In this paper we derive the asymptotic properties of within groups (WG), GMM, and LIML estimators for an autoregressive model with random effects when both T and N tend to infinity. GMM and LIML are consistent and asymptotically equivalent to the WG estimator. When T/N→ 0 the fixed T results for GMM and LIML remain valid, but WG, although consistent, has an asymptotic bias in its asymptotic distribution. When T/N tends to a positive constant, the WG, GMM, and LIML estimators exhibit negative asymptotic biases of order 1/T, 1/N, and 1/(2N−T), respectively. In addition, the crude GMM estimator that neglects the autocorrelation in first differenced errors is inconsistent as T/N→c>0, despite being consistent for fixed T. Finally, we discuss the properties of a random effects pseudo MLE with unrestricted initial conditions when both T and N tend to infinity.

All empirical models of earnings processes in the literature assume a good deal of homogeneity. In contrast to this we model earnings processes allowing for lots of heterogeneity between agents. We also introduce an extension to the linear ARMA model that allows that the initial convergence to the long run may be different from that implied by the conventional ARMA model. This is particularly important for unit root tests which are actually tests of a composite of two independent hypotheses. We fit our models to a variety of statistics including most of those considered by previous investigators. We use a sample drawn from the PSID, and focus on white males with a high school degree. Despite this observable homogeneity we find much greater latent heterogeneity than previous investigators. We show that allowance for heterogeneity makes substantial differences to estimates of model parameters and to outcomes of interest. Additionally we find strong evidence against the hypothesis that any worker has a unit root.