The sample autocorrelation function of I(1) processes

In case of a random walk the theoretical autocorrelations tend to one asymptotically. The sample autocorrelations, however, may decline rather fast even with large samples. We will explain this observation by deriving the asymptotic distribution that turns out to be closely related to the Dickey-Fuller (1979) distribution. Moreover we discuss the behaviour of the sample autocorrelations of integrated MA(1) and AR(1) processes. In order to prove our results we consider more general I(1) processes and apply the functional central limit theorem injected to time series analysis by Phillips (1987). We obtain unit root tests that are based on autocorrelation estimators of higher lags. We discuss their finite sample behaviour experimentally.