Estimating from a More General Time-Series Cum Cross-Section Data Structure

Abstract

This note deals with the problem of parameter estimation from combined time-series-cross-sec tion data. A few works exist on the subject, e.g., Kuh [7], Mundlak [9], [10], Balestra and Nerlove [2], Hoch [6], Hildreth [5], Wallace and Hussain [13].1 Some of these [2], [5], [10], [13], used structures identical to those of error component models. In [13], the error was assumed to consist of three independent components?one associated with time, another with cross-sectional units and the other with the interaction between time and cross-sectional units. In this note we generalize that structure to include four independent com ponents consisting, ceteris paribus, of two, rather than one, cross-sectional components. Examples of one component (of these two) may be countries or regions, of the other may be areas within the coun tries or regions, being especially meaningful in international or interregional studies. For instance, one may be interested in studies of the contributions of multinational corporations in the regional and area developments of the world through their investment participation. Or, in the national con text, one may be interested, for example, in the variation of the parameters of the consumption function over the different regions of a country as well as provinces or smaller divisions of a region. In the Canadian case, differences are known to exist in the consumption pattern of people of Ontario and the Maritime provinces as for two of the regions and between the North and the South of Ontario as for two of the areas within a region. As in the three (error-) component model, the last component in the four component model is an interaction between time and all the cross-section units. But now this is to be understood as a com posite, rather than a unique, entity. It implicitly includes two first order interactions, namely, those between time and either of the cross-section dimen sions and one second order interaction, that is, that between time and the first and the second cross section dimension. On the whole, the composite interaction component as well as the other com ponents (due to time and cross-sections) of the error variable are random. We assume however, unlike some studies of the past, that time and each of the cross-section variables give rise to some fixed effects, knowledge of which might be important in certain contexts. With this introduction, we go on to lay bare (Section II) the model, then estimate (Section III) its parameters on the basis of Aitken's Generalized Estimators and Covariance Estimators, the assump tion being that the components of the error variance are known. When these are not known, we use alternative methods. At the end (Section IV) we conclude by comparing briefly the various esti mators.