Asymptotics of regressions with stationary and nonstationary residuals

A comprehensive description is given of the limiting behaviour of normalised pseudo-MLEs of the coefficients in a discrete-time autoregressive process, with nonstochastic regressors, for all cases: stationary, unit root and explosive situations. The residuals are assumed to be independent and identically distributed, with finite variance, and we allow a wide class of regressors: they need only be uniformly asymptotically negligible and not too regular, in a certain sense. Under these assumptions, the normalised estimator of the regression coefficient is shown to be asymptotically normal, regardless of the value of the autocorrelation coefficient, and asymptotically independent of the normalised estimator of the autocorrelation coefficient, which also has a proper, nondegenerate limiting distribution. The normalisation for the estimators can be based on the sample information matrix. Limiting distributions for likelihood ratio test statistics of hypotheses of interest are also given under the same assumptions.