On the distributions of Augmented Dickey–Fuller statistics in processes with moving average components

This paper considers the distributions of augmented Dickey–Fuller statistics to test for a unit root in the presence of errors that have a moving average part. We examine the dependence of the augmented Dickey–Fuller statistic on the order, k, of the approximating autoregression, by deriving the asymptotic distribution of the statistic for finite k. The results provide a direct indication of the rate of growth of k with sample size that is necessary in order to avoid non-negligible size distortions, and are illustrated by simulations that provide empirical densities for various cases of interest. The size distortions are particularly difficult to control when the moving average part contains a root near unity; for such cases we examine the distribution of the ADF statistics in models estimated by feasible GLS, rather than OLS, to control for the moving-average component. We find that both size distortions and sensitivity of the test statistic to k are greatly reduced in this case.