Estimating and testing the compensated double-log demand model

Applied Economics, 2002, 34, 1177 ±1186

Estimating and testing the compensated double-log demand model J U L I A N M . A L S T O N , J A M E S A . C H A L F A N T * and N I C H O L A S E . P I G G O T T{ University of California, Davis and { North Carolina State University

In spite of the proliferation of ¯exible functional forms for consumer demand systems, the double-log demand model continues to be popular, especially in applied work calling for single-equation models. It is usually estimated in uncompensate d form. It can also be estimated in compensated form, by de¯ating the income variable alone using Stone’s price index. The compensated form has the same right-hand side as a single-equation version of the popular linear approximation to the Almost Ideal demand model, facilitating the construction of a test for choosing between the two alternatives. This paper demonstrates these results, develops the speci®cation test, and illustrates its application using US meat consumption data. Simulations suggest that the test is well-behaved with good power in typical applications.

I. INTRODUCTION In recent years, the academic literature on empirical demand analysis increasingly emphasized the use of ¯exible functional forms to estimate systems of demand equations that are consistent with demand theory. These models have largely displaced traditional, single-equation models for many purposes, such as testing the stability of preferences, but the common ¯exible forms are integrable only for subsets of possible parameter values, and their ¯exibility is typically local, not global.1 Although the double-logarithmi c demand model is not integrable, it continues to be popular, especially in applications calling for single-equation models. Consequently, it is still a very widely used model in studies of the demand response to commodity promotion (e.g., see Kinnucan et al., 1992) and in other studies of individual commodity markets. The theoretical de®ciencies of the double-log model are well known. When estimating the double-log demand model, only the homogeneity restriction can be

imposed; the model must be viewed as an approximation and cannot satisfy the other restrictions from consumer theory globally. Apparently, in some contexts, the bene®ts from ease of estimation and the easy interpretation of parameters as elasticities outweigh the costs of using a model of per capita demand that is not completely consistent with the theory of the individual consumer. This paper makes three points: (1) it clari®es how to impose the homogeneity condition in a double-log demand model so as to yield parameter estimates that can be interpreted as Marshallian, uncompensated elasticities; (2) it shows, alternatively, how to impose homogeneity and obtain estimates that can be interpreted as Hicksian, compensated demand elasticities; and (3) it develops and illustrates a test for the double-log functional form against a single-equation form of the Almost Ideal demand model. The ®rst point is well known and is covered mostly for completeness, although some studies still fail to impose homogeneity properly. The second point is not new, but is apparently not widely appreciated, and the option of

* Corresponding author. E-mail: [email protected] 1 By ¯exibility, it is usually meant that the model includes enough parameters so that, at a particular price vector, any set of elasticities could be obtained. Local ¯exibility is an improvement over functional forms such as the Cobb±Douglas or CES, which rule out complements. The ¯exibility of models such as the translog is local in the sense that, once the elasticities at one price vector are known, they are known at another ± changes in elasticities as prices change are dictated solely by the changes in shares from one price vector to another. See Despotakis (1986) for an illuminating discussion of this point. It is worth noting that, when it is viewed as an approximation to an arbitrary demand equation, the double-log model satis®es this de®nition of local ¯exibility. Applied Economics ISSN 0003±6846 print/ISSN 1466±4283 online # 2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080 /0003684011008600 3

1177

J. Alston et al.

1178 directly estimating compensated price responses in a double-log model has been neglected, probably as a result. The third point makes use of the ®rst two, but it is the substantial original contribution of the paper. Although the double-log model has limitations, its widespread use demonstrates its practical value. Hence, it is useful to be able to interpret the model correctly and to test its empirical validity as an approximation of the data generating process. The contribution of the paper, then, is to show how to estimate and interpret a compensated doublelogarithmic demand model and test it against an alternative.

This invalidates the interpretation of the coe cients as uncompensated price and income elasticities. The only exception is when N represents a subset of goods and the CPI can be thought of as an index of the price of all the other goods in the model, in which case the individual prices of those other goods should not be included as separate regressors. More often, the CPI will be an inappropriate de¯ator, since it is not an index of the price of a relevant subset of goods ± it includes the prices of goods that are already included separately in the model, and it may include prices of goods that do not belong in the model, if a separable group is being modelled.

II. DEFLATION AND HOMOGENEITY IN D O U B LE - L O G D E M A N D E Q U A T I O N S

III. COMPENSATED DOUBLE-LOG DEMAND MODELS

Consider the following `double-log’ demand equation for good i, in which ²ij is the Marshallian elasticity of demand for good i with respect to the jth price, and ²iI is the income elasticity of demand:

In some studies, such as those concerned with obtaining Hicksian measures of welfare change, it is desirable to obtain Hicksian measures of demand elasticities. These can be deduced from Marshallian elasticities, using the Slutsky equation, but they can also be estimated directly if the model is constructed properly. To obtain compensated demands, Stone suggested making use of the Slutsky equation in elasticity form:

ln Qi ˆ ¬i ‡ ²i1 ln P1 ‡ ²i2 ln P2 ‡ ¢ ¢ ¢ ‡ ²iN ln PN ‡ ²iI ln I

…1†

I denotes income (or expenditure on the N relevant goods) and Pj denotes the price of good j. The homogeneity condition is treated as a maintained hypothesis and can be expressed as an adding-up condition on the elasticities of demand for good i: …²i1 ‡ ²i2 ‡ ¢ ¢ ¢ ‡ ²iN ‡ ²iI † ˆ 0

…2†

Homogeneity corresponds to the familiar proposition of `no money illusion’ and can be imposed directly, as a parametric restriction, or by de¯ating all of the monetary variables by any one of the N prices or total expenditure.2 The remaining coe cients are still Marshallian elasticities, and the elasticity with respect to the variable used as a de¯ator can be recovered using the homogeneity restriction. The homogeneity restriction ensures that only real income and relative prices are relevant for explaining consumption. The noption that income and prices should be de®ned in real terms has led many to de¯ate prices and income in double-log demand models prior to estimation. However, the common practice of using a general price index such as the Consumer Price Index (CPI) to de¯ate the prices and income is usually inappropriate. De¯ating by the CPI does impose a type of homogeneity restriction but, at the same time, introduces a new price into the demand model, and the result is that a modi®ed homogeneity condition holds: …²i1 ‡ ²i2 ‡ ¢ ¢ ¢ ‡ ²iN ‡ ²iI † ‡ ²i;CPI ˆ 0 2

²ij ˆ ²ij¤ ¡ Sj ²iI

…3†

²¤ij

where Sj denotes good j’s budget share and is the compensated price elasticity of demand for good i with respect to Pj .3 Substituting (3) into (1) yields ln Qi ˆ ¬i ‡

N X

ˆ ¬i ‡

N X

ˆ ¬i ‡

N X

jˆ1

jˆ1

jˆ1

‰²ij¤ ¡ Sj ²iI Š ln Pj ‡ ²iI ln I ²ij¤

"

ln Pj ‡ ²iI ln I ¡

n X jˆ1

²ij¤ ln Pj ‡ ²iI ln …I =P¤ †

Sj ln Pj

# …4†

where P¤ is Stone’s geometric price index: P¤ ˆ

N Y

S

Pj j

jˆ1

Clearly the last line of Equation 4 represents a compensated (Hicksian) demand function, at least in the sense that its price coe cients are compensated elasticities. This is a double-log model in which the prices are unde¯ated and the nominal group expenditure is de¯ated by Stone’s price index for the group of goods included in the model.

De¯ating all monetary variables by any monetary variable ± relevant or not ± imposes an absence of money illusion, but need not make economic sense. 3 Stone’s motive in working with compensated price elasticities was to obtain a parsimonious demand equation, making it possible to omit unimportant prices as regressors. See Deaton and Muellbauer (1980) for a discussion.

The compensated double-log demand model To impose homogeneity in Equation 4, the compensated price P ¤ elasticities must be restricted to sum to zero ± i.e., j ²ij ˆ 0. This restriction could be imposed directly, or alternatively the same e€ ect could be achieved by expressing all of the included prices relative to one of the included prices as, follows: ³ ´ ³ ´ N ¡1 X Pj I ¤ ln Qi ˆ ¬i ‡ ‡ ²iI ln ¤ …5† ²ij ln P P N jˆ1 It is important to note that, although (5) is identical to (1) in terms of its economic content, as an empirical matter there may be virtue in estimating (5), to measure the Hicksian responses directly. In particular, estimating (5) yields measures of precision of the estimates of the compensated price responses directly, which may be useful in welfare analysis (e.g., Alston and Larson, 1993). To summarize, the interpretation of the parameters in double-log demand models depends on the procedures used for de¯ating and imposing homogeneity. The doublelog model can be required to satisfy the homogeneity restriction and can be made to yield parameters that can be interpreted as income elasticities and either Marshallian, uncompensate d price elasticities or Hicksian, compensated price elasticities so long as appropriate decisions are made about de¯ation and parametric restrictions. If one de¯ates all of the price and income variables by one of the included prices, one has merely imposed homogeneity in a Marshallian demand equation. If one de¯ates only income by Stone’s price index, one has a Hicksian demand equation, in the sense that the price coe cients can be interpreted as compensated price elasticities.4 Some studies may have correctly de¯ated prices and income, imposing homogeneity, but misinterpreted the implications for the interpretation of parameters. Such mistakes are not con®ned to unpublished applied studies. Even some standard references have misconstrued the e€ ect of de¯ation on the interpretation of demand elasticities. For instance, in his price theory text, Becker (1971: 35±40) suggested that when the variables in a demand function are all measured in `real’ terms, the measured price responses correspond to Hicksian, compensated demands. But de¯ating every nominal price and income variable in (1) either by any one of those nominal variables (i.e., imposing homogeneity) or by some other nominal variable (e.g., the CPI ± not a correct way to impose homogeneity) does not yield compensated elasticities.5 More recently, Greene’s econometrics text, for instance, contains an example (1997: 699) where de¯ating all nominal prices and income by Stone’s price index is said to yield 4

1179 compensated elasticities. In general, this interpretation is also incorrect. Only if the price coe cients were restricted to sum to zero, as must be the case for compensated elasticites, could they then be interpreted as compensated elasticities. But in such a case, de¯ating prices by Stone’s index is redundant. If the price coe cients do not sum to zero, however, de¯ating by Stone’s price index does not produce compensated elasticities.

I V . TE S T F O R T H E C O R R E C T FU N C T I O N A L FO R M De¯ating income by Stone’s price index means that the right-hand side of the compensated double-log model in (5) is identical to that of an equation from the common linear approximation of the Almost Ideal demand system (the LA model), which has the expenditure share of good i as the dependent variable, rather than ln Qi : Si ˆ ¬i ‡ ®i1 ln …P1 =PN † ‡ ¢ ¢ ¢

‡ ®iN ¡1 ln …PN¡1 =PN † ‡ ®iI ln …I =P¤ †

…6†

where Si ˆ Pi Qi =I and homogeneity is imposed by using the Nth price to de¯ate the other N ¡ 1 prices. This observation suggests a test of the two speci®cations. While the typical double-log model in (1) seems somewhat dissimilar to the LA model, transforming the double-log model to a `compensated’ demand, as in Equation 5, leads to the insight that each is a special case of the compound model: …1 ¡ ¶† ln Qi ‡ ¶Si ˆ ¬i ‡ ®i1 ln …P1 =PN † ‡ ¢ ¢ ¢ ‡ ®iN¡1 ln …PN¡1 =PN † ‡ ®iI ln …I =P¤ †

…7†

The results from estimating ¶ along with the parameters on the right-hand side could be interpreted as a test of either speci®cation. Subject to the quali®cation that testing one equation of a system is not the same as testing the whole system, the results from estimating ¶ could also be interpreted as a test of the adequacy of the speci®cation of the LA model for demand systems. There are two problems with this interpretation . First, an OLS estimate of ¶ (which could be obtained after moving the term ¶…ln Qi ¡ Si † to the right-hand side of the model) depends upon the scaling of the dependent variables. Second, the estimated ¶ can be shown to be biased by the correlation of that term with the error term.6 Elaborating below, this paper suggests two alternative tests.

This speci®c result is true in the double-log model, but need not hold exactly in other models. Di€ erent functional forms for demand equations imply di€ erent functional forms for the price index necessary to hold real income constant. 5 Wohlgenant (1985) made a similar point, in a similar context. 6 LaFrance (1998) discussed an equivalent problem, in the context of choosing betwen the Almost Ideal and Rotterdam models, and proposed an approach for choosing between those two demand systems.

J. Alston et al.

1180 Note that (7) can be rearranged to yield ln Qi ˆ ¬i ‡ ®i1 ln …P1 =PN † ‡ ¢ ¢ ¢ ‡ ®iN¡1 ln …PN¡1 =PN † ‡ ®iI ln …I =P¤ † ‡ ¶…ln Qi ¡ Si †

…8†

While (8) can be estimated using linear regression, the right-han d side variable in ln Qi ¡ S i is likely to be correlated with the error term. As a result, ¶^ is likely to be biased. It is tempting to deal with this problem by replacing the two possible dependent variables with predicted values d (ln Qi and S^i ), but since each would be a linear combination of the other right-hand side variables, a singular model would result and ¶ could not be estimated. Alternatively, the rest of the right-hand side variables could also be replaced with one predicted value. Under the null hypothesis that the double-log model is correct, for instance, replacing the variables in (8) with the predicted values of ln Qi and S i yields: ln Qi ˆ ¬ ^ i ‡ ®^i1 ln …P1 =PN † ‡ ¢ ¢ ¢ ‡ ®^iN¡1 ln …PN¡1 =PN † d ‡ ®^iI ln …I =P¤ † ‡ ¶…ln Qi ¡ S^i †

d d ˆ ln Qi ‡ ¶…ln Qi ¡ S^i †

…9†

E€ ectively, this study would be using di€ erences in the two predictions to explain the prediction error in the double-log model: d ln Qi ¡ ln Qi

The last speci®cation closely resembles the C-test of Davidson and MacKinnon, with two di€ erences. First, the variable de®ned as the di€ erence between alternative predictions represents predictions of di€ erent transformations of the underlying quantity variable, rather than two competing predictions of the same variable. Second, the sign of ¶ is the opposite of the sign of the usual C-test. A second problem with the test implied by (7) concerns the scaling of dependent variables. Estimating (7) by OLS turns out to imply the minimization of a sum of squares of the linear combination of two vectors of residuals, e1 from the double-log equation and e2 from the share equation (e.g., Maddala): min e30 e3 ˆ min …1 ¡ ¶†2 e10 e1 ‡ ¶2 e20 e2 ‡ 2¶…1 ¡ ¶†e10 e2 where e3 denotes the linear combination of e1 and e2 implicit in Equation 7 above. It is easy to show that when (7) or (8) is estimated, the resulting estimate of ¶ is given by: ¶^ ˆ

e10 e1 ¡ e10 e2 e10 e1 ‡ e20 e2 ¡ 2e10 e2

Thus the estimate of ¶ depends on scaling: if this study rescales shares by multiplying by 100, e2 changes while e1 7

does not, so that ¶^ will not be invariant to scaling. This problem is not avoided by estimating (9) instead of (7). So, another adjustment in the test is required. Instead of the direct use of a predicted value from the share equation, in testing the null hypothesis that the double-log model is g correct, one can use ln Qi , a prediction of the logarithm of quantity consumed that is implied by the share model. In other words, one can use the share model to predict shares, and transform each predicted share to a prediction of ln Q, and compare the result with the predictions obtained directly from the double-log model. This is, in essence, the Davidson and MacKinnon C-test. It is calculated based on d d g ln Qi ˆ ln Q i ‡ ¶…ln Qi ¡ ln Qi †

…10†

d If the double-log model were correct, ln Qi ¡ ln Qi d g should be uncorrelated with the di€ erences ln Qi ¡ ln Qi , so a statistically signi®cant ¶^ is evidence against the null hypothesis that the double-log model is correct. As Davidson and MacKinnon (1991) noted, the C-test tends to be too conservative : its asymptotic behaviour is such that it does not reject the null hypothesis with a frequency as great as the stated size of the test. They suggested the Ptest as one alternative to the C-test. Both the P-test and the same authors’ J-test coincide for models such as those tested in this paper, which are linear in the parameters. A ®nal adjustment that could be done thus involves adding the right-hand side variables from Equation 7 to the regresg sion in (10), which is possible since ln Qi is not linear in these variables. This converts the C-test to the P-test. Both alternatives are tried in the simulations described below. To test the null hypothesis that the share model is correct, reverse the roles of the two models. From the double-log model speci®ed in the alternative hypothesis, a prediction of the share of good i is obtained by transforming the predicted ln Qi into a predicted share. The test then compares the predicted shares from the share model and from the double-log model.

V . A N EX A M P LE U S I N G M E A T CONSUMPTION DATA The estimation of the compensated double-log demand models, and the tests described above, are illustrated using the US meat consumption data from Piggott (1997) comprising prices and quarterly per capita consumption of beef, pork, and poultry (pounds per capita) for the period 1970 to mid-1995.7 Total expenditure on the three meats is used as the `income’ variable. Uncompensated double-log demand models, as in Equation 1, were estimated by OLS for the three meats,

The data set used in Piggott (1997) begins in 1979, so we have extended the sample back in time to include 9 additional years of quarterly observations.

The compensated double-log demand model

1181

Table 1. Uncompensated and compensated double-log demand models for US meat: OLS estimates Uncompensated Beef ²i1 ²i2 ²i3 ²iI t t2 qd1 qd2 qd3 ¬i R2 D-W

71.1194* (0.0457) 70.0495 (0.0278) 1.1689* (0.0623) 1.3305* (0.0718) 1.26E-03* (2.85E-04) 73.07E-05* (2.40E-06) 0.0610* (0.0061) 0.0672* (0.0057) 0.0564* (0.0052) 71.9322* (0.2794) 0.9755 0.6126

Compensated Pork 0.1649* (0.0767) 70.8786* (0.0467) 0.7136* (0.1046) 0.8408* (0.1206) 72.88E-03* (4.79E-04) 2.69E-05* (4.03E-06) 70.0272* (0.0102) 70.0707* (0.0096) 70.0691* (0.0087) 70.7356 (0.4691) 0.8946 0.6138

Poultry 0.1211 (0.0819) 70.0685 (0.0499) 70.0525 (0.1116) 0.1259 (0.1287) 2.06E-03* (5.11E-04) 4.34E-05* (4.30E-06) 70.1640* (0.0109) 70.1059* (0.0103) 70.0736* (0.0093) 1.9265* (0.5006) 0.9810 1.2572

Beef 70.2741* (0.0286) 0.3318* (0.0324) 70.0577 (0.0402) 1.3365* (0.1085) 1.73E-03* (3.86E-04) 75.08E-05* (3.51E-06) 0.0888* (0.0100) 0.0907* (0.0090) 0.0770* (0.0077) 72.0416* (0.4289) 0.9564 0.5841

Pork 0.7143* (0.0333) 70.6201* (0.0378) 70.0942* (0.0469) 1.0706* (0.1266) 72.44E-03* (4.50E-04) 1.11E-05* (4.09E-06) 0.0057 (0.0117) 70.0435* (0.0105) 70.0480* (0.0090) 71.6963* (0.5004) 0.9093 0.64994

Poultry 0.2180* (0.0371) 70.0130 (0.0421) 70.2050* (0.0523) 0.3783* (0.1410) 2.26E-03 (5.01E-04) 3.82E-05 (4.56E-06) 70.1442* (0.0130) 70.0898* (0.0117) 70.0626* (0.0100) 0.9227 (0.5573) 0.9821 1.25912

Notes: i ˆ 1 for beef, 2 for pork, and 3 for poultry; qdi is the seasonal dummy variable for the ith quarter, t is a linear trend (where t ˆ 1 for 1970(1)), and t2 is a quadratic time trend. Estimates in parentheses are standard errors. * Denotes a coe cient that is signi®cantly di€ erent from zero at the 5% signi®cance level.

with the homogeneity condition (2) imposed as a maintained hypothesis. Following previous studies using these or similar data, we incorporate d quarterly seasonal dummies (qd j for j ˆ 1; . . . ; 3) and linear and quadratic time trends (t and t2 , where t ˆ 1 for 1970(1)), but did not include advertising variables, for which previous studies have found mixed results. In addition, using the same data, the corresponding compensated double-log demand models, as shown in Equation 4, were estimated, also with homogeneity imposed.8 Table 1 reports the results. In the models in Table 1, most of the price and income elasticity coe cients are statistically signi®cant and plausible for beef and pork. In the poultry equation, the estimated elasticities seem plausible, but they are mostly not statistically signi®cantly di€ erent from zero. In all three models, the Durbin±Watson statistics indicate some problem with the speci®cation, either autocorrelation or some other speci®cation error. Table 2 shows the results from estimating the same models with corrections for ®rstorder autocorrelation. The elasticity estimates were a€ ected but only slightly, and not in ways that would materially change anyone’s views about meat demand. 8

Considering both the models with and without autocorrelation corrections, there is no clear evidence that either the compensated model or the uncompensated model is generally statistically superior. This seems to imply that, whatever the merits or shortcomings of the double-log model, imposing the Slutsky equation on the uncompensated version is a mild restriction. Another issue of interest is the implied values of the income elasticities, and the compensated and uncompensated own- and cross-price elasticities of demand. All of the own-price elasticities of demand are negative, signi®cant, and plausible (noting that we are holding constant the expenditure on meat, rather than total income, and so we would expect the elasticities to be generally larger in magnitude than if total income were being held constant) with one exception ± the estimate of the unconpensated own-price elasticity of demand for poultry. Most of the cross-price elasticities of demand are positive, indicating that all meats are substitutes, especially in the case of the compensated demands (when the compensated cross-price elasticities are negative numbers, the magnitudes are very small).

Moschini (1995) noted that the scaling of the prices in Stone’s index can a€ ect estimated elasticities. The correction he suggested had relatively small e€ ects on our estimated elasticities, so we simply use the unadjusted P¤ , as is standard practice; this more commonly used model is the version more likely to be tested in applications. However, it is worth nothing that even these small di€ erences can and did have large e€ ects on speci®cation tests.

J. Alston et al.

1182

Table 2. Uncompensated and compensated double-log demand models for US meat: models corrected for autocorrelation Uncompensated Beef ²i1 ²i2 ²i3 ²iI t t2 qd1 qd2 qd3 ¬i » R2

71.0679* (0.0512) 70.0361 (0.0362) 1.1040* (0.0642) 1.2516* (0.0663) 1.02E-03 (5.78E-04) 72.83E-05* (5.23E-06) 0.0568* (0.0043) 0.0634* (0.0042) 0.0553* (0.0029) 71.6276* (0.2556) 0.7209* (0.0687) 0.9873

Compensated Pork 0.0803 (0.0858) 70.8712* (0.0606) 0.7910* (0.1077) 0.9661* (0.1114) 72.84E-03* (9.56E-04) 2.57E-05* (8.64E-06) 70.0218* (0.0073) 70.0653* (0.0071) 70.0677* (0.0048) 71.2270* (0.4291) 0.7159* (0.0691) 0.9454

Poultry 0.0657 (0.0990) 70.1092 (0.0643) 0.0436 (0.1328) 0.1872 (0.1463) 2.48E-03* (7.24E-04) 4.08E-05* (6.23E-06) 70.1588* (0.0101) 70.1015* (0.0099) 70.0721* (0.0075) 1.7170* (0.5671) 0.3900* (0.0912) 0.9836

Beef 70.2683* (0.0424) 0.3323* (0.0439) 70.0640 (0.0425) 1.2449* (0.0964) 1.20E-03 (8.04E-04) 74.45E-05* (7.72E-06) 0.0813* (0.0077) 0.0.839* (0.0070) 0.0743* (0.0048) 71.6887* (0.3777) 0.7400* (0.0666) 0.9781

Pork 0.7041* (0.0490) 70.5803* (0.0514) 70.1239* (0.0514) 1.1244* (0.1182) 72.54E-03* (8.46E-04) 1.05E-05 (7.77E-06) 0.0086 (0.0094) 70.0401* (0.0086) 70.0471* (0.0059) 71.9130* (0.4636) 0.6848* (0.0772) 0.9503

Poultry 0.1988* (0.0502) 70.0377 (0.0553) 70.1611* (0.0637) 0.4522* (0.1591) 2.73E-03* (7.14E-04) 3.43E-05* (6.47E-06) 70.1367* (0.0130) 70.0835* (0.0118) 70.0597* (0.0088) 0.6638 (0.6268) 0.3968* (0.0910) 0.9846

Notes: See Table 1.

The compensated and uncompensated demand models for each good tend to suggest the same story for the uncompensate d elasticities. Consider the results for the model without autocorrelatio n corrections as shown in Table 1. From the compensated demand, the compensated own-price elasticity of demand for beef in ¡0:27 and the elasticity of demand for beef with respect to meat expenditure is 1.34. Given a mean expenditure share of beef of around 0.56, the implied value for the uncompensated elasticity is ¡1:02. From the uncompensated demand model for beef, the uncompensated own-price elasticity of demand for beef is ¡1:12, and the elasticity of demand for beef with respect to meat expenditure is 1.33. Similar relationships hold to a comparable degree for the other elasticities. A ®nal point about interpretation concerns the expenditure variable. As noted by LaFrance (1991), when the expenditure variable is constructed from the price and quantity data, it is correlated with the error term in a quantity-dependen t demand equation. This means that it may not be appropriate to treat the expenditure variable as exogenous. Replacing quantity demanded with its logarithm or converting to the budget share will not avoid this problem. This study tested for the endogeneity of the expenditure variable using the Hausman test, and three of the six equations (beef and poultry for the double-log model and the poultry share equation) had statistically

signi®cant Hausman test statistics. This may re¯ect other speci®cation errors, or may lead one to prefer instrumental variables estimates for those equations, or else the use of total, rather than group expenditure. Since the evidence concerning endogeneity was mixed, this study treats the expenditure variable as exogenous. However, if this assumption is inappropriate, the estimated results reported above and the speci®cation tests below might be more reliable for pork than for beef and poultry. The speci®cation test proposed above was applied to each of these six models ± three meats with and without autocorrelation corrections. For each meat, the compensated double-log model was tested against a model with the same right-hand side, but with expenditure shares as the dependent variable (i.e., a single equation from the linear approximate Almost Ideal demand system, the LA model). The results are reported in Table 3. In each case, the C- and P-tests yielded very similar results. The doublelog model was rejected for beef and pork, but not poultry, using OLS, and for beef and poultry, but not pork, when the model was corrected for ®rst-order autocorrelation. In addition, when the test was reversed, it rejected the single-equation LA model for beef and pork, but not poultry, regardless of whether a correction for autocorrelation was made. As noted above, it is possible to reject both of the alternative models. The results for the LA model are consistent with those of Piggott (1997), who

The compensated double-log demand model

1183

Table 3. C- and P-tests for the Almost Ideal and compensated double-log models for US meats OLS Beef

Corrected for autocorrelation Pork

Poultry

Beef

Pork

Poultry

H0: The double-log model is correct 73.5501* C-test 1.6241* (0.2537) (0.4538) 73.6321* P-test 1.6462* (0.2628) (0.4783)

70.3246 (0.2849) 70.3484 (0.3091)

72.9294* (0.1745) 72.9463* (0.1809)

0.7466 (0. 4264) 0.7476 (0.4491)

70.7301* (0.2333) 70.7530* (0.2475)

H0: The Almost Ideal model is correct 72.6268* C-test 2.5130* (0.2517) (0.4504) 72.6593* P-test 2.5919* (0.2636) (0.4738)

70.2948 (0.2608) 70.3268 (0.2875)

1.9368* (0.1734) 1.9635* (0.1807)

71.7267* (0.4241) 71.7277* (0.4460)

70.0101 (0.2025) 70.0128 (0.2188)

Notes: The reported values correspond to the point estimates of ¶ for the C-test and P-test. Estimates in parentheses are standard errors. * Denotes a rejection of the null hypothesis that the model is correct at the 5% signi®cance level.

found that more ¯exible functional forms dominated the AI model.

V I . P R O P ER T I E S O F T H E S P E C I F I C A T I O N T ES TS To examine the sampling properties of the test developed above requires data consistent with either the double-log model or the share model. For each meat this study followed a procedure similar to what Alston and Chalfant (1991) suggested for Monte Carlo studies of the demand for meats. The model being treated as the `true’ model was estimated using the actual data; predicted values were saved. Then, generating random errors and adding these to the predicted values leads to a new data set that follows the originally hypothesized data-generatin g process. In this fashion, for any of the four meats, it is possible to generate new data sets consistent with the share or double-log models. By repeating this process 10 000 times ± at each replication, adding randomly generated errors to predicted values ± 10 000 new data sets that correspond to the same true model were generated. Then, both share and doublelog models were estimated for each of the 10 000 data sets generated from a true double-log model as well as for each of 10 000 data sets generated from a true share model. Then the C- and P-tests were applied to each data set.9

9

The above process implies a total of four separate Monte Carlo experiments: the C- and P-test alternatives for both the double-log and share models as null hypotheses . All of this was done for the case where the double-log model was used to generate the data. Having no expectation one way or the other about how the particular form chosen to be the true model would in¯uence the results, we then repeated the four experiments using data generated from a share model ± simply reversing the roles of the two models. The combination of 3 meats and 4 experiments per meat, for each `true’ model, means a total of 24 di€ erent cases. This experiment includes one more design parameter. When errors are added to the original predicted values, to generate Monte Carlo data sets, the power of the test being studied depends on the variance of the randomly generated errors. Letting ¼ denote the estimated standard deviation of the error term in the in-sample results, the standard deviation was speci®ed in the Monte Carlo errors to be µ¼ and let µ vary from 0.25 to 1.5 by increments of 0.25. This means that power and size calculations for each of these 24 cases are conducted for each of 6 levels of variance. Tables 4 through 7 contain the results of these experiments. The even-numbered tables contain results for the Ctest; the odd-numbered tables are corresponding P-test results. Tables 4 and 5 correspond to a `true’ double-log model; Tables 6 and 7 to a `true’ share model. Taking beef as an example, Table 4 shows that the C-test

When testing the model known to be incorrect, a second false model could also have been tried as the alternative, to examine the power of the test at rejecting an incorrect model when the true model was not in the set of models considered. As Davidson and MacKinnon (1981) noted, the alternative model need not be correct to be able to cause rejection of the one given in the null hypothesis. It seems reasonable to believe that when the power is greater, the better is the approximation to the true data-generating process provided by the model implied by the alternative hypothesis, but no formal analyses of the sensitivity of the test’s power to such choices have been conducted.

J. Alston et al.

1184 Table 4. Size…¬† and Power…1 ¡ ­ † of the C-test when a `true’ double-log model is tested against a `false’ share model 100¬ Random error 0:25¼ 0:50¼ 0:75¼ 1:00¼ 1:25¼ 1:50¼

100…1 ¡ ­ †

Beef

Pork

Poultry

Beef

6.93 (0.0025) 6.61 (0.0025) 6.28 (0.0024) 6.71 (0.0025) 6.18 (0.0024) 6.10 (0.0024)

6.67 (0.0025) 6.35 (0.0024) 6.44 (0.0025) 6.31 (0.0024) 6.32 (0.0024) 6.15 (0.0024)

6.06 (0.0024) 5.74 (0.0023) 6.30 (0.0024) 5.77 (0.0023) 5.98 (0.0024) 5.68 (0.0023)

100.00 (0.0000) 99.29 (0.0008) 83.69 (0.0037) 61.10 (0.0049) 43.56 (0.0050) 32.45 (0.0047)

Pork 100.00 (0.0000) 97.86 (0.0014) 78.64 (0.0041) 54.92 (0.0050) 40.21 (0.0049) 29.15 (0.0045)

Poultry 100.00 (0.0000) 100.00 (0.0000) 99.19 (0.0009) 91.48 (0.0028) 78.23 (0.0041) 64.62 (0.0048)

Notes: ¬ is the estimated probability of incorrectly rejecting the true model (i.e., ¬ is the probability of a type I error). (1 ¡ ­ ) is the estimated probability that the test correctly rejects the false model (i.e., ­ is the probability of a type II error). The numbers in 2 parentheses are estimated standard errors of the corresponding numbers in the table. The random error is drawn from a N…0; …µ¼† † distribution where ¼ is the actual in-sample estimated standard deviation for each underlying true model, and where µ ˆ 0:25, 0.50, 0.75, 1.00, 1.25 or 1.50. All estimates are based on 10 000 replications. Table 5. Size…¬† and power…1 ¡ ­ † of the P-test when a `true’ double-log model is tested against a `false’ share model 100¬ Random error 0:25¼ 0:50¼ 0:75¼ 1:00¼ 1:25¼ 1:50¼

100…1 ¡ ­ †

Beef

Pork

Poultry

Beef

5.72 (0.0023) 5.41 (0.0023) 5.17 (0.0022) 5.57 (0.0023) 5.17 (0.0022) 5.25 (0.0022)

5.45 (0.0023) 5.30 (0.0022) 5.29 (0.0022) 5.26 (0.0022) 5.40 (0.0023) 5.40 (0.0023)

5.19 (0.0022) 4.93 (0.0022) 5.63 (0.0023) 5.12 (0.0022) 5.64 (0.0023) 5.54 (0.0023)

100.00 (0.0000) 99.03 (0.0010) 81.45 (0.0039) 57.77 (0.0049) 40.54 (0.0049) 30.06 (0.0046)

Pork 100.00 (0.0000) 97.44 (0.0016) 75.92 (0.0043) 52.02 (0.0050) 37.51 (0.0048) 27.23 (0.0045)

Poultry 100.00 (0.0000) 100.00 (0.0000) 99.03 (0.0010) 90.59 (0.0029) 76.53 (0.0042) 62.89 (0.0048)

Notes: See Table 4.

has a size near its nominal size of 5% when the double-log model is true. The numbers in the table are percentages of Monte Carlo replications for which the true model was rejected, corresponding to 100¬.10 There is no obvious pattern to particular di€ erences from 5%, as the standard deviation of the Monte Carlo errors varies from 0:25¼ to 1:5¼, and none of the di€ erences appears to be of much concern for applications of the test. Using a critical value based on a 5% rejection probability seems to imply an approximately 6 or 7% true rejection probabilty. The percentage of replications for which the false, share model for beef is rejected was also computed; for each 10

standard deviation, this gives an estimate of the power of the C-test. The power is quite large when there is not much noise in the data; the incorrect share model is rejected in 100% of the replications for the smallest variation. As the variance in the errors is increased, it becomes harder to tell a false model from a true one, so that by the time the standard deviation is 1.5 times its `in-sample’ value, the power has decreased to 32%. Inspection of the other columns in Table 4 indicates that the C-test behaves similarly for the other meats, although power is better for the poultry equation. Table 5 gives the P-test results for the same replications,

The absolute value of the calculated `t’-statistic for ¶ is compared with Z ¤ ˆ 1:96, since the test is only asymptotically well behaved and since this study has a relatively large number (102) of observations.

The compensated double-log demand model

1185

Table 6. Size…¬† and power…1 ¡ ­ † of the C-test when a `true’ share model is tested against a `false’ double-log model 100¬ Random error 0:25¼ 0:50¼ 0:75¼ 1:00¼ 1:25¼ 1:50¼

100…1 ¡ ­ †

Beef

Pork

Poultry

Beef

6.96 (0.0025) 6.66 (0.0025) 6.21 (0.0024) 6.65 (0.0025) 6.11 (0.0024) 6.18 (0.0024)

6.56 (0.0025) 6.35 (0.0024) 6.39 (0.0024) 6.43 (0.0025) 6.21 (0.0024) 6.07 (0.0024)

6.08 (0.0024) 5.80 (0.0023) 6.18 (0.0024) 5.78 (0.0023) 6.07 (0.0024) 5.60 (0.0023)

100.00 (0.0000) 100.00 (0.0000) 95.36 (0.0021) 78.50 (0.0041) 61.13 (0.0049) 46.76 (0.0050)

Pork 100.00 (0.0000) 96.29 (0.0019) 70.44 (0.0046) 47.09 (0.0050) 32.63 (0.0047) 24.45 (0.0043)

Poultry 100.00 (0.0000) 100.00 (0.0000) 98.87 (0.0011) 89.64 (0.0030) 73.40 (0.0044) 57.62 (0.0049)

Notes: See Table 4. Table 7. Size…¬† and power…1 ¡ ­ † of the P-test when a `true’ share model is tested against a `false’ double-log model 100¬ Random error 0:25¼ 0:50¼ 0:75¼ 1:00¼ 1:25¼ 1:50¼

100…1 ¡ ­ †

Beef

Pork

Poultry

Beef

5.56 (0.0023) 5.46 (0.0023) 5.02 (0.0022) 5.63 (0.0023) 5.19 (0.0022) 5.20 (0.0022)

5.47 (0.0023) 5.17 (0.0022) 5.27 (0.0022) 5.30 (0.0022) 5.41 (0.0023) 5.51 (0.0023)

5.30 (0.0022) 5.04 (0.0022) 5.44 (0.0023) 5.27 (0.0022) 5.62 (0.0023) 5.34 (0.0022)

100.00 (0.0000) 99.98 (0.0001) 94.49 (0.0023) 76.11 (0.0043) 57.97 (0.0049) 43.81 (0.0050)

Pork 100.00 (0.0000) 95.57 (0.0021) 67.60 (0.0047) 43.73 (0.0050) 30.16 (0.0046) 22.94 (0.0042)

Poultry 100.00 (0.0000) 100.00 (0.0000) 98.59 (0.0012) 88.33 (0.0032) 71.27 (0.0045) 55.74 (0.0050)

Notes: See Table 4.

also generated with the double-log model. For all three meats, the estimated size of the P-test is always smaller than that of the C-test and closer to the nominal value of 5%, suggesting the opposite of what Davidson and MacKinnon (1981) shows for asymptotic size. Across rows and across meats, the size measures appear to behave in a similar manner to what was observed in Table 4. The same is true for the estimates of power, which never di€ er from the C-test results by more than four percentage points. Tables 6 and 7 show that the behaviour of the two tests is similar when the share model is used to generate the data. The use of a share model does not appear to a€ ect either the similarity of the two tests or the overall pattern for power and size. 11

The main results can be summarized quite simply. Both the C-test and P-test alternatives appear to have actual sizes quite comparable to their nominal size of 5%. In no case did the frequency of rejection of the true model exceed even 7%. The power of the tests can be very good, with power decreasing, as expected, as the variance of the errors increases. 11 It is not clear what can be made of di€ erent rates of decrease in the probability of rejecting the false model for di€ erent meats, as variance increases. The results are no doubt related to the amount of trend and the quarter-to-quarte r variation in the predicted values themselves, to di€ erent responses to changes in exogenous variables, and perhaps to the goodness-of-®t of the underlying model used to generate the data. In any event, the results are quite encouraging, overall. They show a high probability of

Similar results were found when a similar analysis was made using Australian meat consumption data from Piggot et al. (1996) and using Moschini and Meilke’s (1989) data for US meat consumption, which ended in 1998.

J. Alston et al.

1186 rejecting a false model when it is tested. The fact that the results also vary somewhat across the di€ erent meats and the di€ erent variances suggests that such a power calculation might be useful as a routine part of applying these tests, especially since the experiment is easy to program in a standard package. This study uses SHAZAM (White (1993)). These results also suggest a slight preference for the P-test. Given the relatively poorer performance of the C-test, although only for relatively large standard deviations, the Davidson and MacKinnon conclusion that the P-test is preferable seems to be supported.

VII. CONCLUSION In double-log demand equations, de¯ating income (but not prices) by Stone’s (1954) price index holds real income constant, so that estimated coe cients on logarithms of prices may be interpreted as compensated price elasticities. When income is de¯ated by Stone’s price index, the way to impose homogeneity is to impose the restriction that the coe cients on the prices sum to zero, since they are compensated price elasticities. This can be done by de¯ating all of the prices by one of the included prices, but not by Stone’s price index nor any other price index that does not already appear in the equation. The result of de¯ating income by Stone’s price index is a modi®ed double-log model, but one that is just as easy to estimate as the more common Marshallian form. Estimating such a model provides direct estimates of compensated price responses and measures of precision of those estimates. The modi®ed model has the added virtue of sharing a right-hand side with the linear-approximat e version of the Almost Ideal demand system. This permits the construction of a speci®cation test appropriate for either model. In Monte Carlo experiments, it was found that both versions of the test behaved well in moderatesized samples, typical of those commonly encountered. There seems to be no impediment to the routine application of such speci®cation tests to the typical single-equation double-log model. It could also be applied, equation by equation, to test the LA model. In an application to US meat demand data, it was found that both simple models were always rejected for beef. For pork, both models were rejected when estimated by OLS, but only the share model was rejected when autocorrelation corrections were applied. Conversely, for poultry, the share model was never rejected and the double-log model

was rejected only when the autocorrelation corrections were applied.

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