This paper presents a family of simple nonparametric unit root tests indexed by one parameter, d, and containing Breitung's (2002) test as the special case d = 1. It is shown that (i) each member of the family with d > 0 is consistent, (ii) the asymptotic distribution depends on d, and thus reflects the parameter chosen to implement the test, and (iii) since 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 or Dickey-Fuller type tests are characterized by tuning parameters (bandwidth, lag length, etc.), i.e. parameters which change the test statistic but are not reflected in the asymptotic distribution, and thus 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 timetrend. Furthermore, GLS detrending is shown to improve power when d is small, which is not the case for Breitung's (2002) test. Simulations demonstrate that, apart from some size distortion in the presence of large negative AR or MA coefficients, the proposed test has good finite sample properties in the presence of both linear and nonlinear short-run dynamics. When applying a sieve bootstrap procedure, the proposed test has very good size properties, with finite sample power that is higher than that of Breitung's (2002) test and even rivals the (nearly) optimal parametric GLS detrended augmented Dickey-Fuller test with lag length chosen by an information criterion.
Augmented Dickey-Fuller test fractional integration GLS detrending nonparametric nuisance parameter tuning parameter power envelope unit root test variance ratio