The asymptotic size and power of the augmented Dickey–Fuller test for a unit root

It is shown that the limiting distribution of the augmented Dickey–Fuller (ADF) test under the null hypothesis of a unit root is valid under a very general set of assumptions that goes far beyond the linear AR(∞) process assumption typically imposed. In essence, all that is required is that the error process driving the random walk possesses a continuous spectral density that is strictly positive. Furthermore, under the same weak assumptions, the limiting distribution of the ADF test is derived under the alternative of stationarity, and a theoretical explanation is given for the well-known empirical fact that the test's power is a decreasing function of the chosen autoregressive order p. The intuitive reason for the reduced power of the ADF test is that, as p tends to infinity, the p regressors become asymptotically collinear.