P^, the following formula is given:

Ag - LAg - A4 - A^ - A5

^ " -0.001% - (i^ii/id) - 10.3 • ^^^^^

Once a value for P^ is obtained, equations (258) to (261) can be solved

directly, after inserting values for L and I^, to obtain the k price-determined utilizations.

Allowing for an assumed maximum on exports.—Because of the effect of institutional forces in the world today, it is believed that exports from this

country that exceed specified levels will result in retaliatory action on the

pai-t of other governments. So long as our exports remain below these levels.

It is likely that the same kind of economic forces will apply as those in the

pre-World War II years on which the analysis was based. This adjustment in

the system of equations can be inade easily. The equations are first solved
- 192 -

with no restriction on exports. If the indicated figure for Ce is higher than

the specified maximian, the following formxila is xxsed to estimate P^. In this

formula, the symbol E is used to indicate the assumed maximum for exports.

^d = -0.0015L - (^11/Id) - 2.5 (262.1)

The reader can easily verify that this formula is obtained by substituting Ce = E for equation (260), then deriving the formula for P¿ by the same

algebraic process as that used in the previous case. The other utilizations

are obtained in the same way as previously.

Including the price of com as an endogenous variable in the system for

wheat.—In most years, the quantity of wheat fed is so small in relation to

the total supply of feed concentrates that the price of corn can be assumed to

be determined independently of the price of wheat. If price supports.and

acreage control programs for wheat and corn were dropped, however, the quantity of wheat fed might increase significantly. Under these circumstances,

the quantity of wheat fed depends partly on the price of corn, and the price

of com depends to some extent on the quantity of wheat fed. Hence, it seemed

desirable to modify the system of equations for wheat so that the price of

corn could be included among the simultaneously detemiined variables.

If the analysis for com had been based on a linear, rather than a logarithmic, relationship, this could have been done easily. In the next few

paragraphs we discuss how a linear relationship was derived from the logarithmic one for corn. The linear relation can be used as an approximation for the

logarithmic if changes in X^ from the initial value are small.

To simplify the discussion, we first rewrite equation (256.I) by substituting the letter b for the nimierical value of the regression coefficient.

Thus b = -1.82. The equation then reads:

log X¿ = log Al + b log Xi. (256.2)

If we translate this equation into actual nimibers (rather than logarithms) we

obtain:

X¿ = AiXi^ . (256.3)

We now borrow a notion from differential calculus. To get the slope of a

curve at any given point, we need to evaluate the first derivative at that

point. The first derivative of the function (256.3) with respect to Xj^ is:

-^ = bAiXi^-l . (263)

Inserting the value for b, we get:
193 -

dX,

^^ = -1.82àiXi-2-82 (263.1)

We wish to evaluate the slope of the line when X^—total supply of feed

concentrates—is at its expected level, for the particular analysis, making

use of the appropriate value of Ai. As of the start of the analysis, we know

all values that enter into X^ except the quantity of wheat to he fed during

the crop year, and that we can estiioate approximately. In most instances, an

error of as much as 100 percent in our advance estimate of the quantity of

wheat fed will affect X^ by only a few percentage points (as the quantity of

wheat fed normally constitutes only about 2 percent of the total supply of

feed concentrates) and will affect the estimate of the slope of the line even

less- If the initial estimate of the quantity of wheat fed is found to he

badly off, after making the computations for the system of equations, so that

the computed linear relationship is a poor approximation to the true curve, we

can always make a better approximation by using a revised value for X]_ and

then making a new set of computations for the system. Let us designate the

answer obtained from equation (263.I) as B. The reader shoiad note that logarithms are needed to evaluate the expression Xi"'2.82.

We now wish to obtain a linear equation that has the slope B and that

passes through the point on the original logarithmic cuirve at the chosen value

for X^. By substituting the estimated value of X^ in equation (256.I), we can

obtain an estimated value for XQ at that point. Let us designate these

numbers by the symbols XQ, X^. We now write the eqxaation of the desired

linear relation as:

X¿ = (XQ - BXi) + BXi. (264)

t

The reader who remembers his elementary analytical geometry will see that this

is the equation of a line for which we know the slope and 1 point.

We must now effect some further transformations to make eqiiation (264)

apply to the variables included in the system of equations for wheat. For the

combined analysis, all of X^ is assumed to be given except the quantity of

wheat fed. This can be allowed for in the eqxiation by letting

Xi ^Xi + Of'. (265)

BXi then can be combined with the other constant terms in the equation. The

symbol Of' is used because this is in terms of million tons, while Cf, as used

in the system of eqixations for wheat, is in million bushels. The relationship

between Cf * and Cf is given by:

In the system of equations for wheat, the price of corn, TQ, relates to

60 pounds of No. 3 Yellow at Chicago, average for Jxaly-December, in cents,

whereas XQ is the average price received by farmers per standard or 56-pound
- 19^ -

"bushel, average for November-May, in cents. A relationship between PQ and XQ

can he developed in several ways, one of which follows: (l) Based on the computation discussed hy Foote (.26, p. 12), the season-average price received by

farmers for corn equals approximately Xo/O.95- (2) Based on an analysis referred to by Foote, Klein, and Clough (jO, p. 65), the annual average price of

No. 3 Yellow corn at Chicago equals the annual price received by farmers for

all corn times 1.05 plus 1.11 cents. (3) Based on index nimibers of normal

seasonal variation for No. 3 Yellow corn at Chicago as shown on page 50 of

that biilletin, the July-December price at Chicago equals I.017 times the

annual price, {k) The price of 60 pounds of corn naturally equals 60/56 times

the price of a standard bushel. By combining these relationships, we find

that

Xo = 0.83Pc - 1.004. (267)

If we make the three substitutions implied by equations (265), (266), and

(267), we can rewrite equation (26U) as

Pc = 1.2 (XQ - BXi + BX{ + 1.004) + 0.036 BCf. (264.1)

By letting Aj = 1.2(Xo - BXi + BXÎ + 1.004) and byi = O.036B, we can rewrite

this as

P^ = Ay + b^iCf. (264.2)

The equation in this form is used in the rest of the discussion. In following

it, one should keep in mind the substantial number of computations involved in

obtaining A^ and by^.

We are now ready to consider the system of equations that includes

(264.2). Referring to page 191, if equations (258), (260), and (261) are subtracted from equation (257), equation (268) shown below is given. Equation

(259) now must be modified to show PQ as a separate variable. This is done by

removing 2.5Pc from A4 and transposing this term to the opposite side of the

equality sign. The modified equation is designated as equation (259.l) in the

system shown below, and the modified A4, as A¿. Equations (268), (259.I), and

(264.2) can be written conveniently as follows:

Cf - (0.0015L + 7.8 + 4ll/ld)Pà = A2 - LA3 - A5 - Ag (268)

Cf + 2.5PcL - 2-5Pc = K (259.1)

-byiCf + Pc = Ay. (264.2)

If equation (264.2) is multiplied by -2.5 and subtracted from equation (259.I),

the following equation results:

(l-2.5b7i)Cf + 2.5Pd = Ai; + 2.5A7. (269)
- 195 -

If equation (268) is miiltiplied by (1-2.5^71) and subtracted from equation

(269), a formula for P¿ can "be derived directly. To write this in algebraic

symbols is somewhat complicated but^ when working with numbers in an actual

problem, it would be veiy simple. A value for Cf then can be obtained from

equation (268), P^ can be obtained from equation (264.2)> and the other pricedetermined utilizations for wheat can be obtained easily from the initial

eqiaations.

Allowing for a fixed negative differential between wheat and corn

prices.—^A linear relationship is believed to apply between the quantity of

wheat fed and the price of wheat when the spread between the price of wheat

and the price of corn used in the analysis is between zero and kO cents per 60

pounds (the weight of a bushel of wheat).

For larger price spreads, requirements for wheat in poultry and other

rations is more than the quantity indicated by the linear analysis. Thus,

when use for feed is plotted on the vertical scale, a slope that becomes less

steep is required. When the price of wheat approaches or falls below the comparable price of corn, use of wheat for feed increases rapidly and by more

than that suggested by the linear relationship. For this part of the curve, a

slope that becomes increasingly steep is required. When the price spread is

outside the specified range, the quantity of wheat fed frequently can be estimated approximately by making use of a logarithmic relation between prices of

wheat and quantity fed. The logarithmic relation can be modified for inclusion in the system of linear relations for wheat by the same general approach

as described for the price equation for corn (see page 192).

When the price of wheat falls near or below that for corn, the demand for

wheat for feeding is much more elastic than when the price is considerably

above that for corn. The logarithmic analysis for wheat fed cannot be used

with negative price differentials because the logarithm of a negative number

is xHidefined. The following method was used instead: A 20-cent negative differential between prices received by farmers for wheat and corn seemed like a

maximum, and the regression coefficient for (P^ - Pc) in equation (259.l) was

adjusted in such a way as to reduce the negative price differential to this

level.

The algebra involved in obtaining the adjusted coefficient is rather complicated and need not be shown in detail here. The general approach is as

follows: (1) By making use of relationships between prices received by farmers

and prices at specified terminal markets, the algebraic value for P¿ - PQ that

is equivalent to a negative spread of 20 cents at the farm level can be

obtained. Let this algebraic value equal M. (2) Equation (259.1) is modified

to substitute a regression coefficient for P^ - Pc that is unknown for the

value of 2.5 used under normal circiamstances. Call this coefficient K. (3) M

is substituted for P^ - Pc in equation (259*1) and P¿ - M is substituted for

Pc in eqmtion (264.2). This eliminates Pc from the equations. (4) Equations

(268), (259.1), and (264.2) now contain 3 unknowns—Cf, P^, and K. As some of

the eqxiations may be nonlinear, part of the solution of them may need to be
- 196 -

laade graphically. Once values for Cf, Pa, and K have been obtained^ the other

desired unknowns can be obtained easily. A regression coefficient of 11 instead of 2.5 was obtained in one set of analyses.

Effects of a multiple-price plan for wheat.--Meinken (71, pp. ^9-50)

describes how his system of equations can be used to study the effect of

multiple-price plans. Suppose a 2-price pleui is in effect under which wheat

used for domestic food consumption is sold at a price equivalent to 100 percent of parity, while the remaining wheat sells at a free-market price. The

amoimt of wheat used for food could be estimated from equation (258) (see page

191) based on a value for P¿ equivalent to the parity price. Suppose this

aiaount is C^. The equation 0^^ = Cj^ is substituted for eqioation (258), and the

system is solved for the other variables in the same way as described on page

191.

If the Government were to place a floor imder the "free" price at say 50

percent of parity, an estimate of the quantity of wheat going under the support program at this price, if any, could be obtained as a residual after computing the expected utilizations and commercial carryover. If the Government

established a price for wheat used for food and a lower price for wheat used

for feed, an approach similar to that described above could be used to solve

for the expected utilizations for food and feed and the free-market price at

which the reznaining wheat would sell.

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(62) Kuh, Edwin, and Meyer, John R.

1955- Correlation and Regression Estimates When the Data are Ratios.

Econometrica. 23:400-4l6, illus.

(63) Ladd, George W.

1956. Effects of Shocks and Errors in Estimation: An Empirical Comparison. Jour. Farm Econ. 38:485-^95.

{6k) Lindley, D. V.

1947. Regression Lines and the Linear Fxmctional Relationship. Jour.

Royal Statis. Soc. Suppl. 9:218-244.

(65) Lowenstein, Frank, and Simon, Martin S.

1954. Analyses of Factors That Affect Mill Consumption of Cotton in

the United States. Agr. Econ. Research. 6:101-110, illus.

(66) Marschak, Jacob

1953. Economic Measurements for Policy and Prediction. In Hood, Wm.

C, and Koopmans, TjallingC, ed.. Studies in Econometric

Method, Cowles Commission for Research in Economics Mongr. l4,

324 pp., illus.

(67) Marshall, Alfred

1948. Principles of Economics, 8th ed. 871 pp., illus. London.

First published I92O.

(68) May, K.

1946. The Aggregation Problem for a One Industry Model. Econometrica.

14:285-298.

(69)

1947. Technological Change and Aggregation. Econometrica. 15:51-63.
- 201 -

(70) Meinken, Kenneth W,

1953. The Demand and Price Structure for Oats, Barley, and Sorghum

Grains. U, S. Dept. Agr. Tech. Bull. IO8O, 102 pp., illus.

(71)

1955. The Demand and Price Struct\ire for Wheat. U. S. Dept. Agr.

Tech. Bull. 1136, 93 PP»^ illus.

(72) 9 Rojko, Anthony S., and King, Gordon A.

1956. Measurement of Substitution in Demand from Time Series Data—^A

Synthesis of Three Approaches. Jour. Faiin Econ. 38:711-735^

illus.

(73) Meyer, John R., and Miller, Henry Lawrence, Jr.

1954. Some Comments on the ''Simultaneous-Equations" Approach. Rev.

Econ. and Statis. 36:88-92, illus.

(7^) Mood, Alexander McFarlane

1950. Introduction to the Theory of Statistics. 433 PP«^ illus. New

York.

(75) Nerlove, Marc

1957. A Note on Lpng-Run Automobile Demand. Jour, of Mkt. 22:57-64,

illus.

(76) ^

1958. Distributed Lags and Demand Analysis for Agricultural and Other

Commodities. U. S. Dept. Agr. Agr. Handb. l4l, 121 pp.,

illus. [Processed.]

(77) Orcutt, G. H.

19^8. A Study of the Autoregressive Nature of the Time Series Used for

Tinbergen^s Model of the Economic System of the United States,

1919-1932. Jour. Royal Statis. Soc. Supp. 10:l-53> illus.

(78) , and Cochrane, Donald

19^9- A Sampling Study of the Merits of Autoregressive and Reduced

Form Transformations in Regression Analysis. Jour. Amer.

Statis. Asso. 44:356-372.

(79) Ostle, Bernard

1954. Statistics in Research. 487 pp., illus. Ames, Iowa.

(80) Pubols, Ben H.

1954. Factors Affecting Prices of Apples. Agr. Econ. Research.

6:77-84, illus.

(81) Rojko, Anthony S.

1957. The Demand and Price Structiore for Dairy Products. U. S. Dept.

Agr. Tech. Bull. II68, 252 pp., illus.

(82) Sackrin, S. M.

1957. Income Elasticity of Demand for Cigarettes: A Cross-Section

Analysis. Agr. Econ. Research. 11:1-9, illus.

(83) , and Conover, Arthur G.

1957• Tobacco Smoking in the United States in Relation to Income.

U. S. Dept. Agr. Mkt. Research Rpt. I89, 2l6 pp., illus.

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(84) Shuffett, D. Milton

1954. The Demand and Price Structure for Selected Vegetables. U. S.

Dept. Agr. Tech. Bull. IIO5, 133 PP-, illus.
- 202 -

(85) Siegel, Sidney

1957. Nonparametric Statistics. Amer. Statls. 11:13-19•

(86) Snedecor, George W.

1940. Statistical Methods. Ed. 3, 422pp., illus. Ames, Iowa.

(87) Stone, Richard

1954. The Measurement of Consumers' Expendítiares and Behavior in the

United Kingdom, 192O-I938. 448pp., illus. Cambridge,

England.

(88) Suits, Daniel B.

1955. An Econometric Model of the Watermelon Market. Jour. Farm Econ.

37:237-251, illus.

(89) , and Ooldberger, Arthxir

1955. A Statistical Model for 1955- Mich. Business Rev. 7:25-28.

(90) Then, H.

1953. Estimation and Simultaneous Correlation in CoBxplete Equation

Systems. Netherlands Centraal Plan - Bxireau. The Hague,

Netherlands. [Processed.]

(91) ^

1951+. Estimation of Parameters of Econometric Models. Internatl.

Statis. Inst. Bull. 3^:122-128.

(92)

19547 Linear Aggregation of Economic Relations. 205 PP*^ illus.

Amsterdam.

(93) Tobin, James

1952. A Sixrvey of the Theory of Rationing. Econometrica. 20:521-553.

(94) United States Bureau of Agricxiltural Economics

1953. Coûs\miption of Pood in the united States, 1909-52. U. S. Dept.

Agr. Agr. Handb. 62, 249 pp., illus.

(95) , and Production and Marketing Administration

1949. The Agricultural Estimating and Reporting Service of the United

States Department of Agric\ilture. U. S. Dept. Agr. Misc.

Pub. 703> 266 pp., illus.

(96) Wagner, Harvey M.

1954. A Monte Carlo Study of Estimates of Simultaneous Linear Structural Equations. Tech. Rpt. 12, Dept. of Econ., Stanfo3rd

Univ., 56 pp. [Processed.]

(97) Watson, G. S.

1955. Serial Correlation in Regression Analysis, I. Biometrika.

42:327-3^1.

(98) ^, and Haiinan, E. J.

1956. Serial Correlation in Regression Analysis, II. Biometrika.

43:1^36-448.

(99) Waiagh, Frederick V.

1956. A Partial Indifference Surface for Beef and Pork. Jour. Farm

Econ. 38:102-112, illus.

(100) ^

1957. Graphic Analysis in Agricultural Economics. U. S. Dept. Agr.

Agr. Handb. 128, 69 pp., illus. [Processed.]
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(101) , and Been, R, 0,

1939. Some Observations About the Validity of Miatiple Regressions.

Statis. Jour., N. Y. City College. 2:6-lk.

(102) Wold, Herman

1938. A Study in the Analysis of Stationary Time Series. 21^^- pp.

Uppsala.

(103) ^

1949. On Least Squares Regression With Autocorrelated Variables and

Residuals. Trans. Internatl. Inst. Statis. 32(2):277-289.

(lOi^)

1956. Causal Inference from Observational Data: A Review of Ends and

Means. Jour. Royal Statis. Soc. 119:28-61, illus.

(IÖ5) , and Faxer, P.

1957. On the Specification Error in Regression Analysis. Annals Math.

Statis. 28:265-267.

(106) , and Jureen, Lars

1953. Demand Analysis. 358 PP«^ illus. Stockholm.

(107) Working, Elmer

1927. What Do Statistical "Demand Curves" Show? Quart. Jour. Econ.

4:212-235^ illus.

(108) Yxile, G. Udney, and Kendall, M. G.

1937. An Introduction to the Theory of Statistics. Ed. 11, 570 pp.,

illus. London.

APPENDIX

Proof That the Elasticity of Price Transmission Is

Less Than One in Specified Cases 107?

Case 1: Constant dollar margin.—In this case

Pp - Pv = c, (270)

where c is a constant. It follows that

oPp ^^W

9Qc 9Qc

= 0 (271)

and, therefore, A = 0; and

Consequently

3Pv

TP; = 1- (272)

107/ This section was written "by Marc Nerlove, agricultural economic statistician. Agricultural Marketing Service.
- 204 -

^PrI'w = ^--^< 1^ (273)

since both c and Pj^ are positive. It follows from equations (271) and (273)

that

I-HB < ^' (274)

so that

Case 2: Constant percentage margin.—In this case

Pr - Pw = a Pv , (276)

where a is a constant. It follows that

BPr 3Pw

ajid

BPp

= 0 (277)'

9P = 1 + a , (278)

so that

E

''^" =1. (279) 1 - A • B

In this case, therefore,

ÏQ^Pw = ^cPr • ^^^^

Case 3« The margin is a positive monotonie increasing function of Qe»—^^

this case

Pr - Pw = f(Qc) . (281)

where f * >: 0. It follows that

= f' > 0 , (282)

BPiBQc " ^

so that A > 0, and further that

^Pw

BPr

so that

= 1 , (283)
- 205 -

EprPc = 1 - ^ < 1 ' (284)

since f(Qc) ^^^ ^w ^^^ positive. From eq\xations (282) and (284) we have,

therefore, that

Ep p

^^^^ < 1 . (285) 1 - A • B

Consequently,

Charts that Indicate Approximately the Degree of Partial Correlation

The fo3:Tnulas in this section directly apply to a 5-variable problem.

However, they may be applied to a 3- or 4-variable problem if the following

adjustments are made:

(1) Subscripts which apply to variables not included in the analysis and

which are to the right of the point may be omitted, that is, bi2.3^1-5 i'or a 5-

variable problem is written as bi2.34 ^^ ^ ^-variable problem and as bi2.3 ^^

a 3-variable problem.

(2) Terms which include coefficients having subscripts to the left of the

point and which apply to variables not included in the analysis shoTild be

omitted, that is, the coefficient corresponding to b2^c^234 i^ ^ 5-variable

problem does not appear in a 3- or 4-variable problem.'

(3) Charts which involve variables having subscripts to the left of the

point and which apply to variables not included in the analysis should be

omitted.

In obtaining the charts, computational time is saved by first computing

the unexplained residual, di, for each observation. As this residual is the

same in each chart, the adjusted values of the dependent variable can be obtained from the following formulas:

Xl-345i " ^i + ^12.3^5 ^2i + (Xi - ^12.3^5 ^2) (287)

Xi^2if5i = ^i + '»^13.245 X3i + (Xi - bi3.245 X3) (288)

Xi.235i = ^i + ^14.235 ^J+i + (^1 - ^14.235 X4) (289)

Xi.234i = di + bi5.234 X5^ + (Xi - bi5.234 %) • (290)

The following scatter diagrams are made:
- 206 -

!• ^1.3^5 i^ plotted on the vertical scale and X2 on the horizontal

scale_and a line hased on the following equation is drawn throiigh the point

(X2. Xi):

Xi.3i,5 = xi + bi2.3i,5 (X2 - xa)- (291)

2. '^\.2\\^ is plotted on the vertical scale and X3 on the horizontal

scale_and a*line based on the following equation is drawn through the point

(X3, Xi):

Xi.245 = Xi + hi3.245 -(X3 - X3) • (292)

3. X3^^235 is plotted on the vertical scale and Xi,. on the horizontal

scale_and a'line based on the following equation is drawn through the point

(X4. Xi):

X1.235 = xi + bii^.235 (X1+ - xii.). (293)

h. Xi^234 is plotted on the vertical scale and X5 on the horizontal

scale_and a*line based on the following equation is drawn through the point

(X5. Xi):

Xi.23¿^ = Xi + bi5.234 (X5 - X5). (29^)

The charts obtained in this manner are the required ones^ the first indicating approximately ^\2L.'^^y "^^ second indicating ^Y^^^^> the third indicating rii|.235 ^^^ *^® fourth indicating ri5,234^

When working with analyses based on logarithms or first differences of

logarithms^ if the variables in the charts are retained in their logarithmic

form no modifications are required. Frequently, it is preferred to show the

variables in the charts in their original form. In such cases, all computations up to the point of plotting should be made with the variables given as

logarithms. The anti-logarithms of the values to be plotted are then

obtained. In working with analyses based on first differences of logarithms,

the number 2 should be added to the calculated values before finding the antilogarithms. This eliminates any negative niombers. The anti-logarithms are

then given in teirms of a percentage of the preceding year. The regression

lines in the charts become curves when plotted in terms of the original

values.

In connection with such charts, the final residiials frequently are

plotted against time in a separate chart. When working with logarithmic analyses the di are computed with the variables given as logarithms. The nuinber 2

is added to the calculated value and the anti-logarithm obtained. This then

shows the percentage which the actual value is of the calculated values. The

same procedure is used for analyses based on first differences of logarithms.
- 207 -

GLOSSARY 108/

Aggregates."Totals or index numbers of microvariables.

Aggregative variables.--See agggregates.

Analogy approach.—A method of studying relations between microtheory, aggregates^ and macrotheory by use of analogies. (See page 86.)

Arg\3ment of a function.—One of the independent variables upon whose value

that of a fimction depends. For example, in the function y = f(xi, X2, ..•

x^) the arguments are x^, xg, ... x^^.

Autocorrelation.—See serial correlation.

Autoregressive process.—A sort of inverse of a process of moving summation.

(See page 153.)

B^st estimates.—See contrasting definition given vinder efficient estimates.

Chi-square.—A coefficient that can be used to determine whether a series of

frequencies differ significantly from a theoretical or expected set of frequencies. (See page 178.)

Coaiplete model.—A model that contains one equation for each endogenous variable. In general, complete models are required if we wish to derive from them

eqimtions to be used for analytical purposes or prediction.

Consistency approach.—A method of studying relations between microtheory,

aggregates, and macrotheory. Given any two of the foregoing, the third is

determined in such a way that it is consistent with the other two. (See page

85.)

Consistent estimates.—Estimates of statistical coefficients obtained in such

a way that the average value for many large samples equals the value that

woiad be obtained from a similar calculation based on the coinbined evidence of

all possible samples. For imbiased estimates, the same property holds when

estimates are made from samples of any size.

Correlogram.—The set of autocorrelation coefficients generated by letting the

time lag range over the integers from - 00 to co .

Degree of identification. —See identification.

108/ Definitions given here refer only to terms included in this handbook.

In cases where an exact definition woiad require a large amount of space, a

condensed explanation is given instead, possibly with a cross-reference to the

text. The reader is presumed to be acquainted with terms covered in a first

course in statistics that includes multiple and partial correlation and regression and with elementary terms that relate to matrices.
- 208 -

Derived demand eguation.—^A demand equation that applies at a different level

vithin the market from that which is believed to hold in the structural model.

Derived demand equations noi^mally are estimated statistically by making use of

partially-reduced form equations. (See page 23.)

Derived elasticity coefficient.—^A coefficient that relates to a derived

demand or supply equation.

Distributed lag.--If an economic cause produces its effect over a number of

time periods, we say that the effect occurs with a distributed lag. If the

entire effect takes place within a single time period, this can be considered

a limiting case.

Economic model.—^A set of stmactures or equations consistent with the assumptions developed by a research analyst from economic theory and existing

factors that relate to a particular commodity area.

Efficient estimates.—Estimates of statistical coefficients obtained in such a

way that their average standard error for many large samples is as small as

possible. For "best" estimates, as defined in this handbook, the same property holds when estimates are made from samples of any size.

Elasticity of expectations.—The proportional change in expected future prices

of a commodity in relation to the proportional change in its current price.

(See page 112.)

Elasticity of price transmission.—The elasticity of the retail price with

respect to the price paid producers.

Elasticity of substitution.—^A combination of direct and cross price elasticities and of income elasticities which appears to be of dubious value as a

measure of the ease of substitution between two commodities. (See page 89.)

Endogenous variables.—Variables whose values are assumed to be correlated

with the unexplained residuals in the structural equation in which they occur.

Endogenous' variables frequently are referred to as those that are simultaneously determined by the system of equations.

Error models.—Statistical models in which the variables are assumed to be

subject to error but in which the true variables are assimied to be related in

a functional way.

Evolutionary process.—^A stochastic process for which the sets of distribution

functions depend on the particular point from which we measure time. All stochastic processes which are not stationary are evolutionary.

Exogenous variables.—Variables in a particular economic structure other than

those that are assumed to be endogenous.
- 209 -

Exponential fxmction.—^A mathematical function or its statistical equivalent

in which at least one variable appears as an exponent.

Fajnily of curves,--A group of curves of a specified type for which the value

of a specified constant differs from curve to c\arve.

First differences.—Transformation of variables to express them as a change

from the preceding time period. The section beginning on page 29 discusses

circumstances under: which they should be used.

Fisher*s z transformation.—See z-transformation.

Formal approach.--A method of studying relations between microtheory, aggregates^ and macrotheory. Given a microtheory and some distributional assumptions concerning some or all of the relevant variables which serve to differentiate the individxial units, both the appropriate aggregates and the macrotheory are derived. (See page 86 .)

F-test.—^A test used to determine whether the assembly of data in various

classes defined by certain restrictions exhibit on the average greater differences between members of different classes than between members of the same

class. (See pages 178-183.)

Full information approach.—A maximum likelihood method for obtaining estimates of the structural coefficients for each equation in a system of equations. Estimates of all coefficients in all. equations in the system are

obtained simultaneously. In general, the computations are formidable, so the

method is seldom used.

Identification.--A mathematical property of an equation that indicates whether

the structural coefficients can be estimated by statistical means. Degree of

identification refers to whether the equation is imderidentified, just identified, or overidentified. (See page6l.)

Identities. —Struct\iral equations which are presumed to hold exactly.

Instrumental variables.--Predetermined variables from an equation system that

are chosen for use as multipliers when fitting equations by the instrumental

variable approach. (See page 65 .)

Just identified equation.--An equation that contains just enough variables in

relation to all of the variables in an equation system so that the structural

coefficients can be determined uniquely from the regression coefficients in

the reduced form equations. (See page 62.)

Laiorent matrix.--A matrix which not only is symmetric but for which the elements are equal along any northwest-southeast diagonal.

Limited information approach.—A maximum likelihood method for obtaining estimates of the structural coefficients for equations that are overidentified.
- 210 -

The coefficients usually are estimated for one equation at a time^ with the

simultaneity implied "by the system taken into accoxmt in the computations, but

information on the particular variables that appear in each of the other equations in the system is ignored• The estimates are statistically consistent

and as efficient as any other based on the same amount of information. (See

page 63•)

Linear homogeneous function.—When we consider linear relationships between

two variables, this is a mathematical function for which one variable is a

constant proportion of the second variable. The term also is applied to statistical relationships that involve random errors provided the regression line

passes through the zero intercept.

Linearized form.—As used in this handbook, this term refers to nonlinear combinations of endogenous variables that have been transformed into linear

approximations. (See page 71•)

Macrotheory.--That body of economic theory that deals with relationships that

should hold between aggregative variables.

Markov process.--A commonly discussed stochastic process. (See page 15^.)

Maximum likelihood.--A commonly used mathematical procedure for obtaining

formizlas to estimate statistical coefficients. Coefficients are derived in

such a way as to maximize a likelihood function. The results are known to be

statistically consistent and efficient. Statistical coefficients obtained

from such formulas are called maximum likelihood estimates.

Microtheory.—That body of economic theory that is concerned with the behavior

of individuals, either consumers or production or marketing firms.

Microvariables,--Variables that relate to individuals, either consumers or

production or marketing fiims.

Model.--A set of structures or eqxiations that are compatible with the investigator's advance assumptions about the statistical universe from which a set of

economic data are drawn. (See "economic" and "statistical" models.)

Monotonie fxmction.—A mathematical f\inction of the relation of one variable

to another such that the values of the first variable increases or decreases

steadily as the second variable increases or decreases.

Moving average random process.—^A moving summation process for which the siammation is taken over a finite number of terms. The name is unfortunate,, as

this is not an average in the usual sense, but it has become established.

Moving suTTTmation random process.—^A process obtained by adding together the

elements of a purely random process. (See pa^e 153.)
- 211 -

Nonparaaiietríc tests.—Tests of significance that are independent of the natxire

of the distribution from which the sample was drawn.

Hull hypothesis,—^A hypothesis^ commonly used in connection with tests of significance^ that a specified coefficient or value equals zero.

One-tailed test.—A test of a hypothesis for which the region of rejection is

wholly located at one end of the distribution of the test statistic as, for

example, when we test only for positive serial correlation.

Overidentified equation.—^An equation having mathematical properties such that

alternative estimates of its structural coefficients can be obtained from the

regression coefficients in the reduced form equations. (See page 63.)

Partial indifference curve or siurface.—^An indifference ciirve that represents

substitution possibilities of two commodities for which consumers are indifferent after adjusting for the effects of changes in income or other variables

that might affect the relationship.

Partially-reduced fomi equation.—^An equation obtained by the algebraic substitution of an equation for a variable in a second equation. Equations of

this sort normally are used only when data are lacking on the variable for

which the substitution is made.

Polynomial.—See power function.

Power fxmction.—^A mathematical function or its statistical equivalent in

which at least one variable is raised to a power greater than 1.

Predetermined variables.—Variables that are believed to be uncorrelated with

the unexplained residuals in the structural equations in which they appear.

They include exogenous variables and lagged values of endogenous variables.

Purely random stochastic process.--A stochastic process for which the error

terms are assxmied to be completely independent of one another. (See page 152.)

Random stochastic process.—See purely random stochastic process.

Realization of a process.—^Any particiilar sequence of a process. For example,

the error terms of an economic relationship are a realization of a stochastic

process.

Recursive approach.--A method of fitting recursive systems such that consistent estimates of the coefficients are obtained. (See page 6k.)

Recursive system.--A system of equations of a form such that by successive

steps each of the equations can be transformed into one that contains only a

single endogenous variable other than those which have been treated as dependent in prior analyses. (See page 6k.)
- 212 -

Reduced eguatlons,--Equations that do not contain distributed lags that have

been obtained by algebraic manipulation from equations that do contain such

Reduced form equations > —Equations that result when each endogenous variable

in a system of equations is written as a linear fxmction of all of the predetermined variables in the system. Depending on the circumstances, they may be

(l) algebraically derived from the structural coefficients or (2) fitted by

least squares*

Reduced form method.-^A method that yields estimates of structural coefficients that are statistically consistent and efficient for equations that are

just identified. Reduced form equations are fitted by least squares and the

structural coefficients obtained by an algebraic transformation. (See page

62.)

Sector model.—A model that relates to a particular sector of the economy. A

sector analysis for hogs might be included in a model of the livestock economy

and a sector analysis for all of agriculture might be included in a model of

the general economy.

Serial correlation.—The correlation between- a series of observations and the

same series lagged by one or more units of time.

Shock models.—Statistical models in which the variables are assumed to be

known without error but are related in a stochastic way.

Spatial equilibrium.--A technique for studying geographical price equilibrirons

and flows of commodities betvreen regions or countries. (See page 169.)

Stationary process.--A stochastic process for which the probability laws depend on time in such a way that if we replace time as measured from a fixed

point by a time variable measured from another time point, the probability

laws remain the same. (See page I50.)

Stationary process to the second order.—^A process for which the mean, variance, and correlogram are all independent of time. Stationarity implies stationarity to the second order, but a process stationary to the second order

may be evolutionary. (See page I51.)

Statistical model.—A set of structures or equations consistent with both economic and statistical specifications of the research analyst. (See pages 7-8.)

Stochastic process.--A generalization of the notion of a random variable. It

expresses the ordered development (usually although not necessarily through

time) of a series of events subject to random influences. (See pages lii-9-15i|-.)

Stochastic relations or equations.—Equations that include a set of unexplained residuals or error terms whose direction and magnitude are usxially not

known exactly for any particular set of calculations, but whose behavior on

the average over repeated samples can be described or assumed.
- 213 -

stochastic vector process > —A stochastic process for which each component of x

is interpreted as a vector.

Struct\iral change.--A change in the process by which a set of economic variables is "believed to be generated.

Stinict-ural equations.—Individual equations which define the process by which

a set of economic variables are believed to be generated.

Structure.—The process by which a set of economic variables is believed to be

generated.

Student's t.—See t-test.

Substitution elasticity.—See elasticity of substitution.

Trace of a matrix.—The sum of its diagonal elements.

Transformed variables.—Variables that have been modified in some way. Common

transformations include (l) deflation, (2) conversion to logarithms, and

(3) conversion to first differences. A particular series may be transformed

in more than one way.

t-test.—Student's t is defined as the difference between any particular value

assiamed to be the true value in the imiverse for the statistical measure

examined and the value determined from the sample divided by the standard

error of this coefficient. (See page 178O

Two-rounds estimates.—A method of obtaining estimates of the coefficients in

equations that contain two or more endogenous variables that represents a

modification of the instrumental variables approach so that all the predetermined variables in the system are used in the instrumental set. (See page

65.)

Two-tailed test.—^A test for which the region of rejection comprises areas at

both extremes of the ssunpling distribution of the test function as, for

example, when we test for both positive and negative serial correlation.

Unbiased estimates.—See contrasting definition given under consistent estimates.

Underidentified eguation.--An equation having mathematical properties such

that the structural coefficients cannot be estimated by statistical means.

Variance.--The square of a standard error.

Zero-one variables.—Variables that have a value of 0 in one period and a

value of 1 in another period. For a description of cases in which they are

used, see pa^e 22.

z-transformation.--A transformation developed by Fisher for use in testing

hypotheses that relate to correlation coefficients.
- 2l4 -

INDEX

Page

Aggregation of economic variables 8^-87^ 207

Aitken ' s generalized least squares I63-I67

Analogy approach to aggregation ••..•.....« 86, 207

Apples 20, 29

Argument of a function ••...••... 207

Asparagus •..••..•.•••• 15-19^ 21

Autocorrelation. See serial correlation.

Automobiles 87, 11^4-116

Autoregressive process 153; 15^^ 159> 207

Barley 127, 128

Beef 3, h, 9, hi, 97-99

Best estimates 57, 207, 208

Chi-square 178, 184, 185, 207

Cigarettes 28, 39

Coffee 23, 117; 118

Competing commodities. See substitute commodities.

Complementary commodities • 98, 100

Complete models 9, 10, 207

Consistency approach to aggregation ...•..,•.•••.••..., 85, 207

Consistent estimates 58, 207

Constant term 43

Corn 2, I9, 20, 38; 12if-128, I89-I96

Correlogram • *..... 151, 207

Cotton ...., 22, 28, 118-122

Cottonseed , 3-5

Cyclical factors • 116-122

Dairy products 2, 3> 6, 12, 13/21

Deflation 27, 28, 33-35, 37

Derived demand equations 2-4-27, 100-110, 203-205, 208

Diagrams of~

commodity flows ...••.••• ,., , 1, 2

economic relations i, 3-7

Distributed lags III-II6, 208

Durbin-Watson test . i 173, 174

Economic models 7, 8, 208

Efficient estimates 58, 208

Eggs Ik, 15, 171

Elasticity—

general concepts 78-84, 103-110, 112, 208

of expectations .•.....••.•••. 112, 208

of substitution 89, 208
- 215 -

INDEX~Continued

Page

Endogenous variables 7, Q, 38^ ^^-51^ 208

Errors in data . • lif3-1^7, 208

Evolutionary processes 151^ 208^ 212

Exogenous variables • 7^ ^^> 208

Expected—

income •... 2?, 112

price • 112

Exports^ demand for 29, I9I, 192

Extrapolation 181^^ I85

Fats and oils 33, 3^^ II6, 11?, 17^-176

Peed-livestock economy 18, 19, kk, ^5, 50, 171

First differences 29-33; ^3^ ^+8, k9, 15> 76, 123, 124, 172, 206, 209

Fisher ' s z-transformation • 178

Forecasts, error of 18^1-

Formal approach to aggregation 86, 87, 209

F-tests 122-128, 178-183, 209

Full information estimates 63, 209

Hogs k3, k6

Homogeneous functions .•••••.••• 34, 35, 210

Identification 12, 53-55> 61.6Í+, 209

Identities 7^ 38, 209

Indifference curves or surfaces 96-98, 211

Instrumental variables 65, 209, 213

Joint relations. See multiplicative relations.

Just identified equations 62, 90-9^, 209, 212

Least squares in matrix form I54-158

Lettuce , li|.í¡^^ 11^.5

Limited information estimates 63, 64, 209, 210

Logarithms 29, 35, 37, 38, 40-^3, 79, 122-128, I92-196, 206

Marketing margins 24-26, 203-205

Markoff—

Pï'ocess 154, 159, 160, 210

theorem , 58

Maximum likelihood estimates • 63, 210

Meat 82

See also feed-livestock economy and specific kinds of meat.

Models 7.19^ 210

See also economic models and statistical models.
- 2l6 -

INDEX—Continued

Page

Moving—

average process . • 153, 159, 210

siimmation process 153, 159^ 210

Miiltiplicative relations • 37, 38

Non-linear—

relations 20, 21, 37-39^ 17^-176, 192-I96

variables .• 71-76, 210

N\xll hypothesis \ 177, 178, 211

Oats •. 127, 128

One-tailed test • I78, 211

Overidentified equations • 63, 6h^ 179, 211

Part correlation 17^*-, 176

Partial correlation charts ••... 17^-176, 205, 206

Partially reduced form equations 2^-27, IOI-IO6, 208, 211

Peas • 5, 6, 22

Per capita data 28

Percentage change 29

Plums 110

Pork 3> h, 9. ^5. ^6, 48, 97-99

Potatoes 94-96

Predetermined variables 7, 8, 38, 44-51, 211

Price support prograjns 21, 83, 186-189, 192, 196

Promotional activities 182, 183

Random stochastic processes 152, 153, 159,» 210, 211

Recursive systems o 64, 65, 211

Reduced—

equations 112-114, 212

form equations 6I-63, 91-94^ 209, 211, 212

See also partially reduced form equations.

Residuals. See unexplained residuals.

Sector models 18, 19, 212

Serial correlation 30-32, l48-l69, 212

Soybeans 2

Spatial equilibrium I69-171, 212

Standard error of a forecast. See forecasts.

Stationary process 15O-I54, 163-168, 212

Statistical models 7, 8, 212

Stochastic—

processes 149-154, 208, 211, 212

relations 57, 212
- 217 -

INDEX—Continued

Page

Storage^ deimnd for • • • • • • 29

Structural—

change • 21-23, 56, I89-196, 213

relations 7, 212, 213

Structures 7^ 213

Student's t-test I78-18O, 213

Substitute commodities 22, 87-98, 208

Suga:^» .• 98, 100

Tests of significance ; I77-I83

Theil's approach. See two-rounds estimates.

Timeas a variable 32, 39-^3, 176, 206

lags. See distributed lags.

imits 19^ 20

Total value-supply cvirves 37

t-test. See Student's t-test.

Two—

rounds estimates , 65-67, 213

tailed test I78, 213

Unbiased estimate , 57^ 58, 207

Unexplained residuals—

general nature 172, 173, 212

homoscedasticity • 35^ 37

in structural systems kh, 64, 65, 67-69, 169, 173

relation to variables not included in the initial

analysis 39, ii-O, 116-122, 1^3, 172, 174-176

serial correlation 30-32, l48-l69, 173, 174

Watennelons ,,, 11

Wheat 5, 6, 11, 12, 20, 21, 82-84, I89-I96

Wool 3, 4

Zero-one variables 22, 213

z-transformation. See Fisher's z-transformation.

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