The Use of Error Components Models in Combining Cross Section with Time Series Data

TLDR

The authors propose a mixed regression model with error components to combine cross‑sectional and time‑series data. They compare Aitken and covariance estimators when variances are known, and Zellner‑type iterative versus covariance estimators when variances are unknown, within this mixed‑model framework. The results show that Aitken estimators are more efficient in small samples but asymptotically equivalent to covariance estimators, and that for unknown variances covariance and Zellner‑type estimators share equivalent asymptotic distributions and moment limits, leading the authors to favor traditional covariance estimators and caution against careless data combination.

Abstract

A mixed model of regression with error components is proposed as one of possible interest for combining cross section and time series data. For known variances, it is shown that Aitken estimators and covariance estimators are in one sense asymptotically equivalent, even though the Aitken estimators are more efficient in small samples. Turning to unknown variance components, Zellner-type iterative estimators are compared with covariance estimators. Here, few small sample properties are obtained. However, it is shown that covariance and Zellner-type estimators have equivalent asymptotic distributions and equivalent limits of sequences of first and second order moments for weakly nonstochastic regressors. For the model analyzed, the theoretical results obtained, as well as ease of computation, tend to support traditional covariance estimators of the regression parameters. An additional interesting result presented in an appendix is that ordinary least squares estimates of the fl's (ignoring the error components) have unbounded asymptotic variances. On efficiency grounds, this argues rather strongly for some care in combining data from alternative sources in regression analysis.