ASYMPTOTIC DISTRIBUTIONS OF THE LEAST-SQUARES ESTIMATORS AND TEST STATISTICS IN THE NEAR UNIT-ROOT MODEL WITH NONZERO INITIAL-VALUE AND LOCAL DRIFT AND TREND

This paper considers the distribution of the Dickey-Fuller test in a m odel with non-zero initial value and drift and trend. We show how stoc hastic integral representations for the limiting distribution can be d erived either from the local to unity approach with local drift and tr end or from the continuous record asymptotic results of Sorensen [29]. We also show how the stochastic integral representations can be utili zed as the basis for finding the corresponding characteristic function s via the Fredholm approach of Nabeya and Tanaka [16,17]. This ''link' ' between those two approaches may be of general interest. We further tabulate the asymptotic distribution by inverting the characteristic f unction. Using the same methods, we also find the characteristic funct ion for the asymptotic distribution for the Schmidt-Phillips [26] unit root test. Our results show very clearly the dependence of the variou s tests on the initial value of the time series.