Model selection when estimating and predicting ... - Springer Link

Empirical Economics (2003) 28:353–364

EMPIRICAL ECONOMICS ( Springer-Verlag 2003

Model selection when estimating and predicting consumer demands using international, cross section data* J. A. L. Cranfield1**, James S. Eales2, Thomas W. Hertel2 and Paul V. Preckel2 1 Department of Agricultural Economics and Business, University of Guelph, Guelph, Ontario, CANADA, N1G 2W1 (e-mail: jcranfi[email protected]) 2 Department of Agricultural Economics, Purdue University, West Lafayette, Indiana, U.S.A., 47907-1145 First Version Received: November 2000/Final Version Received: February 2002

Abstract. This paper assesses the ability of five structural demand systems to predict demands when estimated with cross sectional data spanning countries with widely varying per capita expenditure levels. Results indicate demand systems with less restrictive income responses are superior to demand systems with more restrictive income e¤ects. Among the least restrictive demand systems considered, An Implicitly, Directly Additive Demand System (AIDADS) and Quadratic Almost Ideal Demand System (QUAIDS) seem roughly tied for best, while the Quadratic Expenditure System (QES) is a close second. Given di¤erences in the characteristics of AIDADS and QUAIDS, it is concluded the former is better suited to instances where income exhibits wide variation and the latter to cases when prices exhibit considerable variation. Key words: Consumer demand, model selection 1. Introduction Demand analysts often face the question of which functional form to use when analyzing international consumption patterns. While technical in nature, the choice of functional form is nevertheless a very important consideration. The major concern in picking an appropriate functional form is that it be flexible in representing a broad range of income and price responses and * The authors acknowledge the insightful comments of two journal referees and Baldev Raj. Bettina Aten kindly provided the data used in this study. Any errors or omissions remain the responsibility of the authors. Partial financial support of the United States Department of Agriculture – National Research Initiative Grant #97-35400-4752 and the Purdue Research Foundation is gratefully acknowledged. An expanded version of this paper is available from the authors upon request. ** Contact author

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therefore of greater use for projecting consumer demand or incorporation in a policy model (e.g., Chambers; Klevmarken). Yet, many of the previously used demand systems are limited in their ability to capture realistic income responses. In this regard, An Implicitly, Directly Additive Demand System (AIDADS), developed by Rimmer and Powell (1996), and the Quadratic Almost Ideal Demand System (QUAIDS), developed by Banks, Blundell and Lewbel, prove useful as both systems allow for more general income responses than other parametric models of consumer demand. This paper provides a small step towards understanding the role of demand system choice when analyzing international consumption patterns. This is accomplished by comparing the predictive ability of the AIDADS, QUAIDS, Quadratic Expenditure System (QES), Linear Expenditure system (LES) and Almost Ideal Demand System (AIDS) when estimated using cross-sectional data spanning a range of countries with widely varying per capita expenditure levels. The value of our analysis, compared to studies such as Klevmarken and Chambers, lies in the comparison of two new demand systems, QUAIDS and AIDADS, to well established demand systems. Furthermore, the use of cross-section data spanning a broad spectrum of countries, rather than time-series data for a single country, allows for an alternative perspective in conducting a comparative study of di¤erent demand systems. In the next section, the demand systems are presented. This is followed by a discussion of the data, estimation framework and comparison criteria. Results are then presented with a focus on the comparison of in-sample fit and out-of-sample predictions. Finally, results are briefly summarized and conclusions drawn. 2. Empirical models Since this study uses cross-section data spanning countries with widely varying expenditure levels, every country is assumed to have identical preferences over goods, and therefore identical demand functions, and each country acts as a single, representative consumer. Thus, issues related to aggregation over individual consumers and the role of di¤erent preferences across countries have been assumed away. While such assumptions are limiting, as di¤erent cultural, social and economic factors a¤ect the structure of preferences within a particular country, they reflect previous analysis of international consumption patterns (e.g., Theil and Suhm; Barten). Moreover, without these or similar assumptions, the problem would be intractable with the available data. 2.1. An implicitly directly additive demand system Rimmer and Powell (1996) developed An Implicitly Directly Additive Demand System (AIDADS) from an implicitly directly additive utility function. Key properties of AIDADS are that i) the fitted budget shares are restricted to the unit interval by construction, ii) the marginal expenditures1 are non-linear in 1 Marginal expenditure is also referred to as the marginal budget share in the literature. It is ‘‘. . . the fraction of an additional dollar of expenditure spent on each good . . .’’ (Pollak and Wales 1992, p. 5). We find the marginal budget share terminology confusing and so this response to expenditure changes is called marginal expenditure.

Consumer demand model selection using international, cross section data

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real expenditure, and iii) AIDADS is a rank-three demand system2 (Rimmer and Powell 1996, p. 1614). Moreover, implicit direct additivity reduces the number of parameters, while allowing for flexible Engel e¤ects. However, the implicit nature of the AIDADS utility function makes it di‰cult, if not impossible, to aggregate demands across income levels. Despite this limitation, AIDADS is well suited to capturing the pattern of demand’s response to expenditure changes in a realistic manner. In budget share form, AIDADS is written as: pit gi ai þ bi expðut Þ pt0 g 1 þ Ei; t ð1Þ wit ¼ 1 þ expðut Þ yt yt where wit is a budget share, i ¼ 1; . . . ; n indexes goods, t ¼ 1; . . . ; T indexes observations (which are countries in this paper), pt is a vector of prices with typical element pit , yt denotes expenditure, ai , bi and gi are unknown parameters, with gi interpreted as the subsistence level for the ith good, qit is consumption of the ith good, ut is the utility level associated with the optimal consumption bundle for country t, and g is a vector with typical element gi . By construction, AIDADS satisfiesP symmetryP and homogeneity properties, while n n ai ¼ i¼1 bi ¼ 1, 0 a ai , b i a 1. adding-up is satisfied provided i¼1 Note that utility appears as an argument in this demand system. Since utility is not observable it is treated as an unknown value to be estimated for each observation in the sample. This not only complicates estimation, it also assumes utility is cardinal. The assumption of cardinal utility is the cost one must pay to have a demand system with the Engel flexibility of AIDADS. Given AIDADS’ parsimonious specification (in terms of estimated parameters) and ability to capture general Engel e¤ects, AIDADS may be favored, a priori, as a model of consumer behavior when expenditure shows considerable variation and goods are aggregated at a broad level (e.g., food). 2.2. Quadratic expenditure system A Quadratic Expenditure System (QES; Howe, Pollak, and Wales) is also estimated and included in the analysis. Following Pollak and Wales (1992), the QES is defined as: n 2bl pit gi p 0g pit di p0 d Y plt p 0g 2 þ bi 1 t þ bi t 1 t Ei; t wit ¼ yt yt l¼1 yt yt yt yt ð2Þ 2 Demand system rank is the maximum rank of a matrix of coe‰cients associated with functions of income (or expenditure). Gorman proved the rank of an exactly aggregable demand system is at most three; thus, such demand systems are referred to as ‘‘full rank demand systems.’’ Lewbel defined demand system rank as the ‘‘. . . maximum dimension of the function space spanned by the Engel curves of the demand system,’’ (Lewbel, p. 711). Note that Lewbel’s definition applies to all demand systems, not just those which are exactly aggregable. The concept of rank is useful in developing a taxonomy of demand systems according to Engel curve shape. Rank one demands, the most restrictive demand systems, are independent of income; rank two demand systems are less restrictive, allowing linear Engel curves not necessarily through the origin; while rank three (i.e., full rank) demand systems are least restrictive, allowing for non-linear Engel responses.

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where di is an unknown parameter, d is a vector with typical element di and all other variables have been previously defined. The QES satisfies P n symmetry and b i ¼ 1, b i > 0. homogeneity by construction, while adding-up requires i¼1 Two factors motivate inclusion of the QES: it is a rank three demand system and it nests the LES as a special case when di ¼ 0 for all i. Nevertheless, its Engel flexibility is somewhat limited by the fact that marginal expenditures are linear in expenditure. 2.3. Linear expenditure system While the QES nests the LES, so too does AIDADS. If one imposes ai ¼ b i for all i in AIDADS, it collapses to the LES. Given these nested forms, it seems natural to consider the LES when comparing demand systems. The LES estimated in this paper appears as: pit gi pt0 g wit ¼ þ bi 1 Ei; t: ð3Þ yt yt Symmetry and homogeneity arePsatisfied by construction, while adding-up n is satisfied with the restriction i¼1 b i ¼ 1, b i > 0. The LES is a rank-two demand system which means Engel e¤ects associated with this demand system are not as flexible as the AIDADS or QES models. 2.4. Quadratic almost ideal demand system Banks, Blundell and Lewbel developed the Quadratic AIDS (QUAIDS) model to allow for Engel curves that are non-linear in the log of expenditure. This is done by considering the preference structure needed to obtain a share based demand system that is linear in the log of real expenditure and a general function of real expenditure. By assuming the demands are rank three, exactly aggregable, and derived from utility maximization, they show that the resulting demand system is: !1 n n Y X yt yt 2 bl zil lnð plt Þ þ bi ln þ li plt ln Ei; t ð4Þ wit ¼ ai þ Pt Pt l¼1 l¼1 Pn Pn Pn ai lnðpit Þ þ i¼1 where l ¼ 1; . . . ; n indexes goods, lnðPt Þ ¼ a0 þ i¼1 l¼1 zil lnðpit Þ lnð plt Þ is a Translog price index, which is also used in the AIDS unknown parameters. Adding-up requires model, and aP i , b i , zil and P n li are P Pn n n a ¼ 1, b ¼ z ¼ i i¼1 i¼1 i i¼1 il i¼1 li ¼ 0, symmetry zil ¼ zli and homoPn geneity l¼1 zil ¼ 0. Without further restrictions, predicted budget shares for the QUAIDS can stray outside of the unit simplex – an e¤ect that may be exacerbated at higher expenditure levels given the inclusion of the quadratic term in this demand system. However, the quadratic term means the marginal expenditures for QUAIDS are quadratic in the logarithm of expenditure (when evaluated at the fitted values). QUAIDS allows for luxury goods at low expenditure levels but normal goods at high expenditure levels, with the precise characterization


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depending on the sign of bi and li (see Banks, Blundell and Lewbel for further discussion). Finally, when li ¼ 0 for all i, QUAIDS collapses to Deaton and Muellbauer’s AIDS model. 2.5. Almost ideal demand system As AIDS has been widely used in modeling consumer demands, it is also considered in this analysis. Following Deaton and Muellbauer, the AIDS model is written as: wit ¼ ai þ

n X

zil lnðplt Þ þ bi ln

l¼1

yt Pt

Ei; t:

ð5Þ

Pn Pn Pn ai ¼ 1, Pi¼1 bi ¼ i¼1 zil ¼ 0, while symAdding-up is imposed with i¼1 n metry requires zil ¼ zli and homogeneity l¼1 zil ¼ 0. AIDS is a rank-two demand system, so it possesses less flexible Engel e¤ects compared to the QUAIDS, QES, and AIDADS models. Furthermore, marginal expenditures for AIDS vary linearly in the log of expenditure (when evaluated at the fitted values). Like QUAIDS, predicted budget shares for AIDS may stray outside the unit simplex. 3. Data & estimation We use a cross section sample of countries from the 1985 International Comparisons Project (ICP).3 These data are useful in analyzing international demand patterns since they are provided in identical units (i.e., international dollars). The raw data are composed of real and nominal expenditure on 113 final goods and services in 64 countries (which range in expenditure levels from Ethiopia to the USA). For estimation, the 113 final goods and services are aggregated into the following six goods: food, beverages and tobacco, clothing and footwear, gross rent and fuel, household furnishings and operations, and other expenditure. Expenditure on each of the aggregate goods is computed as the sum of nominal expenditure on each good in the aggregate group. Total per capita expenditure equals total nominal expenditure on the six aggregate goods divided by population. The price of each aggregate good equals nominal expenditure on the good divided by real expenditure. Nominal expenditure is defined in exchange rate converted US dollars while real expenditure is defined in purchasing power parity converted international dollars. Finally, budget shares are computed as the ratio of nominal expenditure on the good to total nominal expenditure.4 To evaluate out-of-sample performance of the estimated systems, the 3 Others have used previous ICP data sets for analyzing international demand patterns. For example, Kravis, Heston and Summers; Theil and Clements; Rimmer and Powell (1992) used the 1975 ICP data set, Theil, Chung and Seale used the 1970, 1973, 1975 and 1980 releases, while Wang used the 1985 data. A version of the 1990 release is available at the Penn World Tables. Unfortunately the 1990 release does not include developing countries. 4 A table of summary statistics of the data is available upon request.

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original 64 countries in the 1985 ICP data set are partitioned into two subsamples by sorting the data from lowest to highest per capita expenditure, then eliminating every third country. The resulting sub-sample of 43 observations is used for estimation, while the remaining 21 observations are withheld during estimation and used for out-of-sample comparison. While the choice of eliminating every third country along the expenditure spectrum is arbitrary, it is appropriate in the current setting. For example, if one used the 43 poorest countries for estimation and the wealthiest 21 countries for outof-sample comparison, the resulting Engel elasticities would only reflect the consumption-expenditure relationship in the poorer countries – the adjustment embodied in the wealthier countries will be absent and one may expect poor out-of-sample performance. Our partitioning of the data roughly equalizes the distribution of expenditure in our two sub-samples. From a long-run forecasting perspective, it would be prudent to assess the models estimated with the 43-country sample using data from a di¤erent year. Consequently, we employ the 1975 ICP data set used by Theil and Clements but aggregated to the six good level of aggregation described above. The demand systems estimated using the 43-country sample from the 1985 ICP data set are used to ‘‘backcast’’ the 1975 budget shares for all six goods in the 30 countries contained in both samples. Each demand system is estimated using maximum likelihood methods. To do so, an additive error term is appended to each equation of each system and the last equation dropped to avoid singularity of the covariance matrix. While the structure of the estimation framework is well beyond the scope of the paper, details can be found in Cranfield et al. (2000) and the expanded version of this paper. 3. Model comparison criteria Our approach to model comparison and selection is based on goodness-of-fit measures and statistical comparison of in-sample residuals and out-of-sample prediction errors. Goodness-of-fit measures include the root mean squared error (RMSE), a system-wide RMSE (SRMSE) and the information inaccuracy (IIA) measure. If an exact fit is attained, IIA will equal zero. The larger the absolute value of IIA, the greater the deviation between the actual and fitted budget shares. To account for potential over fitting of models with many parameters, the multivariate Akaike’s information criterion (AIC) and multivariate Schwartz’s criterion (SC) are also reported. The model with the lowest AIC, or SC, is the preferred model. Since the LES is nested in both the AIDADS and QES, a likelihood ratio test (LRT) can be used to test the restrictions consistent with the LES (i.e., ai ¼ b i for all i in the case of AIDADS and di ¼ 0 for all i in the case of the QES). Likewise, since QUAIDS has AIDS as a nested case, a LRT can also be used to test the restriction consistent with AIDS (i.e., li ¼ 0 for all i). However, since the LRT tends to over-reject in finite samples, Italianer’s size correction is used to adjust the likelihood ratio test statistics. Lastly, Pollak and Wales’ (1991) likelihood dominance criteria (LDC) is used to compare non-nested models (e.g., AIDADS to the QES, AIDS and QUAIDS, and the LES to AIDS and QUAIDS). In using the LDC, the model with fewer (more) parameters is selected if the LDC statistic is below (above) a


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context dependent critical range (see Pollak and Wales (1991) for details). If the competing models have an equal number of parameters, the model with the higher log-likelihood value dominates the other. 4. Results All models were estimated with homogeneity and symmetry imposed (these restrictions are imposed automatically for the LES, AIDADS, and QES). Negativity of the substitution matrix was checked by calculating the eigenvalues of the matrix of compensated price e¤ects, evaluated at the in-sample means of the data. In all cases, the concavity property was satisfied at the point of evaluation. 4.1. In-sample evaluation Root mean squared error (RMSE) values from the maximum likelihood estimation of the AIDADS, QES, LES, QUAIDS, and AIDS models are presented in Table 1. In-sample, the QUAIDS model has the lowest RMSE for four of the demands while the AIDS and AIDADS models have lowest RMSE for one commodity each. The LES is the only model which stands

Table 1. Root mean squared error for budget shares based on estimated models and in- and out-of-sample errorsa Model

Food

1985 In-Sample: AIDADS 5.687* LES 10.884 QES 6.389 AIDS 5.998 QUAIDS 5.790 1985 Out-of-sample: AIDADS 8.546 LES 9.514 QES 8.196* AIDS 9.638 QUAIDS 9.939 1975 Out-of-sample: AIDADS 5.567 LES 9.256 QES 7.229 AIDS 5.513* QUAIDS 5.891

Beverage & Tobacco

Clothing & Footwear

Gross Rent & Fuel

Household Furnishings & Operations

Other Expenditure

2.871 2.855 2.947 2.791 2.693*

2.304 2.472 2.319 2.352 2.285*

4.199 4.895 4.183 4.279 3.914*

2.124 2.174 2.144 1.862* 1.864

5.902 9.525 6.051 6.010 5.844*

2.149* 2.199 2.188 2.410 2.467

3.016 2.701* 2.813 2.725 2.733

3.828 4.281 3.673* 3.983 3.933

3.356 3.243* 3.344 3.296 3.321

8.675 9.320 8.366* 8.751 8.934

2.448 2.439 2.481 2.283 2.235*

2.225 2.365 2.223 2.065 1.856*

4.944 5.544 4.908 5.275 4.733*

2.215 2.149* 2.224 2.240 2.243

6.414 8.275 7.571 6.148* 6.562

a Root Error for the ith equation in the respective sample is calculated as P TMean Square îtj Þ 2 Þ 0:5 , where t ¼ 1; . . . ; T indexes countries in the respective sub-sample, wit ðwit w ðT 1 t¼1 îtj is the fitted budget share for the is the actual budget share for the ith good in country t, while w ith good in country t from the jth model. All RMSE values have been multiplied by 100. * Denotes the lowest RMSE across models for that good and sample.

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out with poor RMSE performance, being out-performed by the best model by 63%, and 91% for Other Expenditure and Food, respectively. The LES appears to be the loser in-sample and the QUAIDS the winner, but this is exactly what one would expect, since the LES has the fewest parameters, while the QUAIDS has the most. Also note the di¤erences between the QUAIDS and the other models are not large. While not reported, the standard error of the prediction error, corrected for potential heteroskedasticity, was also calculated for each equation in each model (and for each sub-sample of data). In general, the RMSE rankings appear robust to this alternative measure of goodness-of-fit.5 As illustrated in Table 2, the in-sample system RMSE ranks the demand

Table 2. Log-likelihood function value, system root mean squared error, information inaccuracy, Akaike’s information criterion, and Schwartz’s criterion for budget shares based on estimated models and in- and out-of-sample errorsa

1985 In-Sample: AIDADS LES QES AIDS QUAIDS 1985 Out-of-sample: AIDADS LES QES AIDS QUAIDS 1975 Out-of-sample: AIDADS LES QES AIDS QUAIDS

LLFb

SRMSEc

IIAd

AICe

SCf

737.5 701.7 732.6 740.2 750.1

4.91 7.94 5.19 5.04 4.85*

3.06 5.81 2.81 3.02 2.78*

33.51* 32.17 33.28 33.27 33.49

32.82* 31.76 32.59 32.24 32.37

6.80 7.35 6.55* 7.15 7.30

4.71 5.34 4.45* 5.01 5.13

5.12 7.03 6.07 5.01* 5.21

3.34 4.85 3.97 3.33 3.20*

a The abbreviations are defined as follows: Log-Likelihood Function Value (LLF), System Root Mean Squared Error (SRMSE), Information Inaccuracy (IIA), Akaike’s Information Criterion (AIC), and Schwartz’s Criterion (SC). ^j where T is the number b The LLF values reported in this column are computed as: T 0:5 lnjS ^ is the respective covariance matrix. of observations and S Pn wi RMSEij , where wi is the sample mean of the ith good’s c System RMSE is calculated as i¼1 budget share in the respective sample and RMSEij is the RMSE for the ith good from the jth model in the respective sample. SRMSE values have been multiplied by 100. PT Pn wit w ln d Information inaccuracy is calculated as T 1 t¼1 . These values have been it i¼1 îtj w multiplied by 100. e AIC is calculated as lnjS^j þ T 1 2m, where m is the number of parameters in the respective model. f SC is calculated as lnjS^j þ T 1 lnðTÞm. * Denotes preferred model.

5 Detailed results of the comparison based on the standard deviation of the prediction error are available upon request.


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models (from most to least preferred) as follows: QUAIDS, AIDADS, AIDS, QES, and LES, while the information inaccuracy measures rank the models as: QUAIDS, QES, AIDS, AIDADS and LES. Again, this is as expected. Insample, the most highly parameterized model, QUAIDS, should fit best, while the least parameterized model, LES, should fit the worst. The AIC, which accounts for over fitting, ranks the models as follows: AIDADS, QUAIDS, QES, AIDS, and the LES, while the SC results in the following ranking: AIDADS, QES, QUAIDS, AIDS, and LES. LRT statistics in Table 3 show that at the 5% significance level, the restrictions for the LES are rejected for both the AIDADS and QES models, as are the restrictions for AIDS in the case of the QUAIDS model. Thus, in-sample evidence supports AIDADS and QES over LES and QUAIDS over AIDS. Likelihood dominance criteria comparisons are also shown in Table 3. Since the QES and AIDADS models have the same number of parameters, direct comparison of their respective log-likelihood values indicates a preference for AIDADS over QES. The LDC test statistic for the AIDS-AIDADS comparison is less than the lower bound of the critical range, which indicates a preference of AIDADS over AIDS (since AIDADS has fewer parameters than AIDS). In all other cases LDC test statistics are greater than the upper bounds, which means the model with more parameters is preferred to the model with fewer parameters. Consequently, the LDC comparison suggests QUAIDS is preferred to AIDADS, the QES, and LES, while AIDS is preferred to the LES and QES.

Table 3. Likelihood ratio and likelihood dominance criteria test resultsa m2 versus m1

LRTb

AIDADS versus LES QES versus AIDADS QES versus LES AIDS versus AIDADS AIDS versus LES AIDS versus QES QUAIDS versus AIDADS QUAIDS versus LES QUAIDS versus QES QUAIDS versus AIDS

62.16d

LDC

Critical value rangec

4.95

n.c.f

2.73 38.56 7.68 12.63 48.46 17.58

5.01, 6.54 8.93, 10.58 5.01, 6.54 8.06, 9.93 11.98, 13.78 8.06, 9.92

53.57d

15.89e

a The abbreviations in the table are defined as follows: m2 is the model with the larger number of parameters in the model comparisons, m1 is the model with fewer parameters in the model comparison, Likelihood Ratio Test (LRT) and Likelihood Dominance Criteria (LDC). b The Italianer size correction has been applied to the LRT statistics. c In each cell, the first value is the lower bound of the LDC critical range, computed as: 0:5 ½Cðm2 þ 1Þ Cðm1 þ 1Þ, the second value is the upper bound of the LDC critical range, computed as: 0:5 ½Cðm2 m1 þ 1Þ Cð1Þ, where CðuÞ denotes the critical value of a w 2 statistic with u degrees of freedom at the chosen significance level. d The value in this cell is the likelihood ratio test statistic when testing the restrictions for the LES within the AIDADS model. The number of restrictions is 5 and the 5% critical value equals 11.1. e The value in this cell is the likelihood ratio test statistic when testing the restrictions for the AIDS within the QUAIDS model. The number of restrictions is 5 and the 5% critical value equals 11.1. f Not computed since LDC comparison of models with equal number of parameters depends on the di¤erence in log-likelihood values.

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4.2. Out-of-sample evaluation When the estimated demand systems are compared using the 21 out-of-sample observations from the 1985 ICP data set, a preference for the QES and AIDADS emerges. RMSE values in the 1985 out-of-sample period are the lowest for the QES in three cases, for the LES in two cases and for AIDADS in one case (see Table 1). Interestingly, the QUAIDS, which is a rank-three demand model, does not perform as well in the 1985 out-of-sample data set. The system-wide RMSE and information inaccuracy scores in Table 2 echo the preference for the QES and AIDADS, with the following ranking of the models: QES, AIDADS, AIDS, QUAIDS, and the LES. Results from the 1975 out-of-sample backcasting exercise (see Table 1) show that QUAIDS has the lowest RMSE in three cases, followed by AIDS in two cases, and the LES in one case. Based on these results, it does not appear as though one particular model stands out. When the system-wide RMSE is used to compare the models (see Table 2), the models are ranked (from most to least preferred) as: AIDS, AIDADS, QUAIDS, QES, and LES. The information inaccuracy measures indicate the following rank: QUAIDS, AIDS, AIDADS, QES, and LES. 4.3. A further comparison of the rank three demand systems Results from the in- and out-of-sample comparison generally favor the rank three demand models (AIDADS, QES, and QUAIDS) over rank two demand models (AIDS and LES). However, none of the rank three models is the clearly preferred model. To gain additional insight, we focus on food as its budget share exhibits considerable variation across the expenditure spectrum and is likely to be more di‰cult to fit than other goods. Specifically, the predicted in-sample budget share for food from the QUAIDS model is closest to the actual in-sample average, while those from the QES and AIDADS models are lower than the QUAIDS predictions.6 If one were only concerned with ‘‘average’’ behavior, it would be tempting to conclude a preference for QUAIDS. However, behavior outside the neighborhood of the means is equally important. For instance, for the 22 poorest in-sample countries, AIDADS does a better job of fitting the means of food budget shares than the QES and QUAIDS. When the richest 21 in-sample countries are used for a similar comparison, it appears as though the QES fits better than AIDADS and QUAIDS. 5. Conclusions In this paper, we compare the predictive ability of five structural demand systems when estimated with cross-section data spanning a range of countries with widely varying expenditure levels. Results indicate that, of the models considered, rank three demand systems are superior to rank two demand systems. Among the rank three demand systems considered, An Implicitly, 6 A table showing the breakdown of predicted budget shares is available upon request.


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Directly Additive Demand System (AIDADS) and Quadratic Almost Ideal Demand System (QUAIDS) seem roughly tied for best, while the Quadratic Expenditure System (QES) is a close second. Amongst these rank three systems the choice is very much context dependant. For instance, the QUAIDS model, which is a flexible functional form and exactly aggregable, is better suited to instances where aggregation or cross-price e¤ects are important. Another advantage of the QUAIDS model over AIDADS is its ease of estimation. Nevertheless, despite the fact that AIDADS is not exactly aggregable, it has fewer price related parameters to estimate and is designed so that budget shares lie between zero and one at all expenditure levels. The limited number of price related parameters would suggest that AIDADS be preferred in instances where price variation is limited (and so economy in specifying these e¤ects in important). The theoretical property that budget shares lie between zero and one at all expenditure levels suggests a preference for AIDADS when expenditure shows substantial variation (or when extrapolations would involve large changes in expenditure) but prices are anticipated to experience little change. A number of issues posing limitations to this study also suggest areas of further research. First, we have assumed a representative world consumer. Alternatively, one could aggregate individual demands and estimate these as a function of statistics of the distribution of expenditure. While such a procedure is theoretically possible for the LES, QES, AIDS and QUAIDS, it would be di‰cult, if not impossible for the AIDADS model. Nevertheless, this highlights the importance of continuing to address the issue of aggregation of individuals, especially as the functional forms employed become more complex. Further research is also needed to develop tests that allow one to test the null hypothesis of implicit direct additivity versus other separability structures. Lastly, a comparison of the implicit directly additive preference structure to fractional demand systems would be particularly useful as fractional systems possess desirable regularity properties, are flexible functional forms, and are full rank. References Banks J, Blundell R, Lewbel A (1997) Quadratic engel curves and consumer demand. Review of Economics and Statistics 79:527–539 Barten A (1989) Towards a levels version of the Rotterdam and related demand systems. In: Cornet B, Tulkens H (eds) Contributions of Operations Research and Economics: The Twentieth Anniversary of CORE. MIT Press, Cambridge Mass., pp. 441–465 Chambers M (1990) Forecasting with demand systems – a comparative study. Journal of Econometrics 44:363–376 Cranfield J, Preckel P, Eales J, Hertel T (2000) On the estimation of ‘an implicitly additive demand system’. Applied Economics 32:1907–1915 Deaton A, Muellbauer J (1980) An almost ideal demand system. American Economic Review 70:312–326 Gorman W (1980) Some engel curves. In: Deaton A (ed) Essays in the Theory and Measurement of Consumer Behaviour. Cambridge University Press, New York, pp. 7–30 Howe H, Pollak R, Wales T (1979) Theory and time series estimation of the quadratic expenditure system. Econometrica 47:1231–1248 Italianer A (1985) A small-sample correction for the likelihood ratio test. Economics Letters 19:315–317 Klevmarken N (1979) A comparative study of complete systems of demand functions. Journal of Econometrics 10:165–191

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