Bootstrap Unit Root Tests

We consider the bootstrap unit root tests based on finite order autoregressive integrated
models driven by iid innovations, with or without deterministic time trends.
A general methodology is developed to approximate asymptotic distributions for the
models driven by integrated time series, and used to obtain asymptotic expansions for
the Dickey-Fuller unit root tests. The second-order terms in their expansions are of
stochastic orders Op(n-'I4) and Op(n-'/2), and involve functionals of Brownian motions
and normal random variates. The asymptotic expansions for the bootstrap tests
are also derived and compared with those of the Dickey-Fuller tests. We show in particular
that the bootstrap offers asymptotic refinements for the Dickey-Fuller tests, i.e.,
it corrects their second-order errors. More precisely, it is shown that the critical values
obtained by the bootstrap resampling are correct up to the second-order terms, and
the errors in rejection probabilities are of order o(n-'I2) if the tests are based upon
the bootstrap critical values. Through simulations, we investigate how effective is the
bootstrap correction in small samples.
KEYWORDS:Bootstrap, unit root test, asymptotic expansion.
1. INTRODUCTION
IT IS NOW WELL PERCEIVED that the bootstrap, if applied appropriately,
helps to compute the critical values of asymptotic tests more accurately in finite
samples, and that the tests based on the bootstrap critical values generally
have actual finite sample rejection probabilities closer to their asymptotic nominal
values. See, e.g., Hall (1992) and Horowitz (2001). The bootstrap unit root
tests, i.e., the unit root tests relying on the bootstrap critical values, seem particularly
attractive in this respect. For most of the commonly used unit root
tests, the discrepancies in the actual and nominal rejection probabilities are
known to be large and often too large for the tests to be any reliable. It has
indeed been observed by various authors including Ferretti and Romo (1996)
and Nankervis and Savin (1996) that the bootstrap tests have actual rejection
probabilities that are much closer to their nominal values, compared to the
asymptotic tests, in the unit root models.