We model earnings processes allowing for lots of heterogeneity across agents. We also introduce an extension to the linear ARMA model which allows the initial convergence in the long run to 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 to a variety of statistics including most of those considered by previous investigators. We use a sample drawn from the Panel Study of Income Dynamics (PSID), and focus on white males with a high-school degree. Despite this observable homogeneity, we find more 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.