Bootstrap Unit‐Root Tests: Comparison and Extensions

Abstract

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 or on first differenced data and the use of the stationary bootstrap or sieve bootstrap. 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 show that two sieve bootstrap tests based on residuals remain asymptotically valid. In contrast to the literature which 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.This study leads to the following conclusions: (i) augmented DF-tests are always preferred to standard DF-tests; (ii) the sieve bootstrap performs better than the block bootstrap; (iii) difference-based tests appear to have slightly better size properties but residual-based tests appear more powerful.

Detrending Bootstrap Unit Root Tests

“…This indeed shows in the simulations of Palm et al (2006), who have an ARMA(1,1) process as DGP for u t . The empirical size of the block bootstrap tests is seen to depend heavily on the values of the AR parameter and the MA parameter, whereas this sensitivity is much less for the sieve bootstrap tests.…”

“…In the simulations conducted by Palm et al (2006), a difference in power is indeed found. However, this difference in power is often accompanied by a difference in size.…”

“…Nevertheless, the simulations reported in Palm et al (2006) indicate that the assertions about asymptotic refinements are valid for unit root testing as well, as the ADF tests considered in the paper all have considerably smaller size distortions than the DF tests.…”

“…Parker, Paparoditis, and Politis (2006) propose a stationary DF test based on residuals. Paparoditis and Politis (2005) and Palm, Smeekes, and Urbain (2006) propose residual-based sieve bootstrap tests 1 for both DF and ADF statistics. The latter paper also compares the finite sample performance of all the tests mentioned here by simulation.…”

Maximum Non-Extensive Entropy Block Bootstrap for Non-stationary Processes

“…Bootstrapping unit-root tests is thus one of the potentially most interesting application of the entropy-based bootstrap method constituting an alternative to standard residual-based bootstrapped unit-root tests (see Palm et al 2007 for a review) which are not easy to apply when the dependence structure of the residuals is difficult to ascertain.…”