Empirically Weighted Indexes for Import Demand Functions

Abstract

T HIS paper presents an empirical approach to the aggregation problem. An aggregate dependent variable is properly thought to depend on a long string of disaggregated explanatory variables. There are two extreme ways of analyzing an equation such as this. We may use all of the explanatory variables individually, or we may summarize them in an index and use only the index as an explanatory variable.1 Both of these approaches have serious shortcomings and the purpose of this paper is to consider methods that lie between these extremes. The difficulties with the two extreme approaches are well known. On the one hand data limitations in the form of degrees of freedom inadequacies or multicollinearity usually completely rule out the purely empirical approach. On the other hand, the use of indexes as explanatory variables implies a specification error with unhappy implications worked out by Theil (1954 and 1971), Klein (1946), Allen (1965) and others. It is useful to consider the role of prior judgement in these two approaches. The one analysis excludes prior information about the coefficients and estimates would depend entirely on data evidence if that were possible. The other approach is based on the assumption that the coefficients are known with certainty (up to a factor of proportionality) and no data evidence can alter that constraint. In this paper we will pursue an intermediate course that involves weakening the assumption that the index weights are known with certainty but not to the point that the weights are totally unknown. More formally, this means formulating a proper prior distribution on the coefficients in the unconstrained equation and updating that prior via Bayes Rule. We will analyze import demand functions. One phenomenon specific to import functions is that the domestic indexes that are typically used have weights that are quite unlikely to be adequate for analyzing competition between imports and domestic goods. For example, services substitute little if any with imports and should have low weight in the domestic price index. We hope from our analysis to identify the prices and incomes that are most critical in determining imports and also to obtain better estimates of the aggregate elasticities.