بررسی اثرات خطای تجمیعی در برآورد کشش تقاضای مصرف کننده

Errors introduced by using aggregate data in estimating a consumer demand model have long been a concern. We study the effects of such errors on elasticity estimates derived from AIDS and QUAIDS models. Based on a survey of published articles, a generic parameterization of the income distribution, and the range of Gini coefficients reported for 28 OECD countries, we generate and analyze a large number of “observations” on the differences between elasticities calculated at the aggregate level and those calculated at the micro level. We suggest a procedure for evaluating the likely range of aggregation error when a model is estimated with aggregate data.

In earlier days there was frequently no alternative to the use of aggregate time series data in estimating consumer demand models (Stone, 1954, to cite an early pioneering paper). Underlying the models was the notion of a representative consuming unit that maximized utility but aggregation blurred the relationship between micro theory and econometric practice. The likelihood of “aggregation bias” was well known but there was not much one could do about it. Later, as survey data for individual households became increasingly available (and increasingly rich in content), opportunities opened up for estimating micro-theoretic models using actual micro data. Nevertheless, it remains true today that micro data are not always available in particular contexts, or appropriate for particular research objectives. Survey data may be available in one country but not another, or available for broad categories of goods but not at a detailed level that may be required (food in total but not types of food, for example); a survey may fail entirely to provide certain variables of importance for a particular purpose; trends and dynamics may be of interest, thus necessitating the use of time series available only at the aggregate level. Whatever the reasons it is still the case that aggregate data are often used in estimating consumer demand models, and hence that aggregation bias remains on the list of concerns (we report on a survey of 21 articles containing estimated models; 15 of the articles used aggregate data). Other things equal (and sampling variability aside), elasticities calculated at the aggregate level will generally differ from those calculated at the micro level, even if the same model is used in both cases. The differences, how to calculate them, and what to do about them, are the subjects addressed in this paper. We restrict our attention to two widely used models, Deaton and Muellbauer's (1980) “almost ideal demand system” (AIDS) and the quadratic extension of it (QUAIDS) proposed by Banks et al. (1997). Aggregation of an AIDS micro model over households requires the introduction of an “aggregation parameter” that depends on the distribution of household total expenditure — on the “income distribution,” as we shall call it for convenience, with slight inaccuracy; aggregation of a QUAIDS model requires two such parameters. We consider expenditure elasticities and own-price elasticities in the paper and there is, for each, a micro form and a corresponding macro form. This allows us to do a search for articles with AIDS/QUAIDS models that provide either micro or macro elasticities, calculate the corresponding macro or micro elasticities (under alternative assumptions about income distribution), and thus create a data set reflective of the types and magnitudes of aggregation effects actually found in the empirical literature. Along the way we introduce some procedures for characterizing the income distribution in a generic form and (using data for OECD countries) establish a range of distributions according to degree of inequality. On that basis we are then able to arrive at what we think is a reasonable range for the aggregation parameters and study the effects on elasticities over that range.

Errors resulting from the use of aggregate data in model estimation have long been a concern in econometric consumer demand analysis. Such errors arise from the interaction of a model's parameters with the underlying distribution of incomes. In the QUAIDS framework, two parameters of the distribution determine the links between elasticities calculated with micro data and corresponding elasticities calculated with aggregate data; in the AIDS framework there is one such parameter. Assuming a lognormal function as a generic representation of the income distribution, and using the Gini coefficient as a summary measure of inequality, we have derived the distribution of the two aggregation parameters over a range of Gini values generated with data for 28 OECD countries. Based on a survey of the empirical AIDS and QUAIDS model literature we have extracted a large number of estimated expenditure and price elasticities and calculated the implied aggregation effects in those estimates for alternative Gini values. We conclude that on average the effects are relatively small, even for large Gini values (to view them in a broader context, they may well be no greater than the effects of misspecifying the underlying theoretical model; see Denton et al., 2006). However, there are situations (model parameter configurations) in which the effects can in fact be large, and one would want to be on guard for such situations. We have proposed a simple procedure for evaluating the likely sensitivity of elasticity estimates to aggregation effects after estimating an AIDS or QUAIDS model with aggregate data. The procedure can be applied by someone who has estimated the model, or by the reader of a study in which the model is reported.