Estimation and Prediction from Aggregate Data when Aggregates are Measured More Accurately than Their Components

THE CLASSIC PAPER by Grunfeld and Griliches 3 contains many instructive insights into question of circumstances under which an aggregate dependent variable may be forecasted more precisely with a model based on aggregate variables as opposed to an aggregate of forecasts from individual equations. The statistical model they employ is standard regression framework, a single at macro level and a set of seemingly unrelated regressions (to use Zellner's term) at level. Under assumptions of perfect model specification and nonstochastic regressors, a result of Theil's supports superiority of equations. It is Grunfeld and Griliches' main contention, however, that equations are likely to be more poorly specified than is macro equation. Perhaps, therefore, an aggregation will be realized in prediction of aggregate dependent variable by use of macro equation. While intuitively appealing, one soon finds that to articulate notion that micro equations are likely to be more poorly specified than is macro equation is difficult. Grunfeld and Griliches provide an illustration 3, pp. 7-9 that more than anything else points up elusive character of this notion. Recently, Orcutt, Watts, and Edwards 7 and Edwards and Orcutt 2 published papers that ostensibly support prediction from disaggregated data. Edwards and Orcutt note that a more basic difficulty with equations is that suitable data are scarce. Grunfeld and Griliches had earlier observed in passing that the poor quality of data may be another source of aggregation gain 3, p. 10. It is precisely this consideration which we propose to examine in present paper. More particularly, we examine virtues of estimating or macro equations when independent variables in equations are observed with error, but corresponding aggregate variables have a smaller (or possibly no) observation error. There are many economic applications in which such offsetting errors may be plausibly hypothesized. For example, there may be some arbitrariness in classification of products (or industrial breakdown), so that while total sales figures for a given firm may be well established, their components may be subject to error. Alternatively, we may have common situation in which an aggregate figure is collected on a regular basis from a relatively complete sample, but components are calculated from benchmarks provided from a smaller or an older sample.