A limit theory is established for autoregressive time series that smooths the transition between local and moderate deviations from unity and provides a transitional form that links conventional unit root distributions and the standard normal. Edgeworth expansions of the limit theory are given. These expansions show that the limit theory that holds for values of the autoregressive coeﬀicient that are closer to stationarity than local (i.e. deviations of the form = 1 + (c/n), where n is the sample size and c < 0) holds up to the second order. Similar expansions around the limiting Cauchy density are provided for the mildly explosive case.
Phillips, Peter C.B.; Magdalinos, Tassos; and Giraitis, Liudas, "Smoothing Local-to-Moderate Unit Root Theory" (2008). Cowles Foundation Discussion Papers. 1963.