A POWERFUL TEST OF THE AUTOREGRESSIVE UNIT ROOT HYPOTHESIS BASED ON A TUNING PARAMETER FREE STATISTIC

This paper presents a family of simple nonparametric unit root tests indexed by one parameter, d, and containing the Breitung (2002, Journal of Econometrics 108, 342-363) test as the special case d = 1. It is shown that (a) each member of the family with d > 0 is consistent, (b) the asymptotic distribution depends oil of and thus reflects the parameter chosen to implement the test, and (c) because the asymptotic distribution depends on d and the test remains consistent for all d > 0, it is possible to analyze the power of the test for different values of d. The usual Phillips-Perron and Dickey-Fuller type tests are indexed by bandwidth, lag length, etc., but have none of these three properties. It is shown that members of the family with d < 1 have higher asymptotic local power than the Breitung (2002) test, and when d is small the asymptotic local power of the proposed nonparametric test is relatively close to the parametric power envelope, particularly in the case with a linear time trend. Furthermore, generalized least squares (GLS) detrending is shown to improve power when d is small, which is not the case for the Breitung (2002) test. Simulations demonstrate that when applying a sieve bootstrap procedure, the proposed variance ratio test has very good size properties, with finite-sample power that is higher than that of the Breitung (2002) test and even rivals the (nearly) optimal parametric GLS detrended augmented Dickey-Fuller test with lag length chosen by all information criterion.