United States 

Department of 

Agriculture

Economic

Research

Service

Technical

Bulletin

Number 1932

January 2012

A Quarterly Econometric Model 

for Short-Term Forecasting 

of the U.S. Dairy Industry

Roberto Mosheim
www.ers.usda.gov

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Recommended citation format for this publication:

Mosheim, Roberto. A Quarterly Econometric Model for Short-Term 

Forecasting of the U.S. Dairy Industry, TB-1932, U.S. Department of 

Agriculture, Economic Research Service, January 2012.
United States

Department

of Agriculture

www.ers.usda.gov

A Report from the Economic Research Service

Abstract

This research evaluates the econometric approaches employed by USDA’s Economic 

Research Service (ERS) to contribute to the dairy sector forecasts published in the 

monthly USDA World Agricultural Supply and Demand Estimates (WASDE) report. 

To generate the estimates, a quarterly model for the U.S. dairy industry is specifi ed 

using data for fourth-quarter 1998 (Q4/1998) to fi rst-quarter 2009 (Q1/2009), and it is 

estimated and validated employing data for Q2/2009 to Q1/2010. Different forecasts 

are generated using a variety of single equation and system methods, and then evaluated in terms of forecasting precision or predicting turning points in the data. Different 

approaches, however, more effectively forecast different variables. Vector autoregression 

with exogenous variables outperforms structural regression models when forecasting 

prices, but single and system estimations of structural models are superior to time series 

models when forecasting some items on farm supply and commodity balance sheets.

Keywords: U.S. dairy industry, forecasts, simultaneous equations, vector autoregression 

Acknowledgments

Various versions of this study benefi ted from the comments from USDA, Economic 

Research Service economists Don Blayney, Paul Westcott, and Joe Cooper and four 

anonymous reviewers. Erik Dohlman and Amy Hagerman (both from ERS) provided 

valuable editorial advice. The author also thanks USDA, Economic Research Service 

administrators for their feedback and support, and others who helped complete this 

research. Special thanks also to Angela Anderson for editorial support and Cynthia A. 

Ray for design assistance.

Roberto Mosheim

A Quarterly Econometric Model 

for Short-Term Forecasting 

of the U.S. Dairy Industry

Technical

Bulletin

Number 1932 

January 2012
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Economic Research Service/USDA

Contents

Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Salient Features of the U.S. Dairy Industry . . . . . . . . . . . . . . . . . . . . . . . 2

Comparison of a Selection of Relevant U.S. Dairy Sector 

Forecasting Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Model Specifi cation, Estimation, and Validation . . . . . . . . . . . . . . . . . . . 8

 Data Sources and Variable Construction. . . . . . . . . . . . . . . . . . . . . . . . . . 8

 Specifi cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

 Estimation and Validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Summary and Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Appendix B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Appendix C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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Summary

What Is the Issue?

This report documents the ongoing forecasting activities by USDA’s 

Economic Research Service (ERS) that combine judgment-based forecasting 

with rigorous econometric estimations and data construction that can provide 

better forecasts than a mostly judgment-based system alone. The formal 

modeling process gives the forecasting activity at ERS transparency and full 

documentation on the specifi cation, estimation, and validation procedures 

employed. Specifi cally, the econometric models generate the monthly dairy 

sector forecasts that contribute to the USDA World Agricultural Supply and 

Demand Estimates (WASDE) report. In addition, the model’s estimates 

potentially can be used to examine the structure of the sector and the infl uence of policy-relevant variables. The merit of various econometric and time 

series models, however, is more about their ability to forecast effectively 

than for their potential contribution to policy analysis. 

What Did the Study Find? 

Various estimation methods successfully forecast different endogenous variables, a situation that might change as the sector evolves or as additional data 

become available and the applied econometric model improves. The ERS 

model generated projections that outperformed the consensus forecast by 

USDA’s Interagency Commodity Estimates Committee (ICEC) in roughly 

half of the instances in terms of accuracy and predicting turning points in the 

data of interest. 

The results demonstrate how different methods are preferable, depending 

on the variable. To produce forecasts in the dairy sector, we cannot rely on 

a single estimating method. Moreover, the fi ndings suggest that composite 

forecasting or forecast blending will play a signifi cant role in this process. 

Careful econometric specifi cation and data development will ensure a 

successful transition from the mostly judgment-based forecasting system 

previously used at ERS.

The ERS forecasting model relies on specifi c characteristics not seen in 

previous studies of this kind. Specifi cally, it:

• Uses a variety of methods to estimate endogenous variables;

• Employs both time series and structural models; and

• Uses quarterly data and ex-post forecasting (seldom seen in this type of 

research).

The estimations highlight certain characteristics of the U.S. dairy industry:

• Milk production per cow is seasonal and increases over time.

• Herd size movements are cyclical and tied to fl uctuations in the all-milk 

price.

• As the margin (all-milk price minus feed cost) decreases (increases), herd 

sizes decrease (increase) after a number of periods.
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• Price movement in the all-milk price is correlated to the price of cheese 

and butter more than whey and nonfat dry milk (NDM). 

• The dairy sector is highly interlinked (refl ected in block recursive structure where variables at one stage serve as determinants for the next).

How Was the Study Conducted?

The model is divided into 4 blocks comprised of 15 behavioral equations and 

1 block comprised of 5 identities. Most of the behavioral equations are specifi ed in logarithmic form that permits interpretation of estimated coeffi cients 

as elasticities. Each equation within a block forecasts a variable required by 

ICEC. The blocks are linked in a block-recursive fashion such that the fi rst 

one generates estimates of variables that are then employed as predetermined 

variables in the second, third, and fourth. The resulting structure produces 

consistent forecasts across different sections of the dairy sector.

Economic theory typically defi nes the structural equations in models, 

although that practice is not as useful in the case of time series models where 

all variables within each block infl uence each other. The ERS model is based 

on quarterly data, beginning with the fi rst quarter when all necessary variables are available (fourth-quarter 1998 or Q4/1998). Possible limitations, 

with respect to degrees of freedom for estimation, were the main reasons that 

the system of equations were divided into blocks and also explains why some 

modeling choices, such as the econometric specifi cation of dairy product 

prices by means of inverse (price dependent) product supply equations, were 

made. 

Prior to estimating, the various blocks were identifi ed by the rank condition to ensure that unique values of the structural parameters could be 

derived from the reduced form of the system. Estimations of the model were 

conducted by various simultaneous equations and time series methods that 

generate different values for the endogenous variables. These results were 

validated by withholding four quarters of known data and estimating the 

model, generating an ex-post forecast. The ex-post forecasts were compared 

with the known values of the withheld data to determine how well the models 

performed based on data available at the end of fi rst-quarter 2009 (Q1/2009). 

These projections were also compared with those agreed upon by USDA in 

early April 2009 for the four successive quarters ending in fi rst-quarter 2010 

(Q1/2010).
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Introduction 

The econometric model supports the ongoing forecasting activities performed 

by USDA’s Economic Research Service (ERS) in preparation of the monthly 

USDA World Agricultural Supply and Demand Estimates (WASDE) report. 

ERS analysts combine judgment-based forecasting with both formal econometric modeling and qualitative elements to produce quarterly projections of 

commodity balance sheets (in this case supply, use, and price variables for 

the dairy sector). These quarterly estimates are then presented and discussed 

at the monthly meetings of the Interagency Commodity Estimates Committee 

(ICEC), the members of which, representing several USDA agencies, vet 

the quarterly projections. ICEC’s consensus forecast is then presented in the 

monthly WASDE report. 

This study documents the specifi cation, estimation, and validation (as well 

as the data development) of the ERS dairy sector forecasting model. Singleand multiple-equation models were employed to estimate the entire system, 

including ordinary least squares (OLS), two-stage least squares (2SLS), 

three-stage least squares (3SLS), seemingly unrelated regression (SURE), 

and two time series models (unrestricted and restricted vector autoregression with exogenous variables or VARX). These methods produced different 

values for the endogenous variables. 

The forecasts produced by the various time series (TS) and structural models 

(SM) were validated by withholding four quarters of data when specifying 

and estimating the model. This process generated an ex-post forecast (see 

Greene, 2008, p. 101) to determine how well the models predicted variables, 

the values of which were known from information available at the end of 

fi rst-quarter 2009 (Q1/2009). These projections were also compared with the 

four quarters from Q2/2009 to Q1/2010, as agreed upon by ICEC members at 

the beginning of April 2009. 
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Salient Features of the U.S. Dairy Industry

The relationship between aggregate milk supply and aggregate milk demand 

(use) drives the dairy industry toward equilibrium based on changes in the 

all-milk price (AMP). AMP is a weighted average of the price of milk used 

in fl uid milk products and the price of milk used in manufactured dairy 

products, such as cheese, butter, nonfat dry milk (NDM), and whey. In 

general, the incentive price to dairy farmers is AMP. Total quantity of farm 

milk supplied is derived as the product of total number of cows and milk 

production per cow. Recent trends in cow productivity (milk produced per 

cow) illustrate the seasonality of production per cow (Q2 is spring) and an 

increasing trend in production per cow over time (fi g. 1). The number of 

cows, in contrast, has shown a cyclical pattern related to, among other things, 

movements in AMP and feed costs (FC). A strong inverse correlation (significant at the 1-percent level) can be seen between the number of cows and the 

real difference (in 2005 dollars) between the AMP and feed costs (calculated 

from corn, soybean meal, and hay prices) (fi g. 2). Changes in this real margin 

are associated with movements in the number of cows over time. A positive 

correlation (signifi cant at the 10-percent level) can be seen after fi ve quarters—the most signifi cant positive correlation found in the data.

The all-milk price is linked to manufactured dairy product prices, as noted by 

Alston et al. (2006), by market clearing conditions. A strong positive association can be seen between the movement of AMP and changes in the prices 

of cheese and butter (less so for NDM and whey) (fi g. 3). These relationships 

are complemented by the estimates of the calculated pairwise correlations 

between prices (table 1). All of the correlations between AMP and product 

prices are positive and signifi cant at the 1-percent level. Cheese has the strongest correlation with the all-milk price, and whey has the weakest. These four 

Figure 1

U.S. dairy production per cow and season

Regression of trend: PPC = 439.8 + 17.9* quarter, R2 = 081.

PPC=Production per cow (pounds of milk).

Source: USDA, Economic Research Service calculations based on USDA, National Agricultural Statistics Service data.

I 1999

II 1999

III 1999

IV 1999

I 2000

II 2000

III 2000

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I 2001

II 2001

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II 2009

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0

1,000

2,000

3,000

4,000

5,000

6,000

-150

-100

-50

0

50

100

150

200

Production per cow

Season PPC − average annual PPC

Linear trend (production per cow)

Quarter

Production per cow (pounds of milk) Season PPC - average annual PPC
3

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Figure 2

Herd size and margin (all-milk price minus feed cost), Q1/1999-Q4/2009

AMP=All-milk price.

FC=Feed cost.

HS=Herd size.

Correlation: Herd size at t (quarter) and margin (AMP-FC 2005 $) at t = -0.37, significant at 1-percent level. 

Correlation: Herd size at t and margin at t-5 = 0.27, significant at 8-percent level. 

Source: USDA, Economic Research Service calculations based on USDA, National Agricultural Statistics Service data.

Quarter

Herd size (thousands of cows) Margin (real 2005 $)

I 1999

II 1999

III 1999

IV 1999

I 2000

II 2000

III 2000

IV 2000

I 2001

II 2001

III 2001

IV 2001

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II 2002

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8,800

8,900

9,000

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9,200

9,300

9,400

$0

$2

$4

$6

$8

$10

$12

$14

$16

AMP-FC

COWS

Figure 3

U.S. dairy product prices and all-milk price

AMP=All-milk price.

NDM=Nonfat dry milk.

GDP=Gross domestic product.

Source: USDA, Economic Research Service calculations based on USDA, National Agricultural Statistics Service data.

Quarter

Log [real prices (GDP deflator 2005=100)]

I 1999

II 1999

III 1999

IV 1999

I 2000

II 2000

III 2000

IV 2000

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II 2001

III 2001

IV 2001

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$0.10

$1.00

$10.00

$100.00

AMP

BUTTER CHEESE NDM

WHEY
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products (butter, cheese, NDM, and whey) are especially important because 

they are used to calculate Class III and Class IV prices,1 which are part of the 

ICEC’s forecasting activities reported in the WASDE.

Managing milk production and use becomes a crucial industry objective 

when milk and dairy product prices move in unexpected ways. Public policy 

focuses on the balance of production and use through Government entities 

like the Commodity Credit Corporation (CCC), which buys cheese, butter, 

and NDM, and recently, with concerted action by producers, such as with the 

herd retirement program of Cooperatives Working Together (CWT).2 

The dairy sector—from production to fi nal use, including trade and inventory balance—is complex (fi g. 4). The analysis highlights fi ve blocks within 

the dairy industry. The fi rst block deals with the production of farm milk, 

processing fl uid products and manufacturing cheese, NDM, butter, and whey 

from different economic agents in the sector—farmers and processors. The 

use of milk for dairy products beyond these four commodities is taken into 

account in other parts of the model when analyzing aggregate dairy sector 

components. Both farm milk and dairy products are measured in pounds. 

The second, third, and fourth blocks model aggregate quantities of dairy 

products, which help defi ne equilibrium in the dairy sector as a whole (fi g. 

5). These aggregates are measured on a milk equivalent fat or skim solids 

basis as pointed out previously. Notably, the milk equivalent measure, which 

is employed to construct the dairy product aggregates, also allows one to 

“deconstruct” dairy products back into the original milk they came from. 

Two important defi nitions or identities are usually employed when analyzing 

the dairy sector as a whole (these variables are measured on a milk-equivalent basis). 

• Total supply: Milk marketing (cow inventory multiplied by production 

per cow minus farm use of milk) plus beginning stocks and imports of 

dairy products. 

• Domestic commercial disappearance: Total supply minus ending 

commercial stocks, exports, and Government net removals. 

Combining these two identities leads to an expression for the difference 

between overall marketed production (aggregate domestic quantity supplied) 

and domestic commercial disappearance (aggregate domestic quantity 

demanded). Excess aggregate supply (or aggregate demand) becomes a 

1Information on class prices and 

specifi c price formulas for the year 

available at http://www.ams.usda.gov/

AMSv1.0/ams.fetchTemplateData.do?

template=TemplateG&navID=Industry

MarketingandPromotion&leftNav=Indu

stryMarketingandPromotion&page=Mil

kPrices&description=Prices&acct=dmk

tord.

2Balancing milk supply with demand 

(use) as a public policy objective means 

converting dairy products into milk 

equivalent units to aggregate them 

and to derive aggregate supply and 

demand for the dairy sector as a whole. 

Critics of aggregating dairy products 

into milk equivalent units point to 

issues like double counting because 

there is not complete agreement about 

which products can be considered dairy 

products and uncertainty about the milk 

content in some. However, because aggregation is performed by formula and 

the various components are updated 

regularly, any systematic errors will be 

corrected by statistical agencies eventually (Blayney, 2010). Nevertheless, 

distortions are implied in the aggregation. For example, variable construction procedures assume that imported 

products have the same composition 

of fats and skim solids as domestically 

manufactured ones. A classic exposition for the rationale behind aggregating into milk equivalents is offered in 

Miller (1989).

Table 1

Pairwise correlation of all-milk price and dairy product prices*

Dairy product All-milk price Cheese Butter NDM

Cheese 0.92* NA NA NA

Butter 0.56* 0.62* NA NA

NDM 0.68* 0.48* -0.003 NA

Whey 0.38* 0.21 0.05 0.60*

NDM=Nonfat dry milk. NA=Not applicable.

*Signifi cant at the 1-percent level.

Source: USDA, Economic Research Service estimates.
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Calves Farm milk

PAMP

Fluid grade Manufacturing 

use AMP

PM

PF

PB PCH PW

PN PB

PD > PI PD < PI

Separate Separate

Fluid milk Skim Cream Curds Whey Other

Nonfat dry 

milk

Butter Cheese Dry whey Class III

Class IV

Change in commercial inventories and Government purchases

IM Domestic commercial disappearance EX

Figure 4

Structure of the U.S. dairy sector

PAMP =All-milk price.

PF =Price fluid milk.

PM =Price milk used for dairy products. 

PN =Price nonfat dry milk.

PB =Price butter.

PCH =Price cheese.

PW =Price whey.

IM=Dairy product, commercial imports.

EX=Dairy product, commercial exports.

PD=Dairy products, domestic price.

PI=Dairy product, international price.

Source: USDA, Economic Research Service.
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combination of increases (decreases) in stocks, exports, and/or Government 

removals. Figure 5 illustrates aggregate demand and supply for the dairy 

sector as a whole together with the decomposition of excess aggregate 

demand and supply. The economic relations and patterns are used to specify 

a model of the dairy sector, which will be discussed after reviewing a critical 

selection of dairy forecasting models where the analysis goes beyond correlation and trends to formal modeling in which movements of variables are 

considered within a ceteris paribus (all other things equal) framework.

QS-QD = (EST-BST) + (EXP-IMP) – GOVNR1

Excess supply

Excess demand

QD-QS = (BST-EST) + (IMP-EXP) + GOVNR2

P*

Q*

S (Domestic aggregate supply)

D (Domestic aggregate demand)

Figure 5

U.S. dairy industry aggregate supply-demand

Price

Quantity

P=Price.

Q=Quantity.

S=Supply.

D=Demand.

EST=End of period stocks.

IMP=Imports.

GOVNR=Government net removals.

BST=Beginning of period stocks.

1

Excess supply ends up as increased stocks, trade surplus, and/or Government Commodity 

 Credit Corporation purchases.

2

Excess demand is met by decreases in stocks, trade deficit, and/or Government Commodity 

 Credit Corporation sales. 

Source: USDA, Economic Research Service.
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Comparison of a Selection of Relevant U.S. Dairy 

Sector Forecasting Models 

The ERS model can be distinguished from others, such as that of the USDA’s 

Agricultural Marketing Service (AMS) (2006), in that it is intended to 

provide short-run forecasts. AMS (2006), in contrast, approximates long-run 

equilibrium conditions of supply and demand and thus can be used for longrun projections. This type of model is more appropriate for policy analysis 

where the derived equilibrium multipliers determine the long-term effect 

associated with a permanent change of an exogenous variable.3 Bailey et al. 

(2006) and Bailey (2009) do, however, provide short-term models similar 

to ERS’s in that the dynamic system implied is not solved until the system 

achieves long-run equilibrium. 

The ERS model also differs from Jesse and Schuelke (2002), AMS (2006), 

Westcott (1986), and Premakumar and Chaudhary (1996) in that it uses 

a variety of methods to estimate the endogenous variables of interest and 

selects a “best” forecasting method, or one that most successfully forecasts 

the variable. In ERS’s model, the estimating methods employed are not 

assumed to be the best a priori; they could vary with changes in market 

conditions or the structure of the sector. Also, ERS uses SM and TS, specifi -

cally vector autoregression with exogenous variables (VARX).4 In VARX 

models, all variables included in a system of equations (both endogenous and 

exogenous) affect a particular endogenous variable. In a structural model, by 

contrast, economic theory plays a key role in the specifi cation of any particular equation in the system. 

ERS’s model is validated using ex-post forecasting, a procedure that permits 

the evaluation of the out-of-sample forecasting performance. ICEC requires 

forecasts for up to eight quarterly periods ahead of the current period. This 

report forecasts four quarters. Among studies reviewed here, only Westcott 

(1986) uses out-of-sample or ex-post forecast evaluation. The current ERS 

model only uses 1 year to validate the estimations and, unlike many other 

forecasting models of the U.S. dairy sector, the ERS model is quarterly 

rather than monthly. However, the quarterly timeframe captures the implicit 

seasonality of the sector and reduces the volatility that is more likely to 

appear when employing monthly data. Nevertheless, the use of a quarterly 

model limits the degrees of freedom to specify and estimate the model.5

As a result, ERS’s model specifi es only those variables required for the 

intended purpose—providing estimates required for the WASDE report—as 

endogenous.

ERS’s forecasting model is recursive, as is Jesse and Schuelke’s (2002), in 

that endogenous variables determined at one stage serve as determinants for 

the next stage. However, the recursiveness is in a block form where sets of 

endogenous variables determined simultaneously serve as determinants for 

the next set of equations. Thus, it is an integrated model that encompasses all 

major sectors of the dairy industry while remaining tractable. 

3Formally, a simultaneous system of 

equations (structural model) is one in 

which the endogenous variables in one 

equation may appear as an explanatory variable in another equation. A 

so-called reduced form is obtained 

when this system is solved so that all 

endogenous variables are expressed 

only in terms of the exogenous variables. The system, however, can have 

lagged endogenous variables, which are 

considered exogenous in this context. 

When the system is further solved so 

that every equation is expressed only 

in terms of the current values of the 

endogenous variables and a set of initial 

conditions, we have a long-run model 

(see Greene, 2008, pp. 389-94). A 

broader meaning of a structural model 

is that it is one derived from economic 

theory. In contrast, a more data-driven 

model, such as vector autoregression 

with exogenous variables (VARX), uses 

time series where any variable can be 

introduced just because it works. 

4In general, in vector autoregression models (both with and without 

exogenous variables or moving average components), all variables in a 

system of equations affect a particular 

endogenous variable. The restriction 

can be imposed so that variables with 

insignifi cant coeffi cients are set to zero. 

In an “econometric model,” in contrast, 

specifi cation of an equation plays an 

important role, and it is easier to test 

the particular equation’s conformity 

with economic theory. However, as 

Greene (2008, p. 695) notes, a vector 

autoregression model can be viewed as 

a particular kind of seemingly unrelated 

regression model with all variables 

included or as a reduced form of a 

simultaneous equations system. The 

distinction between the two approaches 

is blurred since one approach can be 

made to look exactly like the other. A 

more general distinction is one between 

behavioral (derived from a choice problem) and nonbehavioral models. 

5Models, such as generalized conditional autoregressive heteroskedasticity 

(GARCH), that Engle (2001) employed 

to analyze patterns of volatility clustering in fi nancial and similar types of 

data may be more appropriate to analyze monthly data in the dairy sector. 
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Model Specifi cation, Estimation, and Validation

Data Sources and Variable Construction 

Table 2 provides descriptive statistics for the variables employed in the 

econometric model, and table 3 presents the associated data sources. The 

endogenous variables constitute the set of variables required by ICEC to 

carry out its role. An ICEC-identifi ed data source is provided shortly before 

each monthly meeting. These variables constitute information that goes 

directly into the quarterly model to generate forecasts. Another set of exogenous variables not provided by ICEC also must be forecasted forward for 

the ERS model. For example, the model uses supply equations to forecast 

prices, which require forecasts of market quantities of cheese, whey, NDM, 

and butter. The model does not estimate price and quantity simultaneously6

for dairy products but it does estimate overall supply and demand for the 

dairy sector (see fi g. 5). The model specifi cation identifi es supply equations 

for manufactured dairy products so as to forecast their prices. The model also 

requires international prices to forecast forward exports, imports, and stocks 

that are crucial components of aggregate supply and demand (see fi g. 5). 

Except for the energy index (see Appendix A), the variables in table 2 require 

little manipulation from the sources specifi ed in table 3. The VARX methodology (see Appendix B) is employed to forecast this latter set of variables. 

Specifi cation

The SM model is composed of fi ve blocks of behavioral equations and identities (table 4). The fi rst block contains farm milk supply (equations 1.1 and 

1.2); dairy product—cheese, NDM, butter, and whey—inverse supply functions (equations 1.3-1.6); and an all-milk price equation (equation 1.7), which 

is an inverse-derived demand equation for farm milk. The second, third, and 

fourth blocks provide estimates of stocks, imports, exports, and net removals 

of aggregate dairy products on a fat basis and a skim solid basis (see fi g. 5) 

(equations 2.1, 2.2, 2.3, 3.1, 3.2, 3.3, 4.1, and 4.2). Finally, the fi fth block 

defi nes milk production and marketing (equations 5.1 and 5.2), aggregate 

supply (equation 5.3), domestic commercial disappearance (equation 5.4), 

and the overall balance or equilibrium (equation 5.5) (see fi g. 5). 

The VARX model structure is similar in structure to the SM. Endogenous 

and exogenous variables in each block are the same for both approaches. For 

VARX models, however, every variable directly infl uences every other variable. Two time series models are estimated—a restricted and an unrestricted 

version. In the restricted model, the insignifi cant coeffi cients from the unrestricted model have been eliminated.

The blocks are interrelated in a variety of ways in both the SM and the 

VARX models. At the specifi cation stage, for example, the same butter and 

NDM domestic prices used in block 1 are used in blocks 2 and 3 to calculate 

domestic-international price differentials or are used in block 4 to calculate 

domestic-support price differences. In the forecasting stage, the prices forecasted in block 1 are connected in the same way to blocks 2, 3, and 4 as in 

the specifi cation stage, (i.e., they are employed to calculate price differentials). Overall, there are 15 behavioral equations and 5 identities. 

6As more data becomes available, 

the model will be respecifi ed to contain 

both supply and demand for the dairy 

products in block 1.
9

A Quarterly Econometric Model for Short-Term Forecasting of the U.S. Dairy Industry / TB-1932

Economic Research Service/USDA

Table 2

Summary statistics for variables used in specifying quarterly dairy forecasting models, Q4/1998-Q1/2009

Defi nition and (variable) Mean

Standard 

deviation Defi nition and (variable) Mean

Standard 

deviation

Endogenous variables

Number of cows

(COW)

9143 88 Beginning stocks fat

(STBFAT)

9.98 2.19

Production per cow

(PPC)

4758 252 Imports fat

(IMPFAT)

1.21 0.21

Cheese price

(CHP)

1.31 0.23 Exports fat

(EXPFAT)

0.63 0.30

Butter price

(BTRP)

1.25 0.28 Net removals fat

(GOVNRF)

0.07 0.12

Nonfat dry milk (NDM) price

(NDMP)

0.94 0.22 Beginning stocks skims

(STBSS)

9.13 1.07

Whey price

(WHP)

0.24 0.10 Imports skims

(IMPSS)

1.15 0.18

All Milk Price

(AMP)

13.85 1.89 Exports skims

(EXPSS)

3.64 1.56

Net removals stock skims

(GOVNRS)

1.23 1.14

Exogenous and Auxiliary Variables

Price 16 percent Pr. Ration

(FC) 4.99 1.08 Gross domestic product

(GDP) 10617 762

Slaughter cow price

(SCP) 42.44 4.49 First quarter

(DQ1) 0.26 0.44

Cheese quantity

(CHQ) 2.46e+08 5.10e+07 Second quarter

(DQ2) 0.24 0.43

Whey quantity

(WHQ) 1.38e+08 2.65e+07 Third quarter

(DQ3) 0.24 0.43

NDM quantity

(NDMQ) 2.51e+08 7.39e+07 Time trend

(t) 24 13

Butter quantity

(BTRQ) 4.89e+07 8.83e+06 Time trend2

(t2) 728 651

Food industry wage

(WG)

8.57 0.69 Butter domestic supply price 

difference

(PRSPBTR)

0.38 0.25

Energy index

(EI)

1.56 0.41 Cheese domestic supply price 

difference

(PRSPCH)

-0.64 0.30

T-Bill

(TB)

2.86 1.85 Farm use of milk 

(FUSE)

0.30 0.02

Butter domestic

Int. price difference

(BTRPDI)

-0.21 0.85 NDM domestic

Int. price difference

(NDMPDI)

0.05 0.33

Source: USDA, Economic Research Service.
10

A Quarterly Econometric Model for Short-Term Forecasting of the U.S. Dairy Industry / TB-1932

Economic Research Service/USDA

Table 3

Data sources and variable units of measurement for quarterly dairy forecasting models1

Variable 

(units) Source (website)

Variable 

(units) Source (website)

COW 

(thousands)

Milk Production

(http://usda.mannlib.cornell.edu/usda/current/MilkProd)

STBFAT 

(ME)

Dairy at a Glance

(http://www.ers.usda.gov/Publications/ldp/)

PPC 

(pound/

head)

Milk Production

(http://usda.mannlib.cornell.edu/usda/current/MilkProd)

IMPFAT 

(ME)

Dairy at a Glance

(http://www.ers.usda.gov/Publications/ldp/)

CHP 

($/pound)

Dairy Product Prices

(http://usda.mannlib.cornell.edu/usda/current/

DairProdPr/)

EXPFAT 

(ME)** (http://www.fas.usda.gov/)

BTRP

($/pound)

Dairy Product Prices

(http://usda.mannlib.cornell.edu/usda/current/DairProdPr/)

GOVNRF 

(ME)

Dairy at a Glance

(http://www.ers.usda.gov/Publications/ldp/)

NDMP 

($/pound)

Dairy Product Prices

(http://usda.mannlib.cornell.edu/usda/current/

DairProdPr/)

STBSS 

(ME)

Dairy at a Glance

(http://www.ers.usda.gov/Publications/ldp/)

WHP 

($/pound)

Dairy Product Prices

(http://usda.mannlib.cornell.edu/usda/current/

DairProdPr/)

IMPSS 

(ME)

Dairy at a Glance

(http://www.ers.usda.gov/Publications/ldp/)

AMP 

($/CWT)

Agricultural Prices

(http://usda.mannlib.cornell.edu/usda/current/AgriPric/) 

EXPSS 

(ME)**

(http://www.fas.usda.gov/)

GOVNRS 

(ME)

Dairy at a Glance

(http://www.ers.usda.gov/Publications/ldp/)

TB 

(%) ***

ICEC GDP Def. 

**

ICEC

MILC 

($/cwt) *

Price Support—Milk Income Loss Contract

(http://www.fsa.usda.gov/FSA/

webapp?area=home&subject=prsu&topic=mpp-mi)

BTRPP 

($/mt) **

Dairy at a Glance

(http://www.ers.usda.gov/Publications/ldp/)

FC 

($/cwt)***

ICEC NDMPP 

($/mt.) **

Dairy at a Glance

(http://www.ers.usda.gov/Publications/ldp/)

SCP ($/

cwt)***

ICEC GDP2005

 ($) ***

ICEC

CHQ

 (lb.) **

Dairy Product Prices

(http://usda.mannlib.cornell.edu/usda/current/

DairProdPr/)

DQ1 

(D. Var.)

N/A

WHQ 

(lb.) **

Dairy Product Prices

(http://usda.mannlib.cornell.edu/usda/current/

DairProdPr/)

DQ2

(D. Var.)

N/A

NDMQ 

(lb.) **

Dairy Product Prices

(http://usda.mannlib.cornell.edu/usda/current/

DairProdPr/)

DQ3

(D. Var)

N/A

BTRQ

(lb.) **

Dairy Product Prices

(http://usda.mannlib.cornell.edu/usda/current/

DairProdPr/)

t 

(trend)

N/A

WG

($/hour)**

Average Hourly Compensation Production Workers

(www.bls.gov)

t2 

(trend sq)

N/A

EI (N/A) ** ERS from data from www.eia.gov

mt=Metric ton. ME=Milk equivalent.

* External forecast (documented in report). ** Internal forecast and/or calculation (documented in report). *** Forecast provided by the Interagency Commodity Estimates Committee. 

Source: USDA, Economic Research Service.
11

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Economic Research Service/USDA

Table 4

Specifi cation of U.S. quarterly dairy model, Q4/1998-Q1/2009

Milk production and inverse supply for dairy products block system statistics

Method Equation 

name

RMSE 

(in sample) FSTAT/ CHI2 Probability

OLS

2SLS

3SLS

SURE

1.1 COW

0.0020

0.0020

0.0018

0.0018

130.07

130.07

1191.41

1191.41

0.0000

“

1.2 PPC

0.012

0.012

0.011

0.011

122.45

122.45

791.28

791.28

“

“

1.3 CHP

0.12

0.12

0.11

0.11

5.14

5.14

51.35

51.35

“

“

1.4 BTRP

0.11

0.11

0.10

0.10

10.38

11.38

133.99

120.96

“

“

1.5 NDMP

0.13

0.13

0.12

0.12

8.98

8.98

91.44

90.70

“

“

“

1.6 WHP

1.7 AMP

0.17

0.17

0.15

0.15

0.027

0.028

0.027

0.024

16.29

16.29

219.80

218.56

110.17

103.39

1148.56

1163.64

“

“

RMSE=Root-mean-square error.

FSTAT/CHI2= F-Statistic (OLS, 2SLS)/Chi Squared (3SLS,SURE).

OLS=Ordinary least squares.

2SLS=Two-stage least squares.

3SLS=Three-stage least squares.

SURE=Seemingly unrelated regression.

Note: Rank condition is satisfi ed for each of the seven equations in this block (i.e. the system 

is identifi ed and hence unique values of the structural parameters can be derived from the 

reduced form of the system).

Source: USDA, Economic Research Service estimates.

Continued—
12

A Quarterly Econometric Model for Short-Term Forecasting of the U.S. Dairy Industry / TB-1932

Economic Research Service/USDA

The specifi cation of the SM and the VARX models only employed data from 

Q4/1998 to Q1/2009. Model equations were specifi ed in logarithmic form, 

permitting the determination of elasticities from the estimated parameters and 

a way to assess whether their signs made theoretical sense, especially in the 

case of the SM. Additionally, the log transformation of the data may have a 

variance stabilizing effect (Lutkepohl and Xu, 2009).7 Once specifi ed, the 

entire system of 15 equations grouped in 5 blocks was estimated recursively 

using 6 different estimation methods that generated 6 different point forecasts for each endogenous variable. The limited number of observations used 

in the estimations did not permit estimation of the entire structure simultaneously. Rank conditions were used to identify each block so that unique 

values of structural parameters could be derived from the reduced form of the 

system.

These estimates were compared across methods as well as with the point estimates agreed upon by ICEC. The fi nal specifi cation (estimation) of the SM 

was performed using SURE,8 the method with the lowest root mean squared 

error (RMSE) compared with the three other model estimation techniques—

OLS, 2SLS, and 3SLS. The VARX model was also estimated in fi ve blocks 

parallel to the SM (i.e., the same endogenous and exogenous variables were 

employed to estimate each block). Moreover, the VARX model was used to 

estimate two models—one restricted9 and one free. The forecasting performance of each method was subsequently determined.

Table 4 presents the SM specifi cation of the model. The endogenous variables in this table constitute the monthly forecasting needs mentioned previously. The forecasts are quarterly estimates of these variables. The variables 

in the different equations were selected according to the role each equation 

plays within the overall model. Each block was estimated by OLS, 2SLS, 

3SLS, and SURE. The results presented correspond to the method that 

minimized in-sample RMSE in a majority of cases, SURE, relative to other 

methods. Parameter estimates are also shown for comparison purposes across 

methods for equations 1.1 and 1.7. The parameters in these two important 

equations are fairly stable across methods. 

Equation 1.1 represents cow inventory. Total milk production is a refl ection of the number of cows and their productivity (defi ned as production 

per cow) (see table 4). As in any production system, inputs are transformed 

into outputs using a given technology, in a given environment, and within a 

specifi c set of constraints. Two important factors were modeled explicitly in 

this set of equations—feed costs and the all-milk price. Coeffi cients with a 

t-value greater than 1 were considered for specifi cation purposes to decrease 

type II errors as suggested in Kennedy (2008, p. 90). Also, on theoretical 

grounds, some variables with t-values less than 1 were considered.

The number of cows is expected to be a function of lagged cow inventory 

(COW), lagged price of milk (AMP), and lagged feed costs (FC). Real values 

for these price and cost variables are determined by dividing current magnitudes by the GDP defl ator. In the case of the AMP variable, direct payments, 

such as those from the Milk Income Loss Contract (MILC) program, were 

added before converting the all-milk price variable to a real value to model 

the farmer’s incentive or effective price. The variable AMP is expected to 

have a positive lagged effect on the number of cows. It can be expected that 

7Unit root test resulted in some 

variables being nonstationary. Typical 

characteristics of nonstationary data 

include: mean that varies widely, 

variances that explode, and shocks 

that appear permanent are not, with 

the implication that forecasts might be 

misleading. Improvements to the ERS 

forecasting model will evaluate the 

costs and benefi ts of addressing this 

issue by implementing nonstationary 

econometric methods.

8For variance structure, see Appendix 

C.

9Note that this exclusion could 

compromise the ceteris paribus context 

if the model were developed further and 

used for policy analysis.
13

A Quarterly Econometric Model for Short-Term Forecasting of the U.S. Dairy Industry / TB-1932

Economic Research Service/USDA

as FC (or approximately 60 percent of dairy farm operating costs according 

to the Agricultural Resource Management Survey) increases, all other things 

being equal, there will be a lagged decrease in herd size. Both AMP and FC 

effects are related to biological lags, culling, and replacements apart from 

the other variables. In the double logarithm specifi cation of the equations, 

the coeffi cients can be interpreted as elasticities. Most importantly, the coeffi cient for AMP is 0.01, meaning that a 1-percent increase in AMP will result 

in a 0.01-percent increase in cow numbers after two quarters. The coeffi -

cient FC is -0.007, implying that a 1-percent increase in FC will result in a 

0.007-percent reduction in cow numbers after two quarters. Also, a 1-percent 

increase in slaughter cow price will result in a 0.0009-percent decrease in 

herd size after two quarters. Cow numbers then follow a cyclical behavior 

related to the AMP/FC price ratio (see fi g. 2) using a ceteris paribus assumption and not just as a correlation. The supply response is quite inelastic, 

refl ecting the changing structure of U.S. dairy farms. Larger farms with more 

capital costs need more milk to cover expenses and therefore have a lower 

supply response. 

Equation 1.2 captures the combined effect of new technologies and factors 

like scale economies on cow productivity. Generally, larger operations have 

higher cow productivity, represented by the variable production per cow 

(PPC). The effect of AMP and FC on PPC is positive and negative, respectively. A 1-percent rise in AMP will result in a 0.03-percent increase in 

production per cow, and a 1-percent increase in FC leads to a 0.04-percent 

decrease in PPC. The variable DQ2 captures the effect of the spring fl ush 

on cow productivity. This seasonal effect means that spring productivity is 

higher by about 0.03 percent relative to the fourth quarter—the excluded 

“base” quarter—and holding other factors constant. The t term indicates that 

PPC has been increasing throughout the period at about 0.006 percent per 

year (see fi g. 1). 

Equations 1.3, 1.4, 1.5, and 1.6 represent inverse (price dependent) supply 

equations for cheese, NDM, butter, and whey, respectively. As such, they 

have some theoretical properties. An inverse supply equation is positive 

with respect to its own quantity and positive with respect to input prices.10

The sign of related products will depend on whether they complement each 

other in production. A positive sign, as is the case in all the inverse supply 

equations in this study, refl ects that the products are either byproducts or that 

there is some interrelation in their production as seen in the dairy industry. In 

the estimation and validation stage, a simultaneous system structure acknowledges explicitly that the production processes of these goods are related. 

The macro input prices—wages, interest rates, and energy—as well as the 

dairy commodity quantities have a strong positive effect on dairy commodity 

prices, though only the cheese quantity is signifi cant. 

The equation for AMP (equation 1.7) is an inverse-derived demand for farm 

milk. This type of demand results from the fi nal demand (for cheese, NDM, 

etc.) that farm milk produces. Hence, a higher dairy product price indicates 

a short dairy product supply relative to demand, which in turn signals a need 

for more milk to make dairy products. Hence, commodity prices (inverse to 

dairy product quantity) are positive derived demand shifters. The effect of the 

price of cheese on AMP is the largest among all dairy products, followed by 

NDM and butter. Whey, a byproduct of cheese production, enters the equa10According to Varian (1992, p. 216), 

an inverse supply function “measures 

the price that must prevail in order for a 

fi rm to fi nd it profi table to supply a given amount of output.” In a one output 

and one input case, for example, if the 

price of the input increases, the price 

of the output needs to increase for the 

fi rm to keep providing the same level of 

output, ceteris paribus. Also, given that 

the output price is greater than the average variable cost, and that marginal cost 

is increasing, an increase in the level of 

output will need a higher output price, 

all other things equal. Hence, the output 

quantity and input price coeffi cients are 

positive in an inverse supply function. 
14

A Quarterly Econometric Model for Short-Term Forecasting of the U.S. Dairy Industry / TB-1932

Economic Research Service/USDA

Table 4

Specifi cation of U.S. quarterly dairy model, Q4/1998-Q1/2009—continued

SURE Estimates of behavioral equations1 block 1

1.1 Cow inventory (COW):

2

1 12 23 24 25 26 7 ln ln ln ln ln ln COW COW COW AMP FC SCP t t t t t tt t = + + + + ++ + β β β β β ββ β − − −− − ο

OLS 1.18 -0.44 0.01 -0.007 -0.0009 -0.0007 0.00002 2.38

 (8.11) (-2.95) (2.54) (-1.90) (-2.16) (-2.48) (2.93) (2.84)

2SLS 1.17 -0.44 0.01 -0.007 -0.009 -0.0007 0.00002 2.38

 (8.11) (-2.95) (2.54) (-1.90) (-2.16) (-2.48) (2.93) (2.84)

3SLS 1.15 -0.44 0.01 -0.005 -0.01 -0.0007 0.00002 2.69

 (9.60) (-3.67) (2.81) (-1.68) (-2.71) (-2.83) (3.37) (3.94)

SURE 1.17 -0.45 0.01 -0.005 -0.01 -0.0006 0.00002 2.56

 (9.80) (-3.74) (2.81) (-1.70 (-2.69) (-2.60) (3.13) (3.75)

1.2 Production per cow (PPC):

1 22 23 24 5 ln ln ln ln 2 PPC PPC AMP FC DQ t tt tt = + + + ++ β β β β ββ − −− ο

 -0.36 0.03 -0.04 0.03 0.006 11.39

 (-4.82) (1.58) (-2.62) (7.19) (16.29) (17.89)

1.3 Inverse cheese supply (CHP):

1 12 23 24 25 46 ln ln ln ln ln ln 1 CHP BTRP WG EI TB CHQ DQ t t tt t t = + + + + ++ β β βββ β − −− − −

 0.56 3.47 0.52 0.11 0.43 -0.07

 (5.75) (2.86) (1.64) (2.56) (2.25) (-2.22) 

2

7 8 o ββ β t t + +

 -0.05 0.001 -15.60 

 (-2.86) (3.31) (-3.88)

1.4 Inverse supply butter (BTRP):

1 2 3 456 4 ln ln ln ln ln ln ln BTRP CHP NDMP WG EI TB BTRQ t t t ttt t = + + +++ + β β β βββ −

1.22 -0.58 3.71 1.42 0.18 0.07

 (8.55) (-4.28) (3.07) (4.35) (3.12) 0.72)

2

7 8 o ββ β t t + +

 -0.05 0.0009 -9.48

 (-3.38) (2.46) (-2.90)

1.5 Inverse supply NDM (NDMP): 2

1 12 13 14 15 46 7 ln ln ln ln ln ln NDMP CHP WG EI TB NDMQ t t t t tt t t = + + + + ++ + β β β β β ββ β − −− − − o

0.47 6.09 0.73 0.39 0.13 -0.07 0.002 -16.22

 (3.66) (4.36) (1.86) (7.61) (1.43) (-4.02) (5.23) (-4.69)

1.6 Inverse supply whey (WHP):

1 22 13 14 15 16 1 ln ln ln ln ln ln WHP WHP BTRP WG EI TB WHQ t t t tttt = + + ++++ β β β βββ − − −−−−

 -0.43 0.78 13.16 1.66 0.93 0.33

 (-4.16) (6.18) (7.29) (3.65) (11.97) (1.58)

2

7 8 o ββ β t t + +

 -0.07 0.003 -38.59

 (-3.16) (5.72) (-7.53)

Continued—
15

A Quarterly Econometric Model for Short-Term Forecasting of the U.S. Dairy Industry / TB-1932

Economic Research Service/USDA

Table 4

Specifi cation of U.S. quarterly dairy model, Q4/1998-Q1/2009—continued

SURE Estimates of behavioral equations1 block 1

1.7 All-milk price (AMP): 

1 12 13 4 5 1 ln ln ln ln ln ln AMP COW PPC NDMP CHP WHP t t t ttt = + + ++ + β β β ββ − − −

OLS -2.37 -0.52 0.20 0.48 0.04

 (-2.59) (-3.68) (4.50) (8.35) (1.70)

2SLS -2.23 -0.53 0.16 0.53 0.06

 (-2.44) (-3.86) (3.16) (8.46) (2.14)

3SLS -1.46 -0.58 0.15 0.60 0.06

 (-1.98) (-5.17) (3.74) (12.04) (2.77)

SURE -1.66 -0.56 0.18 0.52 0.05

 (-2.22) (-4.84) (5.17) (11.22) (2.54)

 

OLS 0.04 0.16 28.30

 (1.24) (3.04) (3.35)

2SLS 0.02 0.14 27.55

 (0.55) (2.65) (3.27)

3SLS -0.009 0.10 20.91

 (-0.28) (2.33) (3.08)

SURE 0.03 0.13 22.30

 (1.15) (2.97) (3.22)

1For OLS and 2SLS, t statistics are shown in parentheses; for 3SLS and SURE, z statistics.

Note: Parameter estimates for other than preferred method are shown.

Beginning inventory, imports, and exports (fat basis) block 

System statistics

Method Equation name RMSE 

(in sample) FSTAT/ CHI2 Probability

OLS

2SLS

3SLS

SURE

2.1 STBFAT 0.12

0.12

0.11

0.11

27.60

27.60

96.30

96.30

0.0000

0.0000

0.0000

0.0000

“

2.2 IMPFAT 0.16

0.16

0.17

0.17

7.45

7.45

16.10

16.10

0.0009

0.0003

0.0001

0.0001

“ 2.3 EXPFAT 0.30

0.30

0.29

0.29

4.94

4.94

10.34

10.34

0.0090

0.0090

0.0060

0.0060

RMSE=Root-mean-square error.

FSTAT/CHI2= F-Statistic (OLS, 2SLS)/Chi Squared (3SLS,SURE).

OLS=Ordinary least squares; 2SLS=Two-stage least squares; 3SLS=Three-stage least squares.

SURE=Seemingly unrelated regression.

Continued—

6 71 ln ββ β BTRP FC t to + + −
16

A Quarterly Econometric Model for Short-Term Forecasting of the U.S. Dairy Industry / TB-1932

Economic Research Service/USDA

tion with a lag. The elasticities’ magnitudes parallel the correlations shown in 

table 1 and fi gure 3 in a non ceteris paribus framework, a likely refl ection of 

their share of use as well as the underlying market effects like the interaction 

of farm supply and processor demand for dairy products. This trend translates 

into an increase in derived demand for milk (rightward shift). The supply 

curve in this context comes from equations 1.1 and 1.2. 

In equation 1.7, total farm milk processed is decomposed into PPC and 

number of cows, and their respective price (AMP) elasticities show the 

effect of productivity on AMP as being very inelastic and that of the number 

of cows as very elastic (i.e., an increase in cow productivity has a larger 

downward effect on AMP than a comparable percentage change increase in 

the dairy herd, all other things equal). Feed price increases, ceteris paribus,

decrease supply (less milk supply at the same price—leftward supply shift), 

requiring a higher AMP in this equation to equilibrate supply with derived 

demand. 

Blocks 2, 3, and 4 represent aggregate quantities of dairy products and, as 

such, discussing them in connection to fi gures 4 and 5 is useful. Equations 

2.1, 2.2, and 2.3 have the variable BTRPDI in common, which is the real 

difference between domestic and international butter prices and which affects 

movements in stocks, imports, and exports (see fi g. 4). An increase in the 

difference between domestic and international prices may result in a tightening of the domestic market relative to the international market, which will 

lead to a reduction in stocks on a fat basis. The same analysis could be done 

for imports, where an increase in BTRPDI will lead to an increase in imports 

and a decrease in exports. A parallel analysis could be done for the effect 

of NDMPDI—the real difference between domestic and international NDM 

prices—on stocks, imports, and exports when examining equations 3.1, 3.2, 

and 3.3. By using domestic prices for butter and NDM, an explicit connection is made to block 1 of the model. 

Net removals for fat and skims are estimated using a Tobit regression with 

lower and upper bounds equal to the maximum positive observation and 

maximum negative observation respectively.11 Government net removals 

could be viewed as a “corner solution” as are purchases, such as cars and 

houses, in which an individual’s consumption might be positive one year 

and zero at other times. Since net Government purchases of dairy products 

can be positive or negative, a two-limit Tobit model is employed. The variables PRSPBTR and PRSPCH represent the differences between domestic 

and support butter and NDM prices. Equations for both these variables show 

that an increase in GDP leads to a decrease in removals, and that PRSPCH is 

signifi cant in explaining a decrease in net removals of skim solids. Details of 

this estimation are presented in Appendix D. Results of an alternative way to 

model net removals—hurdle or two-part model—are presented as well. 

Figure 5 summarizes the aggregate supply and demand system for the dairy 

sector. Net export, inventory, and Government purchases bring the system 

into balance in each period. If equilibrium is not desirable because of low 

average prices for dairy products and AMP, then a policy like the most 

recent herd retirement program from the National Milk Producers Federation, 

Cooperatives Working Together (CWT)12 would shift aggregate supply to 

the left and dairy prices would increase. Similarly, an increase in net exports, 

11The censoring values for the lower 

and upper limit are the minimum and 

maximum observed magnitudes in 

the data. This is STATA’s default if no 

numerical limits are specifi ed.

12CWT is a voluntary farmers’ 

organization whose stated objective is 

to “strengthen and stabilize milk prices 

by balancing supply and demand.” See 

http://www.cwt.coop for more information.
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Table 4

Specifi cation of U.S. quarterly dairy model, Q4/1998-Q1/2009—continued

SURE estimates of behavioral equations block 2

2.1 Beginning stocks (STBFAT):

1 1 21 33 ln t t o STBFAT BTRPDI DQ DQ = +++ β βββ −

 -0.21 -0.22 0.20 2.41

 (-5.73) (-5.23) (4.55) (83.31)

2.2 Imports (IMPFAT):

1 4 2 1 ln ln t tt o IMPFAT IMPFAT BTRPDI = ++ β ββ − −

 0.36 0.17 0.04

 (1.95) (3.05) (0.73)

2.3 Exports (EXPFAT):

1 1 2 3 ln ln EXPFAT EXPFAT BTRPDI t tt = ++ β ββ − − o

 0.31 -0.20 -0.27

 (2.07) (-2.02) (-2.85)

Beginning inventory, imports, and exports (skim solids basis) block

System statistics

Method Equation name

RMSE 

(in sample) FSTAT/ CHI2 Probability

OLS

2SLS

3SLS

SURE

3.1 STBSS 0.072

0.072

0.069

0.069

11.20

11.20

25.21

25.21

0.0000

0.0000

0.0000

0.0000

“

3.2 IMPSS 0.138

0.138

0.132

0.132

8.35

8.35

19.75

19.75

0.0004

0.0004

0.0001

0.0001

“ 3.3 EXPSS 0.146

0.146

0.140

0.140

124.94

124.94

272.39

272.39

0.0000

0.0090

0.0000

0.0000

RMSE=Root-mean-square error.

FSTAT/CHI2= F-Statistic (OLS,2SLS)/Chi Squared (3SLS,SURE).

OLS=Ordinary least squares.

2SLS=Two-stage least squares.

3SLS=Three-stage least squares.

SURE=Seemingly unrelated regression.

Note: Rank condition is satisfi ed for each of the three equations in this block.

Continued—
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Table 4

Specifi cation of U.S. quarterly dairy model, Q4/1998-Q1/2009—continued

SURE estimates of behavioral equations block 3

3.1 Beginning stocks (STBSS):

1 1 2 ln ln 3 t to STBSS NDMPDI DQ = ++ β ββ −

 -0.16 0.10 2.20

 (-2.87) (3.86) (174.36)

3.2 Imports (IMPSS):

1 4 2 1 ln ln ln tt t o IMPSS IMPSS NDMPDI =+ + ββ β − −

 0.62 0.21 0.04

 (4.20) (1.92) (1.27)

3.3 Exports (EXPSS):

1 1 2 3 ln ln ln EXPSS EXPSS NDMPDI tt t =+ + ββ β − − o

 0.79 -0.33 0.29

 (8.51) (-1.88) (2.42)

Continued—

Table 4

Specifi cation of U.S. quarterly dairy model, Q4/1998-Q1/2009—continued

Net removals fats and skim solids block 4

4.1 Net removals fat basis (GOVNRF):

1 12 23 4 15 2 ln GOVNRF GOVNRF GOVNRF PRSPBTR PRSPBTR GDP t t t t tt = + + + ++ β β β β ββ − − − − o

 0.68 -0.43 -0.09 -0.06 -0.00007 0.86

(4.21) (-2.87) (-1.40) (-0.79) (-2.92) (3.08) 

Censored: lower bound -0.1, upper bound 0.6

Log-likelihood (LL) = 39.22

Alternative two-part specifi cation:

 Part 1:

 Decision: Decision to remove fats (DRF = 1) if GOVNRF  0

 Probit regression

11 12 13 2 ln DRF GOVNRF PRSPBTR GDP = + ++ β β ββ t tt − −− o

 12.22 2.50 -0.001 0.86

 (1.45) (1.26) (-1.87) (3.08)

LL= -12.12 

 Part 2:

 Outcome level (given DRF = 1) 

GOVNRF GOVNRF GOVNRF PRSPBTR t t t to = + ++ β β ββ 1 12 23 − −

 0.76 -0.36 -0.09 0.10

 (4.03) (-1.97) (-1.40) (2.48)

Joint decision and outcome LL = 15.30. 

Continued—
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Table 4

Specifi cation of U.S. quarterly dairy model, Q4/1998-Q1/2009—continued

Net removals fats and skim solids block 4

4.2 Net removals block skim solids basis (GOVNRS):

Tobit regression

1 12 23 4 2 ln GOVNRS GOVNRS GOVNRS PRSPCH GDP t t t tt = + + ++ β β β ββ − − − o

 0.72 -0.48 -2.17 -0.0003 4.28

 (5.00) (-3.69) (-4.08) (-1.61) (2.46)

Censored: lower bound -0.4, upper bound 3.5

LL = -38.23

Alternative two-part specifi cation:

 Part 1:

 Decision: Decision to remove skim solids (DRS = 1) if GOVNRS  0

 

 Probit regression

1 12 2 ln DRS PRSPCH GDP t tt = ++ β ββ − − o

 -3.27 -0.002 27.88 

 (2.01) (-2.01) (2.43)

LL= -6.53 

 Part 2:

 Outcome level (given DRS = 1) 

GOVNRS GOVNRS GOVNRS PRSPCH tt t o = + ++ β β ββ 1 22 23 1 − −

0.66 -0.34 -2.52 1.57 

(4.28) (-2.47) (-3.78) (5.18)

Joint decision and outcome LL = -38.29

Continued—

Table 4

Specifi cation of U.S. quarterly dairy model, Q4/1998-Q1/2009—continued 

Milk production, marketing, total supply, domestic commercial disappearance and overall balance (fat and skim solid basis) 

identities block 5 

5.1 Milk production (Milk):

* MILK COW PPC t tt =

5.2 Marketing (MKT):

MKT MILK FUSE t tt = −

5.3 Total supply (TS):

TS MKT STB IMP ti i FAT SS ( ,) = = =+ + t ti i FAT SS ( ,) ( ,) ti i FAT SS =

5.4 Domestic commercial disappearance (DCD)

DCD TS STE EXP GOVNR ti i FAT SS ti i FAT SS ti i FAT SS ti i FAT SS ti i FAT SS ( ,) ( ,) ( ,) ( ,) ( ,) == = = = =− − −

5.5 Overall balance

( ,) ( ,) ( ,) ( ,) 0 ( )( ) ( ) MILK FUSE DCD GOVNR t t ti i FAT SS ti i FAT SS ti ti i FAT SS ti ti i FAT SS = − − − +− +− == = = STB STE IMP EXP

Note: Total supply, domestic commercial disappearance, and overall balance are in both a fat and skim solids basis and are calculated independently.

Source: USDA, Economic Research Service estimates.
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which programs like the current CWT’s export promotion program are 

designed to encourage, would shift aggregate demand right. The sum of the 

production consumption balance, net inventory change, net exports, and net 

removals is zero in every period. The increase in domestic production over 

domestic consumption results in a combination of increases of product inventories, net exports, or net removals (fi g. 5). 

Estimation and Validation

The following methods were employed to estimate the dairy quarterly 

model: Ordinary least squares (OLS), two-stage least squares (2SLS), seemingly unrelated regression (SURE), three-stage least squares (3SLS), and 

constrained and unconstrained vector autoregression with exogenous variables (VARX). The OLS, 2SLS, SURE, and 3SLS approaches rely on the 

least-squares method. The VARX procedure is estimated by iterated seemingly unrelated regression. OLS and 2SLS are single equation methods. 

SURE, 3SLS, and VARX use a system of equations, VARX specifi cally uses 

a time-series method. The only approach that clearly dominates is VARX, 

although this superiority is not complete, and it might change. The robustness 

of the result has been maintained through several iterations of this report, 

where the data has been revised as more observations were added.

Single equation methods do not account for potential error correlations across 

equations, making them asymptotically less effi cient than full information 

methods. Although single equation methods are not necessarily inferior in 

all situations, system methods are more sensitive to specifi cation error since 

errors are carried across equations. Moreover, 2SLS may be preferred to 

3SLS when a small number of observations is available. Because ERS’s 

model is quarterly as opposed to monthly, this last consideration is especially 

important. 

As mentioned above, the ex-post forecast evaluation is conducted by 

comparing projections for Q2/2009 to Q1/2010 endogenous variables to 

actual values for them. Forecasts are evaluated two ways: in terms of their 

accuracy and their ability to predict turns in the data. Root mean square error 

(RMSE) and Theil’s U (another measure of forecasting quality) are two 

statistics employed to assess accuracy:13

1 2 ( ) ˆ t t t

RMSE y y

nο = − ∑

2

2

(1/ ) ( )ˆ

(1/ )

t t

t t

n yy

U

n y

ο

ο

∑ −

= ∑

The variables n yy , , ˆ ο correspond to the number of periods being forecasted, 

actual values, and forecast values, respectively. The formulation of Theil’s U

is equivalent to U2 (Theil, 1966, chapter 2). 

13Accuracy and turn equations, 

except percentage of correct turn measures, come from Greene (2008, pp. 

101-2).
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Turns in a given variable occur when the sign of the difference from quarter 

t to quarter t+1 changes. So if, for variable y, the signs of the difference 

1 1 ( ) t tt Δy yy + + = − and 1 ( ) t tt Δy yy = − − are different, we have a turn in 

the data. A measure of percentage of correct turns is the rate of correct turn 

predictions to total turn predictions. In addition, a Theil U measure that 

tracks the turns in the data is: 

2

2

(1/ ) ( ) ˆ

(1/ ) ( )

t t t

t t

n yy

U

n y

Δ

Δ Δ

Δ

∑ −

= ∑

where 1 1 ( )/ t tt t Δy yy y = − − − and 1 1 ˆ ˆ ( )/ t tt t Δy yy y = − − − .

Tables 5 and 6 present the validation results for the quarterly dairy model, 

illustrating the comparison of the forecasts for the various variables derived 

from the different models and the ICEC based on the accuracy and turn 

statistics presented previously. The accuracy and turning statistics are 

computed for four periods. A lower RMSE and a lower Theil’s U point to 

a better—more accurate—forecast. A higher RMSE and a lower Theil’s U 

indicate a better performance in predicting turning points in the data. 

The various models are “coded” respectively as: 

• A (restricted) and B (unrestricted): vector autoregression with exogenous 

variables (VARX); 

• C: ordinary least squares (OLS); 

• D: two-stage least squares (2SLS); 

• E: seemingly unrelated regression (SURE); 

• F: three-stage least squares (3SLS); and 

• I: the ICEC forecasts. 

Forecasts by the ICEC, as mentioned above, are a combination of expert 

judgment and econometric model results. Forecasts for A, B, C, D, E, and 

F are derived only from ERS’s econometric analysis. AV represents the 

average of the three best forecasts. In this connection, Timmermann (2006, 

p.181) noted that simple combination schemes like arithmetic averages often 

do better “...than more sophisticated rules relying on estimating optimal 

weights.” 

Tables 5 and 6 show the forecast evaluation results for the various blocks 

of the dairy quarterly model. The three best forecasts, their average, and 

the ICEC combined forecast are ordered from best to worst for each variable. The unconstrained VARX tends to forecast prices and the number 

of cows better (both in terms of accuracy and correct turns predicted) than 

other models and combining forecasts, in some instances, improve forecasts 

as shown for whey (see table 5). Other methods (both single- and multipleequation) outperform VARX when forecasting elements of the commodity 

balance sheets (see table 6). 
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Table 5

Block 1 forecast evaluation

Variable Method1 RMSE Theil U-accuracy Method1 Percent correct turns Theil U-turns 

Number of cows B 1238 0.000015 B 67 0.30

I 1552 0.000019 I 67 0.38

A 3826 0.000046 A 67 0.92

AV 3918 0.000047 AV 33 0.95

E 12254 0.000146 E 33 2.98

Production per cow C 136 0.0000051 C 100 0.0072

AV 175 0.0000066 AV 100 0.1958

D 186 0.0000070 D 100 0.0099

E 220 0.0000082 E 100 0.012

I 3920 0.0001465 I 67 0.21

Cheese price I 0.002 0.0012 I 100 0.03

AV 0.010 0.0054 AV 67 0.12

B 0.018 0.0099 B 33 0.28

A 0.034 0.0182 A 33 0.50

F 0.107 0.0578 F 67 1.47

Butter price I 0.0002 0.0002 I 33 0.04

AV 0.0036 0.0022 AV 33 0.41

A 0.0124 0.0075 A 33 1.34

B 0.0142 0.0086 B 33 1.53

E 0.0154 0.0093 E 33 1.84

NDM price AV 0.005 0.005 AV 100 0.30

I 0.011 0.011 I 100 0.56

B 0.019 0.019 B 33 0.94

A 0.042 0.042 A 33 1.90

F 0.057 0.057 F 67 2.51

Whey price B 0.0020 0.020 B 67 0.41

AV 0.0022 0.022 AV 33 0.64

C 0.0043 0.042 C 33 1.12

D 0.0045 0.044 D 33 1.15

I 0.0072 0.071 I 67 1.27

All-milk price I 1.03 0.005 I 33 0.31

B 1.71 0.009 B 67 0.60

A 2.08 0.011 A 33 0.67

AV 2.81 0.012 AV 67 0.85

E 7.32 0.038 E 67 2.46

NDM = Nonfat dry milk. RMSE = Root mean squared error. AV = Average of three best forecasts.

1Methods’ key corresponds to A, constrained vector autoregression with exogenous variables (VARX); B, unconstrained VAR; C, ordinary least 

squares (OLS); D, two-stage least squares (2SLS); E, seemingly unrelated regression (SUR); F, three-stage least squares (3SLS); and I, 

forecasts by ICEC. The specifi c version of the VARX model employed here contains no exogenous variables; previous versions of this research, 

as well as ERS’s regular forecasting activities, usually employ them.

Sources: USDA, World Agricultural Outlook Board and USDA, Economic Research Service estimates.
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Table 6

Blocks 2, 3, and 4 forecast evaluation

Variable Method1 RMSE Theil U-accuracy Method1 Percent correct turns Theil U-turns

FATS

Beginning Stocks E 0.40 0.0023 E 100 0.16

F 0.40 0.0023 F 100 0.16

AV 0.41 0.0024 AV 100 0.17

D 0.44 0.0025 D 100 0.18

I 1.51 0.009 I 100 0.66

Imports AV 0.0150 0.0146 AV 0 0.63

E 0.0151 0.0152 E 0 0.63

F 0.0151 0.0052 F 0 0.63

C 0.0154 0.0155 C 0 0.64

I 0.0161 0.016 I 67 0.70

Exports I 0.03 0.02 I 67 1.0

C 0.0628 0.051 C 0 2.25

D 0.0628 0.051 D 0 2.25

AV 0.0638 0.052 AV 0 2.28

E 0.0659 0.052 E 0 2.35

Net Removals B 0.039 0.40 B 33 0.466

A 0.0407 0.42 A 33 0.488

AV 0.0414 0.43 AV 33 0.493

F 0.045 0.47 F 33 0.530

I 0.048 0.50 I 33 0.577

SKIM SOLIDS

Beginning Stocks I 0.2 0.002 I 33 0.77

B 0.6 0.004 A 33 2.4

A 0.8 0.006 B 33 1.9

AV 1.3 0.009 AV 33 3.9

C 3.2 0.02 C 33 10

Imports I 0.015 0.020 I 67 0.37

A 0.018 0.024 A 67 0.44

B 0.020 0.026 B 67 0.48

AV 0.022 0.030 AV 33 0.52

C 0.045 0.061 C 33 1.10

Exports E 0.396 0.0143 E 33 0.91

F 0.396 0.0143 F 33 0.91

AV 0.398 0.0144 AV 33 0.92

C 0.40 0.0145 C 33 0.93

I 0.49 0.0178 I 33 1.14

Net Removals C 0.027 0.114 C 67 0.8

I 0.028 0.117 I 100 0.2

D 0.03 0.14 D 67 0.9

AV 0.04 0.19 AV 67 1.3

E 0.1 0.42 E 67 2.7

NDM = Nonfat dry milk. RMSE = Root mean squared error. AV = Average of three best forecasts. 

1Methods’ key corresponds to A, constrained vector autoregression with exogenous variables (VARX); B, unconstrained VAR; C, ordinary 

least squares (OLS); D, two-stage least squares (2SLS); E, seemingly unrelated regression (SUR); F, three-stage least squares (3SLS); 

and I, forecasts by ICEC. The specifi c version of the VARX model employed here contains no exogenous variables; previous versions of this 

research, as well as ERS’s regular forecasting activities, usually employ them.

Sources: USDA, World Agricultural Outlook Board and USDA, Economic Research Service estimates.
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Table 7 presents results for the forecast evaluation both in terms of accuracy 

and percentage of correct turn prediction in the data. A Theil’s U measure 

is used to measure accuracy. The labels Q2/2009, Q3/2009, Q4/2009, and 

Q1/2010 represent the quarters for which the measure is calculated. The best 

predictor by quarter is the one with the highest cumulative Theil’s U. Turn 

predictors were calculated as the difference between the value of a given 

variable in a quarter versus the value in the next period and then determining 

if this difference was positive or negative. The value of the sign of the difference in the data was compared with the sign of the difference in the predictions. A correct turn is when the sign of the difference in the data is the same 

as the sign of the prediction. Correct signs of differences were added from 

one period to the next. The best predictors by quarter are the ones that have 

the highest cumulative percentage at that point in time.

When forecasting dairy product prices, the VARX unrestricted and restricted 

models tend to be the best among the ERS forecasting models, performing 

even better when forecasting the prices of whey and NDM than the 

consensus forecast by ICEC. Constrained or unconstrained VARX models 

have the highest percentage of correct turn predictions for butter, whey, or 

all-milk prices in Q2/2009-Q1/2010. Structural models are also good turn 

predictors for these prices. ICEC is better at predicting turns for cheese and 

NDM in this forecasting activity.

Forecasting performance can be very fragile, and good performance in one 

period does not imply good performance in the next. When the analysis was 

done using different years of data (Q4/1998 to Q4/2008, withholding the year 

2008 for the ex-post forecasting exercise), however, forecasting reliability 

measures exhibited a similar performance structure in the time series models 

vis-à-vis single- and multiple-equation models. Moreover, simple arithmetic 

averages of forecasts did show some performance improvement but not in 

every instance. Expert judgment is important in this context to extrapolate 

from the observed trends and to complement the results of this model. As 

such, forecasting performance measures are useful tools for continually 

improving forecasting models. 
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Table 7

Robustness of the best performing forecasting methods1

Theil U and Share of Correct Turns for Q2/2009 to Q1/2010

Most accurate quarterly cumulative predictor (Theil U)

Variable/quarter Q2/2009 Q3/2009 Q4/2009 Q1/2010

Number of cows B E I B

Production per cow E E C C

Cheese price A I I I

Butter price I E I I

NDM price I E AV AV

Whey price F B B B

All-milk price I I I I

Beginning stocks F D E D F

Imports F F E F F

Exports F AV E I I

Net removals F I I B B

Beginning stocks S I E I I

Imports S B E AV I

Exports S D E F E

Net removals S AV E D I

Best cumulative turn predictor (correct turn/total predictions)

Variable/quarter Q2-3/2009 Q3-4/2009 Q4/2009-Q1/2010

Number of cows A/B/D/AV/I A/D/I A/B/I

Production per cow A/B/C/D/E/F/AV/I A/B/C/D/E/F/AV/I C/D/E/F/AV

Cheese price A/B/C/D/E/F/AV/I C/D/E/F/AV/I AV/I

Butter price C/D/E/F C/D/E/F C/D

NDM price A/C/D/E/F/AV/I C/D/E/F/AV/I AV/I

Whey price B B B/I

All-milk price I A/B/C/D/E/F/AV/I B/C/D/E/F/AV

Beginning stocks F C/D/E/F/AV/I C/D/E/F/AV/I C/D/E/F/AV/I

Imports F A/B/C/D/E/F/AV/I A/B/D/I A/B/D/I

Exports F A/B/AV A/B/AV/I A/B/I

Net removals F A/B/C/D/E/F/AV/I A/B/C/D/E/F/AV/I A/B/C/D/E/F/AV/I

Beginning stocks S A/B/C/D/E/F/AV/I A/B/C/D/E/F/AV/I A/B/C/D/E/F/AV/I

Imports S C A/B/C/D/E/F/AV/I A/B/I

Exports S A/B/C/D/E/F/AV/I B/C/D/E/F/AV/I B

Net removals S B/C/D/E/F/AV/I C/AV/I I

NDM = Nonfat dry milk. RMSE = Root mean squared error. AV = Average of three best forecasts. 

1Methods’ key corresponds to A, constrained vector autoregression with exogenous variables 

(VARX); B, unconstrained VAR; C, ordinary least squares (OLS); D, two-stage least squares (2SLS); 

E, seemingly unrelated regression (SUR); F, three-stage least squares (3SLS); and I, forecasts by 

ICEC. The specifi c version of the VARX model employed here contains no exogenous variables; previous versions of this research, as well as ERS’s regular forecasting activities, usually employ them.

Sources: USDA, World Agricultural Outlook Board and USDA, Economic Research Service 

estimates.
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Summary and Conclusion

A model for the U.S. dairy industry using quarterly data for 

Q4/1998-Q1/2009 was specifi ed and estimated as a tool to generate forecasts 

for USDA. The model was validated using data for Q2/2009-Q1/2010 and 

employs a variety of single-equation and system methods to estimate values 

for several variables. Clearly, different methods better forecast different variables. Vector autoregression with exogenous variables (VARX) outperforms 

structural regression models when forecasting prices, but single-equation 

and system estimations of structural models outperform time series models 

when forecasting some crucial items in farm supply and commodity balance 

sheets. This study has shown that it is better not to rely on any one particular 

estimating method. Careful econometric specifi cation and data development 

should ensure that greater weight is given to a model-based dairy forecasting 

system relative to a judgment-based forecasting system. 
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U.S. Department of Agriculture, Agricultural Marketing Service (AMS) 

Dairy Programs. National Econometric Model Documentation: For Model 

Calibrated to USDA Agricultural Baseline Projections to 2015, 2006.

Varian, H.R. Microeconomic Analysis, Third Edition, New York: W.W. 

Norton & Company, 1992.

Westcott, P. C. A Quarterly Model of the U.S. Dairy Sector and Some of Its 

Policy Implications, TB-1717, U.S. Department of Agriculture, Economic 

Research Service, 1986.
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A Quarterly Econometric Model for Short-Term Forecasting of the U.S. Dairy Industry / TB-1932

Economic Research Service/USDA

Appendix A: Constructing the Energy Index

The energy variable used in the ERS dairy forecasting model is constructed 

using a Törnqvist index (see Coelli et al., 2005, p. 90 for details) that can in 

general aggregate different types of commodities consistently across time, 

space or both. In particular, prices and quantities for industrial consumption 

for four types of energy—natural gas, propane, gasoline, and electricity—

from the Energy Information Administration’s Short-Term Energy Outlook 

databases are employed. The index compares the geometric mean of the four 

energy prices in time s with time t, and it is weighted by the m cost shares of 

the various energy components:

(A1)

4

1

ln (ln ln ), 2

T ms mt

st mt ms m

P p p

ω ω

=

⎛ ⎞ + = − ∑ ⎜ ⎟ ⎝ ⎠ 

where t= 1, 2, 46, and s= 1 is the base period which corresponds to 

 Q4/1998,

m = 1, 2, 3, and 4 are the energy cost components, and

1

ms ms

ms M

ms ms i

p q

p q

ω

=

=

∑

 is the value of the m-th component for the base

 period.
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A Quarterly Econometric Model for Short-Term Forecasting of the U.S. Dairy Industry / TB-1932

Economic Research Service/USDA

Appendix B: The VARX Model

In general, and following Lütkepohl (2006, p. 387), a VARX model with 

p and s lags for the endogenous and exogenous variables, respectively, is 

written as:

(B1) *

1 0

 p s

t i ti i ti t

i i

y yx δΦ Θ ε − − = ==+ + + ∑ ∑ ,

where 1 (, , ) k δδ δ = ′ are the constants for the k equations,

 1 (,, ) t t Kt yy y = ′ is the K-dimensional vector of endogenous

 variables,

 is a Φi K K× matrix for the p order autoregressive parameters,

 1 (,, ) t t Mt xx x = ′ is the M-dimensional vector of exogenous

 variables,

 is a Θi K M× matrix for the s order exogenous parameters,

 1 (,, ) t t kt εε ε = ′ is a vector of white noise

 so that () 0 E t ε = and ( ) E t t εε Σ ′ = and ( ) 0 for E ts t s ε ε′ = ≠ .
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A Quarterly Econometric Model for Short-Term Forecasting of the U.S. Dairy Industry / TB-1932

Economic Research Service/USDA

Appendix C: Estimate of Correlations Between 

Equations’ Disturbances

The estimatedΣˆ matrices (symmetric) for blocks 1, 2, and 3 are:

Block 1

Equation 1.1 1.2 1.3 1.4 1.5 1.6 1.7

1.1 3.15e-06

1.2 6.57e-06 .00011

1.3 -2.82e-06 -.00020 .011

1.4 5.89e-06 -.00027 -.00039 .0097

1.5 -4.53e-07 -.00010 .0058 .00098 .014

1.6 -.000028 -.00014 .0084 .0049 .0096 .022

1.7 -5.39e-07 -.000082 -.00042 .000036 .00027 -.00038 .00057

Source: USDA, Economic Research Service estimates.

Block 2

Equation 2.1 2.2 2.3

2.1 .012

2.2 .0020 .025

2.3 .0072 .0034 .08

Source: USDA, Economic Research Service estimates.

Block 3

Equation 3.1 3.2 3.3

3.1 .0048

3.2 -.0033 .017

3.3 .00015 .0012 .020

Source: USDA, Economic Research Service estimates.
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A Quarterly Econometric Model for Short-Term Forecasting of the U.S. Dairy Industry / TB-1932

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Appendix D: Hurdle or Two-Part Model 

Versus Tobit Tests

The hurdle or two-part model is presented as an alternative to the Tobit 

model specifi cation of net removals as it relaxes some of the stronger 

assumptions that the latter makes. Specifi cally, the hurdle model splits the 

selection from the outcome mechanisms, and it can be employed for different 

variables at these two different stages. Here, a Probit model is employed to 

model the censoring mechanism and then a regression equation is used to 

characterize the outcome. Both parts are assumed to be independent (see 

Cameron and Trivedi (2009, pp. 538-541) and Kennedy (2008, p. 269) for 

details). 

Formally: 

Pr( 0 ) 0 ( ) Pr( 1 ( 1, 0d x if y f yx d x f y d x if y

⎧ ⎫ = = = ⎨ ⎬ ⎩ ==> ⎪⎭

,

where d is an indicator variable, ( If d = 1, y > 0 and if d = 0, y = 0) 

(Cameron and Trivedi, 2009). 

The sample period (Q4/1998-Q1/2009) has 11 cases for fat and 7 for skim 

solids where net removals are zero. From the assumption of independence, 

estimates of the two-part model’s joint likelihoods for fat and skim solid net 

removals are 15.28 and -38.29, respectively. In contrast, the Tobit model’s 

likelihoods for fat and skim solids are 39.22 and -38.23, respectively. A 

larger number for the log likelihoods indicate that the model fi ts the data 

better. Hence, the Tobit model is better in this particular sample.