1 Forecast Evaluations in Meat Demand Analysis Zijun Wang Private Enterprise Research Center, Academic Building West, Room 3028, Texas A&M Univers...

Author:
Amos Lester

0 downloads 24 Views 159KB Size

Severity: Notice

Message: Undefined index: description

Filename: shared/document_item_2.php

Line Number: 14

Backtrace:

File: /home/zdoc.pub/public_html/application/views/shared/document_item_2.php

Line: 14

Function: _error_handler

File: /home/zdoc.pub/public_html/application/views/document.php

Line: 109

Function: view

File: /home/zdoc.pub/public_html/application/controllers/Document.php

Line: 142

Function: view

File: /home/zdoc.pub/public_html/index.php

Line: 355

Function: require_once

ABSTRACT This article offers a comparison of short-term forecasting ability of five demand systems with an application to U+S+ meat consumption+ Four static demand systems ~AIDS, Rotterdam, AIM, and DGM! and a dynamic Vector Error Correction Model ~VECM! are considered+ We tested the equality of mean square forecast errors+ We also investigated the possibility of forecast encompassing among competing models+ In general, the dynamic VECM model performed best, followed by the simple causal DGM model+ Among three static systems, the AIDS model slightly leads the competition+ Furthermore, this article provides the first evidence in literature on whether imposition of homogeneity restrictions on a cointegration space can improve the forecast accuracy of a VECM model: it does when it holds+ @EconLit citations: Q10; C53#+ © 2003 Wiley Periodicals, Inc+

1. INTRODUCTION U+S+ meat demand has long attracted enormous effort of the profession+ The literature is still accumulating+ Questions of central concern have been: what is the income ~expenditure! elasticity? Has there been a structural change in meat demand? If so, what has caused the change? Relatively less attention has been paid to demand forecasts and even less to comparing the forecast performance of alternative models on demand+ Kastens and Brester ~1996! and Chambers ~1990! are two exceptions+ Another exception, that of Heien, Chen, Chien, and Garrido ~1996!, compares various meat demand models on the basis of postsample fit+ In this article, we study the forecast performance of five static and dynamic demand systems based upon a wide spectrum of modeling philosophies+1 First, Deaton and Muellbauer’s ~1980! AIDS model is chosen for its popularity in empirical studies+ As a com1 Although univariate ARIMA models have been found useful for forecasting, they do not explicitly take into account the influence of other observable variables known or suspected to be related to the series of interest+ Recent empirical studies have concentrated on demand systems in which theoretical restrictions ~e+g+, homogeneity! can be imposed or tested+

Agribusiness, Vol. 19 (4) 505–524 (2003) © 2003 Wiley Periodicals, Inc. Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/agr.10074

505

506

WANG AND BESSLER

parison, Barten ~1966! and Theil’s ~1965! Rotterdam model is also analyzed, along with Barnett’s ~Barrett & Jonas, 1983; Barrett & Yue, 1988! Asymptotic Ideal model ~AIM!+ We then apply directed graph methodology to the AIDS model to remove variables from the system that are conditionally independent+ We call this resulting system a Directed Graph Model ~DGM!+ Finally, we consider dynamic Vector AutoRegression ~VAR! and cointegration analysis+ To compare the forecast performance of these models, we implement Diebold and Mariano’s ~1995! test on the equality of forecast errors+ We also investigate the possibility that one model encompasses the other+ Our contributions are that: ~1! we introduce the use of the AIM and DGM models in meat demand analysis; and ~2! we focus on out-of-sample forecast evaluation+ Both Diebold and Mariano’s ~1995! test on the equality of forecast errors and the forecast encompassing test have been studied in the economics literature in recent years ~Ericsson & Marquez; 1993; Wu, 1999!+ However, we have not found applications of the ideas in agricultural forecasting+ The article is organized as follows+ Following a brief discussion of data, we estimate all five models and test various hypotheses with 1975–1989 U+S+ quarterly meat consumption+ Based upon estimated models, we generate one- and two-steps-ahead out-ofsample recursive forecasts for the period 1990:I to 1997:IV+ Extensive tests are conducted to evaluate the accuracy of these forecasts+ The final section is a summary of major results of the article+

2. MODEL SPECIFICATION 2.1 AIDS After it was first derived in 1980 by Deaton and Muellbauer, the AIDS model has been applied in many empirical demand analyses+ The model is given by N

wit ⫽ ai ⫹ ( gij ln~ pjt ! ⫹ bi ln~X t 0Pt ! ⫹ u t , for all i ⫽ 1 to N, t ⫽ 1 to T,

~1!

j⫽1

where N is the number of commodity categories considered in the whole system, T is the number of time periods ~e+g+, quarters!, wit is the expenditure share of the ith commodity at time period t, satisfying wit ⫽ ~ pit * qit !0X t , with pit being the price for commodity i at time t and qit quantity demanded, X t is the representative agent’s total consumption expenditure on all N commodities, a, b, and g are parameters to be estimated, and Pt is an aggregate price index given by N

ln~Pt ! ⫽

( wit ln~ pit !, i⫽1

~2!

The theoretical properties of homogeneity in prices and income, and the Slutsky symmetry of the crossprice effects of demand functions, imply both within and across equation restrictions on parameters g and b in ~1!:

FORECAST EVALUATIONS IN MEAT DEMAND ANALYSIS N

N

N

( ai ⫽ 1, ( bi ⫽ 1, ( gij ⫽ 1

i⫽1

507

i⫽1

~adding-up!

i⫽1

N

( gij ⫽ 0

~homogeneity!

j⫽1

gij ⫽ gji

~symmetry!

~3!

2.2 Rotterdam Another often-used differential form demand system is the Rotterdam model, which has a longer history than AIDS ~Barten, 1966; Theil, 1965!+ The absolute price version of the Rotterdam model is N

wit D ln~qit ! ⫽

( gij D ln~ pjt ! ⫹ bi D ln~Qt ! ⫹ ut ,

~4!

j⫽1

where Dln~Qt ! ⫽ S j wjt D ln~qjt !+ Parameter restrictions are the same as in the AIDS model+ Notice that we use the same parameter notation over different models for simplicity, although they may not have identical meanings in each context+

2.3 AIM Almost all flexible functional forms including AIDS are only locally flexible+ This is because the Taylor series expansion underlying most of these functional forms only “weights deviations of the sample from a expansion point by their derivatives at that point” ~Havenner & Saha, 1999!+ The approximation tends to be less accurate for data sets with more variation+ In contrast, the Müntz-Szatz series approximates the unknown functions over the entire data set, and hence, is able to minimize approximation error over the whole sample observations+ Deriving from the Müntz-Szatz series expansion, Barnett and Jonas ~1983! and Barnett and Yue ~1988! estimated a demand system and named it the Asymptotically Ideal Model @AIM~K !#, where K is an index of the expansion function related to the accuracy of the approximation+ The derivation of a general formula of the AIM model is mathematically straightforward yet very tedious; for example, see Barnett and Yue ~1988!+ ~For more discussions on the AIM model, see Havenner & Saha, 1999!+ Unlike in the AIDS or Rotterdam model, there is no way to impose or test the triad restrictions from consumption theory in the framework of the AIM+ More problematically, it is heavily parameterized+ In a five-commodity system there are 27 free parameters for K ⫽ 1, excluding any other exogenous variables in the system+ This figure soars to 243 when K ⫽ 2, which will use up all degrees of freedom in most available time series+ This might limit its potential application in food demand system analysis where usually many disaggregate commodities are involved+ The first share equation in our fivecommodity AIM~1! system ~omitting the time script! is given by

508

WANG AND BESSLER

w1 ⫽ ~a 1 v1 ⫹ a 6 v1 v2 ⫹ a 7 v1 v3 ⫹ a 8 v1 v4 ⫹ a 9 v1 v5 ⫹ a 16 v1 v2 v3 ⫹ a 17 v1 v2 v4 ⫹ a 18 v1 v2 v5 ⫹ a 19 v1 v3 v4 ⫹ a 20 v1 v3 v5 ⫹ a 21 v1 v4 v5 ⫹ a 26 v1 v2 v3 ⫹ a 27 v1 v2 v3 v5 ⫹ a 28 v1 v2 v4 v5 ⫹ a 29 v1 v3 v4 v5 ⫹ a 31 v1 v2 v3 v4 v5 !0 ~a 1 v1 ⫹ a 2 v2 ⫹ a 3 v3 ⫹ a 4 v4 ⫹ a 5 v5 ⫹ 2a 6 v1 v2 ⫹ 2a 7 v1 v3 ⫹ 2a 8 v1 v4 ⫹ 2a 9 v1 v5 ⫹ 2a 10 v2 v3 ⫹ 2a 11 v2 v4 ⫹ 2a 12 v2 v5 ⫹ 2a 13 v3 v4 ⫹ 2a 14 v3 v5 ⫹ 2a 15 v4 v5 ⫹ 3a 16 v1 v2 v3 ⫹ 3a 17 v1 v2 v4 ⫹ 3a 18 v1 v2 v5 ⫹ 3a 19 v1 v3 v4 ⫹ 3a 20 v1 v3 v5 ⫹ 3a 21 v1 v4 v5 ⫹ 3a 22 v2 v3 v4 ⫹ 3a 23 v2 v3 v5 ⫹ 3a 24 v2 v4 v5 ⫹ 3a 25 v3 v4 v5 ⫹ 4a 26 v1 v2 v3 v4 ⫹ 4a 27 v1 v2 v3 v5 ⫹ 4a 28 v1 v2 v4 v5 ⫹ 4a 29 v1 v3 v4 v5 ⫹ 4a 30 v2 v3 v4 v5 ⫹ 5a 31 v1 v2 v3 v4 v5 !

~5!

where w1 is the consumption share of the first commodity, vi is the square root of expenditure-normalized price of commodity i, namely, vi ⫽ ~ pi 0X !102, a j s are parameters to be estimated+ 2.4 DGM It is well known that causal conclusions drawn from ~1! or ~4! require controlled experiments, which is in sharp contrast to the reality that almost all economic data are observational+ Although correlation can be estimated directly in a single uncontrolled study ~as we mostly do!, correlation does not imply causation+ This could be one of the fundamental reasons that traditional structural equation modeling has been incurring questions, in particular, in terms of its usually inferior forecast performance+ Emerging from artificial intelligence and computer sciences, the directed graph method may help to deal with the inconsistence between the statistical theory and observational data+ A directed graph is an assignment of causal flow ~or lack thereof ! among a set of variables ~vertices! based on observed correlation and partial correlation+ Briefly, one forms a complete undirected graph showing an undirected edge ~line! between every variable on the vertex set+ These undirected edges represent all possible paths of causal flow within the system+ Edges are then removed sequentially based on zero correlation or partial correlation ~conditional correlation!+ In applications, Fisher’s z statistic is used to test whether conditional correlations are significantly different from zero+ The edges remaining connected, after all possible correlation and partial correlation tests have been considered, are to be directed for causal flows using the notion of sepset+ The conditioning variable~s! on removed edges between two variables is called the sepset of the variables whose edge has been removed+ Edges are directed by considering triples X ⫺ Y ⫺ Z, such that X and Y are adjacent as are Y and Z, but X and Z are not adjacent+ Direct the edges between triples: X ⫺ Y ⫺ Z as X r Y R Z if Y is not in the sepset of X and Z+ If X r Y, Y and Z are adjacent, X and Z are not adjacent, and there is no arrowhead at Y, then orient Y ⫺ Z as Y r Z+ If there is a directed path from X to Y, and an edge between X and Y, then direct ~X ⫺ Y ! as: X r Y+

FORECAST EVALUATIONS IN MEAT DEMAND ANALYSIS

509

Pearl ~1995!, Spirtes, Glymour, and Scheines ~1993!, and Swanson and Granger ~1997! are useful citations regarding theories and applications of directed graphs+ Empirically, Akleman, Bessler and Burton ~1999! applied directed graphs to model the relationship between corn exports and exchange rates+ Bessler and Akleman ~1998! revisited Gardner’s ~1975! model on the causal relations on farm level and retail level prices for beef and pork+ 2.5 Multivariate VAR/VECM Like many other topics in agricultural economics that involve aggregate supply and demand, the simultaneity problem is found to be present in meat consumption analysis+ Eales and Unnevehr ~1993! found that both prices and quantities appear to be endogenous within the entire meat market+ LaFrance’s ~1991! analysis also indicated that expenditure seldom is strictly exogenous in a system of conditional demands+ However, the simultaneity problem can be avoided in a VAR that does not include any current values of dependent variables ~endogenous variables! among the regressors+ The Granger Representation Theorem implies that for a nonstationary vector of series yt with dimension k, any VAR~ p!: yt ⫽ A 1 yt-1 ⫹ + + + ⫹ A p yt-p ⫹ Bx t ⫹ «t

~6!

has an equivalent Vector Error Correction Model ~VECM! form: p⫺1

Dyt ⫽ Pyt⫺1 ⫹

( Gj Dyt⫺j ⫹ Bx t ⫹ «t ,

~7!

j⫽1

where yt is a k vector of endogenous variables, x t is a d vector of exogenous variables, for example, quarterly dummy variables, P, G and B are matrices of coefficients to be estimated, and «t is a vector of innovations that may be contemporaneously correlated with each other but are uncorrelated with their own lagged values and uncorrelated with all of the right-hand side variables ~see Engle & Granger, 1987!+ If P has a rank r, r ⬍ k, then yt is cointegrated with r cointegrating vectors, reflecting long-run relationship among variables in the system+ The matrices Gj ~ j ⫽ 1,+ + +, p ⫺ 1! consist of short-run dynamics+ The homogeneity constraint in the cointegration space ~P! can be implemented by testing whether the parameters of prices and total consumption expenditure sum to 0+ The VAR model was introduced into agricultural economics as early as 1984 by Bessler, although we have found only a few applications in meat demand+2 It is also interesting to observe that although literature on testing cointegration is explosive in academic areas, not much has been done to examine the forecast performance of a VAR with cointegration restriction imposed ~Lin & Tsay, 1996, is an exception among a handful of articles!+ There has been even less work on the forecast performance of a cointegrated VAR with homogeneity or other theoretical restrictions imposed on cointegration spaces+ ~For a recent example of this last topic, see Wang & Bessler, 2002!+ 2 Karagiannis, Katanidis, and Velentzas ~2000! estimated a dynamic AIDS model incorporating error correction terms in the regression+ The endogenous variables are the shares of different meat products, which is different than the specification of this article+

510

WANG AND BESSLER

3. DATA The data set consists of 92 quarterly observations from 1975:I to 1997:IV for U+S+ per capita consumption ~retail weight! of beef, poultry, pork, other nonmeat food, and all other nonfood goods, their retail prices ~dollar per pound! or price indices, per capita consumption expenditures and food expenditures+ The historical series of beef and pork prices and consumption come from USDA’s Livestock and Meat Statistics and Red Meat Yearbook for 1975 to 1995+ Various issues of on-line Livestock, Dairy, and Poultry are the sources for 1996 and 1997 observations+ The per capita consumption of beef is the sum of beef and veal consumption+ To reduce product aggregation bias, we generate a Divisia price and quantity index based upon the disaggregate indices of chicken and turkey+ Quarterly data for total personal consumption expenditures and food consumption expenditures ~not seasonally adjusted! are available in on-line publications of Bureau of Economic Analysis, Department of Commerce: NDN-0209 in National Income and Product Accounts+ The U+S+ residential population is collected from Statistical Abstract of the U.S+ Following Eales and Unnevehr ~1988!, we form a complete demand system which, in addition to beef, poultry, and pork consumption, also includes two aggregate commodities: nonmeat food and all other goods+ Subtracting total meat expenditures from consumer’s food expenditures gives the nonmeat food expenditures+ The nonmeat food quantity is food quantity ~which, in turn, is the ratio of food expenditures to food’s consumer price index! minus the sum of the meat quantities+ Similarly, the ratio of all other nonfood goods expenditure to the consumer price index for all items less food is used as the quantity of all other goods+ 4. ESTIMATION AND HYPOTHESIS TEST The validity of traditional t- and F-test statistics requires that underlying data generating processes are stationary+ As the first step of estimation, we investigate the stationarity nature of each series+ Graphs of these series show that all meat price and quantity series seem to have either an upward or downward time trend+ Pork consumption series appears to be an exception in which trend is not very clear+ Beef consumption has a downward trend, especially in 70s and 80s+ The presence of seasonality is not surprising+ In implementing the Augmented-Dickey-Fuller ~ADF! test, we choose the Schwarz information criterion ~SC! to determine the appropriate lags of the dependent variable in the ADF test+ Following the convention, all tests are carried out in terms of logarithmic transformed data+ It is somewhat interesting to notice that most series require either no lag or four lags to exclude serial correlation in estimated residuals, which seems reasonable given we use seasonal data+ At the 0+05 significance level, a unit root cannot be rejected for all 16 series+ The above procedure is further applied to test the stationarity of the first difference of the original series+ The null hypothesis of a unit root is definitely rejected in all series+ We conclude that all series under investigation are nonstationary and integrated of order 1+ 4.1 Estimation of AIDS, Rotterdam, and AIM(1) With the first 60 observations, we empirically estimate the AIDS model using iterative Zellner’s Seemingly Unrelated regression method+ Because all variables are nonstation-

FORECAST EVALUATIONS IN MEAT DEMAND ANALYSIS

511

ary, we estimate the AIDS model in its first-difference form+ The explanatory variables are: three seasonal dummies, beef price, poultry price, pork price, nonmeat food price, other-goods price, and total consumption expenditure+ The LR test statistic of homogeneity and symmetry is 13+74 with 10 restrictions, which implies that the theoretical restrictions cannot be rejected at any traditional significance level+ Therefore, these restrictions are imposed in estimation and forecasting+ Furthermore, due to the adding-up restriction on parameters, we drop the other-goods equation+ The estimation results are reported in Table 1+ The coefficient estimates are reasonable, and significant for the most parameters, and all own price effects are highly significant+ Based upon these estimates we calculate the price and expenditure elasticities+ The own price elasticities for beef, poultry, and pork consumption are ⫺0+41, ⫺0+25, and ⫺0+73 close to some published results, for example, Brester and Wohlgenant ~1993!, Eales and Unnevehr ~1988!, etc+ Likewise, the joint test of homogeneity and symmetry cannot be rejected within the Rotterdam framework ~the LR test statistic is 14+30!+ So they are imposed in estimation and forecasting+ The restricted model estimation results are reported in Table 2+ More than half the estimates are significantly different from zero+ Compensated own price elasticities are ⫺0+61, ⫺0+16, and ⫺0+84, respectively+ Limited by the sample size, we only estimate the AIM~1! model+ Unlike in the AIDS or Rotterdam model, parameters in the AIM~1! model do not have direct economic interpretations+ Price and expenditure elasticities cannot be calculated from the AIM model estimates either+ To save space, we do not report the parameter estimates of the AIM and the DGM models+ They are available from the authors at request+ 4.2 DGM Estimation We apply the directed graph method described above to identify the causal relationship among the variables in the AIDS model+ Consider, for example, the model associated with beef consummation share+ We form the starting undirected graph by connecting all pairs formed by beef consumption share, the prices of beef, poultry, pork, nonmeat food, other goods, total consumption expenditure, and three seasonal dummy variables ~so there would be 45 edges!+ We choose a 0+20 significance level in removing edges from the graph consistent with the sample size ~Scheines, Spirtes, Glymour, & Meek, 1994, p+ 81!+ Figure 1 is the final directed graph+ Only four of nine edges that were connected to beef share remain in the final graph+ As we might expect, the changes in beef price and total expenditure cause changes in the share of beef consumption+ The price of its substitute poultry is also a causal force+ Innovations in pork price, nonmeat food price, and othergoods price are not causally connected to the innovation in beef consumption+ The above procedure is applied to other three equations of the AIDS model+ No more than four edges survive the removals in each equation; but all own prices remain in their corresponding graphs+ The final DGM model we estimate and use for forecasting are: ~a! Beef share ⫽ f~beef price, poultry price, expenditure, quarter IV dummy!, ~b! Poultry share ⫽ f~beef price, poultry price, quarter III dummy!, ~c! Pork share ⫽ f~pork price, quarter II dummy!, ~d! Nonmeat food share ⫽ f~nonmeat food price, quarter II dummy, quarter IV dummy!+ Although more than half of the coefficient estimates in the DGM model have similar magnitudes to those in the AIDS model, they may not be directly comparable by model construction+

0+019 ~0+048!

0+119 ~0+187!

0+326* ~0+091!

⫺1+849* ~0+400!

Pork Share

Nonmeat Food Share

⫺0+557* ~0+215!

0+240* ~0+069!

0+019 ~0+048!

0+326* ~0+091!

Pork Price

⫺8+951* ~1+448!

⫺0+028 ~0+219!

⫺0+557* ~0+215! 11+237* ~1+510!

⫺0+950* ~0+225!

⫺0+921* ~0+184!

0+119 ~0+187!

⫺9+500* ~1+636!

⫺0+495 ~0+309!

⫺1+196 ~0+607!

0+168 ~0+400!

⫺1+849* ~0+400!

Expenditures

OtherGoods Price

Nonmeat Foor Price

⫺0+945* ~0+125!

⫺0+024 ~0+023!

⫺0+105* ~0+017!

0+043 ~0+046!

Intercept

Note: All coefficients are multiplied by 100 for ease of presentation+ Figures in parentheses are standard errors+ *Indicates significant at the 0+05 level+ d1, d2, and d3 are three dummy variables for the second, third, and fourth quarters, respectively+ The numbers under Q~4! are p-values of modified Ljung-Box-Pierce Q statistics with four lags+

Log Likelihood Function ⫽ 1430+41

0+574* ~0+064!

0+209* ~0+076!

Poultry Share

0+209* ~0+076!

Poultry Price

1+145* ~0+206!

Beef Price

Aggregate Meat Model ~AIDS!

Beef Share

TABLE 1+

1+939* ~0+177!

⫺0+021 ~0+034!

0+162* ~0+024!

⫺0+057 ~0+066!

d1

1+064* ~0+137!

0+030 ~0+026!

0+126* ~0+018!

⫺0+023 ~0+050!

d2

0+731* ~0+222!

0+073 ~0+042!

0+156* ~0+031!

⫺0+142 ~0+081!

d3

0+152

0+009

0+008

0+020

Average Budget Share

0+948

0+697

0+853

0+844

R2

Q~4!

0+370

0+070

0+033

0+007

512 WANG AND BESSLER

⫺0+991 ~1+590!

⫺0+497* ~0+226! 2+794 ~1+540!

0+916* ~0+223!

⫺0+240 ~0+194!

2+205 ~0+391!

⫺1+526* ~0+398! 0+221 ~0+192!

OtherGoods Price

Nonmeat Foor Price

5+321* ~1+890!

0+835* ~0+319!

⫺0+134 ~0+260!

1+390 ~0+583!

Expenditures

⫺0+942* ~0+144!

0+005 ~0+024!

⫺0+100* ~0+019!

0+089* ~0+044!

Intercept

Note: All coefficients are multiplied by 100 for ease of presentation+ Figures in parentheses are standard errors+ *Indicates significant at the 0+05 level+ d1, d2, and d3 are three dummy variables for the second, third, and fourth quarters, respectively+ The numbers under Q~4! are p-values of modified Ljung-Box-Pierce Q statistics with four lags+

⫺0+497* ~0+226!

0+221 ~0+192!

⫺1+526* ~0+398!

Log Likelihood Function ⫽ 1422+19

Nonmeat Food Share

⫺0+750* ~0+075!

⫺0+021 ~0+051!

0+352* ~0+090!

Pork Share

⫺0+021 ~0+051!

⫺0+125 ~0+065!

0+165* ~0+078!

Poultry Share

Pork Price 0+352* ~0+090!

Poultry Price

0+165* ~0+078!

1+196* ~0+197!

Beef Price

Aggregate Meat Model ~Rotterdam!

Beef Share

TABLE 2+

1+929* ~0+204!

⫺0+066 ~0+035!

0+154* ~0+028!

⫺0+125 ~0+063!

d1

1+078* ~0+156!

0+007 ~0+026!

0+121* ~0+026!

⫺0+072 ~0+047!

d2

0+748* ~0+257!

0+016 ~0+043!

0+150* ~0+036!

⫺0+234* ~0+079!

d3

0+152

0+009

0+008

0+020

Average Budget Share

0+973

0+833

0+891

0+610

R2

Q~4!

0+061

0+267

0+021

0+015

FORECAST EVALUATIONS IN MEAT DEMAND ANALYSIS

513

514

WANG AND BESSLER

Note: Beefp, poltp, porkp, nmfp, and nfdp represent logarithmic prices of beef, poultry, pork, non-meat food, and other goods, respectively; expend represents total consumption expenditure; d1, d2, and d3 are seasonal dummies+

Figure 1

Causally sufficient graphical AIDS: Beef share equation+

4.3 VAR/VECM Estimation In specifying the multivariate VAR, we consider three separate demand systems+ The first system includes seven endogenous variables: beef consumption, five commodity prices, and total consumption expenditures+ We call the resulted model the beef system+ The poultry and pork systems are likewise specified+ As our emphasis is on models’ forecast performance, and hence simplicity, is an important concern, we rely on SC to select the final model+ As before, a maximum of four lags in levels or three lags in first difference are used+ In all three systems, SC values reach its minimal when one lag is used, and one cointegration relation is assumed+ Also, the minimal SC corresponds to the case in which a linear trend is allowed in the data for all three systems+ Parameters are estimated using Johansen’s ~1995! maximum likelihood method+ LR test statistics show that the hypothesis of homogeneity cannot be rejected in the beef system with a probability of 0+58+ Homogeneity is rejected in the other two systems+ To improve the stochastic properties of the model, we also test whether some variables are weak exogenous relative to other variables in the system+ Expenditure is found to be exogenous in both the beef and the pork system, and pork price is exogenous in the poultry system+ Furthermore, expenditure also can be excluded from long-run cointegration space in the beef system+ Therefore, we estimate and forecast the beef system as a partial

FORECAST EVALUATIONS IN MEAT DEMAND ANALYSIS

TABLE 3+

515

Estimation Results of VECM ~Cointegration Space!

Cointegration equation in beef VECM: ⫺9+47ln~beef quantity! ⫺ 10+24*ln~beef price! ⫹ 9+07*ln~poultry price! ⫹ 11+44*ln~pork price! ⫺50+59*ln~non-meat food price! ⫹ 43+2*ln~other-goods price! Cointegration equation in poultry VECM: ⫺46+08*ln~poultry quantity! ⫺1+28*ln~beef price! ⫺ 13+17ln~poultry price! ⫹ 10+04 ln~pork price! ⫹ 24+35ln~non-meat food price! ⫺ 31+43*ln~other-goods price! ⫹ 26+13*ln~expenditure! Cointegration equation in pork VECM: 42+35ln~pork quantity! ⫺ 3+35*ln~beef price! ⫺1+62*ln~poultry price! ⫹ 21+62ln~pork price! ⫺ 39+49*ln~non-meat food price! ⫺ 8+55*ln~other-goods price! ⫹ 24+65*ln~expenditure! Note: * Indicates significant at the 0+05 level+

system conditional on expenditure, the poultry and pork systems as ordinary cointegrated VARs, with weak exogeneity restrictions imposed+ Expenditure is not included in the long-run cointegration space of the beef system+ As there are 86 parameters in each of the three VARs, we only report in Table 3 the parameter estimates of the cointegration spaces+ The estimated long-run responses of own price changes have correct signs ~negative!+ A one-percentage poultry price increase will decrease the per capita consumption of poultry by 0+29 percentage, which is close to 0+25, the estimate from the structural AIDS model+ The estimated long-run effect of pork price change is lower than the AIDS estimate ~0+51 vs+ 0+73!+ The beef own price effect is noticeably larger than any previous estimates, although it is still within the range of published studies ~e+g+, Chern, Huang, & Lee, 1993!+ As expected, the three short-run dynamics of beef, poultry, and pork prices ~not reported! are all less than the corresponding long-run parameters in absolute values+ 5. FORECASTS AND EVALUATIONS 5.1 A Preliminary Comparison Based upon the five models identified earlier, we generate one-step-ahead recursive forecasts over the postsample period ~1990:I–1997:IV! and assess the forecast performance on both an equation-by-equation and a system-wide basis+ The number of forecasting periods ~quarters! is 32, approximately half of sample observations used in specifying the models+ To get preliminary forecast values for contemporaneous right-hand-side variables of AIDS, Rotterdam, and DGM, and weak exogenous variables in VECM, we use a random walk model with shift and seasonal dummy variables+3 This is justified by noting that all 3

The difficulty in forecasting with AIDS and other traditional demand systems is that contemporaneous values of price and expenditure are needed to forecast quantities or shares+ The values of these variables are also unknown with the exception of seasonal dummies and constant+ In Chambers ~1990!, Kastens and Brester ~1996!, etc+, the actual values at t ⫹ 1 are simply used, which is essentially out-of-sample fit instead of out-of-sample forecast per se+ However, to make our results comparable, we did generate the forecasts using actual prices and expenditure+ The ranks largely remain the same, although most models’ performance improves somewhat except that for the AIDS model, which has a slightly larger log determinant of forecast errors when actual prices and expenditures are used+

516

WANG AND BESSLER

series are I~1! processes as observed earlier ~for a similar treatment, see Bessler, 1990!+ With parameter estimates and price and expenditure forecasts at hand, it is straightforward to generate out-of-sample forecasts with AIDS, AIM, DGM, and VECM+ Forecasting with Rotterdam is more involved, as the dependent variable to be forecasted is a nonlinear function of quantity+ Following Kastens and Brester ~1996! and Kastens ~personal communication, 2000!, we use a Taylor series approximation+ A summary of four statistics reflecting forecast performance of five models is given in Table 4+ The VECM has the smallest mean square forecast error ~MSFE! ~0+284! in beef forecast, followed by DGM ~0+322!, and the three static demand systems: Rotterdam ~0+341!, AIDS ~0+493! and finally AIM ~0+730!+ The DGM performs best in poultry forecast with a MSFE of 0+341 approximately one-third of the MSFE from Rotterdam ~1+040!+ AIDS and AIM’s performance on poultry forecast is also poor ~0+822 and 0+631!+ Only in pork consumption forecasts do these five models perform relatively closely+ The Rotterdam module produces virtually the same forecast accuracy as the VECM ~about 0+18!+ The AIDS model performs similarly well in this regard+ Noticeably, the AIM model performs worst in two out of three series+ Comparisons based upon the mean absolute percentage error ~MAPE! provide very similar ordering of the models+ To measure the forecast performance for the variables, taken as a system, two statistics are considered: the log determinant, and the log trace of forecast error matrix of each model+ The log determinant measure takes into account the covariance effect among series, while the second measure does not+ This difference occasionally may lead to slightly different orderings ~see results on the two-step-ahead forecasts in Table 4!+ The VECM performs best by both measures, while AIM performs worst+ DGM, AIDS, and Rotterdam fall in between+ Summarizing the models’ one-step-ahead forecast performance, we may roughly group the five models into two categories+ The VECM and DGM perform similarly well, and belong to the first group+ The second group, consisting of three static demand systems, AIDS, Rotterdam, and AIM, perform worse than the first group in general+ Although we concentrate on one-step-ahead forecasts, as we believe it is the most relevant to meat marketing, we did generate two-steps-ahead forecasts+4 The summary statistics are reported in Table 4+ The major findings include: ~1! the division of two groups remains the same; ~2! the VECM model still leads in the model portfolio; ~3! the superiority of the AIDS model over the other two static models are clearer; ~4! surprisingly AIDS and AIM perform well in pork consumption+ Notice that the three static demand systems also work relatively well in two-step-ahead forecast of pork consumption, we suspect that this “inconsistency” might have resulted from the fact pointed out earlier that trend and dynamics in pork consumption series are not as serious of factors with pork as in other ones+ To investigate the effects of imposition of theoretical restrictions on the model’s statistical behavior, we also generate forecasts from the unrestricted VECM model ~no weak exogeneity or long-run exclusion, but cointegration still imposed! and VECM model with homogeneity restrictions imposed+ The results are summarized in Table 5+ The null hypothesis of homogeneity is not rejected in the beef system, and the imposition of the homogeneity restriction improves the model’s performance in both one- and two-steps4 As footnote 2 indicates, to forecast quantities consumed with the static systems, we first need to generate forecasts for contemporaneous price and expenditures+ Errors from this ancillary procedure are likely to compound the final forecast errors+ This is another reason why we decide not to pursue longer period forecasts here+

Beef Poultry Pork Beef Poultry Pork Beef Poultry Pork Beef Poultry Pork Beef Poultry Pork

AIDS

0+493 0+822 0+208 0+341 1+040 0+182 0+730 0+631 0+280 0+332 0+341 0+306 0+284 0+424 0+185

MSFE 3+576 3+605 2+973 2+700 4+034 2+917 4+176 3+118 3+178 2+963 2+300 3+701 2+411 2+457 2+879

MAPE

⫺3+860 ~1!

⫺3+701 ~2!

⫺2+221 ~5!

⫺3+006 ~4!

⫺3+337 ~3!

Log~det!

⫺0+113 ~1!

⫺0+020 ~2!

0+495 ~5!

0+463 ~4!

0+421 ~3!

Log~trace! 0+798 0+914 0+369 0+533 1+065 0+979 0+878 1+362 0+413 0+939 0+465 0+414 0+513 0+434 0+410

MSFE 4+634 3+818 4+167 3+550 4+092 6+217 4+515 4+618 4+014 5+154 2+743 4+307 3+693 2+573 4+066

MAPE

⫺2+634 ~1!

⫺2+122 ~2!

⫺0+876 ~4!

⫺0+855 ~5!

⫺1+406 ~3!

Log~det!

Two-Steps

0+305 ~1!

0+598 ~2!

0+976 ~5!

0+937 ~4!

0+733 ~3!

Log~trace!

Note: MSFE is mean squared forecast error+ MAPE is mean absolute percentage error+ ln~det! and ln~trace! are natural logarithm of determinant and trace of the matrix that consists of beef, poultry, and pork forecast error series from the same model+ Number underneath each value ~in parenthesis! is the model’s performance rank with 1 being the most preferred and 5 least preferred+

VECM

DGM

AIM

Rotterdam

Commodity

One-Step

Comparisons of Forecast Performance on Meat Quantities Consumed

Model

TABLE 4+

FORECAST EVALUATIONS IN MEAT DEMAND ANALYSIS

517

518

WANG AND BESSLER

TABLE 5+

Mean Square Forecast Error ~MSFE! under Different VECM Model Specifications One-step

Beef Poultry Pork

Two-Steps

Unrestricted

Weak Exogeneity

Homogeneity

Unrestricted

Weak Exogeneity

Homogeneity

0+283 0+443 0+181

0+284 0+424 0+185

0+271 0+420 0+195

0+464 0+518 0+283

0+513 0+434 0+410

0+453 0+485 0+296

ahead forecasts+ The imposition of the restriction, where it does not hold in the sample, leads to mixed results: it helps in the poultry system, but degrades the performance of the pork model in both one- and two-steps-ahead forecasts+ The imposition of the weak exogeneity restriction, and long-run exclusion of expenditure in the beef system has similar mixed results ~for an extended discussions related to this subject, see Wang & Bessler, 2002!+ 5.2 Equality Test of Forecast Errors To test the hypothesis of equality of forecast squared errors, we employ the test statistic proposed in Diebold and Mariano ~1995!: DM ⫽ @ V~ Z dN !# ⫺102 d,N

~8!

where dN is the sample mean of a new series dt formed by the difference of two h-stepsahead squared forecast error series ~h ⫽ 1, 2, + + +!, V~ P dN ! is the asymptotical variance of d+N Harvey et al+ ~1997! suggest correcting for small sample bias of the test by a factor of

冋

T ⫹ 1 ⫺2h ⫹ T ⫺1 h~h ⫺ 1! T

册

102

,

and comparing the statistic with critical values for the t-distribution+ The test results on one-step-ahead forecasts are given in Table 6+ Noticeably, the hypothesis of equality of MSFE between DGM and VECM in the first group cannot be rejected at the 0+05 significance level on an equation-by-equation basis+ The p-values are 0+61, 0+50, and 0+07 for beef, poultry, and pork forecast errors, respectively+ The equality hypothesis cannot be rejected between the AIDS and AIM in the second group ~the relevant p-values are 0+22, 0+31, and 0+48!+ This conclusion also holds for the Rotterdam and AIM model, but does not between the AIDS and AIM+ The beef and poultry consumption forecasts generated from the AIDS model are statistically different than those from the Rotterdam model because the null hypothesis can be rejected at virtually a 0% significance level+ Based upon these individual test p-values, we also implement a simple modified Bonferroni procedure ~Simes, 1986! for testing multiple hypotheses+ We cannot reject the null hypothesis of the equality of MSFE’s on a system-wide basis between the DGM and VECM, AIDS and Rotterdam, and Rotterdam and AIM+ This confirms the above test results on individual series+

1 0+00 0+22 0+10 0+00

Beef ~I!

1 0+05 0+92 0+21

Beef ~II!

1 0+01 0+02

Beef ~III!

1 0+61

Beef ~IV!

1

Beef ~V!

1 0+00 0+31 0+02 0+00

Poultry ~I!

1 0+07 0+01 0+00

Poultry ~II!

1 0+04 0+07

Poultry ~III!

Test Results for the Equality of Mean Square Forecast Error ~MSFE!

1 0+50

Poultry ~IV!

1

Poultry ~V!

1 0+33 0+48 0+00 0+66

Pork ~I!

1 0+26 0+01 0+93

Pork ~II!

1 0+81 0+27

Pork ~III!

1 0+07

Pork ~IV!

1

Pork ~V!

Note: I, II, III, IV, and V represent the AIDS, Rotterdam, AIM, DGM, and VECM models, respectively+ Each entry is the p-value of the null hypothesis that the MSFE of meat consumption generated from the model in the column are equal to the MSFE generated from the model in the row+

Beef ~I! Beef ~II! Beef ~III! Beef ~IV! Beef ~V! Poultry ~I! Poultry ~II! Poultry ~III! Poultry ~IV! Poultry ~V! Pork ~I! Pork ~II! Pork ~III! Pork ~IV! Pork ~V!

TABLE 6+

FORECAST EVALUATIONS IN MEAT DEMAND ANALYSIS

519

520

WANG AND BESSLER

5.3 Forecast Encompassing Test In the comparison of the forecast performance of alternative models, a more stringent requirement would be the competing forecasts embody no useful information absent in the preferred forecasts+ A formal approach, namely forecast encompassing test defined in Chong and Hendry ~1986!, is applied here+ Denote the two forecast error series from two competing models, 1 and 2, by eit , i ⫽ 1, 2, and the composite forecast error by «it + We can write e1t ⫽ l~e1t ⫺ e2t ! ⫹ «it +

~9!

The null hypothesis is l ⫽ 0+ When the null hypothesis is true, model 1 forecast encompasses model 2+ Table 7 gives the encompassing test results in probability form+ The null hypothesis is that the forecasts generated from the model in the first column encompass the forecasts from the model in the first row+ For example, the entry 0+02 in the upper-left corner of Table 7 is the probability of beef forecasts from DGM encompassing beef forecasts from AIDS+ Therefore, we reject the null hypothesis at the traditional 0+05 level in this case+ Notice that all diagonal elements have probability 1, because by the reflexivity feature of forecast encompassing a series encompasses itself+ The dynamic VECM model encompasses the AIDS model in terms of all three forecast series ~the p-values associated with the null hypothesis are 0+14, 0+47, and 0+13, respectively!+ The VECM model also encompasses Rotterdam in all three series+ However, it encompasses AIM only in poultry forecasts and DGM only in pork forecasts+ No evidence is found for DGM model to encompass the VECM, although the DGM generated a smaller MSFE of poultry consumption than VECM does ~0+341 vs+ 0+424!, and we cannot reject the hypothesis of equal MSFE of DGM and VECM+ In general, three static models, AIDS, Rotterdam, and AIM, do not encompass VECM or DGM with exceptions on pork consumption forecasts+ The condition for a system as a whole to encompass another system is more restrictive than in the case of individual series test+ Clements and Hendry ~1998! suggest a joint F-test ~p+ 235!: e1t ⫽ g~e1t ⫺ e2t ! ⫹ «it ,

~10!

where e1t and e2t are both vectors of forecast series from two competing systems+ When G ⫽ 0, the first system encompasses the second system+ This, in our case, requires that all three meat forecast series from the competing model do not have useful information, which is absent in each of the forecast series from the preferred system+ No system passes this test in our sample portfolio+ We also run the test in its weaker form where G is assumed to be diagonal ~no crosseffects among different forecast series!+ Only VECM encompasses Rotterdam at a marginal significance level of 0+08+5 The failure of two systems to encompass each other may signal misspecification of fundamental data generating process in both models+ Missing variable~s! ~e+g+, effect of health information!, insufficient dynamics, incorrect functional form or any combinations of these can cause this failure+ It is also possible that the data generating process has 5 There is not much empirical work applying the encompassing test+ The evidence on encompassing is even less abundant, for example, all six U+S+ trade balance models considered in Eriksson and Marquez ~1993! fail the encompassing tests+

1 0+03 0+00 0+02 0+14

Beef ~I!

0+00 1 0+00 0+03 0+14

Beef ~II!

0+00 0+00 1 0+43 0+04

Beef ~III! 0+00 0+00 0+00 1 0+00

Beef ~IV! 0+00 0+00 0+00 0+00 1

Beef ~V!

1 0+00 0+00 0+00 0+47

Poultry ~I!

Forecast Encompassing Test Results ~Individual Series!

0+00 1 0+00 0+00 0+16

Poultry ~II!

0+00 0+00 1 0+10 0+12

Poultry ~III!

0+00 0+00 0+00 0+00 0+00

Poultry ~IV!

0+00 0+00 0+00 0+00 1

Poultry ~V!

1 0+99 0+00 0+00 0+13

Pork ~I!

0+04 1 0+00 0+00 0+08

Pork ~II!

0+00 0+02 1 0+01 0+00

Pork ~III!

0+18 0+91 0+00 0+00 0+23

Pork ~IV!

0+02 0+11 0+00 0+00 1

Pork ~V!

Note: I, II, III, IV, and V represent the AIDS, Rotterdam, AIM, DGM, and VECM models, respectively+ Each entry is the p-value of the null hypothesis that the model in the first column encompasses the model in the first row on the basis of individual series+

Beef ~I! Beef ~II! Beef ~III! Beef ~IV! Beef ~V! Poultry ~I! Poultry ~II! Poultry ~III! Poultry ~IV! Poultry ~V! Pork ~I! Pork ~II! Pork ~III! Pork ~IV! Pork ~V!

TABLE 7+

FORECAST EVALUATIONS IN MEAT DEMAND ANALYSIS

521

522

WANG AND BESSLER

experienced structural change during the forecasting period+ In any case, this last result indicates room for improvement on specification in meat demand systems+ 6. SUMMARY AND CONCLUSION In this article, we identified five different demand systems: AIDS, Rotterdam, AIM, DGM, and VECM+ We recursively generated one- and two-steps-ahead out-of-sample forecasts for U+S+ meat consumption with each system+ In comparing forecast performance of these systems, we find that, in general, the dynamic VECM model performs best in both oneand two-steps-ahead forecasts+ Its strong performance is consistent across three different meat products, beef, poultry, and pork+ The imposition of homogeneity restriction in the cointegration space improves the forecast performance of the beef VECM system ~when it holds!, and the poultry system, but degrades the performance of the pork system ~where it does not hold!+ With exceptions in pork consumption forecasts, the static demand systems’ performance is poor+ The AIDS model, which is most often used in empirical research, seems to have forecasted better than the other two+ The specification of the DGM model is simpler than any other models+ Its general forecast performance is close to that of the best VECM model, even though it involves no dynamics+ This result is not surprising+ The directed graph method explicitly takes into account causal relationship, which should help the resulted model in forecasting new observations+ Although the evidence provided here tends to favor the dynamic VECM and the simple causal DGM model, we are fully aware that our study has limitations+ Other data sets of different frequency or periods might order the models differently+ The five-model portfolio is far from being exhaustive compared to the growing literature on demand system analysis+ However, we do strongly believe that more research should be done to evaluate demand models in terms of their out-of-sample forecast performance+ REFERENCES Akleman, D+G+, Bessler, D+A+, & Burton, D+M+ ~1999!+ Modeling corn exports and exchange rates with directed graphs and statistical loss functions+ In C+ Glymour & G+ Cooper ~Eds+!, Computation, causation and discovery ~pp+ 497–520!, Cambridge, MA: AAA0MIT Press+ Barnett, W+A+, & Jonas, A+ ~1983!+ The Muntz-Szatz demand system: An application of a globally well-behaved series expansion+ Economics Letters, 11, 337–342+ Barnett, W+A+, & Yue, P+ ~1988!+ Semiparametric estimation of the asymptotically ideal model: The AIM demand system+ Advances in Econometrics, 7, 229–251+ Barten, A+P+ ~1966!+ Theorie en empirie van een volledig stelsel van vraegvergelijkingen+ Doctoral dissertation+ Rotterdam: University of Rotterdam+ Bessler, D+A+ ~1984!+ Relative prices and money: A vector autoregression on Brazilian data+ American Journal of Agricultural Economics, 66, 25–30+ Bessler, D+A+ ~1990!+ Forecasting multiple time series with little prior information+ American Journal of Agricultural Economics, 72, 88–98+ Bessler, D+A+, & Akleman, D+G+ ~1998!+ Farm prices, retail prices, and directed graphs: Results for pork and beef+ American Journal of Agricultural Economics, 80, 1144–1149+ Brester, G+W+, & Wohlgenant, M+K+ ~1993!+ Correcting for measurement error in food demand estimation+ Review of Economics and Statistics, 75, 352–356+ Chambers, M+J+ ~1990!+ Forecasting with demand systems+ Journal of Econometrics, 44, 363–376+ Chern, W+S+, Huang, K+S+, & Lee, H+J+ ~1993!+ Food demand models for forecasting+ In L+ Tweeten

FORECAST EVALUATIONS IN MEAT DEMAND ANALYSIS

523

~Ed+!, Japanese and American agriculture: Tradition and progress in conflict ~pp+ 249–279!, Boulder CO: Westview Press+ Chong, Y+Y+, & Hendry, D+F+ ~1986!+ Econometric evaluation of linear macroeconomic models+ Review of Economic Studies, 53, 671– 690+ Clements, M+P+, & Hendry, D+F+ ~1998!+ Forecasting economic time series+ Cambridge, UK: Cambridge University Press+ Deaton, A+, & Muellbauer, J+ ~1980!+ An almost ideal system+ American Economic Review, 70, 312–326+ Diebold, F+X+, & Mariano, R+S+ ~1995!+ Comparing predictive accuracy+ Journal of Business and Economic Statistics, 13, 253–263+ Eales, J+S+, & Unnevehr, L+J+ ~1988!+ Demand for beef and chicken products: Separability and structural change+ American Journal of Agricultural Economics, 70, 521–532+ Eales, J+S+, & Unnevehr, L+J+ ~1993!+ Simultaneity and structural change in U+S+ meat demand+ American Journal of Agricultural Economics, 75, 259–268+ Engel, R+F+, & Granger, C+W+J+ ~1987!+ Cointegration and error correction: Representation, estimation and testing+ Econometrica, 55, 251–276+ Ericsson, N+R+, & Marquez, J+ ~1993!+ Encompassing the forecasts of U+S+ trade balance models+ Review of Economics and Statistics, 75, 19–31+ Gardner, B+L+ ~1975!+ The farm-retail price spread in a competitive food industry+ American Journal of Agricultural Economics, 57, 399– 409+ Harvey, D+I+, Leybourne, S+J+, & Newbold, P+ ~1997!+ Testing the equality of prediction squared errors+ International Journal of Forecasting, 13, 281–291+ Havenner, A+, & Saha, A+ ~1999!+ Globally flexible asymptotically ideal models+ American Journal of Agricultural Economics, 81, 703–710+ Heien, D+M+, Chen, T+N+, Chien, Y+L+, & Garrido, A+ ~1996!+ Empirical models of meat demand: How do they fit out of sample+ Agribusiness: An International Journal, 12, 51– 66+ Johansen, S+ ~1995!+ Likelihood-based inference in cointegrated vector autoregressive models+ Oxford, UK: Oxford University Press+ Karagiannis, G+, Katranidis, S+, & Velentzas, K+ ~2000!+ An error correction almost ideal demand system for meat in Greece+ Agricultural Economics, 22, 29–35+ Kastens, T+L+, & Brester, G+W+ ~1996!+ Model selection and forecasting ability of theory-constrained food demand systems+ American Journal of Agricultural Economics, 78, 301–312+ LaFrance, J+T+ ~1991!+ When is expenditure “exogenous” in separable demand models? West Journal of Agricultural Economics, 16, 49– 62+ Lin, J+, & Tsay, R+S+ ~1996!+ Co-integration constraint and forecasting: An empirical examination+ Journal of Applied Econometrics, 11, 519–538+ Pearl, J+ ~1995!+ Causal diagrams for empirical research+ Biometrika, 82, 669–710+ Scheines, R+, Spirtes, P+, Glymour, C+, & Meek, C+ ~1994!+ Tetrad II: User’s manual and software+ Mahwah NJ: Lawrence Erlbaum Associates, Inc+ Simes, R+J+ ~1986!+ An improved Bonferroni procedure for multiple tests of significance+ Biometrika, 73, 751–754+ Spirtes, P+, Glymour, C+, & Scheines, R+ ~1993!+ Causation, prediction and search+ New York: Springer-Verlag+ Swanson, N+R+, & Granger, C+W+J+ ~1997!+ Impulse response functions based on a causal approach to residual orthogonalization in vector autoregressions+ Journal of the American Statistical Association, 92, 357–367+ Theil, H+ ~1965!+ The information approach to demand analysis+ Econometrica, 30, 67–87+ Wang, Z+, & Bessler, D+A+ ~2002!+ The homogeneity restriction and forecast performance of VARtype demand systems: An empirical examination of U+S+ meat consumption+ Journal of Forecasting, 21, 193–206+ Wu, J+ ~1999!+ A re-examination of the exchange rate–interest differential relationship: Evidence from Germany and Japan+ Journal of International Money and Finance, 18, 319–336+

Zijun Wang is an assistant research scientist at the Private Enterprise Research Center at Texas A&M University, College Station, TX. Dr. Wang received his bachelor and master degrees in eco-

524

WANG AND BESSLER

nomics, in 1988 and 1991, respectively, both from Renmin University of China, Beijing. Dr. Wang received his PhD in Agricultural Economics from Texas A&M University in 2000. His primary research interests include econometric forecasting, empirical demand analysis, and economic policy analysis. David A. Bessler received his PhD in Agricultural Economics from the University of California at Davis in 1977. He earned the MS in Agricultural Economics and a BS in Economics in 1973 and 1971, respectively, both from the University of Arizona. He is currently a professor in agricultural economics at Texas A&M University, College Station, TX.

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & Close