Should Aggregation Prior to Estimation be the Rule?

Abstract

IN a previous article with Professor Harold Watts, the authors demonstrated empirically the loss of information in the parameter estimators when data are aggregated prior to computing least-squares regressions [3]. These results came from simulations with a simple economic model containing identical microcomponents. Specifically, in addition to the error term, each component spent 0.9 of its previous income and 0.2 of its cash balance. The main point of our previous paper was that estimation prior to yielded substantially greater precision in the estimates of the parameters and their standard errors than did estimation of the same parameters after aggregation. The implications of this for hypothesis testing and the development of satisfactory policy response models seemed obvious. On the basis of a variety of evidence, including the paper with Watts and a paper by Orcutt [4], the case for seeking and frequently using disaggregated data seemed strong but one nagging concern remained. Suppose, as seems likely, the microcomponents exhibit different behaviors. In this case it might not be sensible to pool the data and treat it as a single sample from a single universe. However, if estimators from each micro equation are computed separately, would it still be desirable to use disaggregated data instead of data aggregated over all components? This turned out to be the case with identical components but would it be with nonidentical components in which something more than constant terms were different? This paper copes directly with this issue, and we demonstrate the importance of using disaggregated data even when microcomponents exhibit different behaviors. We do not deal with cases where microcomponents have nonlinear relations, but the need for disaggregated data in such cases seems fairly obvious without Monte Carlo experiments. If we wish to compare the accuracy of estimation at different levels of aggregation, we need a measure of merit different from the extent of bias and variance of parameter estimators, which we used in our previous study, because in an aggregate model whose components have different behaviors, the expected values of the estimators may be meaningless or nonstationary [Zellner, pp. 3-5]. Therefore, we use the accuracy of the out-of-sample forecasts to measure the merit of the estimated equations. In particular, we forecast the aggregate expenditure for the eight time periods following the last sample period. The rootmean-square forecast errors from models based on data at different levels of provide the yardstick for comparisons. Our results suggest that models estimated from micro data will give generally superior out-of-sample forecasts. This finding is at variance with the belief that one reaps an aggregation by aggregating the micro data prior to estimation. The concept of a possible gain was formalized in a 1960 article in this Review by Grunfeld and Griliches: