ComputatioÉI Methods

For

Handling

Systems

of Simultaneous

Equations

WITH APPLICATIONS

TO AGRICULTURE

by Joan Friedman and

Richard J. Foote

Agriculture Handbook No. 94

UNITED STATES DEPARTMENT OF AGRICULTURE

Agricultural Marketing Service

Washington, D. C November 1955
For sale by the Superintendent of Dociaoents, U. S. Government Printing Office

Washington 25, D. C. - Price 55 cents
PREFACE

The basic theory underlying the use of systems of simultaneous equations

m economic research was developed about 19^3. Much of the early literature

on this subject assumed a knowledge of higher mathematics and most of the

applications of the method were made by research workers who had such knowledge. More recently, descriptions of this general approach that presume only

a limited knowledge of mathematics have been published; and the method itself

is beginning to be used by research workers who have little or no knowledge

of calculus and matrix algebra.

Descriptions of the computations involved in handling systems of simultaneous equations are given in several books. Most of these, however,

assume some knowledge of matrix manipulation, and all that have come to the

attention of the authors omit many steps and fail to provide adequate coverage

of the many special situations that are likely to arise.

This handbook is designed to provide a complete description of the steps

involved in the more caramon types of problems and to illustrate them in a way

that will be clear to research and clerical workers who have an acquaintance

only with standard methods for handling single equation multiple regression

analyses. Some knowledge of determinants and matrices is required for a

number of problems, but these aspects are discussed in considerable detail

before they are applied.

As some statistical Computing units may prefer to use a comparable

approach for all problems of this sort, \diether they involve a single or a

simultaneous set of equations, a method of handling ordinary least squares

multiple regression analyses is given which utilizes the same initial steps

as those for systems of simultaneous equations. This method is believed to

be more efficient than those now commonly in use; it is easier for beginners

to understand than those based essentially on the Doolittle approach, as no

back solution is required. The description of this method should be clear to

any clerical worker who is acquainted with the obtaining of sums of squares

and cross products.

The general approach used in this handbook for systems of equations is

given in Chapter ^ of A Textbook of Econometrics by Lawrence R. Klein, and

in Chapter 10 by Chemoff and Divinsky of Studies in Econometric Method,

Cowles Commission for Research in Economics Monograph 14. Minor modifications

have been made in methods that these authors suggest. These modifications and

sources for other material are indicated by footnote. Suggestions offered by

Frederick V. Waugh and Glenn L. Burrows, both of the Agricultural Marketing

Service, and by Clifford Hildreth of Michigan State University, were particularly helpful.

The research on which this report is based was made under authority of

the Agricultural Marketing Act of 19^6 (RMA, Title II).
CONTENTS

Page

A 5-vaxiable multiple regression problem 2

Obtaining the augmented sums of squares and cross-products 3

Adjustments to make the sums of squares nearly equal 1 6

Obtaining multiple and partial regression and correlation

measures • 7

Eliminating or adding variables 15

Standard errors of the function and of forecasts 17

Use of an alternative variable as the dependent one 19

Elementary principles regarding matrices and determinants 21

Definitions 21

Addition and subtraction of matrices 23

Multiplication of matrices 23

Determinants 26

Inverse of a matrix 26

Limited information approach for systems of simultaneous

equations 28

Criteria of identification 28

Computations involved in a 2-equation system 30

Modifications with specified numbers of endogenous and

predetermined variables 50

Procedures for handling complex systems 63

Tests to be made after completing the analysis yg

Serial correlation in the residuals [[[ "rj

Overidentifying restrictions [ //.[ 79

Use of simultaneous equation models for analytical purposes 8I

Solution of simultaneous equations 8I

Use of reduced form equations 82

Simplified formulas for the wheat and dairy models 81|.

Coefficients of correlation for systems of equations 86

Literature cited QJ

Appendix 30

Glossary • ^JÖk

ii
COMPUTATIOKAL METHODS FOR HAHDLIKG SYSTEMS

OF SBÍÜLTABEOUS EQUATIONS

With Applications to Agriculture

Joan Friedman, Mathematical Statistician, and Richard J. Foote, Head

Price and Trade Research Section, '

Agricultiiral Marketing Service

One of the purposes of this handbook is to provide a standard m^-hhrvi «*•

coefficients for economic relationships whether they are derived from a siSle

equation or a system of simultaneous equations. In connection witHhe sS£e

equation approach, an example involving 5 variables is shown. The computations required for other numbers of variables are obvious.

When working with systems of simultaneous equations, modifications are

required depending on (l) whether particular equations are just identified

or overidentified and (2) the number of endogenous and predetemined variables

m each. Methods for detemining the degree of identification are given on

page 2Ö; table 11 pn page 51 shows the exact steps to be used for any given

situation. Examples are worked out in detail at each point that it appeared

confusion mi^it arise. In the past, many analysts have used the method of

reduced forms to handle systems of equations for which each equation is just

identified. We suggest, instead, use of a method that represents a slight

modification of that used for equations that are overidentified. The structural coefficients obtained by either method are identical.

In addition to obtaining estimates of the various coefficients as such,

most analysts want some indication of the probable sampling errors in these

coefficients and in forecasts made from the analLysis. If the independent

variables used and the unexplained residuals from the analysis meet certain

rather rigid specifications, such measures are available for single equation

analyses regardless of the number of observations used in the study, and

methods for obtaining them are given here. Exact estimates for such measures

when working with systems of simultaneous equations, or for single equation

analyses when based on the kind of data usually used in economic research,

have been developed only for the case of an infinitely large sample.

Alternative methods for just identified equations which presimiably woiJild

give identical answers with respect to the standard errors of the coefficients

if applied to an infinitely large sample yield quite different answers when

applied to analyses based on samples of the size commonly encountered in

studies relating to economic data. We have outlined one of these methods for

obtaining standard errors of the coefficients for equations that are just

identified. Based on a study of a limited number of empirical examples,

estimates obtained by this method range from 0.25 to 0.9 times as large as

those given by an equally good method as applied to infinitely large samples.

Conventional t-tests cannot be carried out with such estimates but, in a

rather general way, they should give scane indication of the probable magnitude of sampling fluctuations in the coefficients.
- 2 -

In contrast to these resiolts, Wagner (21) l/ used a Monte Carlo approach

to study the sampling variability of coefficients that relate to equations that

are overidentified, based on 100 samples of 20 observations each. These

results suggest that the t-distribution may apply to such estimates and their

computed standard errors. Further research will be required to ascertain

whether this holds in general or only for the particular model that he studied.

In view of these problems with respect to standard errors of structural

coefficients, we do not give a formula for the standard error of forecast from

a system of simultaneous equations.

As many computations are required for complex systems of equations, it

frequently is convenient to adjust the sums of squares and cross products in

such a way as to make the sums of squares nearly equal to 1. This is desirable

for any problem, although less important for simple than for complex ones.

Methods of making this adjustment are discussed on page 6. All examples are

ba^ed on the assumption that such an adjustment is used.

In general, the carrying of 9 decimals, particularly for problems involving many variables, is recommended to avoid the necessity of using a "floating"

decimal point. In all cases, sufficient decimals should be carried for a minimum of k significant figures to appear in any computation. Some calculating

machines do not provide full carryover for this number of decimals unless they

are equipped with a special attachment. Errors caused by not having full

carryover are important chiefly in connection with some types of negative

multiplication. Clerical workers who plan to use 9 decimals in their carç)utations should ascertain iriaether their machines are equipped to provide full

carryover and, if not, should consult their manufacturer's representative. To

save space in the tables included in this handbook, fewer decimals are shown

than were actually used in the computations. Because of this, some computations appear to be slightly in error.

This handbook is designed to show how to make the necessary computations

when working with alternative types of analyses. Material relating to interpretation has in general been omitted. The reader is referred to standard

texts on statistics and econometrics. Specific references are given on certain

topics.

A 5-VARIABLE MULTIPLE REGRESSION PROBLEM

This example is talien from a study by Lowenstein and Simon (15) ; it deals

with factors that affect the domestic mill consumption of cotton. Logarithms

of the following variables for the years 1921-^*^0 and 19ÍI-8-52 were used in the

computations shown here:

l/ Underlined numbers in parentheses refer to Literature Cited, p. 87.
- 3 -

X-L - DcMestic mill consumption of cotton per capita, pounds

X2 - Deflated disposable income per capita divided by 10, dollars

X3 - Change in deflated disposable income per capita from the preceding year, dollars

X4 - Mill consumption of synthetic fibers per 100 persons, pounds

X5 - Deflated price per pound of Middling 7/8 inch cotton at the

10 spot markets, year beginning the preceding July, cents.

Since the regression equation was based on logarithms of the variables, coding

of X2 and Xi^ affected only the constant term. The decoded value is given in

their article vhereas all of the coefficients shown in this section of the

handbook apply to the coded variables expressed in logarithms.

Obtaining the Augmented Sums of Squares and Cross Products

T^ö î^irst step in the solution of any problem of this sort is to compute the "augmented" sums of squares and cross products or moments. Use of

augmented mcments is suggested to avoid rounding errors involved in obtaining arithmetic means. As used in this connection, an augmented moment

equals the actual moment multiplied by the number of observations in the

sample, here designated as N. In working with augmented moments, the s\ms

of squares and cross products in terms of original values are cimiulated

directly on the calculating machine as for any problem of this type. The

total for the observations included in the analysis is then multiplied by N,

or the number of such observations. The correction factor for an augmented

sum of squares equals the square of the sum of the series. The correction

factor for an augmented cross product equals the product of the sum for each

series. Subtraction of the correction factor from the augmented svm gives

the augmented sum in terms of deviations from the respective means. These

computations for the 5-variable regression problem are illustrated in table 1.

It should be noted that X-^, the dependent variable, is written first.

A check column shoxild always be carried in these and other computations.

This is obtained by computing a "new" variable, 2, for each year or observation in the analysis; this variable equals the sum of all of the variables

for that observation. To check the computations involved in obtaining Z,

the sum for each of the variables, including the variable E, over all of the

years included in the analysis is obtained. These sums constitute the first

row of table 1. The sum of these sums for the variables other than 2 should

exactly equal the sxmi of Z. If they do, the computations involved in obtaining 2 are correct. The second row in table 1—the means—is obtained by

dividing the sums in the first row by N, the sample size, which, for this

exançle, equals 25. Cross-products for 2 with the other variables in the

analysis are obtained in the usual way and are shown in the last column.

The check for each row is c€u:Tied out by computing the sum of all the items

in the row, except for the item in the last, or i; column. For example,
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in the second row, the check is obtained by adding the means for Xi, Xp Xo

Xl|., and X5. This should equal, except for rounding errors, the item ^'the

r column, and if true, this is indicated by a V placed next to that item, 2/

The terms in the lower left-hand part of the table are omitted. But in

order to check the cconputations in all sections after the first, these

omitted terms must be included* For example, the computation of the check

for the first row of the third section of table 1 is given by:

0.2060 + 0.2690 + 0.0276 + 0.3222 -»- 0.2035 = I.O28U

The terms omitted from this row in the table, 0.2060 and O.2690, are obtained,

respectively, from the first row of the first section and the first row of

the second section of column (3), the column in which the first written term

of the row, O.0276, appears. In general, for the ith row of any section, the

omitted terms to the left of any given term, call it m, are obtained from

the ith row of each section of the column in which m appears.

If a discrepancy due to a rounding error should occur, the sum across

the row is considered as the correct figure and the figure originally shown

in the Z column is corrected accordingly. This corrected value is used in

further computations. The following tabulation, showing the original computations with nine decimals for the items in the lower right corner of

table 1, illustrates this point:

r colximn for section 5 of table 1

With k decimals ': With 9 decimals

210.3922 V •'. 210.39220317^6

5,259.8050 -y i' 5,259.805079^*00 y

5,21^7.6909 ^/ : 5,2¡+7.690950246 7

12.IIIH J ; 12.IIIH29154 y

r the calcvaation with nine decimals, E in the first row equals

210.392203177. The resiilt obtained by adding across the row is

210.392203176. Therefore the ninth decimal place is changed from a 7 to

a 6. The corrected value, 210.392203176, is multiplied by N to give

5,259.805079400 in the second row. The above exançle illustrates a

further point: since only a limited number of decimals are shown in this

handbook, a /was placed after all items in the 2 column that serve as

checks. However, rounding errors do occur in some of these items. These

result in part because the omitted figures were dropped without rounding.

2; Rounding errors are usually taken to mean a discrepancy in the final

decimal place. In some couçutations, the number of significant figures in

the items operated upon is a furüier consideration.
6 .

If an error is made in computing the sums of squares and cross products,

the following method is more efficient thaa a direct recomputation as a means

of locating the error* Suppose that the checking operation indicates that an

error has been made in obtaining the extensions with Xi. Continue to calculate the extensions with X2 an<i^ if "the check for this indicates that no error

has been made, we know that the augmented moment between Xn and Xp in the

first section is correct. Similarly, if the extensions with X3 check, we know

that the augmented moment between Xi and X3 in the first section is correct.

If all other extensions check, the mistake is in the computation of the sum of

the squares for X-i. If one of the extensions does not check, recccçutation of

the corresponding element in the first section is indicated. A similar procedure is used if the initial error occurs in an extension other than with X^»

Adjustments to Make the Sums of Squares Nearly Equal 1

It is a great convenience in computations to have all the elements on the

main diagonal close to 1. In making this adjustment we are concerned only with

the last row in each section of table 1. A set of values that are powers of

10, the ki, where i is the variable to which it applies, is chosen such that

when the sum of squares for the variable is multiplied by the square of the ki

the answer lies between 0.1 and 10. The value (kj^) is referred to as the adjustment factor. The kj^ are shown in the second column of table 2. They are

determined in the following manner: In table 1, note that the sums of squares

for X]^, X2, and X3, respectively, lie between 0.1 and 10; therefore the adjustment factor equals (l.O)^ or 1.0 and k equals 1.0 for X^^ Xg^ and Xy For X¡^,

however, the sum of the squares equals I6I.I827 and it must be multiplied by

an adjustment factor of (O.l)^ or 0.01 to bring it between 0.1 and 10; therefore kj, equals 0.1. If the sum of squares for X]^, for exaniple, had been

1611.8ä7, the adjustment factor woiad be (O.Ol) or 0.0001 and k^^ would equal

0.01. The adjustment factors for the cross products, the kj^kj, are obtained

by multiplying the k's for the variables involved. For example,

adjustment factor for Exoxii., equals (l.O) (O.l) =0.1. The kj^kj

the right-hand section or table 2.

kok),, the

af<e sho^vn in

Table 2.- Adjustment factors

Variable ¡ Value of kj^ [

kj[kj for -

i ""i ; ^2 .' ^3 i •

X5 .

% ': 1.0 1.00 1.00 1.00 0.10 0.10

X2 [ 1.0 1.00 1.00 .10 .10

X3 1.0 1.00 .10 .10

Xi, i .1 .01 .01

^5 .1 .01
Adjusted augmented moanents are obtained by multiplying the augmented

moment, the terms in the last row of each section of table 1, by its appropriate adjustment factor from table 2.

The adjustment process is important, naturally, only ^en the ki

differ considerably from one. Particularly when working with logarithms or

first differences of logarithms, all of the k^ normally are close to one.

Some computing units may prefer to adopt a general rule that adjustments are

made only when at least one of the k^ lies outside the range 0.1 to 10. Had

this rule been followed, adjustments wovild not have been made for this analysis.

The steps involved in obtaining the adjusted augmented mcÄnents are exactly the same for single and multiple equation analyses.

Obtaining Multiple and Partial Regression and Correlation Measures

The method of determining multiple regression constants discussed in the

following pages differs in these two important ways from that given in some

of the standard statistical textbooks: (l) The use of D, the inverse of the

complete moment matrix, ^ and (2) the canc)utation of the inverse using a

variation of the Doolittle method that omits the conventional back solution. 4/

Steps involved in the forward solution of the Doolittle method are given

here in full detail as an aid to readers who are unacquainted with this method.

Experience with our central computing unit demonstrated this as the easiest

way to learn how to carry out these operations. Once the general approach is

learned, many of the computations shown individually in table 3 can be cimnilated

directly in the calculating machine. Use is made of all possible shortcuts of

this kind in the so-called abbreviated Doolittle method. This is the method

described by Klein (13, pp. 151-155). An exanrple based on it is shown in the

appendix of this handbook. The so-called Grout method makes use of similar

shortcuts and is an equally efficient method for solving systems of simultaneous equations or inverting matrices. This method is described in detail

in the appendix, p. 95*

Computations involved in the forward solution of the Doolittle method are

shown in table 3 and are as follows:

In rows (1) - (5), columns (1) - (5), enter the adjusted augmented moments

computed above. The reader will note that the X's are listed in numerical

order; in the method used by Ezekiel, X^ is placed after the last independent

37 This approach, suggested to the authors by Frederick V. Waugh, Director,

Agricultural Economics Division, Agricultural Marketing Service, substantially

reduces the number of calculations necessary for the estimation of the various

multiple regression coefficients, particularly the partial correlation coefficients. The method explained by Ezekiel (8), for example, is based upon the

computation of the inverse of a matrix using only the moments for the independent variables. Ezekiel refers to the inverse of this as the C matrix, the

elements of which are the Cj^j.

y This approach was suggested to the authors by Daniel B. Suits of the

Department of Economics, University of Michi^n.
. 8 -

variable. Computations involved in obtaining table 2 and the adjusted augmented moments should be carefully checked as no automatic checks are available for these steps.

Additional columns, li, one for each variable in the analysis, are added

in columns (7) - (ll). The makeup of these is obvious from the table.

As an alternative, data shown in the upper section of table 3 can be

recorded directly as the first row of each subsequent section.

In this forward solution we carry two check columns: 2^, column (6),

for that part of the solution concerning the x*s; and 2^, column (12), for

that part of the solution concerning the I ' s. For the upper section of table

3, that is, rows (l) - (5), these colimms are obtained in the following way:

The element in the ith. row of the 2x column is obtained by adding together

the elements in the ith row of columns (l) - (5)^ including the omitted elements. The element omitted in the ith row and jth colxmm can be found in the

jth row of the ith colxjmn: For example, the omitted element in row (^), column (3), is the element in row (3), column (k), namely, 0.09^0. The element

in row (k) of the 2^ column is given by: O.6Ú3O + 2.l64o + 0.09^ + I.6II8 +

(-0.0ÍK)7)= 4.U722. The element in the ith row of the 2j column is obtained by

adding the elements in the ith row of columns (7) - (12;. Because of the

makeup of the columns, however, each element in these rows of the 2j colixmn

equals 1. In the computations outlined below, 2^ and 2j are treated as additional variables, with all the operations performed upon them.

Only the second row in the first section and the last two rows in each

succeeding section of the solution are checked. This is done in two parts,

one for the x's and one for the I's. In order to check the computations in

either of these rows in the x part of the forward solution, sum all the elements in that row for the x columns and compare that s\jm with the element in

the 2j^ column for that row. There is no question of omitted elements here.

These figures should be identical, except for rounding errors. If they are

identical, this is indicated by a ^. Where a discrepancy occurs due to a

rounding error, the sum across the row replaces the element in the 2 coliimn

and is used in further computations. (See p. 5«) The check on the computations in the I section is obtained in like manner; that is, sum the elements

in the ith row, columns (7) to (ll), and compare that svan with the element in

the ith row of the 2£ column.

We now consider computations involved in each row of the lower sections

of the forward solution in table 3.

Row (1).--Copy row (l) from the upper section of table 3.

Row (1").—Divide row (l) by its first term, that is by 2.1088, and perform the check. For computational purposes, it is more efficient to compute

1/2.1088 = O.U7^1, lock it in the calculating machine, and multiply each item

of row (1) by it.

Row (2).--Copy row (2) of the upper section of table 3.
9-
- 10 -

Row (l)(-l,08ol|).-Multiply row (l) by -l-08o4. This factor is the element of row (1"), coliimii (2), with its sign changed. Note that no figures are

Inserted in this section of the table in columns to the left of column (2).

Row (2*) >--Add row (2) and the following line and perform the check.

Row (2").--Divide row (2') by its first term^ that is, by 2.7^7^, and

perform the check. Or, miütiply row (2') by l/2.7^7^ = O.3639.

Row (3) ,-.-Copy row (3).

Row (1) ( -0.2750) . —Multiply row (l) by -0.2750. This factor is the element of row (1"), column (3), with its sign changed.

Row (2')(Q;0780). —Miiltiply row (2») by O.O78O. This factor is the

element of row (2"), column (3), with its sign changed.

Row (3*) .—Add row (3) and the two rows following it and perform the check.

Row (3") .—Divide row (3') by its first term, that is, by 0.^973, and perform the check. Or multiply row (3' ) by 1/O.4973 = 2.0106.

Row (k) .--Copy row (k).

Row (1) (-0.3049). —Multiply row (l) by -O.30U9. This factor is the element of row (1"), column (k), with its sign changed.

Row (2')(-0.53^7) ^"-Multiply row (2') by -0.53^7- This factor is the element of row (2"), column (h), with its sign changed.

Row (30('0-06^1).—Multiply row (3») by -0.0641. This factor is the element of row (3"), column (k), with its sign changed.

Row (4*).—Add row (k) and the three rows following it and perform the check.

Row (U").—Divide row (k') by its first term, that is, by 0.6279 and perfoim

the check. Or, miiltiply row (4») by I/0.6279 = I.592U.

Row (5)*--Copy row (5).

Row (1) (-0.0672). —Multiply row (l) by -O.O672. This is the element of

row (1"), column (5), with its sign changed.

Row (2')(-0.0559)--Multiply row (2') by -0.0559- This is the element of

row (2"), column (5), with its sign changed.

Rov (3')(-0.1599).-Multiply row (3») by -0.1599. This is the element of

row (3"), column (5), with its sign changed.

Row (k') (0.2729) - --Multiply row (i+') by 0.2729. This is the element of row

(V), column (5), with its sign changed.
- 11 -

Roy (50 «—Add row (5) and the four rows following it and perform the

check.

Row (5") >—Divide row (5') by its first term, that is, by 0.0286, and

perform the check. Or, multiply row (5') by 1/0.O286 = Sif.8714-9.

This completes the forward solution.

Unfortunately, the checks do not guarantee that the correct multiplicand has been used; they only prove that the multiplications were carried out

correctly. As a final check, it is suggested that the multiplicands shown in

the stub be examined to make sure that the correct value was used and that

these then be used to recheck the computations in the ^i column (column 12 in

table 3) • Experience in our central computing unit has indicated that

occasionally a statistical clerk is interrupted between the computations

involved in the x and the I part of the table and that the wrong multiplicand

is used in the latter set of computations. It seems unlikely, however, that a

wrong multiplicand would be used in the x part of the table and the correct

one in the I part. When the abbreviated Doolittle solution is used (see

appendix), this final check is not needed, as the computations are carried out

on a column-by-column basis rather than a row-by^row basis.

D Matrix."The D matrix is shown in table 3 immediately following the I

part of the forward solution. Its computation involves the terms in the last

2 rows of each section in the I part of the forward solution. The element in

the ith row and jth column of the D matrix, d. ., is obtained by the following

formula:

d.. = (l,Ii)(r', Ij) + (2% li)(2", ip + (3S Ii)(3", Ij) + i^', li)

ik", Ij) + (5', Ii)(5", Ij)

where the first term within the parentheses refers to the row and the second,

to the column designation of the elements in the forward solution. Therefore:

d^i = (l)(0.U7lH) + (-1.08olf)(-0.3932) + (-0.3593) (-0.7225)

+ (0.2958) (O.i^Tll) + (0.131^) (it. 58i^7) = 1.9008

and di2 = (l)io) + (-l.o8oU)(o.3639) + (-0.3593)(o.l569)

+ (0.2958)(-OL8595) + (0.13lit)(-7.5270)

These sums should he cumulated directly in the calculating machine. A

check column, 2, is also carried in this computation. The elements in the Z

column, diZ, are computed in the same way as any other element in the D

matrix. In the general formula given above, Ij becomes Sj. That the sum

across the ith row of D is identical (except for possible rounding errors

growing out of the carrying of only k decimals) with the element in the ith

Sw of the Z column is indicated by a check mark. This checks the computation

of the ith row. It will be noted that the elements in the lower part of the
. 12 .

D matrix have been omitted. These need not be computed, since d^j = ^ji* ^^

computing the check on the computations in rows after the first, however,

these omitted elements must be included. For example, the check on the computation of the fourth row of the D matrix is given by:

1.7227 + (-2.91^3) + (-1.7920) + if.191^ + 9.5205 = 10.7282

The next to the last (fifth) column of the D matrix need not be conç)uted,

since it corresponds to the last row (5") in the I part of the forward solution.

All the usual measures of partial regression and correlation can be obtained

easily from the D matrix. These calculations are shown in rows (6) - (12) of

colimins (1) - (5); column (6) is a check column.

Partial Regression Coefficients.--The calculation of the highest order

partial regression coefficients, the "b's", is shown in row (6). This is done

as follows:

Row (6): Divide each element of the first row of the D matrix, incliiding

the element in the T colxmin, by the first element in the first row of D, and

change the sign of the resulting quotient. Symbolically, bn j. = "^ij/^H^

where j refers to the subscript of the x's and the column of the D matrix.

^12.3^4.5^ the coefficient of X2, therefore equals - dj^g/^u ^^ " (-1.693^1)/l.9008

= 0.0909. ^13.2^5^ ^14.235^ ^^^ ^15.234^ ^^ coefficients on x^, xi^., and X5,

respectively, are obtained in like manner. That the sum ax^ross row (6) is

identical (except for possible rounding error due to carrying only k decimals)

with the element in the Z column is indicated by a check mark. This checks the

computation of the b's. Since x-i is the dependent variable, no coefficient is

attached to it. The -1 in row (o), column (l), and the figures in the following rows 8Lre written in order to check the computations«

Standard Errors of the Regression Coefficients.—^The calculation of the

standard errors of the highest order partial regression coefficients is shown

in rows (7) - (11) . This is done as follows:

Row (7): Compute ctudjj, that is, the product of the element in the first

row and first colimin of D with the successive diagonal elements of D. For

example, the element in the first or x^^ column is obtained by squaring á^j^;

the element in the second or X2 colimm equals d3^3^d22 = (I.9OO8)(2.^4-6^^7) =

U.685O; and the element in the 2 column is obtained by multiplying djj. by

Sdjj. Sdjj is the sum of the diagonal elements of D and is shown in the last

column of row (12). That the sum across row (7) is identical (except for

possible rounding error) with the element in the 2 column is indicated by a

check mark.

2

Ro\/ (8): Compute d^j, that is, the square of each of the elements in the

first row of the D matrix, excluding the element in the E column. The element

in the 2 column of row (8) is the s\mi across the row. The check on this row is

one of recomputation.
- 13 -

Row (9): Subtract each element of row (8) frcaa the element in the corresponding column of row (?), including those in the Z column. That the sum

across row (9) is identical (except for possible roimding error) with the element in the 2 column is indicated by a check mark. This checks the computation

of row (9) .

2

Row (10): Compute l/^'d^i, where N* equals the sample size minus the total

number of variables, and dy^ is^the square of the element in the first row and

first column of D. The value dfi is given in the first column of rows (7) and

(8). ^In this example, N' equals 25-5 or 20 and d?n equals 3.613O; therefore

l/N'dii = 1/72.2616 = 0.0138. Multiply each element in row (9) by O.OI38, including that in the Z column. That the sum across row (lO) is identical (except for possible rounding error) with the item in the 2 column is indicated

by a check mark.

Row (U): Compute the square root of the element in the corresponding

column of row (lO), except the element in the 2 column. The elements in row

(11) are the standai^ errors of the coefficients in the corresponding column

of row (6). The check is one of recamputation.

Coefficients of Partial Determination.—^The calculation of the higjiest

order coefficients of partial determination (the square of the partial correlation coefficient) is shown in row (12). This is done as follows:

Row (12): Divide each element in row (8) by the element in the corresponding column of row (7), except the 2 columaa. The elements in row (12)

are the coefficients of partial determination. The element in row (12),

column (2), for example, equals r?^ ^i^c« The check on this row is one of

recomputation.

If the coefficients of partial correlation are desired, they can be

obtained by taking the square root of the elements in row (12).

2

Coefficient of Multiple Determination. '-Ri,23¿í-5> ^^^ coefficient of multiple determination, is obtained by the following foimula:

2 <^11°^11 - 1

^1.23^5 " d^i m^i

where d-, n is the element in the first row and first column of the D matrix,

and myi is the adjusted augmented moment of x^^ on x^, which is found in the

first row and first column of the upper part of table |. In this example,

djj^ equals I.9OO8 and m-^^ equals 2.1088. Therefore, Ri.23^15 =

(1.9008)(2.1088) - 1 ^ 0.7505. The coefficient of multiple correlation,

(1.900ö)(2.10bö) ^ ^ ^u ^^•

^.231^5' if desired, can be obtained by taking the square root of the coefficient of multiple determination.

Standard Error of Estimate.— 81^23^5^ *^® standard error of estimate,

is obtained by the following formula: *

^1.23^5 =" yV HITc NN'dii
- Ik.

where d is the element in the first row and first column of the D matrix, H

is the simple size, and N* is N minus the total number of variables. In this

example, dnn equals 1,9008, N equals 25, and N* equals 20. Therefore,

s£2^i^5 = 1 = 0.0010. s, 2QU5' ^^® standard error of estimate,

* 25(20)(1.900Ö)

equals the square root of this value or 0.032^. The N in this formula is

required because of the use of augmented moments.

Regression Equation Based on Deadjusted Data.—Since the regression coefficients and their standard errors are computed on the basis of adjusted data,

they must be deadjusted in order to apply to the original data. This deadjustment, carried out in rows (ik) - (l8), is as follows:

Column (l): Enter the variables in numerical order.

Column (2): Enter the regression coefficients, the b's, obtained in row

(6). Note that no figure is* entered for X-j^.

Column (3): Enter the standard errors of the regression coefficients,

obtained in row (ll).

Column {k): Enter the appropriate values of k^ from table 2.

Column (5): Compute k| = k^/k.j_.

Column (6): The deadjusted b*s are obtained by multiplying the b*s, column (2), by their respective k!.

Column (7): The deadjusted standard errors of the b's are obtained by

multiplying the s-j^, column (3), by their respective kj^.

Column (8): Enter the means of the variables from table 1.

Column (9): Computations in this column are used in obtaining the constant

for the equation. Multiply the deadjusted b's, column (6), by the mean in the

corresponding row of column (8) and add the figures in column (9), or. cumulate

the products directly in the machine. The constant in the equat^ion, a, is

obtained by subtracting the cum^jlated product from the mean of X-j^, the element

in row (1^), column (8). Hence, a » 1,1|065 - 1.2381^ = O.I680. This result

can be recorded directly as the constant in the regression equation shown in

row 19.

The final regression equation, in the following form, is shown in row (19) •

Xi = a + ^12.345 Xg + ^13.21^5^3 * ^1^.235^^ + ^15-23^^5

The figures in the table within the parentheses are the standard errors of the

respective regression coefficients.

The standard error of estimate, s^^231^5> ^^o must be deadjusted. This is

done by dividing Si^23l].5 ^y ^1- The latter is given in row (ik), column {k) .
-15 -

In our example, kl = 1; therefore, for this example, the standard error of

estimate is the same on an adjusted or deadjusted basis. The indicated computation is shown at the end of row (l^).

The coefficient of miatiple determination need not he deadjusted.

The check in this section is one of recomputation.

If all of the kj equal one, columns (6) and (7) can he emitted. In this

case, column (2) is used in place of column (6) in obtaining the constant in

the equation in column (9).

Eliminating or Adding Variables

If one or more variables are to be eliminated or added, the measures of

correlation and regression can be obtained without rerunning the aoalysis.

El^Tninating Variables.--Application of the formula given below, which

applies if one variable is to be eliminated, yields elements of a similar D

matrix, D)k(, for all variables except the omitted one, x¿. 5/ ^^ elements

of this matrix, the ^ij)k(, can be obtained by the formula:

where the d's are the elements of D. These dij)ic( values are used in place of

the corresponding á±¿ values in the computations beginning with row (6) of

table 3. These computations are explained on p. 12.

For example, if % were to be dropped frcan the previous analysis, we

would compute the first row of D)i|(, that is, dii)i^(, di2)U(, ^13)U(i ^^

di5)l^( by the formula:

^lâ)H = 3

If we consider the adjusted augaaented motnents of x-^ with the other variables

given in the first row of table 3 as mij, a check on the computation of the

first row of Dvu is given by computing miidii)l^( + mi2di2)4( + ini3di3)l^(

+ mi^dic)!^/. This STBII should equal 1.

It is not necessary to compute the entire T))U matrix. In addition to

the first row, we need only compute the diagonal elements, that is, djj)i^(,

given by the formula: 2

5/ This foimila was suggested by Frederick V. Waugh, Director, Division of

A^cultural Economics, Agricultural Marketing Service.
- 16 -

The partial regression coefficients can be obtained by:

\i.)H - - J^

Their standard errors ai*e given by:

The coefficients of partial determination equal:*

^iDiiC^jijiti

The coefficient of nrultiple determjnation equals:

^1 235 ' "^^^^ °^i " '

^'^^^ «ill)4( °11

The standard error of estimate is given by:

^.235 y N N' dii)i,(

Similarly, if both Xj. and Xj^ were to be eliminated, we wovild conipute the

elements of Dx^^./, the *iij)kr(> *s follows:

*^r)k(

Thus, if more than one variable is to he eliminated, the computations imxst

be done in steps by eliminating them one at a time«

Use of the above formula is easy if only one variable is to be eliminated;

it becomes more difficult as additional variables are dropped. Sometimes the

analyst laiows fairly well in advance which variables may need to be eliminated.

If so, he should use them as the highest-numbered independent variables. If

X]| and Xc were to be eliminated, this could be done by dropping columns {k),

(5), (10; and (11) and rows (k) and (5) and their corresponding sections in

the forward solution. The D matrix then could be easily recomputed and the

remaining computations carried out as in any 3-variable ancLLysis. New check

sums for use in the computations beginning with row (6) probably would be

advisable.
- 17 -

Adding Variables.—In general. It Is easier to drop variables than to add

them. Hence, as many variables as are likely to be \ised should be Incorporated

In the Initial analysis; some of these can then be dropped If this appears

advisable. At times, however, a v&rlable will need to be added. Assume that the

added variable Is ^^. Use can be made of all of the computations already made

In the forward solution. Colimins are added between the former columns (5) and

(6) and between columns (U) and (12), and a row (6) and a corresponding section

are cwided In the forward solution. Figures in these columns can be filled In by

performing the same sort of coniputatlons as were done previously. An additional

product from the new section 6 will need to be added to each of the elements

in the original D matrix, and a new column (6) and row (6; should be added.

These steps can be checked by recoonputation or by use of a new check sum. All

of the coefficients should be recalculated, wLiring use of the new D matrix and

of new check sxmis.

Standard Errors of the Function and of Forecasts

The standard error of a point on the regression equation, or function, relates to a point on the regression sinrface corresponding to specified values of

the Independent variables. Its value differs from point to point, depending on

the specific values assigned to these variables. This coefficient is used in

two ways: (l) It is a convenient step in confuting the standard error of a specific forecast, €uid (2) when the values assigned to the independent variables are

all zero, it equals the stemdard error of the constant in the regression equation.

For a ^-variable multiple regression problem, the square of the stcmdard

error of a point on the regression equation, or function, is given liy:

2 2 r

^1.23if5 " ^1.23^5 I + NC22(X2 - X2)^ + NC33(X3 - X^)^ + Uc¡^{X¡^^X¡,f

+ NC55(X5 - X5)^ + 2EÎC23(X2 - X2)(X3 - X3) + 2Nc2l^(X2 - X2)(Xl^ - X4)

+ 2NC25(X2 - X2)(X5 - X5) + 2Nc3l^(X3 - X^)(Xk - Xk)

+ 2NC35(X3 - X3)(X5 - X5) + 2Kc45(Xi^ - xi^XXj - X5)]

where sf ^23J^5 ^^ "^^^ deadjusted value of the square of the standard error of

estimate'obtained by squaring the deadjusted standard error of estimate, N is

the nimiber of observations on which the analysis is based, 6/ and the Cj^j are

obtained from the elements of the D matrix shown in table 3 by the formula:

¿Là - d d ^

6/ N is required in the terms after the first within the brackets because of

the use of augmented moments in the computations.
- 18-

If these values of en are substituted in the formula for the square a£ the

staSïïd Sror of thiWtion, M and dn appear in each of the products within

the hrackets. We can, therefore, rewrite the formula as:

1.23U5 1,^ u "-

+ oUx^ - l.f + 2c" (X. - XJ(X, - X ) + 2c" (X - l^KX^ - \)

'55^"5 "5^ '23^ 2 ^2'^ 3 3

+ ac^jCXg - x^)(X^ - X5) + 2c^^(X3 - X3)(Xj^ - \)

. 2c55(X3 - X3)(X5 - X^) + 2c;^(X^ - X^)(X^ . X^)]

Where c^j = cLll^ij - cLü^lJ- ^® coariputed Cij are sJiown in table h.

Table h.- cïj for the 5 variable multiple regression problem

Outline VfiuLues

°24 e"°25 ; 1.8171 0.2898 -2.6222 -6.5U33

33

fs"

3^

c"

35

3.^89 - .7076 -lf.58if5

c" c."It5 •

4.9992 10.1983

°55 .

I 1*5.2709

cop, coo, cili,., and C55 were conputed in row (9) of table 3, columns (2), (3),

(5), ana (5), respectively. The other cj« must he computed directly. For

exaarple, 0^3 = diid23 - di2dl3. Substituting the values from tíie D matrix, we

obtain: ^c'i^ = (l.9008)(l.5U8l) - (-1.693^)(-1.5665) = O.2898. The computation

of the c'ij can he checked by computing the following sums of products:

(a) c^ m22 + c^^ m^ + c^i^ m^^ + c^^ mgj

(b) 0^3 m23 + c^3 »33 + c'^^ 1^^ + c^^ 1^^^

(c) c^^ mgi^ + c^i^ r^ + ci;i^ m^ + cj^ mj^^

(d) «25 "^5-^ «35 "^5 * °ÍÍ5 %5 ^ "55 "55

Where m^i is the adjxisted augmented moment of x^ on xj shown in rows (l) - (5),

columns (l) - (5), of table 3. Each of these sums of products should equal du,

except for possible rounding errors. For example, to check the first row of the
. 19 -

cï^, ve compute (a): (I.8ITI) (5-2090) + (0.2898)(0.1^121) + (-2.6222)(2.10^0)

+ t-6.5^^33)(0.3071) = 1.9008. The second, third, and fourth rows are checked

by computing (b), (c), and (d), respectively. These c^j are in adjusted terms.

For use in the formula for the standard error of a function, the O^A must

be deadJusted. This is done by multiplying c¿^ by kj^k*, the appropriate adjustment factor from table 2. For example, to deacljust Cgi^., -2.6222, multiply by

k2k4, or 0.10. Therefore, the deadjusted value of C2l^ = (-2.6222) (0.10) =

-0.2622. The nature of the formula is such, however, that dj^l ^s never

deadjusted.

The means that are used in the formula are obtained from table (3)> rows

(15) - (18) of column (8). These are given aa a deadjusted basis.

Inserting the deadjusted standard error of estimate, c^j^ means, and the

adjusted du in the formula for the square of the standard error of the function

gives:

sl ^.,. = 0.0010 Jji_ + 25 ri.8171 (Xp - 1.9^31)^ + 3.^689 (Xo.0.0051)^

^1.23^^5 \] 2525 1.9008 1.9008 L ^ ^

+ 0.0lf99(X4-2.l895)2 + 0.if527(X5 - 1.2379)

+ 2(0.2898) (X2 - 1.9i+3l)(X3-0.0051) + 2(-0.2622) (Xg-l.9*^31) (Xi^-2.1895)

+ 2(-0.651^3) (X2-1.91^31 (X5-I.23T9) + 2(-0.0707) (X3-0.0051) (Xi^-2.1895)

+ 2(-0.1*584) (X3-0.0051) (X5-1.2379) + 2(0.1019) (Xj^-2.1895) (X5-1.2379)]}

The standard error of the function is ohtadned "by inserting the specified

values of Xg, Xo, Xi^ and Xc for any given observation and talcing the square

root of the result.

The standard error of a specific forecast is obtained from the following

formula:

^\.23if5 " ^ l-23»v5

s% , + 0.0010

^1.231*5

where s^ is on a deadjusted basis.

I.23U5

Use of an Alternative Variable as the Dependent One

All measures of regression and correlation given in preceding sections are

based on the use of X. as the dependent variable. If, after the analysis is

run, it seems desirable to have one of the other variables, Xi, as the dependent one, the various statistical measures can be obtained from the original D

matrix by use of the following:
- 20 -

The partial regression coefficients equal:

.. - - '-ii 'ij. ~ dii

If, for example, Xp is to be used as the dependent variable in the 5-variable

problem given abovi, ve vould conçnite: ^.31^5» ^23.lli5* ^2U.135' ^^

1)25.13!^, ^*®^ ^.3i*5 = " ^ ' ®*°*

d^a

The standard errors of the regression coefficients are given "by:

s. = / ^22^11 " 21

For example, ^^.3it5 ./ H' ^

\/ «^22

The coefficients of partial determination equal:

,2

o d- • rfj. = xi_

<iii"iJJ

2 ^?. For example, '^21. -

The coefficient of nniltiple determination is given "by:

R^. = ^ii-'ii - ^

^i^^i

For example, R2.I3U5 = <^2g°22 - ^

d22m22

The standard error of estimate equals:

For exaniple, Sp ^^j,c = / ^

It should iDe noted that \^en variables are eliminated, added, or interchangad., the regression coefficients, their standard errors, and the standard

error of estimate must be deadjusted before they can be applied to the original

data. All of the formulas shown apply to adjusted values.
- 21 -

ELEMENTARY PRINCIPLES REC3ARDING MATRICES AND DETERMINANTS

In this section, we briefly discuss those elements of matrix algebra that

axe required for an understanding of the computational methods described later

m thxs handbook. In some sections, discussion is fairly complete; in others

only a bare outline of essentials is given. The discussion is particularly '

incomplete in connection with deteminants and the inverse of a matrix. More

complete summaries of material that relates to matrices and determinants are

given in Klein (l¿, pp. 324-3^1), Tintaer (l8, pp. 331-3^1), and elsewhere.

Definitions

A matrix is an array of numbers arranged in rows and columns. Certain

rules of addition, subtraction, and multiplication that apply to matrices are

described below.

In general terms, the matrix A can be written in the following way:

A = (aij) =

^•11 ^12 ^13-

agi 3^2 823,

ö'ln

*31 ^32 ^33* • • ®'3n

^ ^m2 ^3* ®inn

where the subscripts attached to the a*s indicate the row and column, respec*

tively« For example, the first subscript on aog means that the element is in

the third row, and the second subscript means the element is in the second

column of the A matrix. Since A has m rows and n coliamns, we call A an (m x n)

matrix. When m:^, that is, when the nxzmber of rows is equal to the number of

colimms, we have a square matrix. A square matrix with n rows and n columns is

called a matrix of order n.

Consider the (3 x ^) matrix A:

A =

5321

4213

6-1 0 2

Here e^^==2, 823=1, ^3"^^ ®^^*

Throughout this handbook, computations are perfonaed with augnented moment

matrices. An augmented moment matrix, M^Q^, is a matrix whose elements are the

adjusted augmented sums of squares and cross products (or augmented moments) of

the variables indicated in the subscripts, in this case, the X*s. 7/ If^ ^^^

7/ In future references to this type of matrix, the word "augmented" and the

bar over the M are onitted.
- 22 -

example, we deal with 3 X's designated respectively X-^, Xg and X5 and the m^j

are the elements of MJQJ, then m^j^ is the mcanent of x^ on x^, mi2 is the moment

of X]^ on XQ, m^o is the moment of Xj^ on xg, n^ is the moment of X2 on X]^, 10122

is the moment of X2 on X2, 023 is the moment of xg on xß, m.^ is the moment of

xg on X,, m_2 is the mcanent of x^ on Xg, and m^o is the moment of x^ on xg.

A row vector is a matrix that has one row and n columns.

Example: B = (l -2 O) is a row vector with n=3 columns; here,

^11 = 1, ^12 = -2, bi3 = 0.

A coltmm vector is a matrix that has m rows and one column.

Example:

C =

2

3

-1

h

is a column vector with msk rows; here, CJJL = 2,

C21 = 3^031 = -1* %i = ^•

A scalar is a matrix that has one row and one column; that is, it is a

matrix with one element, or an ordinary number.

Example: D = 5 is a scalar.

A symmetrical matrix is a square matrix in -vdiich all the corresponding

elements above the main, or left to right, diagonal are equal to elements

below the diagonal; that is, aj^j = SLA^.

Example:

E =

3

-1

2

-1

k

1

2

1

-2

is a symmetrical matrix.

It should be noted that the au^ented sums of squares and cross products

used in multiple regression analysis form a symmetrical matrix.

The unit or identity matrix, I, is a square matrix in which all the elements along the main dieigonal are 1, and «.n the nondiagonal elements are zero.

Exas^le:

I =

1

0

0

0

1

0

0

0

1

is a unit matrix of order 3.

The transpose of ttie matrix A, written A', is a matrix in which the rows

of A are the columns of A ', and the columns of A are the rows of A '.

Example: A' = 2

0

-1

1

3

2

is the transpose of A = 2

-1

3

0

1

2
- 23 -

It should te noted that the transpose of a symmetrical matrix equals the

matrix itself.

The transpose of a row vector is a column vector and vice versa. In this

handbook, the term vector used alone always refers to a row vector; a column

vector, therefore, is written as the transpose. This CMivention is cconaonly

used "by mathematicians.

Addition and Subtraction of Matrices

Addition and subtraction of matrices can be performed cmly if the matrices

have the sane numWr of rows and the same number of columns, respectively.

Addition of matrices is performed by adding their elemente tem by term.

Example:

A =

1

3

2

0

2

1

h 3

-1 -6

1 0

C = A + B =

1 2

5 3

3 2

B =

5

-1

2

0

2

1

5

-3

3

2

1

1

1

0

1

2

3

3

Subtraction of matrices is performed by subtracting their elements term by

term.

Example: Using the matrices A and B given above,

D = A - B =

1-2 3

11-1

10 0

1

-9

-3

Multiplication of Matrices

The simplest type of multiplication involves matrices and scalars. The

product of a scalar (which, as defined previously, is an ordinary number) and

a matrix is a matrix whose elements are those of the original matrix, each

multiplied by the scalar.

Example: 1

0

-2

3

2

0

-1^

6

MultiTjlication of two matrices A and B can be performed onljr if i^.f^

of colSnsofl equals the m^ber of rows of B. If A is an (m x ^T^^J^^^«^

B"ls an?n TvT 5aE?37,'nEhe"i?3éEcrE is an m x p) matrix whose elements are

?h^^^f cert^ pï^ucts^f the elements of A and B. The products involved

can be most easily defined by considering some examples.
2k -

Exaniple:

A =

E =

^ ^

B =

^11 \2 ^3

821 ^22 _ ^21 ^22 ^23

®11 ®12 ®13 •where

^21 ^22 "23

®11 = ^"^n + ^V' ®12 = ^\2 + ^2^22' ^13 = ^^13 " ^2^23'

«21 = ^1^11 + ^22^1' ®22 = ^1^12 + ^^22' ^3 = ®2l"''l3 * ^22^23

'[i.î] B= [i a !]

E = A X B =

-1

1

3

0

k

-1

The elements of E are ohtaLned in the following way:

Oil = (1 X 1) + (2 X "^^ = '^^ ®12 = (1 X 3) + (2 X 0) = 3, ei3 = (l x 2)

+ (2x1) = i^

elg^ = (0 X 1) + (-1 X -1) = 1, e^g = (0 X 3) + (-1 x 0) =0, e^^ = (O x 2)

+ (-1 X 1) = -1

In general, e^^^, the ijth elanent of the matrix product E = AB equals the

sum of the products of the elements of the ith row of A with the corresponding

elements of the jth column of B, heginning at the left-hand side and top,

respectively.

As a further illustration of matrix multiplication, consider the following:

A =

_ 2

2.0613 -0.3084 1.2149 0.4268 1.3570 1.7838

.1301 2.4359 .5411 B = -1.2243 .6109 -.6l3i*

1.6210 - .5041 2.1008 .6064 -.2184 .3880

.3825 1.7689 -.4162 • -

The additional colixrm to the right of the B matrix headed "S" is composed of

row Slims from the B matrix and is used to check the computation of E = AB.
25

E = AB =

1.99^1

-2.5986

2.5829

-2.2548

2.3l^3l^

1.5i^65

I.U329

1.6906

l^.3375-y

-1.052JÍ1

lf.015îfô

.56U2V

îî ^i ^ ^^-5 ^! Îï! elements of A and B, respectively, then eij, the element

in the ith row and jth column of E, including the 2 column, is ¿vL hy:

Hù = ^il^J + ^2\j + ^3^3j

For example.

Similarly,

®32 = ^31^12 + ^32^22 + ^33^32

= (1.6210) (1.3570) + (-O.50I1I) (0.6109)

+ (2.1008) (^.2184)

= 1.4329

^IS " ^1^2 -^ ^^2S + ^3*32:

= (2.0613) (1.7838) + (^.3081f) (-0.6134)

+ (1.2149)(0.3880)

= 4.3375

If the sum 8u:ros3 the 1th row of £ is identical with the element in the

ith rov of the 35 column of E, this is indicated by a check mark. This checks

the computation of the ith row of E. Such was the case in the first and fourth

rows. If there is a rounding error, that is, a discrepancy in the final decimal place, the sum across the row is used to "correct" the element in the 2

column. (See p. 50 This was the case in the second and third rows.

It should be noted that these computations were carried out with h decimals instead of the 9 decimals used for most of the computations given in this

laandbodk.

unlike ordinary multiplication, the order in örtlich matrix multiplication

is perforoied is lizrportant; that is, AB does not necessarily equal M. In fact,

for the examples shown above, BA does not even exist. Only for sqiiare matrices

is reverse multiplication possible and here AB in g^ieral is not the same as BA.*

Exacqple:

A =

1 2

3 4

B= fs -1
26

*XB= [¿ :|] B.A = [-i o]

A row vector times a column vector egiials a scalar^ but a column vector

times a row vector is a matrix.

Example: F = ( 1 2 -l) G =

F X G = 5 G X F =

1

3

2

1 2

3 6

2 If

-1

-3-2

Therefore, it is important to perform multiplication in the indicated ojrder.

Determinants

A matrix, as defined above, is an array of numbers; it has no value.

Associated with a square matrix, however, is a numerical value called the

determinant, written det. A = |A| .

The value of a second order determinant is defined as follows:

|A|. "^ "^

®21 ®22

The value of a third order determinant can be obtained as follows:

- ^^22 ^2'a,21

|B| =

Example:

iBl =

^11 ^12 ^3

b2i b22 b23

^31 ^32 ^33

= b-Li(b22Í)33 - ^23^30^ - ^21(^12^33 " ^3^32)

+ ^31^^12^23 - "''13^22)

.6 11 -8

2 -3 0

2 -5 k

= (-6) [(-3 X if) . (0 X -5)1 - 2 [(11 X 1^) . (-8 X .5)]

+ 2 [(11 xO) . (^ X -3)J = 16

Methods for evaluating determinants of higher oixier are given in ttie

appendix, pp. 89, 101.

Inverse of a Matrix

The inverse of the square matrix A, designated as A" , is that matrix \rb±ch

when multiplied hy A equals the imit matrix, that is,

-1

A A = I
- 27

Example:

A =

-1

0

-1 0

-1

2

2

A-^A. 0

1

The inverse was obtained by application of the formula given below.

It should be emphasized that an inverse exists only for a square matrix

and is itself a square matrix. The concept of matrix inversion is analogous'

to division of ordinary numbers.

For a scalar, that is, a single ntanber, the inverse is simply the reciprocal of that number.

Por a second order matrix, the inverse can be obtained directly from the

following formula;

'22 "*12

XT lÂT

-^21 Q-il

LTST ~nn" J

where

and

^1 \2

'21 =^22

|A| 8^822 - *21^*

For a symmetrical matrix that has more than two rows and two colvmms, the

inverse can be obtained by a variation of the Doolittle method. This method of

matrix inversion h£is already been performed in the computation of the D matrix

for the multiple regression problem. Throughout this handbook, most of the

matrices that we invert are symmetrical, so that the inverse also is symmetrical

To invert a symmetrical matrix by the variation of the Doolittle method, the

worksheet is set up in the following form:

Sm

1

0

0

2l

11 "^12 13

n>22 23

lk

°2k

"^3k

0

1

0

0

0

1

0

0

0

"kk 0 ... 1

The matrix on the left of order k is the one to be inverted; the one on the

right is a unit matrix of order k. (Compare this with the upper section of

table 3.) A forward solution is carried out as explained on p. 9^ The inverse

and its check are obtained in exactly the same way as the D matrix sho\jn in

table 3 and explained on p. 11, since the D matrix is the inverse of the complete moment matrix. The inverse matrix is symmetrical and of order k.

Methods for inverting nonsyrametrical matrices are given in the appendix.

(See p. 98.)
. 28 -

LIMITED INFORMATION APPROACH FOR SÏSIEMS OF SIMULTANEOUS EQUATIONS

This section deals with conrputational methods for estimating structural

coefficients and their standard errors in a system of simultaneous equations.

The method used is called that of "maximum likelihood limited information single

equation," commonly referred to as the "limited information" approach. In this

method we proceed to estimate the coefficients and their standard errors for one

equation at a time, with the simultaneily inrplied by the system taken into account

in the computations. This is in contrast to a "full information" method, which

solves simultaneously for all of the coefficients in euLL of the equations of the

system. Although the full information method provides standard errors that are

smaller (that is, statistically more efficient), the computations for most problems are much more difficult. For a description of full information methods,

the reader is referred to Chemoff and Divinsky (2, pp* 252-259) • Limited information estimates have the desirable statisticcú. property of consistency and are

as efficient SLS any other method that utilizes the same amount of information. 6/

In studies that deal with systems of simultaneous equations, it is convenient to divide the variables involved into two groups: (l) Those that are

determined simultaneously within the system, commonly called "endogenous," and

(2) those that affect the endogenous variables but are not directly affected by

them, commonly called "predeteimined." 2/ ^^ endogenous variables are commonly

designated by Y's and the predeteimined variables by Z's. Frequently some other

designation is used in the structural equations. For example, disposable income

is at times designated by Ï in structural equations, as in the problem on p. 29,

but may be considered a predetermined variable. As with single-equation analyses,

lower-case letters are \ised to indicate variables expressed as deviations from

their respective means.

Certain other methods can be used to obtain statistical coefficients in

i^stems of simultaneous equations that also have the desirable statistical property of consistency. One of these, the recursive approach, tends to be less

efficient for most problems to which it is applicable than the limited information approach, and hence may be less desirable. Others, such as the method of

instrumental variables and an approach suggested by Theil (12^20), give alternative estimates for the coefficients, depending on the particular variables \ased.

The authors believe that most research analysts prefer methods that give unique

answers, even though the answers obt8d.ned may be no better, in a statistical

sense, than any one of the several alternative answers given by these other

methods.

Criteria of Identification

Since this handbook is concerned primarily with computational methods, we

give only a rule of thumb that relates to identification. One so-called

'counting rule" establishes the conditions necessary for identiflability of an

equation in a system of linear equations in which, as is commonly the case, the

identifying information consists of a priori knowledge of which variables may

actually enter each equation. This counting rule tells us that if the number of

variables in the system (endogenous plus all predetermined variables, counted

8/ For a brief discussion of the statistical properties efficiency and consist-

^"^^'x^^ Tmtner (l8, p. 86) and Klein (l¿, pp. 52-53).

..¿L^^iri"^ variables include those detemined outside of the system

commonly called exogenous," and lagged valuea.^_«ie endogenous variables.

p
. 29 -

separately) minus the number of variables in the particular equation equals

the number of endogenous variables in the system less one, we have a Just identified equation. If the number of variables in the system minus the mSbeirodT

variables in a particular equation is greater than the number of endogenous

variables in the system less one, we have an overidentified equation. Joist

identified and overidentified equations both are handled in this handbook essentially by the same approach, although slightly different steps are foUowed

If the number of variables in the system minus the number of variables in the

particular equation is less than the number of endogenous variables in the system

less one, we have an underidentified equation. An underidentified equation can

not, in general, be fitted by statistical methods. 10/

This rule applies only when a single variable is multiplied by each struc

tural coefficient • The following system serves as an example of one to which the

counting rule does not apply. Here P^ and Qf are endogenoiis and the other variables are eissimied to be predetermined.

Pr = ai + b^(Q^ + Of ) + \¿í:

Qf = ag + bgiiPj. . T) + bggT

A siiperficial application of the counting rule might suggest that these equations

are each J\ist identified, as there are 5 variables in the system, 2 of which are

endogenous, and h variables appear in each equation. But each of these equations

in fact is overidentified becaxzse we specify in effect that the coefficients for

Qà and Qf and for Pr and T are respectively eqiaal in absolute terms. This is

indicated by the enclosure of each pair of variables in a parenthesis with a

single common coefficient.

If an error is made in determining the degree of identification of an equation, it will be immediately apparent in the computation. If in doubt as to

whether an equation is just identified, it is best to assxome that it is over«

identified, as errors eu:e most likely to resialt from ignoring certain restrictions of the kind mentioned in the preceding example. As indicated in table U,

page 31, the coorputations involved in sections (l) to (8) are the same regardless

of whether the equation is Just identified or overidentified. If the assumption

that the equation is overidentified is incorrect, but instead the equation is

just identified, the computation involved in obtaining X (section (9) when there

are 2 endogenous variables in the eqiaation and sections (9«l) and (lO.l) when

there are more than 2 endogenous variables) is indeterminate, that is, división

by zero is indicated, except for rounding errors. If this occurs, the steps

for a joist identified equation should be followed. If, in fact, the equation is

underidentified, indeterminate results will be obtained in attempting to apply

the methods used for just identified equations.

For a further disciossion of identification, the reader is referred to

Koopmans (JA) and Tintner (l8, pp. 15^-166).

10/ Certain other coimting rules have been suggested. In general, these lead

to the same conclusions as the rule given here. Methods ba^sed on the rank of

certain matrices can be used to establish necessary and sufficient conditions of

Identifiability. The mathematics involved in these is beyond the scope of this

handbook.
- 30 -

Computations Involved In a 2-Eq^uatlon System

This exairrple is taken from an unpublished study by Holland 11/ dealing

with the demand-supply structure for construction lunber. The folloving variables for the years 19l6-4l were used In fitting the equations:

Yj = price per thousand board feet of softwood construction limber,

in dollars

Yg = per capita shipments of softwood construction Ixamber, in

board feet

Z^ = P^r capita expenditures for new construction, including maintenance

and repair, in dollars

Z2 = per capita production of Portland cement, in barrels

Zj = index numbers of cost per thousand board feet of manufacturing

softwood lumber, 1916-41 = 100

u-L , U2 = random error terms for equations (l) and (2), respectively.

The system can be written in the following form:

Yi = a^ + bi2Y2 + ^11^1 + ^12^2 (l)

Y2 = a2 + b2lYi + C23Z3 + ug (2)

For coBçutational purposes, however, it is convenient to rewrite the

equations m the form:

\ ^1 + ^2 ^2 + «1 2i + C2 Zg = Ui (1.1)

\ ^2 * \ ^1 + \ ^3 = ^2 (2.1)

4-^.. "^ superscripts indicate the equation. When working with any given equation the superscripts are dropped. The computations are designed^ofan equaItZ ^\^^°^ a Ixnear cotnbination of the variables is set equal to a random error

tern. Hence, all variables are put on the same side of the eqviality sign, in the

same order as they appear in ^e original equation, and consecutively nSbered b's

and c s are assigned as coefficients to the y's and z's, respectively. Variables

^t.l?ZtTL^ ^ ' °^ deviations from their respective means and hence are

c2 bf^S^-S i^"" case letters. The constant terns then equal zero and hence

can be omitted. The relation between the two sets of coefficients in equations

+3{Jï^^U^^''^\l'^^' f""^ ^^°*°'"^ Affecting the Consumption of Lumber in

the United States, with Emphasis on Demand. Ph.D. thesis, university of Califorîe^«. »nl; c f^n^*^ î?^- ^°"-^ ^^^^^^ *^^* 2p should have been treated

as an endogenous variable. A more complicated model tSat allows for this may be

fitted later. The model as shown here, however, serves as a useful expository

example.
- 31 -

(1), (2) and (l.l), (2.1) is as follovs:

^^ \ . ^12 = -^2 ^ ^U = -^1 ^ ^12 = .C2 ; 1 = b^ , ^23^ = .b| , Cgj = c^.

After the ccmputations are carried out, the valties for the constants in the

original eqtaations are determined and the equations are rewritten in the form

given in (l) and (2).

The counting rule is applied to one equation at a time to determine its

degree of identification. For this st\idy, the total number of variables in the

system is 3 cmd the nuniber of endogenous variables in the system is 2. For

equation (l.l), the number of variables in the equation is k. Applying the rule,

5-lj. a 2-1 ; therefore, the equation is just identified. For equation (2.1), the

nomber of variables in the equation is 3; as 5-3 is greater than 2-1, the equation is overidentified. This information is used at a later stage.

Estimation of Coefficients.—In using the limited information approach,

the coefficients of one equation at a time are estimated. But we first perform

two preliminary computations which are applicable to €LL1 equations in the system.

These are as follows:

(1) C^pqpute adjusted a\;igmented moments for all of the variables in the

system* This computation is explained on p. 3* The worksheet should be set

up so that the predetermined variables, the Z*s ¿come first in numerical order,

followed by the endogenous variables, the Y*s, in numerical order. For this

example, the cooiputation of the augmented moments is shown in table 5, the

adjustment factors in table 6, and the adj\isted augmented moments in table 7*

The adjusted ax^gmented moments foxm the basis for all later contputations, as

they are the elements of the moment matrices. Table 7 is referred to as the

moment matrix of all the variables in the system. Although we refer to this

table in the ejgplanations that follow, it is not necessary to set it up; we do

so only for illustrative purposes. While learning the method, however, it may

be convenient to use the equivalent of table 7« Later on, the parts shown in

tables 8 to 10 may be written there directly. In the following e:q>lanations,

a reference to table 7 indicates \ise of the adjusted augmented moments obtained

by multiplying the a\igmented moments by their appropriate adjustment factors.

(2) Perform a forward Doolittle solution, using the variables shown in

the di€igram on p. 33. These computations first are indicated in a shorthand

matrix notation. Some matrix terms are expledned on p. 21. New notation is

explEdned as needed.

We first set iip the matrices M^z and Mgy. Mzz includes the moments for

all the z's in the system; Mzy includes moments of the z's on the y's for the

system. Certain variables may at times be omitted from M^g amd Mgy (see p. 66

and 7o). In this example, z= (^i^ ZQ, ZO) and y^ (yi, y2;- Therefore, M^^ is

a 3x3 matrix and Mgy^ a 3x2 matrix. The elements of these matrices are the

adjusted augmented moments, which can be obtained from table 7 or, if this is

not used, directly from tables 5 and 6. Two 2 colimins are obtained, one for

M22 and the other for Mgy* These columns consist of row simis for the respective matrices. As Mzz is a symmetrical matrix, terms below the diagonal are
32 -

Item

Sum

Mean

Extensicms

with—

Z-,

Y^

Table 5.- Coniputation of augmented moments for lumber problem l/

2,329.1

89.5

228,371.8

5,937,667.0

5.'t24.893.1

26.2

1.0

2,5'H.8

66,087.6

6I.O7O.O

512,773.» 5,017.6

28.5

7'»2.9

687.'»

ITIT

2,249.1

86.5

203,041.3

5,279,074.8

5.238.468.7

4o,(" ,606.0"

625.3

24.0

58,380.1

1,517,884.4

1.456.504.4

61,380.5^

3,841.0

147.7

363,086.7

9,440,255.9

8'. 946,226.7

lt9'»,029.2

9,070.81^

348.8^

855,421.9^

22,240,970.0 •

21,127,163.1'^

1,113,806."9^

2,267.9

58,965.7

58,971.4

:5:F^

647.7

16,842.0

16,396.4

555:^

4,067.6

105,759.4

100,711.0

5,04ö.it

9,553.7''

248,397.8''

237,836.3'^

10,561.'t«'

198,158.7

5,152,126.9

5,058,450.8

93,676.1

54,540.2

1,418,045.9

1,406,452.1

327,395.3

8,512,279.8

8,638,793.1

11,593.7 -126,513.2

785,403.5^

20,420,493.4/

20,401,136.2 ^

19,357.1-"

15,895.1 95,718.1

413,273.8 2,488,671.5

391,050.1 2,401,930.9

SZ,2S3.1 Ö6,7'»0.5

225,181.4^

5,854,717.8 '

5.672.334.0 ^

102,303.0 •

611,975.6 1,402,243.6^

15,911,367.6 36,458,334.5-^

14,753,281.0 34,840,942.8 '^

1,150,086.6 1,617,391.7-^

1/ Of the decimal places vhich were used in the original computations only one is shown in the

table; therefore some of the computations may appear slightly in error. On the other hand, the

check appears more accurate than is the case when more decimal s are shown.

Table 6.- Adjustment factors for lumber problem

•

•

kik.i for -

Variable ; h ; ^2 ; ^3 I "1 ' V

^1

'! '

yi

^2

1 0.001

; .1

1 .01

1 .01

I .001

0.000001 0.0001

.01

0.00001

.001

.0001

0.00001

.001

.0001

.0001

0.000001

.0001

.00001

.00001

.000001

Table 7. - Adjiisted augmented moments for Iximber problem l/

Variable ! ^1 ; -a ; '3 : ^1 ;

yg

^2

1 / BIU.,. .

I 0.5127 0.5017

.5543

0.4060

-.0056

9.3676

0.6138

.4456

1.1593

2.2223

0.4940

.5048

-1.2651

.8674

1.1580

tat
- 33 -

not written. But they must be included when obtaining the elements of the 2

column for U^^. The worksheet is of the form:

^1 ^2 ^3 S yi y2 S

A forward Doolittle solution is then carried out. Computations involved are

explained on p. 9 and, for this exaniple, they are shown in table 8. Note

that in this diagram, as in those that follow, the omitted elements of symmetrical matrices are indicated by shaded boxes.

We now begin the computations for the single equations, working first with

the overidentified equation (2.1). In the following computations, a new notation, y*- and z*, is used, y* is the vector of y's in the equation under consideration and in the order that they appear in the equation, z* is defined in

like manner. Therefore, in equation (2.1), y* = (y2, y^^) and z» = (ZQ). For

equation (l.l), y* = (y^, y2) and z» = (zj^, Z2).

The first step is to compute My#^2?(zy*« Before beginning the actual

computations, the analyst should determine the y* that are needed in the several equations to be solved. This will provide a clue as to the form that the

computation of My*:^z3M2y* should take. 12/ For the limiber problem the elements of My»^2^zy* ^^^ ^^^ ^^''^ ^^^ each equation. Therefore it need be computed only once. The terms then can be rearranged for use with the second

equation. In the present example, it is more convenient, conQoutationally, to

compute this matrix first for equation (l.l) and then rearrange the terms for

use in connection with equation (2.1). If, however, a system had 8 endogenous

variables and k equations, ^ere for equation I, y* = (yj, y2); equation II,

y* = (y^, yif); equation III, y* = (yj, yg); and equationTV, y» - (y«, yo),

^V^s^zi^zy^ should be computed separately for each equation« This was the case

for the system discussed on pp. 63-67. On the other hand, in the following

system of equations where for equation I, y* = (y^^, yg, y^); equation II, y* =

(yp^ yOf yc:, yg); equation III, y* = (yg, y^, yj, y^); and equation IV, y* =

(yi^ yi^ yv yk» y^)» ^^^ number of calculations could be substantially reduced

by confuting the complete ^gM-^ rather than the separate My#^^2y» ^^^

each equation.

The computation of WL^gMgzMzy* ^^^ equation (l.l) is shown in table 8

iimnediately below the forward solution. The computation of this matrix from

specified terms of the M^y part of the forward solution is analogous to the

computation of the D matrix for the multiple regression problem (see p. U).

Using only the last two rows in each section of the forward solution, the element in the ith row and jth column of M^z*ÇzMj^y» is obtained by cumulating the

12/ The method used in this handbook differs frMi that described in Chemoff

and Divinsky (2, p. 242) where the complete MyaMi*izy is always computed. For

some tyr^ffj^<>niiAtions thia_jinvolves many needless conç)utations
3i^-

HU

IIo

0)O

+>

g* so

•HI

CO

•3

'SS

CMH

ri "^^

jd- CO

>> VD r-j

B; OJcn

>> CO

iS-t^

tío

•HIoH

«

's"

iOica

VI

-PC

•H

0)^o^

Î*

«i^0

•\

ca

<uo

c3 H

ft H1O

(U

»o

ONÄ

** .'

•gs

frt Lf

<U 0) s d o «H ^ w

0^Peg

«ll

Sä0 V

-H ft p ft

43s

b

'^l
- 35 -

products of the terms in the next to last rov of the ith column with the terms

in the last row of the Jth column. For example, if m±A are the elements of

My*25M;^2y*, then:

mii=(0.6l38)(l.l970)+(-0^l5lf9)(.2.1A8l)+(-0.3132)(-O.Otô3)=1.1293

mi2=(0.6l38)(0.963l^)+(-0.15if9)(0.3383)+(-0.3132)(-0.23íf5)=0.6l2lf

mjjj=(0.6l38)(2.l6(A)+(-0.15l^9)(-2.l098)+(-0.3l32)(-0-2828)=l.7^17

The check placed next to the items in the 2 colimin at the bottom of this section

indicates that the computations are correct, except for possible rounding errors.

As this matrix is symmetrical, the terms below the main diagonal are not computed.

In obtaining the check by sijmming across the items in the row, the omitted terms

must be incltided in the sxmi. My^gM^j^zy* for equation (2.1) is obtained by rearranging the elements of that ror equation (l.l). The nature of this rearrangement is clear from the table.

In the following computations, we first work with the overidentified equation (2.1). This is done chiefly for expository reasons, as it is easier to

imderstand the computations for an overidentified equation than for a just

identified one. In the solution of an actual problem, the equations probably

would be fitted in order. In some systems of equations, certain equations may

involve only a single endogenous variable. In such cases, these equations can

be fitted directly by least squares, but the predetermined variables involved

in them should be inclxoded in the Hzz matrix. Equations that can be handled by

least squares always can be fitted by the method described in the first major

section of this handbook. (See p. 2.) In some instances, however, many computations can be saved by what in effect is a simultaneous approach for both the

equations that are to be fitted by the limited information method and those that

csm be handled by least squares. Examples sure discussed in detail on pp. 69-76«

In operations that involve multiplication of symmetrical matrices that have

missing elements, the missing elements should be filled in before performing the

multiplication; otherwise it is difficult to determine which elements are involved in the product. Elements can be omitted from the product matrix, however,

unless it is to be used in further multiplications, as it is always sympo^trical.

Missing elements are not needed in connection with addition and subtraction, as

this is carried out on a term by term basis. Missing elements that are needed

have been shown in tables 9 and 10. This explains wiiy some matrices are shown

in full, while in others the elements below the main diagonal are omitted.

In obtaining the coefficients for equation (2.1) we first compute Wy*y* =

Myiiy* - M5y*aMzJ^zy** ^niis computation is of the form:
-36-

->%

:3>

M

OH

S RÄ 1 ^ ^

orí

1

:»

5?

Ä*

1

^ s-s;

3Sô H

CII :^'>^

J

II

SÄHO

SS

d\ I Ä-i

I

î

ss

Sti' I s^

I Kw

"

9

Í

:i^

Ä

¿S«

s ¿^

o o\ -fto

^ü

«q

¿H

<no (X) u^

Ínui

ll>

lA

X <o9

Oi

H

««^

O

6" en M

1Ó

(n

r

a ff

s-^ '^ O

«n

II

i^

Si

¿'S

S î; s p

iT

58« ifj

H

19

>»^

19

%

S*^

ô

s

,R§

«eu

IsOi-O

1

t

«

>¡* (^

•yy

aÔH

5ï

RfîO

+ ' g

O

I

1

Il

S

S

O

>?

H* t' •

nil

"

¿•PV

1

:5i
- 37 -

Vy* - Vz**zz^y* = ^y*y*

72 yi I 72 yi 2 y2 yi 2

72 V2

H yi H yi

These ccsmputations are shown in sections (l), (2) and (3) of table 9, Section

numbers refer to the order in which the computations are performed. Arranging

them as shown avoids needless copying of data. Steps involved in these 3 sections are as follows: (l) Copy My*zMz^*zy* ^^r equation (2.1) from table 8.

Here, as in other instances of copying matrices, the S colximn is also copied.

The check is obtained by reconiputing the sum across the copied row, including

the omitted figures. This shoxad be identical with the figure in the 2 colimm.

13/ If not, an error WBB made in copying. (2) My^ty«^ a symmetrical moment

matrix of the y*'s, is obtained from table 7. For this exantple the y*s are

reversed in order. A column, S, composed of row sums is computed. (3) ^y*v*

is obtained by siibtracting (l) from (2), element by element, including the items

in the £ columns. The sums across the rows of Wy^tyX- are obtained and the fact

that they are identical with the items in the 2 column of the respective rows

of Wyííy* is indicated by a check mark. This provides a check on the computation.

We now compute P' = M^i^^^^^y*« ^n^ß computation is of the form:

Mz*z*; X Mgily* —

a

72 yi 2 y2 yi 2

See sections (4), (5), and (6) of table 9. {k) z* has only one element. Therefore, M2«2^ is a scalar or ordinary number equal to 9* 3676. Its inverse, Mjiz*,

is simply the reciprocal of that number or 1/9-3676 = O.IO67. If z* had more

than one element, other methods described on p. 26 would be used for inverting

Mz*z* (5) Mz«y« is a moment matrix. It can be copied from the upper section of

the forward Doolittle solution, table 8, or directly from table 7. A column, 2,

composed of row simas, is computed. (6) As explained in the section on matrix

multiplication, a scalar times a matrix equals a matrix whose elements are those

of the original matrix, including 3^ each multiplied by the scalar. That the

sum across the row of P' is identical (except for possible rounding error) with

the element in the Z coltam of P* is indicated by a check mark.

We next compute My*z*P'. See sections (5)> (6), and (7) of table 9-

Sections (5) and (6) were obtained in the previous step. (7) To obtain m^j,

the ijth element of My»z«P', multiply the ith column of Hz^f excluding 2,

12/ Rounding errors occur only when the indicated operation is multiplication

or division.
- 38-

with the jth column of PS including 2. For example:

mil = (-1.2651)(-0.1350) = 0.1708

n,l2 = (-1.265l)(0.ia37) = -0.1565

m^2 = (-1.2651)(-0.0112) = O.Oltó

11122 = (1.1593)(0.1237) = 0.11^31^

ni22 = (1.1593)(-0.0112) - -0.0130.

That the sums across the rovfs of My*z*P' are identical (except for Possible

rounding errors) with the elements'^in the respective rows of the £ column of

U^4^'iB indicated hy a check mark. Use of column-hycolumn multiplication

iTdiscussed on p. 101. As the product matrix is symmetrical,elements below the

diagonal need not be ocnqputed.

form:

We now contpute By^ = My*z"zXy* " *W*^' * ^^ ccnrputation is of the

72 yI 2 72 yi 72 71 Z

72

^1

72

See section (l), (?) and (8) of table 9. Sections (l) and (7) were obtained in

previous steps, (8) By^hr^ is obtained by subtracting (7) from (l), element by

element, including the elements in the Z columns. The s\mis across the rows of

Byífyíí. are obtained and the fact that they are identical (except for eirors

tnat result from the carrying of only k decimals) with the elements in the

Z column of the respective rows of By^y^ is indicated by a check mark.

We now compute X, See section (9) of table 9. Let \r±¡ be the elements

of WyHyíí- in section (3)^ and b^j, the elements of Byj^y)«- in section (8). Then

compute:

pi iw W3JW22 - W12

P2 = ^11^22 + biiW22 - 2bi2Wi2

2

P3 » |B| = biib22 - bi2

1/2P3

p| - I1.P2P3

f.P^ - ltI>iP3
39

form:

We next COntpUte Ryify» = Wy#y» - XBy<,y», This coniputation is of the

72 yi s ygyi s

Vy*

yg yi 2

yi yi

See sections (3), (lO), and (ll) of tahle 9» Section (3) was obtained previously. Section (10) is obtained by multiplying each element of By«y», including Z, obtained in section (8), by the scalar, X., conrputed in section (9).

That the sums across the rows of >3y*y* are identical (except for possible

rounding errors) with the elements in the respective rows of the 2 column of

X^Hjy» is indicated by check marks, (ll) Ry«y» is obtained by subtracting

(ID) frran (3), element by element, including the elements in the 2 columns.

The sums across the rows of Ryiiy« are obtained and the fact that they are

identical (except for errors that result from the carrying of only k decimals)

with the elements in the 2 column of the respective rows by Ryiiy» is indicated

by check marks. An additional check on the ccmputation of By«y«> as well as

on the coniputation of X., is to contputelRl = rjjLrgg - r^l, where the r^^ are

elements of Ry«y*. |R| should equal zero. ''

We now compute:

b« =

b' is the vector of the coefficients of the endogenous variables, the y*'8.

For eqtiation (2.1), and any other equation where the number of endogenous

variables is eqiial to two: _

1

b- = -rii

See section (12) of table 9. "bn = 1 "by definition (see p. 31) and bo = - -=:•

"^ ^12

where r^ and r2^2 ^^® elements of Ry»y*> obtained in section (ll). ih/

We next compute c' = -P'b'. c' is the vector of the coefficients of the

predetermined variables, the z**s. This computation is of the form:

157 This method for obtaining the coefficients, already normalized, differs

trcm that discussed in KLein (13, pp. 179-l8o) and Caiernoff and Divinsky

(2, p. 2^5), where the coefficients are first obtained and then normalized.

Normalization, in this sense, refers to the transformation of coefficients so

that bj^ = 1.
'3 1

-ko .

-P' X b«

72 yi

H 1

H

D

Note that far equation (2.1), z* has only one element, and therefore c* is a

scalar. See sections (6), (12), and (13) of table 9. Sections (6) and (12)

were obtained previously. (13) c' is obtained by multiplying -PS excluding 2, by b*. Computationwise, it is easier to neglect the minus sign prefixed to P*. Instead, multiply P* by b', and then change the sign of the result. This gives c^. The check in this step is one of reccDa¡putation#

In an actiaal problem, ve would continue with equation (2«l) and estimate

the standard errors of the coefficients Just computed. But for illustrative

purposes, we first proceed to estimate the coefficients of the Just identified equation (l.l)« The relevant cooiputations are shown in table 10.

For equation (l.l), z* = (z-^, Zg) and y* = (y^, J2^n The matrix shown

in section (l), table 10, was computed in table 6. Since the same y's, although in reverse order, are involved in equation ^.l), seme of the computations can be eliminated. This always is the case when the same variables

appear in several equations. For exanrple, compare the matrices in sections

(2) and (3) of tables 9 and 10. Those of table 10 can be obtained from those

in table 9 by reversing the order of the y's. In making the computations for

several equations, steps should be copied whenever possible. The copying can

be checked as described on page 37» After copying these matrices, we compute

*' = M^ijjiiM^^iy». This computation is of the form:

M

-1

z*z*

Z2

^*y*

yi 72 2 71 72 2

^1 ^1 ^1

Z2 Z2 Z2

See section (1*), (5), and (6) of table 10. (U) Since M-«^» is a 2x2 matrix,

it should be inverted according to the definition given on page 26. These

computations, shown in sections (ha) and (li-b), are as follows; (Ua) Write

Mg«^», which is obtained from the upper section of the forward Doolittle solution in table 8. A colunm, Z, coniposed of row sums, is also coniputed but

does not enter into the cosoputations until a later stage, (lib) MtriXe the

adjoint of Vi¿^*. This is obtained T?y interchanging the elements in the main

diagon^ of Mj*^« and changing the sim of the elements not in the main diagonal. Rreceeding this matrix, write l/|M| where |Mi is the product of the

diagonal elements of Mz»^» minus the product of the nondiagonal elements.
- kl

-

ï

^s

IsHo1H

Ï^TîS' % s

N IIH H

>?• IIO

S^

S?

551 H

^^

irvirv

t.

H II

HH

^ lAlA O

N^

O

\^

ss

^1H H ^o^

l-i

s« S"

H

rooJ

w fi

i

s«

COH

ô«

îS« If .

A*

ci

^ SI

H

oôc^

H

w

î:«^

s fi^á^

-

M

H

i

>?

d

îi^

d

if\ S'A"

roH ^^vo en H

NO CIR^ ^ rS* lA lf\

HIH

o\vo

H &5>

t- ir\ H H

•°H8

O?

9

¿ •••¿.-v" ^&

W \\

M

vo tn H

w

d

á^ ^5

g

á^

UMA

t

9

«^

^

id

«-^^

s fi* à«

i

%N

IIOo <n oHo

>;*^

N ilH

tf IIO

rf*

1H

¿J^S«

Ml H loi

^ ^9

^

fn CO [^

îS*

d

^

5^^

M

NN

M

<niA H

Í

:^

d

i^r

o OJ

lAH

S-tS01 ^ s S9

i-î

H

>r

S"

rr>OJ

lAlf\

o\••

ii

w C,

^%

SI H

cr>

M

^ î» O^ H 5

>»

OH

>-» CQ

t-OO ITk rH lA I <no\vo o o «A

Vû CM^

O ^0 03

CU-* f- O iOrH

o88r

j cgop

3VO rH

H lA
- k2 -

In this example, |M| = (0.5127) (0.55l»3) - (0.501?) (0.501?) = 0.032H.

(1|.) VL-l^ is obtained ty multiplying the elements of the adjoint of M2;«z» by

1/ M .This computation is the same as that described on page 26, although

in the latter no mention is made of the adjoint matrix. Insert the missing

elements and compute a column, S, contposed of row sums, for M"« „. The check

on the computation of the inverse is obtained by cumulatively multiplying the

elements in the Z column of Vi^*^!* ^ ^^ elements in the £ column of the corresponding rows of M"i ^j; this sum should approximately 15/ eqtial two, the

order of the matrices involved. (5) Mz^y« is obtained from the upper section

of the forward Doolittle solution, table 8. A column, Z, composed of row

sums, is computed. (6) P' is obtained by multiplying M"¿ - by M ^-^. As

explained in the section on matrix multiplication, the IJfh elemeStTof F' is

obtained by summing the products of the elements of the row of H~^ ^ excluding £, with the elements of the Jth column of ^2«^», including têe s column.

If pj . are the elanaents of P', then:

P¿^ = (17.0739) (O.asS) + (-15.1^556) (0.4456) = 3-5925

P^g = (17.0739) (0.4940) + (-15.4556) (0.5048) = O.6323

P^j. = (17.0739) (1.1078) + (-15.4556) (0.9504) = 4.2249

P¿^ = (-15.4556)(0.6138) + (15.794?) (0.4456) = -2.4481

P¿2 = (-15.4556)(0.4940) + (15.794?) (0.5048) = 0.3383

P¿. = (-15.4556)(1.1078) + (-15.794?) (0.9504) = -2.1098

That the sums across the rows of P' are identical (except for possible rounding errors) with the elements in the respective rows of the S column of P' is

indicated by check marks. This checks the computation. An additional

row, 2', composed of column sums, is computed for P« and used in later computations.

Q .+/^ ''?^^*'°TÎ^Vz*^'• ^** sections (5), (6), and (?) of table 10. Sections (5) and (6) fere obtained in previous steps. (?) The ijth element

of My#2«P' is obtained by summing the products of the elements of the ith

colimin of Mz«y», excluding 2, with the elements in the Jth column of P«, including 2 but excluding 2 •. If m^ are the elements of vÇ*^', then:

t^^^^ly^iV "^I^^f«*«ly" i« «sed deliberately in preference to the

iîÎÏ J^t^ possible rounding errors" which has been used in connection

with Checks on matrix operations. In an overaU check of this Spe? M extea

accumulation of rounding errors occurs which has to be SkefLt^ticíSnr^
'hs

mj;! = (O.asô) (3.5925) + (0.1*1*56) (-2.Mrôl) = i.ulu

m^^ = (0.6138) (0.6323) + (0.10*56) (0.3383) = 0.5389

°Í£ = (0-^38) (l*.22l*9) + (0.1*1*56) (-2.1098) = 1.6531

nigg = (0.1*9'*0) (0.6323) + (0.50l»8) (0.3383) = 0.1*832

^2Z= (O.k^) (l*.22l*9) + (0.501*8) (-2.1098) = 1.0221

That the sums across the rows of M^g»?' are Identical (except for possihle

rounding errors) with the elements in the respective rows of the 2 colvnnn of

M - ^^P' is indicated by check marks. It will be remembered that column-byclölumn multiplication was used in the corresptaiding step in table 9.

Although the matrices in sections (5) and (6) are nonsymmetrical, the matrix in section (7) is symmetrical. Hence, in obtaining the product matrix

P', all elements must be conqputed, but in obtaining the product matrix VL^ ^,

the elements below the main diagonal need not be computed. ^

We next compute By*y* = My^^tÇ^y» - My«2«P'. This computation is of

the form:

M „ M'^ ^

y*z zz zy« V^»^' Vy*

yi 72 2

yi ^2 2 yi ^2 2

yi yi yi

72 H 72 H. 72

See sections (l), (7), and (8) of tal>le 10. Sections (l) and (7) were obtained in previous steps. (8) Computations involved in ol^taining By^^Ht sxe

explained on page 35. ^^^

We now compute j^l^ • = - [^JJBJ "^JL • "b ' is the vector of the regression

coefficients of the endogenous variables, the y's, in equation (l.l), that

is, h» =|?ll. but bi = 1 by definition (see page 33). Therefore b' = [ij.

2^b' is b' with its first element omitted, that is, .b' = b 1 ..B is By*yît

with the first row and first column omitted. Bj^ is the first column of By^fy*

with the first element omitted. See sections (9.2), (10.2), and (11.2) of

table 10. (9.2) We copy By*y*, omitting the first row and first column.

Byxy* was obtained in section (8). -.B is therefore a single element, 0.3565;

its inverse, as explained previously, is the reciprocal of that element, so

that i^B-l = 1/0.3565 = 2.8050. (10.2) Before obtaining B^, missing elements

iû By^K^ must be inserted, at least mentally. The first column of ^y*y* with

its first element omitted is a scalar, 0.073^. (11.2) j.^" is obtained by

nwltiplying - GLI^"''' ^y ^1* 1^* ^® ^^^ product of two scalars and is itself

a scalar. The check here is one of recomputation. As in section 13 for

equation (2.1), ccmputationwise, it is easier to neglect the minxis sign
- kk -

prefixed to nB"^ Instead multiply ufi'^ ty B^. Then change the sign of

the result to gpt nh'. If more than two endogenous variables were included

in the equation, Rnhrft would be of an order greater than two and other methods

would be used ^to SÍSt uB (see p. 26). ib« would then be the product of a

matrix times a vector and would itself be a vector.

We nov compute c' = -P' b'. This computation is of the form:

-p« X b* = c'

zi yi

y2

1 21

Z2

2'

See sections (6), (12.2), and (13) of table 10. Section (6) was obtained

previously; in the following computation we make \ise of 2 S Q- column s\im,

rather than 2, a row sum. (12.2) ib*, which is b* with its first element emitted, was obtained in section (11.2;. However, b^^ = 1 by definition. Therefore we can write b' by inserting the omitted element, b^, in ^.b*. (13) C

is obtained by multiplying -PS including Z' and excluding 2, by b'. That the

sum of the elements of c' is identical (except for possible rounding error)

with the element in the 2' row of c' is indicated by a check mark. This

checks the computation. Computationwise, it is easier to neglect the minxis

sign prefixed to P*. Instead, multiply P', including 2' and excliading 2, by

b' and change the sign of the result to get c', including 2'. Then perform

the check in the usual way.

Estimation of standard Errors of the Coefficients. —We now consider the

computations involved in obtaining the standard errors of the coefficients,

beginning with the overidentified equation (2.1) (see table 9).

First we compute bWyj^y». This computation is of the farm:

b

72 yi

X

^y*y*

72 yi 2 ya yi 2

yg

^1

Section numbers are continuations of those used to obtain the coefficients.

See sections (l4), (15), and (16) of table 9. (l4) b is a row vector; it is

the transpose of the column vector b' obtained in section (12). (15) W^*»«
. U5 -

was conputed in section (3). It is copied in section (I5), including the missing element and 2» The check marks next to the items in the 2 column indicate that they are identical with the sums across the rows of the copied

Wyî^. They confirm the copying• If preferred, this computation can be carriea out directly by use of the data in section (3) including the missing

element, (I6) bWy^yjí- is obtained by multiplying b by Wy*y*, including 2.

That the stua across the row of bWy*yif is identical (except for possible rounding error) with the item in the 2 column is indicated by a check mark. The

reader will note that subscript designations for Myi^ are omitted in sections

(16) to (18) of the table.

We now compute tVy^h^ b». See section (1^), (I6) and (I7) of table 9.

Sections (1^) and (I6) were obtained previously. (17) Wy*y* b' is obtained

by cumulating the product of the ith element of bWyXy» excluding 2, with the

ith element of b. (bWy*y* and b are always vectors ana therefore bWyJíy* b'

is always a scalar.) in this example, bWy»y» b* > (0.1002) (l)

+ (.0.680l)(-0.8555) = 0.6820. This is, in effect, row-by-row multiplication

(see p.lOl).

We next compute ll{(bWy*yí(^)'bWy^y*}, See sections (I6) and (18) of

table 9. Section (16) was obtained previously. (I8) ii{(bWy*y*) «bWyjfy^J

is the matrix (bWy*y*) »bWyííy* with its first row and first column omitted.

We do not have to compute the entire matrix, (bWy*y*) •bWyity^, but only that

part in which we are interested, that is, ii{(bWy*y*)'bWy»y»}. In this

example, (bWy*y^) •bWyí^yít is a 2x2 matrix, so that if the first row and first

column are omitted, we are left with a scalar, which is obtained by sqiiaring

the second element of bWy»^. That is, nli^^'y^)'^Wy^y*} * (-0.680l)(-0.680l)

« 0.1f625. The reason for this will be clear to the reader if he writes down

the product matrix that would result from the multiplication of a column vector

by a row vector, where each vector contains the same elements. The check on

this operation is one of reccmputation. If, however, there were more than two

elements in y*, iií(Wyí^y») ''tMy^] would be a matrix.

We now compute

D « 1 / X ( X was obtained in section (9).)

bWyí^yíí-b» (bWy*yifb« was obtained in section (17) )

1 + D

C « (1 + D) bWy^yitb«

C* » C/H', where N« is the sample size minus the total number of

variables in the equation. These conputations are shown in section (19)-
k6

We now compute ^^G = B x |( bWy^^)'bWy,^]. See sections

(18), (19), and (20) of table 9« Sections (18) and (19) were obtained previously. (20) iiG is obtained by multiplying ii{(^^y*y*) '^^y*7*}> ohtalosd.

in section (18), by D , ccnrputed in section (19)«

bWy*y«b'

We next compute VLH = jj^B - 2Jp* See sections (20), (21), and (22) of

table 9. Section (20) was obtained previously. (2l) uB is obtained by

eliminating the first row and first colTimn of By«^», obtained in section (8).

(22) „H is obtiiined by subtracting (20) from (21*. It should be noted that

in this example sections (20), (21), and (22) are scalars. If there were more

than two elements in y*, they would be matrices and the usual checks on matrix

subtraction vould be carried out.

We now ccorpute Fy^ = uH~^. See section (23) of table 9. (23) F^^ is

obtained by taking the inverse of jj^H, computed in section (22). In this

exaaiple 2j,H is a scalar and its inverse is the reciprocal of that number. If,

however, TIH were a matrix, other methods of matrix inversion would be used

(see p. 20).

We next compute F'^^g = oi^'^bb* S®® sections (23), {2k), and (25) of

table 9. Section (23) was obtained previously, (2l*) QJF' is obtained by

eliminating the first column from the matrix P* obtained in section (6).

(25) P^c is obtained by multiplying QI^' by T^-^ . Note again that Q^P«, P^^,,

and F\ç are scalars. If they were higher order matrices, matrix multiplication and the usual checks would be performed.

We now compute U = oi^'^bc* See sections Í2k), (25), and (26) of table

9. Sections (24) and (25; were obtained previously. (26) Since „P* and

P'bc «^« ^°*^ scalars, Ü is obtained by multiplying Q^T* by F' . If such

were not the case, matrix multiplication would be required, keeping in mind

that the product involves the transpose of P'>,e. An efficient method to perform this operation is described on pp. k-J-kS.

We next compute F^ç. = u + içi^*- See sections (26). (2?), and (28) of

table 9. Section (26) was obtained previously. (2?) M:L# was computed in

section (4). (28) Fg^ is obtained by adding (26) and (2?). If higher order

matrices were involved, the usual checks on matrix addition or subtraction

would be performed (see p. 37). The check here, as in the computations above

where only scalars are involved, is one of recoraputation.

^ . ?V^ i??^*? ^'^'b' See section (29) of table 9. (29) C»Pi,>, is obiwiS ^^^^°8 Phb> Obtained in section (23), bj C», cLputS i^ section (19). The diagonal elements of C*F^^ are the variances (square of the

stand^d error) of the b's, excluding ^^T" The variance of b. equals zLo ly

b^ ^r¿*P "^^ ^"""^"-V. "^^^^ ^' ^ «°^- Ti.eretore ihe^varia^ce of"^

niJ ïfL^!^;„of^^ ^^^i^ examples, the coefficients and the variances

lat^ secti^f ^^^ *° ""^^ original data. This is discussed in a
.1^7 -

We next ccampute C*?^^» See section (30) of table 9. (30) C*F is

obtained by multiplying P^ç, obtained in section (28), by C*, conrpuled in section (19)» The diagonal elements of C^F^^ are the variances of the c's. In

this example, there is only one c, that is, a,. Therefore, O^F^^ is a scalar,

and the variance of Cj^ equals C*Fec»

When more than 2 y* or more than 1 z* are involved, only the diagonal

elements of C*Pbb ®^ ^^cc ^^®^ ^® computed.

We now proceed to estimate the variances of the coefficients of the just

identified equation (l*l). The relevant computations are shown in table 10.

Sections (ik), (15)> (I6), and (17) are obtained in exactly the same way as

for the overidentified equation (see table 10 and p. kk)»

For a just identified equation, there is no section (18) comparable to

that for an overidentified equation.

bW .iL.-M.b ' We next compute C* = y*y* , where N' equals the number of observaH' , .

tions minus the total nxamber of variables in the equation. See section (19*2)

of table 10. (19*2) C3* is obtained by dividing bWy^eyjib*, which was computed

in section (l?)^ ^ N*. For this example, bWy^iy^b^ is divided by 26 - 1^ = 22.

Note that the subscripts on Wy^iy* have been omitted in this part of table 10.

Computations comparable to those of sections (20) to (22) of the overidentified equation, table 9, are not obtained for a just identified equation.

We now coapute F^-u « nB-^* See section (23*2) of table 10. (23-2) F^^

is obtained ty talriLng the inverse of JJL^* AS this computation was performed

in section (9.2), section (9*2) is copied into section (23*2)*

We next compute; P' = oi^'^'bb- ^^^ sections (23.2), {2k), and (25)

of table 10. Section (23?2) was obtained previously. (24) QI?' iß obtained

by eliminating the first column of the matrix P' obtained previously in section (6). This column, including 2', is copied from section (6). That the

sum of the elements of the copied column of Q^P* is identical with the item

in the 2' row is indicated by a check mark. 125) F¿c ^^ obtained by multiplying the elements of oi^'> including 2 S by Pbb* ^"^^^ ^^^ 0 ^ ^^ *^® elements

of the column of ¥¿ is identical (except for possible rounding error) with

the element in the 2* row is indicated by a check mark. This checks the computation. It should be noted that F^-^ in this example is a scalar. If there

were more than two elements in y*, P^^ would be a matrix, and the usual matrix

multiplication would be performed.

We now coorpute Ü = oi^'^bc* ^ee sections {2k), (25), and (26). Sections

(210 and (25) were obtained previously. (26) Ü is a matrix whose ijth element

is equal to the product of the element in the ith row of QJ^\ excluding 2',

by the element in the jth row of P¿^, including 2*. In tos example, if we

let Uj^4 be the elements of U:
ujjL « (0-6323) (1.7738) = 1.1217

u^g = (0.6323) (0.9»^90) = 0-6001

xx^ = (0.6323) (2-7228) = 1-7218

U22 = (0.3383) (0.9^90) - 0.3210

U2S = (0-3383) (2.7228) = 0.9212

That the BvanB across the rows of U are identical, except for possible rounding errors, with the elements in the respective rows of the 2 column of Ü is

indicated by check marks- Note that row-by-row multiplication is used hereIf P* and P^e were higher order matrices instead of vectors, as would be

the^case if y* had more than two elements, U would be a matrix whose ijth element would be the cumulative product of the elements in the ith row of Ql^'^

excluding the row Z', by the elements in the Jth row of P^^, including 2

(see p-lOl)-

We now compute F^c = U + M^î^** See sections (26), (27), and (28). Section (26) was obtained previously. (27) Mjez* was obtained in section (4).

It can be copied into section (2T); the usual check on copying matrices is

performed (see p- 37). If preferred, section (27) can be omitted- (28) PQC

is obtained by adding (26) and (27), including their respective 2*s. That

the sums across the rows of Pec are identical, (except for errors that result

frcmi carrying only k decimals) with the elements in the respective rows of

the 2 column of Pec is indicated by check marks.

We next compute C^Pi^ij- See section (29). (29) O^^^tíb is obtained by multiplying P^^, obtained in section (23.2), by C», computed in section (19*2).

^*^bb ®Q>^*^s the variance of b2-

We noir conqcnite C^P^^. See section (30). (30) C*Pçç is obtained by multiplying Pçç, obtained in section (28), by C*, computed in section (19*2)-

As explained before, the diagonal elements of C*?^^ are the variances of the

c's- The element in the upper left-hand corner is the variance of C]^; the

element in the lower right-hand corner is the variance of CQ* Standard errors

of the coefficients are obtained by taking the square root of the variances

of the respective coefficients- If desired, only the diagonal elements of

this matrix need be ccnrputedWe now return to the equations in their original form, that is, equations

(1) and (2)- The conrputations involved are shown at the bottom of tables 9

and 10- The following explanation is in terms of the overidentified equation,

table 9.

Column (1): List the variables in the order that they appear in

equation (2).
.1^9-

Column (2): Enter the estimates of the coefficients in equation (2).

This is done by referring to the relationship between the coefficients in the

original equation (2) and the rewritten form for ccraputationBa purposes,

equation (2.l), given on p. 31, and to the b« in section (12) and the c» in

section (13). For example, the coefficient on Y2 is 1, since the coefficient

of the first variable is always 1. bg^ = -b|- Since bp,^the second element

in bS equals -0.8555. bgn = 0.8555- Similarly, 023 = %?. Since c,, the

element in cS equals 0.2409, C23 = -0.2lf09. ^ ± 1

Column (3): Enter the estimates of the standard errors of the coefficients in eqxiation (2). There is no standard error for the first coefficient.

The standard error of tgi equals the standard error of b2, which is obtained

by taking the square root of C^F^^^, section (29). If there were more than

two elements in y* and hence more than two b*s, their standard errors would

be obtained ly taking the square roots of the diagonal elements of C»F|j^,

which would then be a matrix. The standard error of C23 equals the standard

error of c^, which is obtained by taking the square root of C^F^ç, section (30) • If mere than one element were in z«, as in equation (l.l), the

standard errors of the c's would be obtained by taking the square roots of the

diagonal elements of C*Fç^. (see p. 62).

Column {h): Copy the k^ for the respective variables from table 6.

Column (5): Compute kJ, obtained by dividing ki by the value of k for

the first variable in column {l). That is, kj for YQ^ and ZQ is obtained by

dividing their respective kj^ by 0.001, the value of k for Y2.

Column (6) : Deadjusted coefficients are obtained by multiplying the

elements of column (2) by those of colxmm (5)*

Colimm (7): Deadjtisted standard errors are obtained by multiplying

the elements of colimm (3) by those of column (5).

Column (8): Enter the means of the variables, which are obtained from

table 5*

Column (9): Computations in this colimm are used in obtaining the constant for the equation. Multiply the elements of column (8) by those of column (6) and conrpute their sum. This can be cumulated directly in the machine.

The constant in the equation, a, is obtained by subtracting this sum from the

mean of YQ, the first element in column (8). Hence a = 147-7307 - (-2.6520) =

150.3828. This result can be recorded directly as the first term to the right

of the equality sign in the final regression equation shown in the last row

of this section. The figures within the parentheses are the deadjusted standard errors of the coefficients.

The check in this section is one of recomputation.

DeadJusting the coefficients and their standard errors, and the writing

of equation (l), are accomplished in like manner in table 10.
- 50 -

Modifications vlth Specified Nuiribers of Endogenous

and Predetermined Variables

In the last section ve described computations involved in applying the

limited infonnatlon approach to a 2-eqxiation model of a given type. Here we

discuss modifications in the procedure required when the equations involved

are of a different type. These modifications res\ilt from the fact that we

frequently work with equations where the number of elements in y*, in z*, or

the degree of identification differ from those of the equations in the lumber

model. The modifications may appear in the form of alternate steps—for

example, the case of an overidentified equation where the number of elements

in y* is three or more—or aiaxiliary steps—for exainple, a variation of the

Doolittle method to Invert Mg^z* where the nuniber of elements in z* is three

or more. In addition, the form of many of the matrices shown in tables 9 and

10 may be changed due to an increased number of variables. For example, a

scalar may become a vector, or a vector a larger matrix. Furthermore, the

increased size of the matrices sometimes necessitates the introduction of

additional checks on matrix operations.

Table 11 sxmmarizes some of these modifications; others are e^qplained in

the text. The section numbers in table 11 are the same as those in tables 9

and 10. The section numbers of table 9 are taken as a standard. They represent the necessary steps for an overidentified equation with two elements in

y* and one element in z*. Additional section numbers are indicated by the

following scheme: (l) A number with a digit to the right of the decimal indicates an alternative method necessitated by the identification of the

equation. For example^ (9.I) and (9.2) refer, respectively, to the alternates

to section (9) of table 9 for (a) an overidentified equation whose elements in

y* are three or more, and (b) a just identified equation, regardless of the

number of elements in y* or z*. A section number followed by ".1" refers to

an overidentified equation; one followed by ".2", to a just identified one.

(2) A number followed by a letter Indicates an auxiliary step necessary to

produce the HÄtrix for the section Indicated by the number. For example,

(^c) indicates a variation of the Doolittle method to invert Mz*z* required

for section (k). "c" always refers to a variation of the Doolittle method to

invert a matrix whose order is three or more. (3) A number to the right of

the decimal and a letter coniblne the two previous attributes. For exainple,

(9-20) refers to a variation of the Doolittle method to Invert the matrix required for section (9.2), which Itself is a variation applicable to just

identified equations.

We first discuss table 11, section by section, (l) to note changes in

the size of the matrices with an increasing number of variables, (2) to introduce new checks on the matrix operations and (3) to explain new or additional

steps.

in v»^\í5\^°^r^°^v^^*^°''' ^® ^^**^^ S <i^^°*«s *»»e nuniber of elements

in y«, and h, the number of elements in z». A matrix of order g refers to a

matrix having £ rows and g columns. Purtheim>re, when we state^tLt a maîrîx

has £ rows (columns), we know that the first row (column) corresponds trSf
- 51 -

Table 11.- Alternative steps involved in computations for

equations with specified characteristics

Section

^) ^'^z'^zy*

2) V^y^^

k) ^z^z*

Same procedure for all

h = 2 U

(^a), (Ub), table 10~

h = 1 ly

{h), tabl^F

h > ^.

5) M^^^y»

6) p«

;8) By*y»

Same procedure for eill

Overidentifled equation

= 2 2/

(9), table 9

(10), do.

(11), do.

(12), do.

TXTW

(9.Ï), pp. 53-5*

(10.1), pp. 53-55

(U.l), pp. 55-56

(12.1), pp. 56-57

Just identified equation

g = 2 2/ : g ^ 3 2/ I g > if 2/

(9.2), table 10 ; (9.2a)T (9.2b), p. 57 ;C9-2c), •^735

(10.2), table 10 and pp. 57-5Ô

(11.2), Do.

(12.2), Do.

c'

h

bWy»y*b'

2i

24) 01^'

25) P'bc

26) U

27) M¡»8»

28) ^cc

29) C%v,

30) C*F.^

Same procedure for all

Overidentified eq\aation

(18), table 9 and p. 59

(19), do.

(20), do.

(21), do.

(22), do.

Just identified equation

No comparable section

(19.2), table 10 and p. 60

No comparable section

Do.

Do.

Overidentified equation

g = 2 2/ ; g = j 2/ ~ _

(2^), table 9 I (23a)j23b),pp. 39-60: (23c), p. 6Ö"

Tzoz:

Just identified equation

= 22/; g = 3 2/ ; g 2 4 ¿A

(23:2

2/ ; fí = 3 2/ ; fi

!), copy results from (9-2)

Same procedure for all

2/ £ denotes the nuniber of elements in y*.
- 52 -

first element of y*, 16/ and is so designated; the second rov (column) corresponds to the second element of y*, and so forth; so that, in general, the ith

row (column) corresponds to the ith element of y*. A similar statement can

toe made for a matrix with h rows or columns toy substituting z* for y*.

Section (1) :My*zííÍMzy* .--The computation of this, matrix is explained

on p. 31 .—There are g columns in addition to a 2 column ccanposed of row

sums. It is a symmetrical matrix of order g. As with all symmetrical

matrices, the terms toelow the main diagonal need not toe contputed; however,

they must toe incliaded in ototaining the row sums. The computed matrix, including Z, is copied into section (l) and the check on copying is performed

as explained on p. 37*

Section (2):%Ky* .--The formation of this matrix is explained on p. 37.

It 1« ft ftynwiPti-ri rftl matrix of Order g. With an additional 2 coliJmn.

Section (3):WyXy» >—The computation of this matrix and its check is

e^lained on pT jf. It is a symmetrical matrix of order g, with an additional

S column.

Section (lf):M2Íz* .—The computation of this matrix is determined toy the

number or elements m z*, h:

If h = 1, see section {k), table 9, and p. 37.

If h = 2, see sections (kk) and (Uto) of tatole 10, and p. kO.

If h >, 3, use a variation of the Doolittle method to invert Mz*z*«

This constitutes section (tc). For an e:q)lanation of the computations involved,

as well as the check on this computation, see p. 26. This computation is toest

carried out on a separate worksheet and the restait, Mjiz*, including L, the

colimm of row sums, and the missing elements, entered into section {k). At

times use can toe made of part of the computations necessary to compute Mzi-

(See p. 75-)

Mz*z* is a symmetrical matrix of order h, with an additional 2 column.

Section ($):Mz»y» .—The formation of this nonsymmetrical matrix is

explainea on p. 37 • It has h rows and g colxmms, with an additional 2

column.

.1 Section (6);?' .--This nonsymmetrical matrix is ototained toy multiplying

Mz*z* ^y Mz-Äy*. Scalar m\atiplication and its check are explained on p. kO.

Matrix multiplication and its check are explained on p. 2k. Like Mz-Äy*, P*

has h rows and g columns, in addition to a 2 column. A row, 2', conqposed of

coliamn sums is ototadned if there are two or more rows in P' and is used in

later computations. (See sections (13), (24) and (25).) All elements of this

matrix must toe computed,

16/ See p. 32.
- 53

Section (7);Mür»z»P' .--This coniputation is explained on p. 37, and is

summarized Here, to ot)taln the IJth element of My*z*PS cumulate the products of the tenus of the 1th column of Mz^y*, excluding 2, with the Jth column

of PS Including 2 hut excluding the row Z«. That the sums across the rows

of IM^*P' are Identical (except for possihle rounding errors) with the items

in the respective rows of the Z column of My*2*P' is indicated by check marks.

This provides the check on the computation. My*z»P» is a symmetrical matrix

of order g, with an additional 2 column. Note that column-hy-column multiplication is used in this step.

It is

Section (8);ByWy» .--The computation of this matrix is explained on p. 38.

\ a symmexncal matrix of order g, with an additional 2 column.

Up to this point the steps are exactly the same regai-dless of the degree

of identification of the equation, the only variation arising from the number

of elements in z*. However, for sections (9) - (12), as indicated in

table 11, there are alternative methods according to both the degree of

identification and the nuniber of elements in y*.

Sections (9) - (l2);For Overidentified Equations,"(l) If g = 2, the procedure is exactly that e:q)lalned on p. 38 and shown in sections (9), (lO),

(n), and (12) of table 9. (2) If ^, the procedure of sections (9.I),

(lO,l), (11.1), and (12.1) sho\ald be used. The computations and e:qplanations

which follow are in terms of equation (2.I), where g = 2, as the method is

applicable for any number of elements in y* greater than one. 17/ However,

if g = 2, the method shown in sections (9), (lO), (ll), and (12}" of table 9

is computationally more efficient and hence should be used. The present

exaiople is given only as a simplified case for illustrative purposes*

^^In sections (9-1)> ancL (10.l) of table 12, compute A^the transpose of A

~ B^4(y^y^(y«, uslng another variation of the Doolittle method. The forward

solution is carried out in section (9-1) £^^ a back solution in section (lO.l).

Explanation is as follows: In the upper section of the forward Doolittle

solution, write By*y« and Wyîiy*, obtained from sections (8) and (3), respectively. Note that Wy^jy* is written out in full. Ccanpute also a 2 column,

composed of row svans, Including those terms in B which are omitted because of

symmetry. The encircled numbers are used for illustrative pvirposes in the outline of the computation of A' and its check. Carry out the forward Doolittle

solution in rows 1 - 2" as explained on p. 9» Note that only one 2 column

is used to check the computation of the entire row. In section (10.l), the

computation of A' is in the form of a back solution. This involyes a square

matrix of order g. The outline showing the computation of ^ the aij, the elements of A*, is self-explanatory. The s\ibscripts of the aiJ refer to the rows

and columns of A'; they have no reference to the s\ibscripts of the y*s. The

17/ When there is only one element in y*, we have an ordinary multiple regression equation which can be handled by the least squares method explained

in detail in the first part of this handbook, or by the methods described

on pp. 70-75.
. 5^ -

Table 12.- Computations involved in obtaining sections (9*1) and (10.l)

for an overidentified eqxiation for vhich g=2 l/

Section (9.1) - Forward solution

! Vy* "y*y*

Row

: y2 : yi : 72 ! yi i S

yi

': 0.6688 0.7689

.9858

0.3183

.2550

0.2550

1.0930

2.0112

3.1028

(1)

d")

! 0.6688

: 1.

0.7689

1.11*96(5

0.3183

M590

0.2550

.3812©

2.0112

3.0069(5) v^

(2)

(1) (-1.11*96)

.9858

-.881*0

.2550

-.3660

1.0930

-.2931

3.1028

-2.3122

(2')

(2")

.1017

1.

-.1110

-1.0912(D

.7998^

7.8617®

.7905 _^

7.7701* (TV

Section (10.1) - Back solution

Computation of A' Outline

: a'li

: 0.1*759

: 1.251*6

: 1.7305

a'i2

-1.0912

a'l2(-G))

a'll

©=a'i2

.3^2

-9.0386

-8.6573

a'22

7.8617 (2)

^•22(-(î))

a'2i

©=a'22

^'21

3.OO69

-8.9336

a's2

7.770I**' 0

**S2(-(!D)

^'21

Check:

a|i2 + a'22

a'll + a'21

(!)=a'22

+ 1 = a*vî>

1 - a-g

1/ These computations were performed with 9 decimal places, of which

She s2SSv'? *^" *^^^';. ^^^"^°^ «°^^ °^ *^« ci^uteti^nsm^fSpear to he slightly in error. This method is used only ^en g>3. This siSlified example is shown for purposes of illustratiS. " ^
- 55 -

check on the computation of A' also is self-eijqplanatory. These computations

can be easily extended for any nuniber of elements in y*. (See appendix

P- 92.)

In section (ll.l) we estimate X by an iterative method- 18/ Computations

are shown in table 13. The explanation is as follows: Write A», computed in

section (10.1). Next to it write the vector q(0), which is a column of I's.

q(0} has as many I's as the nimiber of rows in A'. Beneath it write QU) whose

elements, 0^"^^^ are obtained in the following manner: Q^í^) equals the sum of

the elements of the first column of A ', ^-^/ equals the sum of the elements of

the second column of A». In general, therefore, the ith element of QUJ is

obtained by summing the ith colimm of A *. These sums (plus l) were computed

in the check on the conrputations of A'. For our exanrple,

^i^) = (1-7305) + (-8.6573) = -6.9267

Q|I) = (-1.0912) + (7.8617) = 6.7704

Table 13.- Ccmputations involved in obtaining section (ll.l)

for an overidentified equation l/

1 Successive iterations

A» 1 ,<°) ; .(^' ;

: .(2) ; q(3) .'' ,(4) ; 1(5) ;: ,(«)

1.7305 -1.0912!

-8.6573 7.8617

. 1.

: 1.

1.

-.977^^

1.

-.8609

1.

-.8558

1.

-.8555

1.

-.8555

1.

-.8555

; Q(I) ;

; Q^^^ ;

• Q(3) ; Q(4) : Q(5) :

; Q(6) : Q(7)

-6.9267

6.7704

10.1926

-8.7756

9.1843

-7.8600

9.1395

-7.8193

9.1373

-7.8173

9.1372

-7.8172

9.1372

-7.8172

1/ These computations were performed with 9 decimal places, of which only k

appear in the table. Because of this some of the computations may appear to be

slightly in error.

q(l), the elements of q(l), are obtained by dividing each of the ql by

the element of Ç^^ havin«. the largest absolute value. In the present example,

the largest element of Q^^^ is -6.9267. q^l), therefore, equals -6.92677-6.9267

18/ This method differs from that discussed by Chemoff and Divinslsy (2,

p. 244) and KLein (13, p. I83).
- 56 -

- 1. ail) = 6 7704/-6.9267 = -O.mK <l(^^ *^e elements of Q!^^\ are 6b-

¡.kelvin «J-Si¿¿nf,1=«: «(^S e,Jal. t^ .- of t^jP^gfenSLfthe

elements of the first column of A' with the elements of ^^^¿.^¿;;^^

sum of the products of the elements of the second column of A« with the elements of cfi). In general, therefore, the ith element of Q^^; eguals the sum

of the products of the ith column of A' with the elements of qU;. In our

example,

Q(2) . (1.7305) (1) + (-8.6573) (-0.977^^) = 10-1926

Q(2) ^ (.1.0912) (1) + (7.8617) (-0.977^) = -8.7756

(Á^), the elements of <¿^\ are obtained by dividing each of the QÍ^) hy

the element of QV^T having the largest absolute value. In our example, therefore, qpT = 10.1926/10.1926 = 1; 4^) = -8.7756/10.1926 = .0.8609. 0)3;, the

elements of Qv3), are obtained in the following manner: Qj^-^; equals the sum

of the products of the alements of the first column of A' with the elements

of q(^)} QA^/ equals the s\mi of the products of the elements of the second

column of A' with the elements of q(2) ; and in general, Q^3) equals the sum

of the products of the ith column of A' with the elements of q^^f. In our

example,

Q(3) = (1.7305) (1) + (-8.6573) (-0.8609) = 9-1843

Q(3) = (-1.0912) (1) + (7.8617) (-0.8609) = -7.8600

Continue in this manner, that is, q^^^ = QJ^) /^O f where QJ^^ is the

element in Q(^) having the largest absolute value; ana Q.i^+l) equals the sum

of the products of the ith column of A* by the qL^), un-BLl the elements in the

successive Q vectors, that is, CL(^/ and ii^l^+l), îigree to the nxmiber of required

decimal places. Six iterations were required for accuracy to h decimals as

shown in table 13; nine iterations were needed for the required agreement of

nine decimal places when the full computations were carried out. I9/ The

largest element of the final computed Q vector, that is, QÍ^) in table 13, is

the estimate of Xv The only check on the computations needed is that of recomputing the final Q and q vectors; atU other checks are \mnecessary owing

to the nature of the iterative process. The reader will note that, except

for a rounding error due to dropping decimals, the estimate for X. obtained in

table 13 is the same as that shown in section (9), table 9.

In section (12.l) we estimate b', the vector of the coefficients of the

endogenous variables, b' is obtained from the final q vector, computed in

section (11.1), in the following manner: If the first element of the final q

vector equals 1, b' is this final q vector; if the first element of the final

q vector does not equal 1, b' is obtained by dividing each element of the

19/ Some authors recommend raising the matrix to the eighth power before

beginning the iterations. When this was tried on this example, the number of

necessary iterations was reduced to three. But the additional work involved

in raising the matrix to a higher power, which requires use of a "floating"

decimal point on desk calculators, outweighs the advantages.
- 57 -

final q vector by its first element. For example, if the elements of the

final q vector were 0.2 and O.k, the elements of b* woiild be 1.0 and 2 0

respectively. In our case, the first element of the final q vector, q(6)

equals 1 and the second element equals -O.8555. These results are the same

as those shown in section (12) of table 9. If there are more than two elements in b •, compute the stmi of all the elements in the vector and enter that

sum in a S' row. This simi is used later.

Sections (9)'(l2):For Just Identified Eciuations.—d) If g=2, the procedure

is that explained on p. 1^3 and shown in sections (9-2), (10.2), (11.2), and

(12.2) of table 10.

(2) If g=3, sections (9*2a) and (9*2b) are used to invert the matrix

3jB^ which is required for section (9-2). jjß is ^ symmetrical matrix obtained

by omitting the first row and first column of By^^, computed in section (8).

A column, Z, is obtained for jjß by subtracting the first element in each row

other than the first of Byjiy* from 2 of By^iy*. That the sum across the rows

of iiB is identical with the figure appearing in the 2 column of nB is indicated by a check mark. When y* has 3 elements, nB is a 2x2 matrix. The

method used to invert a matrix of order 2, inclxading the ch)eck, is explained

on p. 26, and shown in sections (4a) and (lib) of table 10. Sections (9.2a)

and (9*2b) are analagous to sections (4a) and (4b), respectively, U^^z* being

replaced by uB. nB-l, including the missing elements, is then entered in

section (9*2). A row, 2', consisting of column sums, should be computed; this

enters into later computations.

In section (10.2) write Bi, the first coltmm of Byiiy* with its first element omitted. Before obtaining Bi, the missing elements in By^cy^ must be

filled in, at least mentally. When g»3, Bi is a column vector consisting of

two rows. An additional row, 2', is obtained by stibtracting the element in

the first row and first column of By#y« from the item in the first row and 2

colxmm of ByNy«. That the simi of the items in Bi is identiccúL with the corresponding value in the Z* row is indicated by a check mark. This provides a

check on the copying. Z*, however, is not \ised in later computations.

In section (11.2) canpute jb', which is b', the column vector of the

coefficients of the endogenous variables, with its first element omitted. The

ith element of 3b ' is obtained by cimiulating the product of the elements of

the ith row of -[liB]*^, exdtading 2 but including 2», with the elements of

Bi, excluding 2'. That the sum of the elements of ib' is identical (except

for possible rounding error) with the element in the 2 • row of ib • is indicated by a check mark. Coanputationwise, it is easier to neglect the minus

sign prefixed to nB-l. Instead, multiply n^"^^ including £', by Bi, excluding 2 » ; change the sign of the result, ib », including Z ' ; and perform the

check in the manner indicated previously.

In section (12.2) b' is obtained from ib' by inserting the omitted element, bi, which equals 1 by definition, b* is a coliimn vector having 3 rows

when y* has 3 elements. An additional row, 2', is obtained by adding 1 to the

element in the z* row of jb •. That the sum of the elements in b ' is identical

with the element in the z* row of b' is indicated by a check mark. This

checks the copying.
. 58 -

(3) If g^> a variation of the Doolittle method is used to invert uB,

which is required for section (9.2), and this is considered as section (9-2c).

For an explanation of the conrputations involved, as well as the check on this

operation, see p. 26. This computation is best carried out on a separate

worksheet and the result, nB-l, including the missing elements, entered into

section (9.2) • An additional row, r, consisting of column sums should be

coanputed. jjß ^^^ ^*® inverse are symmetrical matrices of order g-1. In

sections (10,2), (11.2) and (12.2), the computations are the same as those

just given for the case where g=3» Bi (section 10.2) and ib' (section 11.2)

are column vectors with g-1 rows, in additioh to a Z' row. b' (section 12.2)

is a colimm vector with g rows, in addition to a 2' row.

With section (13), see table U, the computations are applicable regardless of the degree of identification of the equation.

Section (13) ; c '. »-This is obtained by multiplying -P', obtained in section

(6), by b', obtained in section (12) for an overidentified equation where g=2,

section (12.1) for an overidentified equation where g^, or section (12.2) for

a Just identified equation. The ith element of c' is obtained by cumulating

the product of the elements of the ith row of -P*, excluding S but including

2', with the elements of b', excluding Z«. That the sum of the elements of c'

is identical (except for possible ro\mding error) with the element in the 2'

row of c' is indicated by a check mark, c' is a column vector with h rows in

addition to a 2' row. Computationwise, it is easier to neglect the minus sign

prefixed to P». Instead, multiply P', including 2' and excluding 2, by b'

excluding 2'j change the sign of the resulting c', including 2', and perform

the check. This procedure is used when P' has a 2' row, that is, \rtien h¿2.

If however, h = 1, as in table 9, c' is a scalar and the check is one of

recomputation.

Beginning with section (ik) conrputations are started for the standard

errors of the coefficients just obtained.

Section (l4);b,—This row vector with g coliimns is obtained by transposing the column vector b', obtained in section (12) for an overidentified

equation where g=2; section (12.l) for an overidentified equation where g>3j

or section (12.2) for just identified equations regardless of the number of

elements in y^. When g>3, the element in the 2' row of b' becomes the element

in the 2 column of b. That the sum of the elements of the transposed vector

is identical with the element in the 2 column of b is indicated by a check

mark. This checks the transposition.

Section (l3):Wyy.--This matrix was computed in section (3). (See p. 37.)

I* is coplea into section (15). The missing elements are included in the

copied version. The usual check on copying should be \ised.

Section (16) ;bWy%*,-^This computation is explained on page ^5. bWy^y*

IS a row vector having g columns in addition to a 2 column. ^^

Seotion (l7):bWy^^b\--Thls scalar is obtained by cumulating the prod-

^l^^4^l "^i® ^^îî element of bWy^(y* excluding 2 (obtained in section (16)), with

the ith element of b, excluding 2 (obtained in section (ik)). Note that bWy^fy*
- 59 -

and b are always row vectors and that Wyîiy* "b' is always a scalar. As with

all scalars^ the check is one of recomputation.

Sections (l8) - (23):For Overldentified Eq\xations Only,->«In section (l8)

we obtain x^^^^y^y*) ^^^yihy^}^y omitting the first row and first column of

(bWyîiy») ^hWjrMy*, a symmetrical matrix of order g. Its ijth element is obtained

by multiplying the ith element of bWy^iy* with the jth element of bWy^fy*. We

do not, however, compute the entire matrix, (bWy*y*) 'bWy^iy*, but only that

part in which we are interested, that is, all rows and columns other than the

first. For example, if y* had four elements, and we denote the elements of

(bWy^iyif)'bWy^iy* as bj[j, we would not compute bn, bi2, bi3, or bilf. We would

not compute b21, b3i, or bi|.i in any event, as it is symmetrical. This computation and its check are explained on page ^5 for the case where g = 2.

Note that the E column of bWjrííy* is not used in these computations. When

g>2, an additionetl sum, j^E, is computed for bWy^ty*. Since 2 is the sum of all

the elements of bWyJiyït, jJL is equal to 2 miniis the first element of bWyity*.

The element in the ith row and 2 colxmaa of i^Wy^yi^)*Wyi^y^y±s obtained

by multiplying the ith element of bWy»y* by ^t. That the sxmis across

the rows of lii^'^'-y^K^ ^'^^y^y*}" ^^^ identical (except for possible rounding

errors) with the items in the 2 column of the respective rows is indicated by

check marks. This checks the computation.

See table 9 and p. 45 for computations involved in section (19)- Except

for X, which is obtained in section (9) when g = 2 and in section (ll.l) (see

p. 55} when ^3, the computations are the same regardless of the nimiber of

e7.ements in y*.

In section (20), we obtain nG by multiplying ii[(^'Wy*y*)*^Wy»y*}

^y P,,' l¿(^Vy*^'^^y*y*) is the symmetrical matrix of order g-l

obtained in section (l8). w.^KHeb' ^^ ^ scalar obtained in section (19).

For an explanation of scalar ÎK3[tiplication and its check, see p. ^. VLG is

a symmetrical matrix of order g-l with an additional 2 column (when g-ST^

In section (21) we obtain nB, a symmetrical matrix, by omitting the

first row and first colimm of By*y*, computed in section (8). A column, 2,

is obtained for uB (when g^3) by subtracting the first element in each row,

other than the first, from 2 of By^fy*. That the sums across the rows of uB

are identical with the elements in the 2 column of the respective rows of

llB is indicated by check marks.

In section (2$) we obtain iiH, a symmetrical matrix, by subtracting nG^

CQiç)uted in section (20), from uB, computed in section (21). Like ilG and

llB, iiH is of order g-l and has an additional 2 column (when g - 3) •

In section (23) we obtain Fbb- This matrix is the inverse of ji.H, computed in section (22). If g = 2, uH is a scalar, and its inverse is obtained

by taking the reciprocal of that number. (See p. 26 and table 90 If 6 = 3>

sections (23a) and (23b) are used to invert the 2x2 matrix, uH, required for

section (23). The method used to invert a 2x2 matrix and its check is e3q)lained
- 6o -

on p. 26, and shown in sections (4a) and {k\>) of table 10. Sections (23a) and

(23b) are analogous to sections (4a) and (I*), respectively, when Mz*z* is

replaced hy viH. uH-l is then entered into section (23). If g ¿ 4, a variation of the Doolittle method is used to invert nH^ ^*lch is required for

section (23). This is done in section (23c). For an explanation of the computations involved and the check, see p. 26. This computation is best carried

out on a separate worksheet, and the resxilt, nH"! and, Z, the column of row

sums, entered into section (23). Note that nH-l, or Pbb> is a symmetrical

matrix of order g-1, vhen g ^ 3. The missing elements should be shown in

section (23).

Sections (19.2) and (23.2) tFor Just Identified Equations Only.—For a

Just identified equation there is no section (1Ö) comparable to that for an

overidentified equation.

In section (19.2) we compute C* = bWyJiyA'/N'. N' equals the number of

observations minus the total number of variables in the equation. bWy^iy*b' is

the scalar obtained in section (17).

Computations comparable to those of sections (20) - (22) for an overidentified equation are not obtained for a just identified equation.

In section (23.2) we obtain Pbb by taking the inverse of 13B, that is,

the matrix obtained by omitting the first row and first column of ByXy«,

obtained in section (8). This ccoxputatlon was performed in section (9*2).

We therefore copy section (9*2), Including the ^ column which is obteiined

when g = 3 or g !1 4, into section (23-2). Any missing elements shoxild be indiaded in the copied version.

With section {2k) (see table ll) the contputatlons are applicable regardless of the degree of identification of the equation.

Section (24) :oiP' .--This nonsymmetrlcal matrix is obtained by eliminating

the first column or f^, computed previously in section (6), and copying the

remaining columns, except the 2 column, into section {2k). When h ¿ 2, the

elements of the 2* row of P' are also copied into section {2k). That the sums

of the elements in the copied columns of QIF' are identiceú. with the elements

in the respective colimáis of the Z* row is indicated by check marks. This

checks the copying, QI^' 1»S h rows, in addition to a 2» row (when h I 2),

and g-1 columns.

Section (2$)tF'i)c.—This matrix is obtained by multiplying oi^S including s', oy j?-bb^ exc±ualng Z. This is a nonsymmetrlcal matrix, so all elements

must be computed. That the sums of the elements of the coltmms of F'-bc are

identical (except for possible rounding errors) with the elements in the

respective columns of the E» row of F'-bc^ is Indicated by check marks. F'^c

has h rows, in addition to 2», and g-1 columns. This procedure holds only

when oiP' has a 2' row, that is, when h ^ 2. If, however, h = 1 and g = 2,

as in table 9, oi^S ^bb and their product F'bc^ are scalars and the check is

one of recomputation. On the other hand, if h = 1 and g i 3, QIP* is a row

vector and Pbb, a symnetrical matrix of order g-1. F'tc is obtained by
61

multiplying QIP* "by Fbt, incltiding 2. That the sum of the elements of the

resulting row vector, F'bc^ is identical (except for possible rounding error)

with the elements in the 2 column of F'be is indicated by check marks. P«>>c

is then a row vector with g - 1 columns in addition to a 2 column.

Many analysts will have an interest only in the standard error (the

square root of the variance) of each coefficient. They will not wish to have

the covariance terns computed. If only the standard errors (or variances) of

the coefficients are desired, only the diagonal elements of each matrix need

be computed starting with Ü in section (26).

Section (26); U.—The ijth element of this matrix is obtained by summing

the products of the elements of the ith row of oi^S excluding 2', with the

elements of the Jth row of F'bc^ including 2'. Note that row4)y-row multiplication is used here. That the sums across the rows of U are identical (except

for possible rounding errors) with the elements in the respective rows of the

2 column of Ü is indicated by check marks. U is a symmetrical matrix of order

h with an additional 2 column. This procedure holds only when h - 2. If,

however, h = 1 and g = 2, as in table 9, QI^S F'be aj^d their product Ü are

scalars, and the check is one of recomputation. On the other hand, if h = 1

and g > 3> U is the scalar obtained by cimrulating the products of the ith

element of the row vector oiP' with the ith element of the row vector F'bo

excluding 2. The check is one of recomputation.

Section (27): M^»g».—This matrix, including the 2 column, which was

obtained in section (¿f;, is copied into section (27). If the covariances are

not desired, only the diagonal elements need be copied. The usual check on

the copying of matrices should be performed.

Section (28); Fee.—This matrix is obtained by adding U and i^*z*> as

eacplainea on p. i|^. It is a symmetrical matrix of order h, with an additional

2 colimm.

Section (29); CTFbb^'^^niis matrix is obtained by multiplying Fbb^ caaputed

in sections (23) and (23.2), by C*, computed in sections (19) and (19.2), for

overidentified and Jxist identified eq\iations, respectively. Since C^ is a

scalar, each element of Fbb> including those in the 2 colimm, is multiplied by

C*. That the sums across the rows of C*Pbb are identical (except for possible

romading errors) with the elements in the respective rows of the 2 column of

C^ï\jb is indicated by check marks. Like Fbb> C^^Fbb is a.symmetrical matrix of

order g-1, and will be of the form

^ ^

yf . . y| z

^

»1 IP

• H

^k
- 62 -

The variance of ti, the coefficient on y|j (the ith element in the y* vector)

is the element in the yf row and y± colvaam of 0*?^^^. Note there is no row or

column corresponding to y*, since the variance of bi equals zero. The nondiagoñal elements of C^P^i, are the covariance terms for the coefficients on

the endogenous variables, the b's. For example, the element in the y* rov

and y¡l column of C*P5|j is the covariance bob]^, where bo and \)¡^ are the coefficients on yg and yj, respectively.

If only the veuriances are desired, only the diagonal elements of C^Pj^^)

need be computed.

Section (30): C*Fcc.—This matrix is obtained by multiplying F^c *y C*.

This computation is analogous to that explained in section (29), p. 6I, with

Fb)j replaced by Fee* Tbe variance of Ci, the coefficient on zf, (the ith

element in the z* vector) is the element in the z* row and z± colimm of C*FccLike Fcci C*Fcc is a symmetrical matrix of order h. The nondiagonal elements

of C*Pcc are the covariance terms for the coefficients on the predetermined

variables, the c's. For example, the element in the Z2 row and z* column of

C*Fcc is the covariance CgC^, where Cg and c^ are the coefficients on zg and

and Z3, respectively.

If covariance terms for the b's with the c's are desired, a matrix

C*(-F^ç) can be obtained by multiplying -F^^ (the negative of the transpose of

F¿c, computed in section (25)) by C*. This is a nonsymmetrical matrix with

g-1 rows and h columns. The element in the yj row and zj column of C*(-Pbe)

is the covariance bicj, where bi and cj are the coefficients on y± and zt,

respectively. *^

The final step is to deadjust the coefficients and their standard errors

and rewrite the equation in its original fom, that is, in terms of one endogenous variable set equal to a linear coobinatlon of the other variables.

This is described on p. 48. ~x<»wj.w.

Equatioris Having No z*'s.^-For the special case in which z* has no elements, that is, h = 0, the method outlined in table U is followed with these

changes: (a) Do not use sections (k), (5), (6) and (7): (b) sectionf8) is
- 63 -

Procedures for Handling Complex Systems

In all systems of equations discussed in detail so far, each coefficient

relates to a variable composed of a single item. Freqxiently, the coefficients

in the structural equations relate to a composite variable that consists of an

arithmetic s\am of several endogenous or predetermined variables. Slight modifications are required vííen, this is so. Other modifications may "be needed when

some of the equations can he fitted by least squares, but others are to be fitted by the limited information approach. Still further modifications are

required when the system involves equations that are nonlinear in the variables.

Methods of handling problems of this sort axe discussed in detail in connection

with the 3 examples included in this section.

A 6-equation Model of the Wheat Economy.—^This example is taken from

Meinken (lo) > This system of equations is designed to give simultaneouslydetermined estimates of prices in domestic and world outlets and the quantity

of wheat utilized domestically in each of the following price-determined outlets:

Use for food, feed, export, and end-of-year stocks. Toted demand for use in

these outlets plus that used for seed and industrial purposes must add to the

total domestic supply at the beginning of the marketing year. The economic

reasoning behind the equations used, and results from the statistical fitting,

are disciissed in detail in Meinken's bulletin. Aspects discussed here are

those which require a different computational procedure from the systems of

equations covered in previous sections.

The following variables are assumed to have been simultaneously determined

by a common set of economic forces during the years included in the analysis.

Thus, they are the endogenous variables in the structural equations.

P^ - Wholesale price of wheat at Liverpool, England, per bushel, converted

to United States currency at par, cents; or P^ - Wholesale price of wheat at

Liverpool per bushel, converted to United States currency at current rate of

exchange, cents.

P^,^ - Wholesale price of No. 2 Hard Red Winter wheat at Kansas City per

bushel, cents.

C« - Domestic use of wheat for feed, million bushels.

C - Domestic net exports and shipments to United States Territories of

wheat and flour on a wheat equivalent basis, million bushels.

C - Domestic end-of-year stocks of wheat, million bushels,

s

Ch - Domestic use of vheat and wheat products for food by civilians,

million bushels.

The following variables are believed to have influenced the values of 1 or

more of the endogíious variables during the years included in the study but not

TLve SL^Lenced by them to a significant degree during any given m^JetIng ylL. Thus they are the predetennined variables in the structural equations.
. 6k .

S - World production of vheat plus stocks about August 1, excluding

Russia^and China but including net exports from Russia, million bushels.

I - Index of wholesale prices of 1+5 raw materials in England (1910-1^

= 100)Y

S¿ - Domestic production plus stocks on July 1 of wheat minus use for

seed and industrial purposes, million bushels.

Pçj - Wholesale price of No. 3 Yellow com at Chicago, July-December, per

60-pound bushel, cents.

A - Poultry units fed on farms during the year beginning October, millions .

N - Ocean freigibt rates from G\llf ports to Liverpool plxxs the united

Kingdom tariff on wheat from this country minus the average export subsidy

paid by our government per bushel, cents.

Iz, - Wholesale price of all commodities in this country as computed by

the Bureau of Labor Statistics (1926 = 100).

F - Indications regarding the domestic winter wheat crop for the following year on December 1 as made by the Crop Reporting BoaixL (planted

axîreage times condition).

L - Total population on January 1, millions.

Q - Average processing tax on wheat per bushel, cents.

M - Wage rates of all factory workers per hour, cents.

D - Consxmiers* disposable income, bill ion dollars.

T - Time (1921 = 1).

Symbolic letters should be used in connection with these "structural"

variables, as the endogenous {Y^) and predetermined (Zj,) variables used in

fitting usually differ from those that appear in the structural equations.

Systems of equations may be fitted for at least 2 alternative reasons:

(1) To determine certain structural coefficients, such as coefficients of elasticity, that are statistically consistent, or (2) to obtain simultaneous estimates of the several endogenous variables. If, in the example being considered in this section, the analyst were interested only in the elasticity of

demand with respect to price for use of wheat as feed, only a single equation

would need to be fitted. This would be fitted by the limited information

approach, since the equation contains 2 endogenous variables. If the price

of wheat were determined by a support program, the fitted equation coxild be

used to estímate probable use of wheat for feed in futvire years. On the other

hand, if simultaneous estimates of all of the endogenous variables in future

years are desired, the number of equations used in fitting should equal the

number of endogenous variables in the system.
- 65 -

Sometimes considerable algebraic manipiAlation Is n^Arie^ +

equations to this number. This manipulati^usuïïlv invSvî ° ''^''"^ ^^

certain identities into the other eqSSion^ T^^sh^d be d^fïT'^'^

determining the particular relation? that are to be fÍSÍL bv Stî íf%

"^^l: .^^ ^ *^« =°^«e Of the algebraic manipilaUon c^tSnltíí^í. "^

^^^^ oTrííí:^ f^Sio^HLe^^r ~~2;jt:i the

For the wheat model, tííly^S ^s or^ÏÏ^aSî w^eí îSi'jS^f^ '^ <=°^^«*«-

L - ^3 + b3i(p¿ + Q) + ^3^ + b33T + h^¡^ (3^1)

Ce = a5 + b5i(P^ - Pd - N) (5)

Cs = a6 + b6i(g) + b62F (6)

Sd = Ci, + Cf. + Cg + Cg (y)

^w = s© + ^8lSw + ^82lv (8)

Ecjuation (8) can be fitted directly by least squares as it contains only

a single endogenous variable. But S^ and I should be included as predetermined

variables in the system. P^ rather than P^ is included in the set of equations

to be fitted by the limited information approach 21/ (see equation (5)), but

^y = ^v* H®û<^®> S^ 8ûd I^ each were multiplied By k to make them consistent

•with the variables \xsed in the other equations.

Equation (7) need not be fitted as it involves no statistical coefficients.

Hence, we have k equations, namely (3*1), (h), (5) and (6), to be fitted directly

by the limited infoimation method.

The predetermined variables for the entire system are obtained by picking

out each predetermined variable included in any one of the structural equations.

For this model, these are: L, Q, D/L, T, M, P , A, N, I¿, F, S^^, kS^ and kl^.

20/ In equation (6) as given by Meinken (l6), PdAd ^^ multiplied by 100.

As this makes for a cumbersome notation, this constant is emitted in the equations as used in this handbook.

21/ If the disturbance in equation (8) is assumed to be independent of those

in the other structural equations, then a least squares fit for this equation

is equivalent to that obtained by the full information approach (see p. 28)-

If the disturbance is not assumed to be independent of those in "the other equations, then a least ôquares fit is equivalent to that obtained by the limited

information approach. Independence among the several disturbances in the

structural equations noimally is not assumed in the limited information approach.
. 66 -

Since D does not appear separately in any equation, D/L can be used as a

single variable. The 13 predetermined variables ordinarily would be designated as Zi, Z2, ..• Z13, respectively, and would be used to fonn the VL^z

matrix of order 13. For this analysis, however, the variable T was

not included in M«. because this time trend is known to represent in only

an imprecise way the true variables that cause the downward trend over time

in the demand for wheat for use in food products. Thus Mgz became a matrix

of order 12. To give determinate results, at least two more observations

than the number of variables included in Mgz should be available 22/.

The endogenous variables to be used in fitting are obtained by using

each cambln^tion of variables that includes at least one endogenous variable.

In this connection, we are interested only in those endogenous variables

that are included in equations to be fitted directly by the limited information approach. Hence, P^ is onitted. For this model, the endogenous

variables used in fitting are: C^/L, Pd^^^ Cf, Pd-Pc> Ce, P^-Pd-N, Cg, BI^

FdAd- These 8 variables are designated respectively as Yi, Y2, ... Y8*

Two of these Yi appear in each of the k equations to be fitted by the limited

information approach. Chemoff and Rubin (2^ pp. 210-212) suggest that the

nonlinear combinations Ch/L and Pd/ld> Q-s veil as the linear combinations,

be treated in this way. An alternative method of handling the nonlinear combinations of variables is discussed on p. 67.

In obtaining the moments, we use a symmetrical matrix of order 12 + 8 = 20,

plus a check sum, making use of the variables other than T noted in the preceding two paragraphs. Some of the moments obtained are not used, but if only

those moments actually to be used are obtained, with a check sum on each,

more work is required than if the entire operation were to be performed with

a single matrix. Moments that involve T with the other variables included

in. equation (3.I) (see p. 65) are obtained as a separate operation*^ (In some

problems, as in the example described on p. 73> it is more efficient to obtain

the moments with T simultaneously with those for the other variables, even

though it is to be omitted from Mg^.)

The forward Doolittle solution required to obtain My^zMzzMzy* is next

performed as a composite operation for all of the equations to be fitted by

the limited information approach. The upper section of the forward Doolittle

solution, in outline form, is as follows. Note that only a single check sum

is used.

22/ Predetermined variables sometimes are omitted from Mg^ to save computational time. Estimates of the coefficients that are statistically consistent still are obtained, provided sufficient predetermined variables are used

to provide identification, but the estimates are less efficient (in a statistical sense) than if all of the predetermined variables in the system are

used. (See Hildreth and Jarrett (12^ pp. 69-70),)
- 67 -

Z2

'12

The items from the last two rows of each section of the M,„ portion of this

operation (see p. 32) axe transferred to another sheet, with a^Sieck sum folloJ!

ing each pair of y's. The boxheads on this sheet are of the form:

Row y^^ 72 ^ ^3 y^ 2 y? y6 ^ y^ yg E

^%i V-z^zz^zy* ^ e<3Piired for the 4 equations axe obtained as described on

p. 32. Use of the several check sums provides a complete check on the opera,-

tions. In some problems, it is more efficient to compute the complete MVJíÍÍM

matrix as a single operation, but in this exançle this method would reqiñre more

than twice as many cvmrulative cross-multiplications.

In fitting each of the h equations, the following variables axe involved:

Equation

(3.1)

W

(5)

(6)

Cf, Pd - Pc

D/L, T, M

A

These eqixations are fitted, making use of the appropriate Y's and Z's aad

the steps given in table 11 for equations having specified numbers of endogenous

and predetermined variables. Each of these equations is overidentified. The

My-ityx- matrices, \rtiich are of order 2 in each case, and the M ^^^ matrices are

copied fl*am the matrix of adjusted moments • In each equation, the variable to

the left of the equality sign in the structural equation is designated as y^^

in the fitting process.

Nonlinear variables.—As noted on p. 66, Chemoff and Rubin (3, pp. 210-212)

suggest treating nonlinear combinations that involve endogenotis variables, such

as G^/L and V^/l^, as though they were each a single variable. Klein (13> PP*

120-121), on the other hand, suggests use of formulas that transform the nonlinear combinations into linear approximations. These approximations are then

substituted for the original variables. The authors believe that the Klein

approach is to be preferred.
. 68 .

The following formulas are given by Klein in this connection:

XY 2^YX + 5nf - XY

X/Y «X/Y + X/Y - (X/Y^)Y.

where X and Y are the means of X and Y, respectively. If either the product or

the quotient is multiplied by a constant, then each term in the transformation

is multiplied by the constant.

Each of the nonlinear variables in the model for wheat involves a quotient. Hence these variables can be rewritten in the following way:

L L L ^e

^d^^d 1, ^d

—- — +—î'd- =-2 Id

Id ^d Id Id

- ^ - ,^2

In computing M , the predetermined variables -(C^/L )L and -(P¿/ld)l¿

are substituted forT and I,, respectively. The change from the M^^ used in

the previous example can be made easily, as the variables L and I^^ are, respectively, merely multiplied by a constant.

For purposes of fitting, the linear combination of endogenous and predetermined variables is used in the same way as the quotient in the preceding

example. In using the structtiral equations for analytical purposes, however,

it is convenient to rewrite these variables so that the endogenous and predetermined variables are separated (see p. 82). This can, be done easily by an

algebraic transformation.

Nonlinear equations. - -All 6 equations used in the wheat model are stated

in linear form. This is a desirable condition when using computational methods

described in this handbook. 23/ In some models, the analyst may prefer to

express his equations as linear in logarithms or a^ some other known function

of the observed variables. This resxilts in a system of equations that are consistent in so far as linearity is concerned so long as none of the equations

are identities like equation (7) in the model for wheat. If identities of this

sort are involved, all other equations must be linear in the actual variables

if consistency among the several equations is to be maintained. If all the

23/ Chemoff and Rubin (3, pp. 210-212) point out, in effect, that systems of

equations that involve some equations for which the variables are expressed in

logarithms and other equations for which the variables are in terms of actual

data can be handled directly by the limited information approach by using the

logarithms of the appropriate variables in M^z in place of the original data.

Estimates of the coefficients obtained in this way are statistically consistent

but probably, under most conditions, are less efficient than if all variables

are either in actual data or in logarithms.
- 69 -

equations are expressed as linear in losarithms, and nonlinear combinations of

variables like those discussed in the preceding section are involved, no modification in the computational procedxire is required as the logarithm of a

product equals the sum of the logarithms of the factors and the logarithm of a

quotient equals the logarithm of the numerator minus the logarithm of the

denominator. Thus such combinations easily can be made linear as a part of

the logarithmic transformation.

In connection with the vheat model, a logarithmic relation for equation

{k), which deals with the demand for wheat for feeding, appears to prevail.

However, this can be approximated by a linear relationship except in those

years for which the spread between the price of wheat and com is negative or

exceeds about ko cents per 60-pound bushel. This price spread exceeded ¿0

cents in 3 of the years used in fitting the equations. Use for feed in these

years was considerably higher than would have been expected if based on a

linear relation due to minimum requirements for wheat in poultry rations. A

scatter diagram was made of this relationship and a free-hand line drawn. Use

of wheat for feed in these years was read from the line and exports were

increased by the difference between the quantity fed and the quantity indicated for feeding from the line. Thus the data used for fitting were trçmsformed in such a way as to be consistent with a set of linear relations. A

similar adjustment in the reverse direction is reqtiired when this relationship is used for analytical purposes. This adjustment is described in detail

in the appendix of Meinken*s bxilletin.

A 9-equation Model of the Demand for Dairy Products.--Rojko (17) describes

a number of S-^quation models of the demand structure for dairy products, including one for the post-World War II years. The following is a natural extension of his post-World War II model. As each regression coefficient relates to

only a single variable, the variables are written down directly as Y's (endogenous) and Z's (predetermined). In the following listing, only enough information regarding the variables is given to indicate in a general way their

economic meaning.

The following variables are involved in the model:

Yi - Quantity of milk used for fluid milk and cream

Y2 - Price of fluid milk and cream

Y3 - Quantity of milk used for butter

Y4 - Price of butter

Y5 - Quantity of milk used for American cheese

Yg - Price of American cheese

Y7 - Quantity of milk used for manufactured products other than butter and

American cheese

YQ - Price of manufactured products other than butter and American cheese

Yo - Quantity of margarine produced or sold

Zi - Disposable consumer income ^ 4i

'Z2 - Price of margarine, which is assumed to be determined by the fats and oils

economy

Z3 - Total quantity of milk available for consumption

Zi^ - Price of meat, pooiltry, and eggs
70

The following equations are used. In each case, the variable on the left

is expressed as a linear function of those included in the parentiiesis on the

right. To save space, full detail for the equations is omitted.

Yl

Y6

Y?

Y8

= f (Y2.

= f (zi.

= f Yl+

= f(Zi.

= f(Y6.

= f(Zl.

= f(Y8.

= f(Zl;

= f(YU,

Zl)

Z2»

Zl^

Z2,

Zl>

Z2^

Zl)

Z2^

Z3) 4)

Z3)

Z3)

Z3) 4)

= Yl + Y3 + ^5 + Y7

(1)

(2)

(3)

ih)

(5)

(6)

(7)

(8)

(9)

(10)

The economic reasoning behind these equations is discussed in detail in the

article by Rojko (17) •

This model has 10 equations and 9 endogenous variables. The 10 equations

can be reduced to 9 by substituting the algebraic value of any one of the Y's

in equation (10), for example Yi, for its value in equation (l). This in no vay

affects the equation as these variables are algebraic equivalents. The chief

importance of equation (10) is to remind the analyst that he cannot express his

other equations as linear in the logarithms of the variables if all equations

are to be consistent with respect to linearity.

In fitting these equations, Zj^ is omitted from M^^ ^o^ ^^'^ same reason that

time \ms omitted from IH^^ for the model relating to wheat. That is, Z¡^ is only

an imprecise measure of prices of itons that compete with American cheese as

alternative sources of protein.

As each of the equations (2), {k), (6), and (8) involves only a single

endogenous variable, they can be fitted directly by least squares. But as

each involves all of the predeteroiined variables included in Vlzzf i°^ch ccmputational time can be saved by using the method described in succeeding paragraphs* Equation (10) need not be fitted. Each of the remaining equations is

overidentified; after the computations described below are conpleted, they are

handled by the methods outlined in table 11.

In obtaining the moments, a symmetrical matrix of order 3^-9

plus a check sum. (Moments involving ^k are computed separately. )

12 is used,

The forward

Doolittle solution is ^sed to obtain (l) Mil i^V making use of an identity

matrix) and (2) My^zMzzMzy* as a composite operation. The upper section of the

forward Doolittle solution, in outline form, is as follows. Note that only a

single over all check sum is used.

M.zz «zy

Zl Zg Z3 n ya ... 79 Il 12 I3 z

21

Z2

^3 ^

100

010

001
71

The items from the last two lines in each section of the M and the I

portions of this operation (see p. 32) are transferred to another"^sheet, with

check Slims computed in the following way:

y^ Y2 2 y^ yi^ ^5 ^6 2 2^7 ye 2 y9 yi^ s ^1 \ h

The 5 ¿ -'^zz *^zy* required - for the 5 equations - to be fitted by. the limited

information approach are obtained as described on p. 32. Use of the several

check s\ms provides a complete check on the operation. MJ^ likewise is obtained

in the way described on p. 2$. The remaining operations for the overidentified

equations are the same as for any system.

For each of the equations to be fitted by least squares, we first obtain

b' = M^gM^yíe- For each equation, Mgy* is a vector, as only 1 y* is involved.

Since j^-1 is symmetrical, this miiltiplication can be carried out either as a

row-by-colxamn or as a column-by-column product; the column-by-colvonn product

( see p • 101) may be preferred as a matter of convenience and the checks shown

in the outline given below are designed for a multiplication performed in this

way. The missing elements in M^^ should be filled in before performing the multiplication. The sum of the 3 h^j in the b' vector should approximately equal

the sum of the products obtained by making use of the check sum coluimi. This

checks the computation. This computation in outline form is as follows:

Mzz M,zy*

The elements of the b* vector are the respective highest-order partial regression coefficients for the equation that involves y^.

These computations for the h equations to be fitted by least sqiiares can

be systematized in the following way:

\ ^2 ^3 y2 ^h ^6 ^8

^'

b^ t)6 bg

In this computation, the product M^^y* can be obtained '^^'Tf'-fl^lJZÂT

equation in turn. Here the Mzy* aM b'i sections each °°^ff ^^^^.f^^fthe

column vectors. Each set of computations is the same as that described in the

preceding paragraph.
- 72 -

The remaining statistical coefficients for these «J"»*^«'^« «^«¿«í* ^J"

tained, again making use of a composite set of computations. To fill in the

first two columns in the following outline, use is made of values obtained in

fitting other equations by the limited information approach. For exemple,

Myiyi for Yo is the element in the lower right comer of My*y» for equation

(ly, and MyTîMi^zyi for yg is the element in the lower right comer of

Myljííi^z/ for eqStion (l). A similar situation holds for y„ and the matrices

for equation (3), y6 and the matrices for equation (5), and yß and the matrices

for equation (7). N, the number of observations in the sample, and N ,. the

number of observations minus the number of variables, are the same for each

equation. For this example, N» = N - 1*. The outline below gives the value of

the multiple coefficient of determination and the standard error of estimate

for each equation. A check on the copying of data in the first two columns is

given by computing the items in column (4) and checking them against the appropriate elements of Wy«^ for each equation. These should check exactly. The

other items are checiea by recomputation.

""i ;

(1)

Vi

(2)

'Viz^^z'^zyi

2 (3)í2) «i = -tl- wyiVi

= (1) - (2)

^ UN'

(6>

y2

yu

y6

ya

The standard errors of the regression coefficients for each equation are

obtained by making use of the following outline. The items in column (l) are

obtained by multiplying by N the corresponding items in column (5) of the preceding outline. If desired^ the diagonal elements tXKm M^| can be written

above the box heads so BA to be conveniently available in performing these

operations. The diagonal elements from M^J are multiplia, respectively, by

the items in column (l), and the square root of the product is obtained. The

items in column (3) are the standard errors of the highest order partial regression coefficients between yi and zj^; the items in column (5) are the

standard errors of the regression coefficients between y^ and zp, and so forth.

The check is one of recomputation.

^2

ys

^'

ZggrVyiT^

= NsJ

(1)

^ii X (1)

Value

J[2I_

Square

root

(?).

.-.

-1

Z22 X (1)

Value Square

root

(5)

-1 Z33X (1)

Value Square

root

f7)
- 73 -

The recession coefficients and their standard errors and the standard

error of estiinate must be deadjusted and constant terns obtained for Se^ua

tions. The procedure for each equation is exactly the same as shovn S the

lower section of table 3. When this approach is used, coefficients of partial

détermination cannot be obtained conveniently. pari^iaj.

The method of obtaining these coefficients from camputations used in

connection with fitting equations by the limited information approach is

derived from material given in Hildreth and Jarrett (12, pp. II17.151). it

should be remembered that this method can be used directly only when M is

the same for both sets of equations. ^^

A Partially-Reduced Form Model for Feeds.—The system of equations

described in this section was developed by Gordon King, Agricxiltural Marketing Service, united States Department of Agricultxire, to study effects of the

price support program for feed grains on prices and use for feeding of feed

concentrates. As in the preceding example, only enough information regarding

the variables is given to indicate in a general way their economic meaning.

The following «idogenous variables are involved in the model:

P- - Price of feed grains that actually prevailed

Pj^ - Price of high protein feeds

Pj^ - Price of feed grains that would have prevailed had there been no price

support program for feed grains

^ - Q\iantity of feed grains going under loan and remaining under loan at the

end of the marketing period plus grain delivered under purchase agreements

The following predetermined variables are involved in the model:

Q„ - Supply of feed grains available for feeding during the marketing period

Qjj - Quantity of high protein feeds used for feed

Pg - Support price for com (In years when no support program was in operation, this variable was set at such a level that no grain would normally

have gone under loan.)

A - Grain-consuming f^r\imfL\ units fed during the year

P^ - Price of livestock and livestock products

T - Time

In a more complete model, A and P^^, and possibly Qj^, would have been treated

as endogenous variables.
-71* -

The following structural equations are involved. In each case, the variable on the left is expressed as a linear function of those included in the

parenthesis on the right. To save space, full detail for the equations is

cMl't'ted«

Pf = f (Qg, Qh» A, Pi) (1)

% = f(Pf-Ps) Í2)

Pg = f(Qg-Qi, %, A, Pi) (3)

P^^ = f(Qg-Qi, Qh^ A^ ^1> ^) (^)

As no data are available for Pf for years in which a support program was

in operation, the right-hand side of equation (l) is substituted for Pf in equation (2). The resulting equation, shown here as equation (5), is called a

partially-reduced form equation.

Ql = f(Qg, Qh> A, Pi, -Ps) (5)

Equations (3), (k), and (5) are used in fitting. The following variables

are used: Yi = Pg, Y2 = Qg-Qi, Y3 = Ph> ^h = ^If ^1 = %> ^ = %' ^3 =

-Pg, 7.l^, = A, Zc = Pi, Z5 = T. The following tabulation shows the Y* and Z*

involved in each equation:

Equation Y^ Z*

(3) Yl^ Y2 Z2, Z4, Z5

i^) Y3, Y2 Zg, Z]^, Z5, Z5

(5) YU Zi, Zg, Z3, Zi^, Z5

As equation (5) involves only a single endogenous variable, it can be fitted directly by least squares. As in the preceding example, however, computational time is saved by handling all computations as a composite unit« The

counting rule does not apply strictly to equations (3) and (h) because each

contains a composite variable, but if the rule were applied to the equations

when they are expressed in Y's and Z*s it would suggest that they are overidentified, which is correct. In the model described in succeeding paragraphs,

Zg (which represents a time trend) is omitted from M • In other fonnulations,

however, Zg migjit be included in M , as the underlying time trend is believed

to be nearly linear over the period included in the analysis.

In setting up the forward Doolittle solution, we consider three sorts of

computations: Those that relate to obtaining (a) M ^f. M"^ -^ for equations

(3) and (i^); (b) M'^ for equation (5); and (c) M¿iJ fo?%|uations (3) and {k).

As in the preceding example, the independent variables in equation (5) are the

same as the predetermined variables in M^j; for the system. Also 3 of the predetermined variables in equations (3) and {k) are identical. These 3 variables

are listed first in the forward Doolittle solution and then the others that are
75

involved in Mzj. All U y's axe next listed, and then the 5-variable identity

matrix. As in the preceding example, a single check sum is used. The upper

section of the forward Doolittle solution has the following form:

Mzz Mzy

yi 72 y^ y^ I2 ^1 5 ^1 ^3

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

By ccsnputing a new check sum for the last two rows of each section of the

forward Doolittle solution, K^z can "be obtained and checked directly from the

I part of the solution. The inverse should be written down on a second sheet,

with space at the right for additional columns. Missing elements from this

symmetrical matrix should be filled in. The vector Mzyi^ can be written in an

adjacent column and the highest order b's obtained directly by perfonning a

column-by-colxBQin multiplication as described on p. 71. As only a single equation is to be conrputed by least squares, R^ and s can be conç)uted directly by

use of the formulas implied by the table on p.72. Myi^zM^^zyn^ can be obtained

by cvmnilating the products from the last two rows of each section of the yi^

column of the forward Doolittle solution in the way described on p. 32. Squared

values of the standard errors of the respective regression coefficients are

obtained by multiplying the diagonal elements of M-¿ by Ns'^. These can be

written down directly in a column next to the b's, and the square roots inserted in the next colxann. Except for deadJusting the coefficients, this completes the coinç)utations for the least squares equation.

Because of the nature of the y's involved in equations (3) and (k), it is

efficient to compute the entire VLy^l^zy n»*rix for y^, 72, eaà Vy This can

be done by placing a strip of paper over the yi^ column and computing a new check

sum for yi, ya and yo for the last two rows of each section of the forward

Doolittle solution. The triple-product matrix then is obtained and checked as

described on p. 32. At the same time, computations involved in obtaining

Mä^* for these equations can be completed. This can be done by placing strips

of ¿aper over the forward Doolittle solution in such a way that columns for

» T -u- v^v VI T-^ and the orieinal 2, and rows and sections for z-^

äd :i'Zi'Z%/â: ^¿ta^^or^lô s^ould\e inserted in place^of those o^ifn^^

shown^for zn, and separate check sums are computed for the z's and the I s that

appear on thé modified table. By completing t^^^^^^^^^f^.£P^^jf°Í^Í°^ J^e

and the new check sums, infomation needed to obtain the ^z*z*J±.^^Jft' f^_

eleiDents of the inverse for equation (3) can be obtained as J P«f* °^.*^^Jr

putations involved in obtaining the elements of the inverse for equation (4).

Se ¿l¿ on the operation for equation {h) automatically checks the computation for equation (3).
. 76 -

These operations having been ccinpleted, the usual computations for an overidentified equation are carried out for each equation as shown in table 11« If

desired, M"¿ ^ can be recorded directly in the space provided for step four in

each tabled ^In carrying out the computations for eq\iation (k), it should be

remembered that the y variables are listed in the order y^^ 72*

TESTS TO BE MADE AFTER CCMPLETING THE ANALYSIS

Poxir tests are commonly made by an analyst after he has run any correlation or regression study, (l) The signs and relative magnitudes of the regression coefficients are checked to make sure that they are consistent with the

analyst's expectations based on economic theory and knowledge of the field to

which the study relates. (2) The coefficients are compared with their standard

errors to see whether they differ from zero or some other value by a statistically significant amount. (3) The unexplained residuals are analyzed, perhaps by plotting them in a time serieô, to see whether they are essentially

random, (k) If the study is an important one, the analyst makes up charts

that indicate the degree of psurtial correlation and studies these to learn

how particular obse37vations affect the analysis and to see if the type of

function fitted appears to be consistent with the data. Tests of this sort

as applied to least squares analyses are described briefly by Foote and Fox

(9^ PP* 25-36). The analysis also would be checked to see how usefxil it is

for prediction for observations not included in the study.

The first of these checks is equally important when working with systCTis

of equations. Before using the system for analytical puiposes, the analyst

must make up his mind whether any gross inconsistencies render the results

entirely useless for such purposes. In some situations, he may be content

to replace certain coefficients with others that on a Judgment basis appear

more logical. In so doing, however, calculated standard errors for other

coefficients lose their validity. As noted on p. 1, t-tests of statistical

coefficients to indicate whether they differ significantly from some assumed

value, normally zero, may not be valid when the analyst is working with the

kind of time series normally used in economic research; but the ratio of the

estimate to its calculated standard error ordinarily should give a rough idea

of the probable sampling variability of the estimate. A test that indicates

the probability that the unexplained residuals are serially correlated is

described in the following section. This test was designed for use with equations fitted by least squares but on intuitive grounds it appears to be fairly

applicable for use with equations fitted by the limited information approach.

Following that, tests are described that indicated whether particular equations

fitted by the limited information approach are, in fact, probably overidentified or Just identified. Graphic studies of the individual equations us\ially

are not made when we are working with systems of equations. Detailed studies

of the usefulness of the system for projecting trends or analyzing alternative

policy questions represent a partial substitute. Formal tests of significance

of the residuals to check whether the estimated relations fit data outside the

sample period as well as shoiild be expected from their fit during the sançle

period are described by Hildreth and Jarrett (12, pp. 119-129). These tests

are not described here; they are only a "crude makeshift" as applied to equations fitted by the limited information approach.
- 77 -

Serial Correlation in the Residuals

Durbln and Watson (6) developed a method by which the unexplained residuals

from an equation fitted by least squares can be tested to see 1? successive

values are correlated. 2U/ This sort of correlation is commonly referred to

as serial' correlation. Use of the limits shown in table Ik must be regarded

as approximate when this test is applied to residuals from equations fitted by

the limited infomation 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

d« = t=2

N 2

S (^t - dt-l)

^ 2

t=l ^

where ij. is the unexplained residual for observation t. If gaps occur in the

data, tne number of observations that enter into the numerator are reduced by

one for each gap. This statistic can be computed for equations fitted by least

squares or for each structural equation in a system of sim\iltaneous equations.

Table lU is used to obtain upper and lower bounds for the critical values

for a 2-talled test at a 5-percent probability level. N is the nimiber of

observations in the analysis and k' is the number of Independent or predetermined variables in the equation. 25/ We compute d' and ^ - d' and find the

appropriate value for d^ and du in the table. The two computed values relate

to the two tails of the sampling distribution, d* relating to positive serial

correlation and if-d', to negative serial correlation. If d* or 4-d' is less

than dr ^ we assume that the residuals may be serially correlated, either positively or negatively. If both d' and 4-d' are greater than d^j^ ve assume that

there is no serial correlation. If neither of the computed values is less than

dL, but one of them lies between dx, and à\j, the test Is inconclusive.

2k/ Klein (13, pp. 89-9O) suggests, in this connection, use of a test

developed by Hart and von Neumann (11) in 19ÍI-2. This test, however, relates

to serial correlation in a variable rather than to serial correlation in the

unexplained residuals as such. The Durbln-Watson test was not published until

1951. On intuitive grounds, it appears to be preferred as a test for residuals.

25/ This procedxrre is suggested by Hlldreth and Jarrett (12, p. 78) based on

an analogy between limited information and least squares estimates. Professor

Durbln, in a discussion of a paper by Morgan and Corlett in the Journal of the

Royal Statistical Society, 0^:355, suggests instead use for N of the number of

observations minus the number of dependent variables less one. He says, For

the overldentlfled case it seems possible to obtain inequalities for the significance points".
. 78 -

Table ik.- Significance points of CIL and dy

N

k'«l

^

k'=2

^

k'«3

h %

k'=h

%

k'=5

^ ^

15

16

17

18

19

20

23.

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

ko

45

50

55

60

65

70

75

80

85

90

95

100

0.95

0.98

1.01

1.03

1.06

1.08

1.10

1.12

1.14

1.16

1.18

1.19

1.21

1.22

1.24

1.25

1.26

1.27

1.28

1.29

1.30

1.31

1.32

1.33

1.34

1.35

1.39

1.42

1.45

1.47

1.49

1.51

1.53

1.54

1.56

1.57

1.58

1.59

1.23

1.24

1.25

1.26

1.28

1.28

1.30

1.31

1.32

1.33

1.34

1.35

1.36

1.37

1.38

1.38

1.39

1.4o

1.41

1.41

1.42

1.43

1.43

1.44

1.44

1.45

1.48

1.50

1.52

1.54

1.55

1.57

1.58

1.59

1.60

1.61

1.62

1.63

0.83

0.86

0.90

0.93

0.96

0.99

1.01

1.04

1.06

1.08

1.10

1.12

1.13

1.15

1.17

1.18

1.20

1.21

1.22

1.24

1.25

1.26

1.27

1.28

1.29

1.30

1.34

1.38

1.41

1.44

1.46

1.48

1.50

1.52

1.53

1.55

1.56

1.57

1.40

1.40

1.40

1.40

1.41

1.41

1.41

1.42

1.42

1.43

1.43

1.44

1.44

1.45

1.45

1.46

1.47

1.47

1.48

1.48

1.48

1.49

1.49

1.50

1.50

1.51

1.53

1.54

1.56

1.57

1.59

1.60

1.61

1.62

1.63

1.64

1.65

1.65

0.71

0.75

0.79

0.82

0.86

0.89

0.92

0.95

0.97

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.13

1.15

1.16

1.17

1.19

1.20

1.21

1.23

1.24

1.25

1.30

1.34

1.37

1.4o

1.43

1.45

1.47

1.49

1.51

1.53

1.54

1.55

1.61 0.59

1.59 0.64

1.58 0.68

1.56 0.72

1.55 0.76

1.55 0.79

1.54 0.83

1.54 0.86

1.54 0.89

1.54 0.91

1.54 0.94

1.54 0.96

1.54 0.99

1.54 1.01

1.54 1.03

1.54 1.05

1.55 1.07

1.55 1.08

1.55 1.10

1.55 1.12

1.55 1.13

1.56 1.15

1.56 1.16

1.56 1.17

1.56 1.19

1.57 1.20

1.58 1.25

1.59 1.30

1.60 1.33

1.61 1.37

1.62 1.4o

1.63 1.42

1.64 l.í^5

1.65 1.47

1.65 1.49

1.66 1.50

1.67 1.52

1.67 1.53

1.84

1.80

1.77

1.74

1.72

1.70

1.69

1.68

1.67

1.66

1.65

1.65

1.64

1,64

1.63

1.63

1.63

1.63

1.63

1.63

1.63

1.63

1.62

1.62

1.63

1.63

1.63

1.64

1.64

1.65

1.66

1.66

1.67

1.67

1.68

1.69

1.69

1.70

0.48

0.53

0.57

0.62

0.66

0.70

0.73

0.77

0.80

0.83

0.86

0.88

0.91

0.93

0.96

0.98

1.00

1.02

1.04

1.06

1.07

1.09

1.10

1.12

1.13

1.15

1.21

1.26

1.30

1.33

1.36

1.39

1.42

1.44

1.46

1.48

1.50

1.51

2.09

2.03

1.98

1.93

1.90

1.87

1.84

1.82

1.80

1.79

1.77

1.76

1.75

1.74

1.73

1.73

1.72

1.71

1.71

1.70

1.70

1.70

1.70

1.70

1.69

1.69

1.69

1.69

1.69

I.Ó9

1.69

1.70

1.70

1.70

1.71

1.71

1.71

1.72

Prom Durbin and Watson (< . p. 174) R*.T^r./vi»«^ v ^ .

vjL* P' J-C*}' Reproduced by permission of Biometrika.
- 79 -

Overldentlfyíng Restrictions

Anderson and Rubin (1, p. 56) developed a test designed to determine the

.alidity of assimptions that certain variables do not appear in a particular

equation, given the validity of the remaining specifications regarding the

system. The number of variables in a system that are omitted from a particular

equation determine the degree of identification; hence this is known as a test

of overidentifying restrictions. As noted by Hildreth and Jarrett (12, p. 79),

in many applications to economics the investigator is likely to have better

grounds for the specification of vhich variables enter particular equations

than for other aspects of the system, such as, for example, whether the relations are linear in actual data or in logarithms. In such cases the interpretation of the test is subject to doubt, but it remains true that an extreme

value for the test statistic ig an indication of difficulty somewhere in the

statistical specification used. Because of the nature of the test, it is used

only for overidentified equations.

The test is carried out in 2 stages. First we determine \diether it is

reasonable to conclude that the equation is overidentified. We do this

by postulating a null hypothesis that the equation is underidentified

or just identified. If this mxLl hypothesis is not rejected, we make a second

test to determine whether it is reasonable to conclude that the equation is just

identified. The first test can be made easily from data computed in the fitting

process; the second test requires a considerable number of additional computations whenever the number of y*^ exceeds 2.

First Phase.--To cairy out the first phase of the test, we compute the

following:

(2.3026)N logj^Q (1 + lA )

where N is the sample size, and X was computed in section (9) (see p. 53).

This statistic follows approximately a X^ distribution with H - h - g + 1

degrees of freedom, where H is the total nmber of predetermined variables in

the system, h is the number of predetemined variables in the equation, and g

is the number of endogenous variables in the equation. If the computed test

statistic exceeds the critical value associated with the number of desees of

freedom at the desired probability level, we have reason to conclude that the

equation is overidentified. Tables of expected values of X^ when the assumed

hypothesis is true are given in most textbooks on statistics and elsewhere.

Probabilities of 5 percent or 1 percent are those commonly accepted by statisticians .

second Phase.-If the computed test statistic for the first phase is less

than the critical value, we proceed with the second phase. The ^^lue of X

used in the first stage'of the test is the largest root of ^^^^^^^^^^^^^

For the second stage of the test, we need to determine ^;^/^;°^t.ÍrSSix to

of the same equation. This can be done by transfoiming f ^/^f^^^^^íi

the w described below and then getting the largest ^°°^ ^f^^^-^J^'f"^

matrix by the iterative process described on p. 55- H°^f^^'Jf ? ' tíoí!

second iLgest root can be detemined by a «^í^^^Í/^I^^Í^Í °^ ^'^^^gst ro^t

The paragraphs that follow indicate the way in which the second largest^oo

is deteSied for a g of specified size. Absolute values of the roots are

used in each phase.
. 8o .

(1) g = 2: The second largest root is given directly by the formula

^2 =, P2 VP2-^PIP.^

2P3

where the p*s are defined as in step (9) (see p. 38). For equation (2.1) of

the lumber problem, application of this formula gives a value for the second

largest root of 0.4550.

(2) g > 3: Use is made of results obtained by the iterative process

described on p. 55. The example used for illustrative purposes in table I3

is that of equation (2.1) of the lumber problem, for ^^Thich g = 2. This example also will be used in this section, although the reader should note that in

actual practice this approach is used only when g i 3- Modifications required

for a larger nimber of y* are obvious.

The first step is to examine the last column for q. shown in table I3 to

determine the row in which the 1 occurs. For this example, this is found in

col\m3n q\Q and the 1 occurs in the first row. This tells us that in the next

step we make use of -che first columm of the A' matrix originally used in table

13. Had the 1 appeared in the second row, we would have used the second column

from the A' matrix. This column vector is multiplied by the transpose of the

column vector for q in the last column of table 13. For this example, the

computation is as follows:

1.7305

-8.6573

Tl -0.8555] 1.7305

-8.6573

.l.lrô05

7.^66

The resulting product matrix then is subtracted from A* to give V.

computation is as follows:

The

1.7305

-8.6573

-1.0912

7.8617

1.7305

-8.6573

-I.U805 0.3893

.4550

A partial check on the computation is given by the fact that a column of zeros

always must be found in V in the coliomn corresponding to the row in which the

1 occurred in the last column of table I3.

We now obtain V» by interchanging rows and columns of V and operate on V*

in table 13 in exactly the same way as we did on the A' matrix. When g = 2,

only a single iteration is required. The second column for Q is as follows:

0.3893

.^550
- 81 .

This indicates that the second largest root equals 0.if550, which is the same as

obtained by use of the formula.

To carry out the second phase of the test, we cmpute the following:

(2.3026)N log-LQÍ^i + X2 - 2)

where N is the sample size, X^ is the largest root found in section (9) and

Xp is the second largest root found by the computation described above. This

statistic follows approximately aX distribution with H - h - g + 2 degrees

of freedcm. If the computed test statistic exceeds the critical value associated with the number of degrees of freedom at the desired probability level

we have reason to conclude that the equation is just identified. If the computed test statistic is less than this value, the test suggests that the equation is underidentified and little confidence can be placed in the values

obtained for the coefficients.

USE OF SIMULTANEOUS EQUATION MODELS FOR ANALYTICAL PURPOSES

After the coefficients in a complete system of equations have been obtained

by appropriate statistical means, the analyst will in general wish to use the

system for predicting trends or for other analytical pxirposes. For this use,

the approach is different from that employed when a single equation is used,

although the same general steps are involved. In this section, we first discuss an approach that can be used for any system of equations, and then

describe some computational shortcuts that frequently can be used. In the

general approach, the n equations in n endogenous variables must be solved for

each period for which an estimate is desired, making use of the coefficients

estimated by the fitting process. In this connection, the reader sho\ild

remember that oa<i ol the reasons for using a system of simultaneous equations

is to ensure that estimates of the endogenous variables made from the system

are mutually consistent. The 6-equation model for wheat serves as a convenient

example for the general solution even though this approach would not be used in

connection with actual computations (see p. 8^). We first consider a general

solution for any system of simultaneous equations.

Solution of Simultaneous Equations

Consider a system of equations of the following form:

^11% "* ^12^2 + • • • ^In^ = ^1

bi oX-i + bpo^p + • • • bppXy^ = a.2

^nl^l + ^n2^2 + • • • ^nn^ "^ ®n

Since we have n equations in n unknowns, in general a unique solution to this

system of equations will exist.

If B is an n by n matrix of the b^j, X' is a column vector of the Xj, and

A* is a column vector of the aj^, this system of equations can be written in the

following matrix notation, BX* = A'. The truth of this is immediately evident

if we consider any element in the product vector.
. 82 -

We now multiply each side of the matrix equation by B . This gives

B - hsX' = B'-^A'

or B"-^A* = X'

Translating the matrix notation back into terms of ordinary algebra, we have

the following system of equations that can be used to solve for the X's:

^11^1 +^12^ + ••• ^In^ = ^1

^21^1 *" ^22^ *"••• ^2n^ = ^

For most systems of equations, the b's are given and we wish to determine the

X's associated with a particular set of a's. This can be done easily by making

use of the second system of equations based on the elements of the inverse

matrix.

Use of Reduced Form Equations

Let us rewrite the structural equations for the model for wh^at (see p. 65)

as shown below. In doing so, we assume that a value for P is obtained directly

from equation (8) in the same way as for any least squares analysis, and that

this is multiplied by the appropriate value of k to give P'. This variable is

then treated in the equations used for analytical purposes as though it were

predetermined.

Ch/L - ^31^d = ^3 •" ^31^ + ^32(I>/L) + b33T + bgi^M = A3 (3.I)

Of - bi^iP¿ = ai^ - bi^iP^ + loi^^ = Ai^ (k)

^s" ^6l(Pd/ld) = a6 + b62F = A5 (6)

Ch -^f-^^^e-^ ^s = S^ = A7 (7)

'e + ^51^d = ^5 + ^5l(^w - N) = A5 (5)

The A*s equal the sum of the predetermined variables in each equation multiplied

by their appropriate coefficients plus the constant term. In general, the A's

differ in each period.

Since we have 5 equations in 5 unknowns, ordinarily we can solve for the

values of the 5 endogenous variables in any given year by making use of the

inverse of the matrix of coefficients of the endogenous variables and an appropriate vector of the A*s. The matrix of coefficients of the endogenous variables equals the following:
o¿ -

"l/L 0 0 0

-^31

0 1 0 0 -Hl

0 0 1 0

^51

0 0 0 1

-^61/1

1 1 1 1 0

This is a nonsymmetrical matrix. Methods of inversion for matrices of order

greater then 2 described so far in this handbook apply only to symmetric^

of thxs matrix for this example because ve can derive the needed formulas

directly from the equations themselves. Methods that could be used to invert

this matrix are described in the appendix, p. 98.

This matrix illustrates one objection to the use of nonlinear endogenous

variables in connection with systems of equations. All of the elements of the

matrix are constants except for the coefficients that relate to the 2 nonlinear

variables. If the inverse of the matrix of coefficients had to be used for

this system, a new inverse for each year would be required. Had we used all

linear variables, or the linear approximation for the nonlinear variables

described on p. 68, the matrix of coefficients for the (linearized) endogenous

variables would have consisted only of constants, and only a single inversion

vould be required. Nonlinear predetemined variables present no problem, as

the A's in any case are recomputed for each year.

Given the inverse of this matrix, each endogenous variable can be expressed

as a linear function of the A*s mailtiplied by appropriate elements of the inverse. If D represen-fes the matrix of coefficients, these equations have the

following form:

Ch = ^11^3 + ^l2^k + ^13^5 + ¿ÍÍA6 + ^il^^

etc.

In these, each endogenous variable is in effect expressed as a function of all

of the predetermined variables in the system. These are Imown as the reduced

form equations. They represent a convenient way to express the equations to

readers not acquainted with matrix notation, particularly when the elements of

the inverse are replaced by a set of actual numbers.

A new set of reduced form equations must be algebraically determined whenever changes are made in the basic structure of the model, as, for example,

when an endogenous price is replaced by a predetermined one based on a price

support program.
- Qk -

Sligplífled Formulas for the Wheat

aod Dairy Models

In the case of hoth the wheat and dairy models^ slisple formulas for

simultaneously projecting the endogenous veurlahles can he developed directly

from the structural eqiiatlons as such. In some cases^ slmlleu: methods can he

xised to reduce the size of the matrix of coefficients to be Inverted, To

Illustrate this point, a modified version of the wheat model Is presented In

the last part of this section. Frequently this general approach can be used

In working with the kinds of eqiiatlons that are common In economic analysis.

We first consider the eq-uatlons for wheat presented on p. 82, after substituting the A*s for the terms that involve only predetermined variables. If

each term in equation (3*l) is multiplied by L, the following system of equations is given:

-b3iLP¿ = A3L

Cf - Hl^á = AU

Ce + ^51^d = ^5

Cs - ^6l(Vld) = A6

Ch + Cf + Ce + Cg = Ay

If each of the first k equations is svibtracted from the last equation, all of

the endogenous variables except P^ are eliminated. P^ then can be expressed

as the following function of L, I¿, the A's, and the b's:

^ 1*31 + Hi - ^51 + (t>6l/Id)

Transposition of the first 4 equations yields the following:

Ch = Li^SlPd + A3)

Cf = bl^iPd + AV

Ce = -^5iPd + A5

•^s = (Wld)l'd + A6

S^cZef i^delal^ït SiHou "^ ^ ^^^^^ ^ -*-^ P-^-^^- ^«
-85 -

Use of the structural equations in the dairy model for analytical purposes is even sinrpler. Here we estimate Y2> YI4., Yg, and Y8 directly from the

given values of the predetermined variables for the particular year, Tnav-tng

use of equations (2), {k), (6), and (8), When these values and those for the

predetermined variables are substituted in the remaining structural equations,

values for the remaining Y's are given directly. Equation (lO) provides a

partial check on the coinputations •

In connection with analyses of the effect of alternative support programs

on prices for wheat and com, it is useful to modify the structural equations

for wheat to include the price of com aa an endogenous variable. The way in

which the additional equation is derived need not concern us here. This

exBDiple, however, provides a useful addition to those discussed in preceding

paragraphs in indicating useful techniques for deriving formulas for projection. In this example we are concerned with the following structural equations,

where the A*s have the same meaning as in the preceding examples:

Ch - "bsiLPd = A3L

Cf - Hi^à - ^kl^c = ^UCe + ^51^d 'H

Cs - *6l(PcLAd) = A6

Ch + Cf + Ce + Cg = ^7

^8lCf + Pe =Ae

When the first, third, and fourth equations are subtracted from the fifth

equation and the terms are reairanged, the following equations are left:

Cf + (bgiL - bji + b5i/l¿)P¿ = Ay- A3L - A5 - A^

teiCf + Pc = Ae

If, for this reduced set of eqiaations, the last equation is multiplied

by -bi^i and subtracted from the first equation, the following system is given:

(1 + bi^ib8i)Cf - bi^i^à = A4 + bi^iAs

Cf . + (bjiL - bjj^ + b5i/ld)Pà »Ay - A3L - A5 - Aö

These 2 equations in 2 unknown can be solved easily by direct substitution or

by inversion of the 2 by 2 matrix of coefficients of the endogenous variables.

(See p. 2d)
. 86 .

COEFFICIENTS OF CORRELATION FOR SYSTEMS OF EQUATIONS

A number of correlation concepts can be considered in connection with

systems of equations. Two sorts of correlation coefficients might be of

interest: (l) Those that indicate the relative accuracy with which the individual endogenous variables can be estimated and (2) those that indicate

the closeness of fit of the entire system. Methods for estimating each of

these are discussed in this section.

Coefficients that are similar to miiltiple correlation coefficients or

their squared values, commonly called coefficients of multiple determination,

can be computed for the years included in the study by making use of the reduced form equations. (See p. 82.) For each year included in the analysis,

an estimate of the particular endogenoixs variable is obtained and compared

with the actual value for that year. If d equals the difference between

these 2 values, the coefficient of determination for the variable i is given

by:

R?i = 1 - ^mii

The N is required in the numerator of the last term because of the use of augmented moments. Considerable computation may be required to obtain coefficients

of this type as, for the wheat model for example, each of the reduced form equations involves 5 A's or (in effect) ik predetermined variables. For computational purposes, it might be desirable in some models to write the reduced form

equations in terms of the predetermined variables as such rather than in terms

of the A*s. For the wheat model, little would be gained by this procedure.

Measures of this sort, when multiplied by 100, show the percentage of

variation in each endogenous variable explained by all of the predetermined

variables in the system. In this sense, they are comparable to a multiple

coefficient of determination. They probably would not, however, be subject

to the same sampling distribution.

Theil (19, 20) has developed a statistic which he calls the coefficient

of simultaneous correlation, S^. This is analogous to a coefficient of multiple determination, R2, in that it indicates the percentage of variation in

the Joint distribution of the endogenous variables included in the system explained by all of the predetermined variables in the system. 26/ Theil has

shown that it can be consistently estimated from the saarple. If in connection

with fitting the equations the entire VLyz^zhky matrix was obtained, as for

the luniber problem, this coefficient is easily obtained in the following way:

26/ This coefficient was called to the authors' attention by Roy Radner of

the Cowles Foundation for Research in Economics. Theil suggests use of a

similar coefficient for each structural equation. This shows the percentage

of variation in the Joint distribution of the endogenous variables in that

equation explained by the predetermined variables in that equation. The

authors of this handbook believe this to be of less interest than the two

alternative coefficients discussed in the text.
-87 -

l«yy|

These 2 determinants always are symmetrical. If of order greater than 3, they

can be evaluated by the Doolittle method as shown in the appendix, p. 89! if

of order < 3, they can be evaluated by the methods discussed on p. 26, If, as

for the wheat and dairy models, only part of the My^M^zM^y matrix was obtained,

the rest of the matrix would need to be computed to obtain this coefficient.

LITERATURE CITED

(1) Anderson, T. W., and Rubin, Hsîrman

19^9« Estimation of the Parameters of a Single Equation in a Complete

System of Stochastic Equations* Annals Math. Statis. 20:46-63.

(2) Chernoff, Herman, and Divinsky, Nathan

1953» The Computation of Maximum-Likelihood Estimates of Linear

Structural Equations. In Hood, Wm. C, and Koopmans, Tjailing

C, ed.. Studies in Econometric Method, Cowles Commission for

Research in Economics Monogr. Ik, 324 pp., illus.

(3) , and Rubin, H.

1953. Asymptotic Properties of Limited-Information Estimates Under

Generalized Conditions. In Hood, Wm. C, and Koopmans,

Trailing C, ed.. Studies in Econometric Method, Cowles Commission for Research in Economics Monogr. ik, 324 pp., illus.

(4) Crout, P. D.

1941. A Short Method for Evaluating Determinants and Solving Systems

of Linear Equations with Real or Complex Coefficients.

Marchant Methods MM-1Ö2. Oakland, Calif.

(5) Doolittle M. H.

1878. Method Employed in the Solution of Normal Equations and the

Adjustment of Triangulation. U. S. Coast Guard and Geodetic

Survey Rpt., pp. 115-120.

(6) Durbin, J., and Watson, G. S.

1951. Testing for Serial Correlation in Least Squares Regression.

Biometrika. 38(1-2):159-177> illus.

(7) Dwyer, Paul S.

1951. Linear Computations. 344 pp. New York.

(8) Ezekiel, Mordecai

1941. Methods of Correlation Analysis. Ed. 2, 531 PP-> illus. New York.

(9) Foote, Richard J., and Fox, Karl A.

1954. Analytical Tools for Measuring Demand. U. S. Dept. Agr.

Agr. Handb. 64. 86 pp., illus.

(10) Gauss, Carl Friedrich

1873. Supplementum Theoria Combinationis Observatorium Erroribus

Minimis Obnoxiae, Werke, 4:31.

(11) Hart, B. I., and von Neumann, J.

1942. Tabulation of the Probabilities for the Ratio of the Mean Square

Difference to the Variance. Annals Math. Statis. 13:207-214.
. 88 -

(12) Hildreth, Clifford, and Jarrett, F. G. w n ^^

1955. A Statistical Study of Livestock Production and Marketing.

Cowles Commission for Research in Economics Monogr. 13,

156 pp., illus.

(13) KLein, Lawrence R.

1953. A Textbook of Econometrics. 355 PP*^ illus. Evanston, lililí) Kbopmans, Tjailing C.

1953. Identification Problems in EconcHnic Model Construction. In

Hood, Vfai. C-, and Kbopmans, Tjailing C, ed.. Studies In

Econcmetric Method, Cowles Commission for Research in Economics

Monogr. 1^,324 pp., illus.

(15) Lowenstein, Prank, and Simon, Martin S.

1954. Analyses of Factors that Affect Mill Consumption of Cotton in

the united States. Agr. Ecou. Research. 6:101-110, illxis.

(16) Meinken, Kenneth W.

1955. The Demand and Price Structure for Wheat. U. S. Dept. Agr.

Tech. Bui. 1136. (In press.)

(IT) Rojko, Anthony S.

1953. An Application of the use of Economic Models to the Dairy

Indxistry. Jour. Farm Econ. 35î83^-8^9> illus.

(18) Tintner, Gerhard

1952. Econometrics. 370 pp. Hew York.

(19) Theil, H.

19^3 Estimation and Simultaneous Correlation in Complete Equation

Systems. Netherlands Centraal Plan-Bureau. The Hague,

Netherlands. [Processed.]

(20)

195^* Estimation of Parameters of Econometric Models. Internatl.

Statis. Inst. Bui. 3^:122-128.

(21) Wagner, Harvey M.

195^1-. A Monte Carlo Study of Estimates of Simultaneous Linear

Structural Equations. Tech. Rpt. 12, Dept. Econ., Stanford

Univ. 56 pp. [Processed.]

(22) Waugh, Frederick V.

1935* A Simplified Method of Determining Multiple Regression Constants.

Amer. Statis. Asso. Jour. 30:69lf-700.

(23) __^^ , and Dwyer, Paul S.

19'<'5» Compact Computation of the Inverse of a Matrix. Annals Math.

Statis. 16:259-271.
.89

APPENDIX

Abbreviated Doollttle Method

The forvard solution of a method outlined by Doolittle (¿) is illustrated

in detail in tables 3 and 8. This scheme for inverting matrices and solving

systems of equations is based upon methods developed earlier by Gauss (lO).

This approach, as it is usually explained, applies only to symmetrical matrices.

In this section of the appendix, we apply it only to symmetrical matrices. In

the next section, however, we show how a modification of the same method applies

to the solution of any matrix, whether symmetrical or not.

Here we outline an abbreviated form of the forward solution of the Doolittle

method. In abbreviated or compact methods we do as much work as possible on the

calculating machine rather than on the worksheet. For example, in the method

outlined here, the sum of two or more products is cimiulated on the calculating

machine; only the cumulated sum is copied. This eliminates the recording of

many entries, and the possibility of making mistakes in the copying.

Outline 1 presents, in symbolic form, the abbreviated forward solution for

a matrix of order n. This method is applicable to symmetrical matrices only.

Methods for handling nonsymmetrical matrices are discussed on p. 9^.

Evaluation of Determ1na.nts.—The abbreviated forward solution can be used

to evaluate determinants. The value of a symmetrical determinant equals the

product of the diagonal •a* elements of the abbreviated forward solution. Referring to outline 1, the value of the determinant of the matrix of order n is

given by: «jj^ «22 ^3 ••• ^^m •

Inversion of Matrices.—To invert a symmetrical matrix, a worksheet is set

up in the fozm:

where A Is the matrix of order n to be inverted and I is the unit matrix of order n. An abbreviated forward solution is ccunried out on this augmented matrix,

and the inverse is obtained by cxmmilating the products of certain elements from

the I part of the forward solution as explained on p. 11.

Table 15 gives the abbreviated form of the coniputations involved in obtaining the inverse, or D, matrix shown in table 3. Note that the abbreviated form

contains only the last two rows of each section of the forward solution of

table 3. The abbreviated solution uses only one check column, that is, column

(U), headed Z. As many of the ccnrputations are carried directly in the machine,

it is better practice to compute one check for the entire row of the forward
- 90 -

Outline 1.- Al)l3reviated forward solution

Computations

Bow

(1) *n ^2 "13 • • • *ln *1S

(2) *22 *23 • • • *2n *22

(3) *33 • • • % *32

Instroctlons

(n)

(1)

(1")

(2«)

(2«)

(3')

(3»)

"11 ^^2 «13

(n«)

(n»)

12

22

13

23

3

23

33

n2

°12

In ^IS

nn

2n

2n

2S

22

3n 32

3n 32

a a

nn n2

n2

Bow

(1) - (n) ínter the matrix of order n«

The elements in the 2 colximn

are obtained by stimming across the

row, including those terms omitted

because of symmetry.

n

That is. a^2 = 2 a.j (i=l,2....n)

(1) ci^j =a^j (j-1.2....n,2)

Check: a^^= 20,^^ (j=l, 2,...n)

^^"^ ^Ij" <^j/aii(J=2,...n,2)

Check: P^^= 2p^j + 1 (j=2....n)

('•^^2r"2J-Pl2^j(J^--'^-^>

Check: a^^ = 2a (j=2.,..n)

(2«) P^j = «gj /o^22 (J"3*-«*»»2)

Check: p^^ = SPgj + 1 (J=3..*.n)

(3')<x^=a -p a -9 a

3J 3j "^13 Ij ^23 2J

(j=3,...n.2)

Check; a^2 = 2a^^ (j=3»...n)

(3") p3j= a3j/a33 (J=^,*..n.2)

Check: P32 = 2P3J + 1 (j=^,,..n)

In ^neral:

(^') \j= V PikV^afcV -- Pk-i.k

^'^'^ Pfcj" \j/\k (J=k+l....n,2) Check:

Until: (»Oa^j^a^^-ß^^o.j-ß^a^^-..

<"'> ^X = «nl /«nn

•^-l.j 0=k:....n,2) Check: o^=2o^j(j»k,,..B)

PfcS = Sßj^ + 1 (J=fcH,...n)

• " ^-l.n Vl,J<J ="'2> «»»««k- «n2= «nn
91

-

CVJ

MH

w

UNS

-4-'

M 00

<MH

^

M VO

5!

5^

M S? §H §S fl:? S^S

a Si

•••••••

VO iH CVJ U\H VO en _ ^ : si ir\0 c^cooD-ci-vo ^-S mr-vo c^cvJ ii\i IfNCO

H

£^âs ->

oovS I CM

HH

OOOOH OO OO OO OOt

OOOHO OO OO

OOo OH

H ITN

S»« • •

HVO

en

H^

en

cvi u\ • •

ON

ES

OOHOO OO OO CVJII

en S-^ GÖ VO ON a\

VO t-UN crn£\

en OH tfNoo

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0)»u

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•H

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^"^

Vl en 0 ^w^

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0) ^0-%

•5 CVJ 0á0

^^H

-^ 00-^om

I•H en

^^oo^

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t^ONt-H • • • •

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CJNH ITNr-en H enn o H

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IT^ON

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enX en ^"^

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•• •••• ••

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f-H t:r lA-zt VO • • • o• •

CVJ LfN

eu

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envo O O

II

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en

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. 92 -

solution than to make separate checks for each part of the row. The elements

in column (12), headed IN-, are obtained hy sumning across only the I columns

of the rovfs. These elements are obtained after the forward solution has been

completed and are used to check the coinputation of the inverse matrix.

The computation of the inverse, or D, matrix frcm the abbreviated forward

solution is the same as that of the full forward solution, as explained on p. 11.

The formula for dij, the ijth element of the matrix D of order 5 given on p. U,

can easily be extended to apply to a matrix of any order.

Cognputation of the Matrix Product B'A~J-B.—The abbreviated forward solution

is also used to obtain the matrix product B'A-lB. In this case, B need not be

symmetrical. In any case, it is written out in full. The worksheet is set up

in the form:

B

An abbreviated forward solution is carried out on this augmented matrix and

B'A-1B is obtained by ciamilating the products of certain elements from the B

part of the forward solution. This computation is analogous to the computation

of the inverse matrix, with matrix B replacing unit matrix I.

Table l6 gives the abbreviated form of the forward solution for the computation where A = Mgz and B = M^y shown in table 8. Note that the abbreviated

form makes use of only one check column, D •

Computation of the Matrix Product A'^B.—The matrix product A'^B can be

efficiently computed with an abbreviated forward and back solution. Table 17

gives the abbreviated forward and back solution for the computation of the transpose of A-J-B, where A = By^ey* and B = W,^Kr*, which is shown in complete detail

in table 12. Again B is written out in full.

Outline 2 presents, in symbolic form, the computation of the transpose of

A"-*"B, where A is the symmetricfiO. n x n matrix shown in rows (l) through (n) and

columns (l) through (n), and B is the n x m matrix, which need not be symmetrical, shown in rows (l) through (n) and columns (rn-l) through (iH-m). A forward

solution is carried out and the transpose of A'^B is obtained in the form of a

back solution beginning with the last column, that is, "v^^^; continuing to the

next to the last column, Y!^ ^^3^; and so forth, until the first column, 'V' is

obtained. Table I7 ill\istAtes the computations for a problem in which n=iB=:2.

If m = n and B is a unit matrix of order n

compute A" . This method actually gives (A"!) '

A-1 is also symmetrical, and (A"l)'= A-1,

., this method can be used to

but since A is symmetrical.
- 93

Table 16.- Abbreviated forward solution for data shovn in table 8 1/

; "« ; ««y

: Z

Row

; ^1 ; H : *3 ; ^1 ; ^2 ^

ais (3) 4

0.5127 0.5017

.55V3

O.J«)60

-.0056

9.3676

0.6138

1-1593

0.if9^

.50if8

-1.2651

2.5284

2.0008

9.6622

U")

0.5127

1.

0.5017

•9785

0.1*060

•7918

0.6138

1.1970

o.if9Uo

.9634

2.5284

4.9308/

1.1078

2,l6o4

Í« .0633

1.

-.U030

-6.3653

-.15V9

-2.W»8l

.02llf

.3383

-.4732^'

-7.4751^

-.1335

-2.1098

(3')

(3")

6.1t807 \

1.

-.3132

-.0W3

-1.5200

-.23^*5

4.6474''

.7171^

-1.8332

-.2828

Confutation of i^X^zy

yi

ys

yi

1.1293 0.6124

.8397

1.7417^

1.4521^

^-1

*V*z**2z ^Sey* for equation (2.1)

y2 yi z

^2 0.8397 0.6124 1.4521V'

yi 1.1293 1.7417/

1/ These conq>utation8 were performed with 9 decimal places of which only 4 appear in the table; therefore,

some of the conputations may appesir slightly in error.

Table I7.- Abbreviated form for computations shown in table 12 1/

Section (9.1) - Forward solution

; By»y* Wy*y»

Row :

: ^2 yi y2 yi

r

"A \ 0.6688 0.7689

.9858

0.3183

.2550

0.2550

1.0930

2.0112

3.1028

!ii. ! 0.6688

: 1.

0.7689

1.1496

0.3183

.^759

0.2550

.3812

2.0112

3.0069/

(2')

(2")

.1017

1.

-.1110

-1.0912

.7998

7.8617

.7905^^

7.7704 v^

Section (10.1) .- Back solution for A» - B""S/

1.7305

-8.6573

-1.0912

7.8617

-5.9267v^ 7.7704y

y These computations were performed with 9 decimal places of which 4 appear in the table; therefore, some

of the computations may appear to be slightly in error.
- 9h

Outline 2.- Abl)reviated forward and l)ack eoliitlon

Computations Instructions

aaa.tt.aa a •••a a

11 12 13 In l.rfl l,n+2*" l,r+m 12

*22*23* • •^2û^2,n^l°^2 ,n+2* ' •^2,n4-in*22

^33*^*%^3.iH-1^3.n+2*"«*3,tt^ni^32:

Carry out a forward solution as indicated

in outline 1.

In general:

( J=k,. • .n.tt^-l,. • .n+m.Z)

nn n,D+l n,c+2*** n.n+in nS

Check: a = Sa (J«k,...n, i:+l,..,xr»-m)

kS kj

^k/ \j^%k (^'^^-'•••••»•o+l»'«-»-*-».^)

Check: ^^^^^^y^A*'^ (j=kí-l,...n,n+l,...n*-m)

Forward solution Back solution

*L1°12°13—"ln°l.itfl°l,cf2—°l.irfm°lS

^ ^23"-W2.iH-l"2.n+2—^,rH-nf2S

(1) Ï' = ß (i=i..,.iD.j;)

in n,n+i

Check: 7' = Sï« + 1 (i=l,,.,r!)

¿n ID

(2) V « ß --Y» ß ^ (i=l,,..!i»,S)

1, n-1 n-l,n+i in n-l.n

Check: 7» = n\ 4 1 (i«l,...in)

2,n""l i,n—l

OL CL ., a ^ • • • a a _

nn n,ü+l n,D+2 n.n+m nZ

^ ^n,n+l^n,n4-2- * •^n.n+m^n2

(3) y* =ß -V ß -Y« ß

i,n-2 n-2,E+i in n-2,n i,n-l n-2,n-l

(i*l,..,ni,Z)

Check: 7» «2nr« + 1 (i«=l....«:)

Z,c-2 i,n-2

In general:

Back solution

^' -... -Y" Y« -Y»

11 l»n-2 l.n-i In

Y« ... Y» Y» Y»

21 2,n-2 2,n-i 2n

'Y«^ ... Y« Y« Y«

31 3.n-2 3,n-l 3n

Yt B B ->Y< S -Yi ñ -

ik Pfc.n+i 'in^kn i.n-l^k.n-l ••'

-n.m\.k.l(^=^--»-^)

Check: >¿j^ * 2^[j^ + 1 (i'l....»)

Until:

Y« =p -vi ß «71 o

il l.cfi in In i,n-l 1,D-1

•Y' .... Y» Vi V»

mi m^n—2 ni,û-l nm

•Y» ,... Y«^ Y» Y»

21 2,n-2 2,n-l 2n

Check: V«.j "2:^11+1 (i«l....«.)

(3) (2) (1)
- 95 -

If m = 1, this method can be used to solve the system of equations:

^ll^i + «la^S + . . . + ai^ = a^^j^l

The back solution provides the solution in the form:

ij ~ ^j (j=l^ ... n)

This eliminates one step in the usual matrix solution of a system of simultaneouB e<3uations-- multiplication of the inverse of the matrix of the coefficients by the vector of the numbers to the right of the equality sign.

An efficient method for conrputing the inverse matrix without a back

solution and without the reduction of the unit matrix is given by Waugh (22).

This approach is not included in this handbook because it does not relate

directly to any specific step in the statistical fitting of systems of simultaneous equations.

A General Method

The abbreviated Doolittle method ats we have described it is applicable

only when A is symmetrical. But a slight modification of the method makes it

applicable to amy matrix^ whether symmetrical or not.

The method outlined below is commonly known as the Crout method (h).

Similar methods have been developed by Waugh and Dwyer (23,). The methods of

Crout and Wa\agh and Dwyer require no more work than the usual Doolittle method.

It is unnecessary to learn two different methods—one for symmetrical matrices

and one for nonsymmetric€LL matrices. The statistician or statistical clerk who

beccanes familiar with the method outlined in the following pages will be able

to solve any set of equations and invert any matrix.

In this handbook, however, this method is xised only for nonsymmetrical

matrices. This was done chiefly because many statistical clerks aind research

workers are familiar with the Doolittle method as applied to symmetrical matrices.

They will not have to learn a new approach for rnsuy of the operations we have

described.
96 -

Solving Systems ^ Equations.—The system of equations:

»11^1 + *12^2 + • • • + «in^n = '10

agjXi + 022^2 +

. + &2x?n = a20

> <^'

ajjjXi + SjjgXa + . . . + anr^n = ^0 /

where ai ^ * a^^ cannot be solved by the methods discussed up to this point.

If n = 2, the equations can be solved easily by direct substitution. If

n = 3, the above equations can be efficiently solved by Cramer's Rule, which says

that:

*11 *10 *13

a a a

21 20 23

a a a

31 30 33

*10 *12 ^13

*20 ^22 *23

^0 *32 "33

X3 =

*11 *12 ^10

*21 *22 *20

V V "30

where A =

»11 *12 »13

*21 *22 *23

*31 ^2 ^3

The value of a third order determinant is given on p. 26. When n > 3> this

method is not efficient, since the evaluation of higher order determinants is

cumbersome. Therefore when n > 3, we use the Grout netlKDd to solve the system

of equations.

Outline 3 presents, in symbolic form, the Crout method for solving a system

of equations. An illustration of this method to solve the following equations

is given in table I8.

C.

o.oWgc

f - 2.5Pc + 2.5P¿ = 203.9

c = 138.1

- 10.1»67Pi = -1866.79

(2)

These equations could be solved easily by Cramer's Rule or by the approach described on p. Ö5; the Crout method is shown for Illustrative purposes.
Outline 3.- Grout method

Computations

Given the system of equations In n unknowns

*11^1 ^ »12^2 + HfA - •- ^ ô^^Xn = a^o

*21^1 -*• ^2^2 ^ ^3X3 -»•...•»• agnXn - ago

»31^1 * *32^2 * *33^3 -^ • • • •»• a3nXn = a3o

®nl^l -*• ^12^2 •»• ön3^3 * • • • «nn^ = *nO

Given mfttrix

*11 *12 *13 ••• »In »10 »IZ

^1 *22 *23 • • • »2n *20 »22

*31 ^32 *33 ••• *3n *30 *32

^ll »n2 ^13 ' • • ^ ^lO »nz

Auxiliary matrix

hi "12 hi ... ^In ^10 hz (2)

"sx b22 b23 ... b2n bgo b22 (M

^31 ^33 ••• *>3n ^30 ^3E (6)

•

»nl »»ns • • • V *>nO ^iZ (2n)

(1) (3) (5) (2n-l)

XJ

Solution

n-2 n-1 n

X» X* ,x»

n-2 n-1 n

(3) W (1)

- 97 -

for solving a system of linear equations

Instructions

Given matrix

n ~~

^±1 " ^ »IJ (i-l,2,...n,0) That is, the sum across the

J*l 1th row.

Auxiliary matrix

(1) b^i » a^ (1=1,...n)

(2) \^ = aLj/»lJL(J'2,...n,0,Z)

Check: bj^jj * r^bjLj + 1 (^=2,...n,0)

(3) ^22 = 822 ->>2it>L2

^2 = «12 -^1^2 (l=3/«..n)

(^') ^J ' («2r^l^j)/^ (j-3,...n,0,z)

Check: b^j, = jb^^^ 1 (j=3,...n,0)

^^^ ^33 ^ '^3"V^3'^32^23

\3^ \3"^A3"^2^23 (i^^---«^)

Check: 32 ^^J * ^

{J=if,...n,0)

In general:

^kk'^-\l^lk-^^2k"\3V"• • • "^k, k-1Vl,k

^k=^k-^l\k-\2^2k-^3V" • • ^,k.lVl,k ^^=^*^' • • •")

\j=Kj-\l^r^^2J-\3^3J- • -^k-lVl,J >/\k

(J=k+l,...n,0,2) Check: b^^=Ebj^+l (j=k+l,. ..n,0)

Until:

(2n-l) V^-^^-^^-----^,n-lVl,n

(^) V^^J-^a\j-\2^J---Vn-lVl,jV^nn ^^^^'"^^

Check: b^ = \o * ^

Solution

U)^=^0 ^? =- h^-nz« Check: ^ = \ + ^

(2) V b_..b_.X. n-1,0 "n-l,n'Vi ^S.l-Vl,z-Vl>n^

Check: X» , = X , 1

n-1 n-1

(3) X

n-2 n-2,0 n-2,n n n-2,n-1 n-1

X» = b -b X<<-h ^ X» , Check: X* .-^^ ^

n-2 n-2, S n-2,n n n-2,n-l n-1 n-2 n-2

In general:

\ = ^kO-^kn^ -\,n-lVl'Vn-2^-2---^,k+A+l

X£ = \z-^kn^-^k,n-l^-l -\,n-2^-2—-\,kfl^*l

Check: X| = X^ + 1

+1

The final check is the substitution of the unknowns in

the original equation.
- 98 -

Table l8.- Grout method for solviug a system of equations 1/

Coefficient for

Cf

Giren matrix:

1.

.<A89

1.

-2.5000

1.

0

Pd

2.5000

0

-lO.tóTO

203.9000

138.1000

-1866.7900

20lf.9000

139.1^

-1876.2570

Auxiliary matrix:

1.0000 -2.5000 2.5000

.Olf89 1.1222 -.1089

1.0000 2.5000 -12.691*6

203.9000

lllf.1717

185.5991

20U.9000^

115.0628 V

186.5991«^

Solution:

75.9r62

76.8762V

134.3896

135.3896^

185.5991

186.5991^

Final check:

203.9OOOV'

138.0999>/

-1866.7899*^

1/ These conrputations were performed with 9 decimal places^ of which only k

appear in the table; therefore some of the conrputations may appear slightly in

error.

Matrix Inversion.—The Crout method also can be used to invert nonsymmetrical matrices. The coniputations involved in inverting a matrix of order n are

indicated in outline k. The matrix to be inverted is given in rows (l) through

in), columns (l) through (n), and the unit matrix is written in rows (l) through

(n), columns (nfl) through (iH-n). The transpose of the inverse matrix is obtidned in the form of a back solution, with the last column, that is, ein, computed first.

The system of equations (l) can "be written, in matrix notation, in the form:

AX' = a'

where

A =

»11 ai2

*21 ^2

^nl an2

»In Xl

a2n X' = X2

•

*nn

•

Xn

and a' =

»10

»20

^0
- 99

Outline ^^,- Crout method for matriz inversion

Comtnitations Instnictions

ftlven matrix

*11*12*13* • '^In^,»*!^,D+2' • •*!,ir*^*12

*21*22*23* • •*2n*2 ,n+l*2 ,D4-2' * • *2 ,rH-n*2S

AtcdLliary matrix

^11^12^13* • "^In^l.n+lh.iH-2* • •h.n+n^l2

^21^22^23-**^2n^2.tti-1^2,nH-2**-^2,iH-n^2S

^31^32^33* • •^3n^3.n+1^3,xH"2- • •^3,n+n^32

^nl^n2^n3* • •^xm^n.tt*"l^n.irf2* • •^n.nrfn^nS

Solution

^11 ••• •l,n-2 ®l»n-l ^In

'21 ••• ®2,n-2 ®2,n-l ®2n

31 ••• ®3.n-2 ®3,n-l ^2n

^nl ••• '^n,n-2 ®n.n-l ^*xm

®S1 ••• ^5:,n-2 ®2,n-l ®2n

(3) (2) (1)

Given matrix

jajLj I (i»J - 1» ••• n) i8 the

L '^J matrix to "be inverted*

n+n)

L J is the unit matrix.

i.« = 2 a. ^ (i=l,...n, n+1,,,.

^^ J=l '•' n+n)

that i9^ the sum across

the ith row.

Auxiliary matrix

The auxiliary matrix is computed

in the manner indicated in

outline 3»

Solution

(1) «in " Vn+i ^^'^' — °»^^

Check: ej,p = Se^^ + 1

(i^l, ... n)

<2) «i.n-l" Vl.xt^-i *inVl,n

(i=l, ... n,2)

Check: ej.n-l'^n.n-l* ^

(i*l. ... n)

Í3) ei.n-2'^n-2,a+l"®in*n-2.n

®i,n-l^n-2,n-l (i=l n.S)

<^«°^= «2,n-2'^i.n-2+l (i=l,...n)

In general: eik=\,^i-«in^kn"®i.n-l\,n-r'"'*i,fc*A.fcfl

(i»l,.,.n,S) Check: e^= Sejjj.+1 (i»l,.,.n)
- 100 -

As shown on p. 82, a solution to the equations can he found by premultiplylng

a' hy A-1.

In using a system of simultaneous equations for analytical purposes, the

matrix A usually consists of a set of constants, hut a' varies from period to

period (see p. 82). In such cases, it is convenient to compute A'J- and then

pos-tamiltiply it hy the varying a' to get the X' values for each period. Equations (2) are solved hy this method in table 19.

this exaniple.

1 -2.5 2.5 Cf

A = .01*89 1 0 X' = Pc and a' =

1 0 -I0.1í67 .^dj

203.9

138.1

-1866.79

Since the Crout method gives [A"!] S X' is obtained hy a coliann-hy-coltamn

multiplication of [A'^Y by a •• See p. 101 for an explanation of this and p. 103

for the check on the coorputation.

If only one or tvo elements of A ai^ siibject to change, as with nonlinear

variables, a new a\uciliary matrix need not be computed. Only those terms of the

auxiliary matrix that are affected by the changing a.±¡ values need be conrputed.

These terms can be deteimined from outline 3 or ^•

Table 19.- Crout method for solving a system of

equations \ising the inverse matrix 1/

Cf ; Pc ; Pd :

•

Il :•

12

• •

: 13 :

• •

z

Given matrix:

1.0000 -2.5000

.OU89 1.0000

1.0000 .0000

2.5000

.0000

-10.1í670

1

0

0

0

1

0

0

0

1

2.0000

2.01*89

-8.1*670

Auxiliary matrix:

1.0000 -2.5000

.Olf89 1.1222

1.0000 2.5000

2.5000

-.1089

-I2.69íi6

1

-.01*35

.0701

0

.8910

.1751»

0

0

-.0787

2.0000«^

1.7385 ^

1.1668 •

Trsuaspose of the inverse matrix:

r

Solution:

0.731^7 -0.0359

1.8367 .9101

.175!^ -.0085

3.7l«69v 1.8656^

0.0701

.1751*

-.0787

1.1668V

0.7689

2.9221*

.0881

203.9

138.1

-1866.79

75.8762

1^1*.3896

185.5991

• 395.865OV

appear in the table; therefore some of the computations may appear slightly in

error«
- 101 .

Evaluation ^ Determinante, »-The Crout method also can be used to evaluate

nonsymmetrical determinants. Referring to outline 3, the value of the determii^^^* I ^ij I (i^J=l^ ... n) is given by b^ b22 1^33 • • • h^n •

Additional Comments on Matrix Multiplication

Alternative Forms.—Matrix multiplication is defined on p.2lf in terms of

a row-by-column operation; that is, e±j, the ij th element of the matrix product

E = AB, equals the s\mi of the products of the elements in the ith rov of A with '

the elements in the Jth column of B, starting at the left and the top respectively.

Matrix miatiplication also can be defined as a row-by-row operation, that

is^ eij, the ij th element in the matrix product, E = AB, equals the sum of the

products of the elements in the ith row of A with the elements in the jth row

of B' (the transpose of B). Or it can be defined as a column-by-colimin operation where eiu the ijth element in the matrix product, E = AB, equals the sum

of the products of the elements of the ith column of A* with the elements in the

jth column of B. If A and B axe symmetrical, the matrix product AB can be obtained by either (l) a row-by-column, (2) a row-by-row, or (3) a column-by-colxmin

multiplication of A with B.

Row-by-row or column-by-column multiplications are useful concepts, especially when either B' or A' has been computed and the matrix product AB is

needed. It saves copying data and its attendent mistakes. Referring back to

the limited information conrputations, in section (7) MUt2*P' ^^ obtained by a

column-by-column multiplication of (My*2*)' = Mg^^« with P' (see p. 37). in

section (26), U = oi^^'^bc ^ß obtfiiined by a row-by-row multiplication of QIP* with

F¿ (see p.lf7). bW could have been computed in section (16; by a column-by-column

multiplication of b* with W (see p.lA). As a meajis of separating the computation

of the coefficients cmd their standard errors, a row-by-column multiplication

of b with W is shown instead, in tables 9 and 10.

Checks.—In the e:q>lanation of the limited information computations, checks

on the computation of the matrix product AB are introduced for certain specified

forms of A axid B. These include: (l) Augmenting B with a S colimm, composed

of row sums. After the m\iltiplication is carried out on the augmented B matrix,

the ith element of the resiilting S column of AB is checked against the sum across

the ith row of AB. See, for example, the computation of Mgig* times Mg-jiy* = P'

in table 10. (2) Augmenting A with a row, composed of column sums. After the

multiplication is carried out on the axagmented A matrix, the jth element of the

resulting Z' row of AB is checked against the simi down the jth column of AB.

See for exaniple, the computation of -P* times b ' = c' in table 10. (3) A reconrputation where the pr'oduct is a scalar. See, for exaaiple, the compitation of

-P' times b • = c' in table %

The same general procedure, that is, the use of a £ column or a 2» row,

can be extended so that a check is available for the coanputation of any matrix

product obtained by any method of matrix m\xltiplication. These results are

sumnarized in outline ^.

This outline is divided into three sections: Computations that relate to

(a) row-by-Column multiplication; (b) row-by-row multiplication; and (c) columnby-column multiplication. Under each of these methods are listed all possible
- 102 -

Outline 5.- Computational forms for checking matriz multiplications

Bov1>y-column multiplication

Tyi» of multiplication

Matrix X matrix

Column z row

- matrix

matrix

Row X matrix row

Scalar x row row

Matrix X column column

Column X scalar column

Bow X column scalar

Scalar z scalar scalar

[ ]

.[ ]

[]

J]

il

1 ]

..[]

Computational form

rr

[ J'

[ ]

[ ]

[]

C D

[Í

Á ]

[Î

d ]

Scalar product-recompute

So.

C 3

C 1

J]

..[]

Bow-by-row multiplication

Matrix X matrix = matrix

Coluinn X column B matrix

Bow X matrix = row

Scalar x column C row

Matrix X row B column

Column X scalar S column

Bow X row - scalar

Scalar x scalar S scalar

[ ] i ] TJ

..[ ]

X I ] .n

[] X ,ü if

A] X Ü z'L .

[ ] X

I

..[ ]

C 1^

C] C 3^=

E'L .

X

X

c :i s«Lr.[] c: .Ü

Scalar product-recompute

Do.

Column-l)y-column multiplication

Matrix X matrix

Bow X row

= matrix

matrix

Column X matrix row

Scalar x row row

Miatrix X column column

Bow X scalar column

Column X column scalar

Scalar x scalar scalar

_ J ^

r -1 2

... g -

r n S

- J ^ ]

li

CD X C 3^ .

if

C 3 X C 1 xH

J X [ ]"^'= C 1

C3 X C 3 = = L l'

r 1 2

J ^ iJ z»LC 3= X Cl s'L.

Scalar product-recompute

Do.
- 103 -

types of mxiltiplication. The word "col\amn" refers to a column vector and the

word "row," to a row vector. Next to each of these Is the computational form

that should be used to check the operation. The symbol r -idenotes a matrix;

L J a column vector; ^ ] a row vector; Q] a scalar; 2 a column of row sums;

euod Z* a row of column sums.

As an illustration of the use of outline 5, consider the following: a

column-by-column multiplication of a matrix times a column vector. (This

method was used on p. 71.) Outline 5 indicates that we augment the matrix

with a £ column. After the column-by-column multiplication is carried out

(as explained on p. 101), the resulting column vector has an additional row,

St ^ Tiiat the sum of the elements of the column vector is equal to the

element in the E' row is indicated by a check mark. This checks the computation.

For several types of matrix multiplication, alternate computational

fonos are indicated in outline 5* The preferable form to use is determined

by: (a) The form of the matrices used to conrpute the product, that is,

whether a 2 column or 2» row has already been computed; or (b) further computations, if any, into which the product enters, that is whether a 2

column of 2» row is desired.
- 1(A -

GLOSSARÏ 27/

Adjoint of a matrix.—In this handbook this is iised as a computational device

for inverting a (2x2) matrix (see p. '^O ). For a general definition^ see

Klein (13, pp. 335-337) or other references on matrices.

Adjusted moments.--Moments that have been adjusted in such a way as to make

the augmented sums of squares for each variable nearly equal to 1.

Adjustment factors. - -Factors used to obtain the adjusted moments from the

augmented moments (see p. 6 )•

Augmented matrix.--Two or more matrices or vectors written adjacently to

facilitate computational operations. The reader is cautioned not to confuse

this with an augmented moment matrix.

Augmented moment matrix.—^A matrix consisting of adjusted augmented moments

(see p. 7 JT

Augmented moments.--Moments are simis of squares and cross-products from which

a correction factor has been deducted to give results that would have been

obtained had the calculations been based on veiriables expressed as deviations

from their respective means. Augmented moments are moments that have been

multiplied by the number of observations included in the analysis. They are

used to avoid rounding errors.

Coefficient of deteimination.—The square of a correlation coefficient.

Column-by-column matrix multiplication.--A variant of the more cammon rowby-colunm cootputation (see p. 101).

Colxmin vector.—A matrix consisting of m rows and one coltmin (see p. 22 )•

Consistent estimates. —Estimates of statistical coefficients obtained in

such a way that the average value for many large samples equals the value

that MToulä be obtained from a similar cfitLculation based on the combined

evidence of all possible samples. For unbiased estimates, the same property

holds when estimates are made from samples of any size.

22/ Definitions given here refer only to terms included in this handbook.

In cases where an exact definition would require a large amoiont of space, a

condensed explanation is given instead, possibly with a cross-reference to

another publication. The reader is presxmied to be acqixainted with terms

covered in a first course in statistics that includes multiple and partial

correlation and regression.
- 105 -

Covarlance.—The Joint sampling variation between tvo or more statistical

coefficients. It is analogous to the variance of a single coefficient.

Cramer's r\ale>—A coniputational scheme for solving systems of equations.

In this handbook, its use is recommended only for nonsymmetrical systems

consisting of exactly 3 equations (see p. 96).

Critical values.—Points in a probability distribution that delineate a

given percentage of the items. Points of 5 percent or 1 percent are

commonly used in statistical tests.

Grout method.—A computational scheme for solving systems of equations

inverting matrices, and performing similar operations. In this handbook,

it is used only when the matrix to be inverted is nonsymmetrical; but it

can be used for any matrix (see p, 95).

readjusted data.--The origina], data on which the eumlysis was based, as

contrasted with the adjusted moments used for coipputational purposes.

Degree of identification.—See identification.

Determinant.--A numerical value associated with a square matrix (see p. 26).

Diagonal of a matrix.--See main diagonal of a matrix.

Doolittle method.—A computational scheme for solving systems of equations,

inverting matrices, and performing similar operations. As commonly used,

it includes a forward and a back solution; in this handbook, in most

instances only the forward part of the solution is used. As described in

this handbook, the method can be applied only when the matrix to be inverted

is symmetrical (see p. 89).

Efficient estimates.--Estimates of statistical coefficients obtained in such

a way that their average standard error for many large sazoples is as small as

possible. For "besf estimates, the same property holds when estimates are

made from samples of any size.

Endogenous variables.--A set of variables that are assimied to be determined

simultaneously by cormoon economic forces. Lagged values of endogenous

variables are included among predetermined variables in an analysis.

Exogenous variables.--A set of variables that are unaffected by the common

economic forces that are assumed to affect endogenous variables in a system

of equations.

Floating decimal point.--The carrying of a fixed nimiber of significant figures

in clerical coniputations, and therefore a varying number of decimal places, as

contrasted with the method, suggested in this handbook, of carrying a fixed

nuoiber of decimal places.
- 106 -

Full infoimatlon approach>-A maxlmua likelihood method for deriving 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 involved 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 underidentified, just

identified, or overidentified.

Identity matrix. - -A square matrix in which all elements on the main diagonal

are 1, and all nondiagonal elements are zero (see p. 22).

Instrumental variable approach.—One of several methods for obtaining

estimates of the structural coefficients that are statistically consistent in

a single equation that contains more than one endogenous variable. A set of

predetermined variables from the entire system is chosen such that one

variable is available for each coefficient to be estimated. These are called

instrumental variables. They are used in the manner described by Klein

(13, prp. 122-125). The objection to this approach is that different answers

are obtained depending on the particular set of instrumental variables used.

Inverse of a matrix.--The inverse of the matrix A is written as A"-^. The

inverse of A is that matrix which when multiplied by A equals the identity

matrix. If the original matrix is symmetrical, the inverse also is symmetrical (see p. 20*

Iterative method.—A computational device or formula in which an initial vcuLue

is assimied or estimated and successive values are derived from the formula.

Such methods are used only when the successive values are known to converge to

the value that would be obtained by a direct computation.

Just identified equation.—An equation that has the mathematical property that

peimits a tinique determination of its structural coefficients from regression

coefficients in reduced form equations (see p. 83 ). Such eq\iations can be

fitted by the method of reduced forms or by a modification of the limited

information approach.

Limited information approach.—^A maYiTmiTn likelihood method for deriving

estimates of the structural coefficients for equations that are overidentified.

The coefficients usually are estimated for one equation at a time, with the

simultaneity ijirplied by the system taken into account in the computations, but

information on the peurbicular variables that appear in each of the other

equations in the system is ignored. The estimates are statistically consistent

and as efficient as any others based on the same amount of information. A

slight modification of this approach is used for equations that are just

identified.

Main diagonal of a matrix.—The elements, listed in order, along the diagonal

of a square matrix starting with a,, and ending with a . ±± nn
- 107 -

Matrix.—An array of nimbers airanged in rows and columns (see p. 2l). An

(m X n) matrix consists of m rows and n colimins.

Maximim likelihood.—A commonly used mathematical procedure for obtaining

formulas to estimate statistical coefficients. Coefficients are derived in

such a vay as to maximize a likelihood fxmction. The results are knovn to be

statistically consistent and efficient. Statistical coefficients obtained frcm

such formulas are called maximum likelihood estimates.

Model.—A system of related structural equations, together with some implied

Joint or combined probability distribution for their error tenas.

Moment matrix.--As used in this handbook, this always means an aiignented moment

matrix.

Moments.—See aiagmented moments.

Monte Carlo approach.—A method used to indicate empirically the amoxmt and

kind of sampling variation that can be expected imder given conditions.

Samples are drawn from a known population, estimates based on these samples are

made, and the results analyzed.

Oixier of a matrix or determinant.—The nimiber of rows or columns in a sqiiare

matrix or in a determinant.

Qveridentified equation.—An equation that has the mathematical property that

alternative estimates of its structural coefficients can be obtained from the

regression coefficients in the reduced form equations (see p. 83). Hence the

method of reduced forms cannot be used to estimate the coefficients; instead

the limited information approach commonly is used.

Parti ally-reduced foim equation.*-In some equation systems data are unavailable

for certain endogenous variables in the structural equations. In such cases,

variables with which this variable is asstmied to be related are substituted

algebraically for it in other equations. The resulting equations are CÉÚLled

"partieilly-reduced form equations" (see p. T**- ).

Predeteimined variables.—A set of variables that are assimied to affect

endogenous variables in a system of equations but not to be directly affected

by them. They may include exogenous variables and lagged values of endogenous

variables.

Rank of a matrix.--A matrix has rank r if the largest nonzero determinant

included in the matrix is of order r.

Recursive approach.—Some systems of equations are formed such that at least

one of them contains only a single endogenous veoriable. Consistent estimates

of coefficients in such equations can be obtained by solving them directly by

least squares. If other equations contain only one endogenous variable other

than those contained in the first set of equations, consistent estimates of
. 108 -

the coefficients in these equations can be obtained if they are solved directOy

by least squares, provided calculated values of the other endogenous variables

are substituted for actual values before making computations. Systems of

equations in which each equation can be fitted by least squares by the successive substitution of calculated values of endogenous variables are known as

recursive systems, and this method of solving them is called the recxarsive

approach. Although the estimates of the structural coefficients are statistically consistent, frequently they have larger standard errors than if estimated

directly as in the limited information approach.

Reduced form equations.--Equations that result when each endogenous variable

in a system of equations is written as a linear function of all of the predetennined variables in the system. In this handbook, they are used as a

computational device when using the system of equations for 8uaalytical purposes

(see p. 83)•

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 solved by least squares and the structural coefficients obtained by an algebriaic transformation. Results obtained

are identical with those given by the modification of the limited infonnation

method described in this handbook.

Row-by-row matrix multiplication.—A variant of the more common row-by-column

computation (see p. lOl;.

Row vector.—A matrix consisting of one row and n columns (see p. 22).

Scalar.--An ordinary number, as contrasted with a matrix or a vector. It can

be thovight of as a matrix with one row and one column.

Simultaneous correlation coefficient.—A coefficient that indicates the

percentage of variation in the joint distribution of endogenous variables in a

system of equations explained by all predetermined variables in the system

(see p. 86).

Square matrix.—A matrix having the same number of rows and columns.

Structural equations.—Equations derived from the basic economic relationships

that are assumed to prevail within a system of equations, as contrasted with

other equations, such as reduced foim equations, that are used chiefly as a

conrputational device.

Symmetrical matrix.—A square matrix in which all the corresponding elements

above the main diagonal are equal to elements below the diagonal; that is.

Hi = a^iCsee p. 22).
- 109 -

Transpose of a matrix.—The transpose of the matrix A is written as A«. The

transpose of A is a matrix in which rows of A are columns of A«, and columns

of A are rows of A«. The transpose of a symmetrical matrix equals the matrix

itself (see p. 22).

Two-tailed test.—A statistical test involving both tails of a probability

distribution. CriticiuL values for a 2-tailed test at a 5-percent probability

level are found such that 2| percent of the items lie to the left and right

of these points respectively in each tail.

Underidentified equation. - -An equation that has the mathematical property that

its structuTEuL coefficients cannot be estimated by statistical means (see p.

29).

Unit matrix.—See identity matrix.

Variance>—The square of a standard error.

Vector.—A matrix consisting of only one row or one column. By convention, the

tena vector alone implies a row vector consisting of one row. A column vector

consisting of one column is written as the transpose of a (row) vector

(see p. 22).