Bootstrap Unit Root Tests: Comparison and Extensions

In this paper we study and compare the properties of several bootstrap unit root tests recently proposed in the literature. The tests are Dickey-Fuller or Augmented DF-tests, either based on residuals from an autoregression and the use of the block bootstrap (Paparoditis & Politis, 2003) or on first differenced data and the use of the stationary bootstrap (Swensen, 2003a) or sieve bootstrap (Psaradakis, 2001; Chang & Park, 2003). We extend the analysis by interchanging the data transformations (differences versus residuals), the types of bootstrap and the presence or absence of a correction for autocorrelation in the tests. We prove that two sieve bootstrap tests based on residuals remain asymptotically valid, thereby completing the proofs of validity for all the types of DF bootstrap tests. In contrast to the literature which basically focuses on a comparison of the bootstrap tests with an asymptotic test, we compare the bootstrap tests among them using response surfaces for their size and power in a simulation study. We also investigate how the tests behave when accounting for a deterministic trend, even in the absence of such a trend in the data. This study leads to the following conclusions: (i) augmented DF-tests are always preferred to standard DF-tests; (ii) the sieve bootstrap performs slightly better than the block bootstrap; (iii) difference-based and residual-based tests behave similarly in terms of size although the latter appear more powerful. The results for the response surfaces allow us to make statements about the behaviour of the bootstrap tests as sample size increases.