HIGHER-ORDER SAMPLE AUTOCORRELATIONS AND THE UNIT-ROOT HYPOTHESIS

Hasza (1980) has derived the limiting distributions of sample autocorrelations for ARIMA(p, 1, q) processes; it appears that the sample autocorrelations minus one, times the sample size, converge in distribution to functions of a Wiener process. In this paper we extend Hasza's results in the following ways. First, we shall not assume a parametric form of the data-generating process, but adopt instead mixing conditions. Second, next to the sample autocorrelation function considered by Hasza we also consider a slightly different form of the autocorrelation function, and it appears that the limiting distributions involved are different. Third, we allow the lag length to grow with the sample size, leading to limiting distributions that are independent of the covariance function of the differenced time series under review. Fourth, we also consider the case of detrended time series. Finally, we construct new tests of the unit root hypothesis, based on higher-order sample autocorrelations.