Bootstrap Unit Root Tests
针对有限阶自回归积分模型，提出自助法单位根检验的一般方法，推导Dickey-Fuller检验的渐近展开，证明自助法能修正其二阶误差，并通过模拟验证小样本下的有效性。
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−1/4) and Op(n−1/2), 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−1/2) if the tests are based upon the bootstrap critical values. Through simulations, we investigate how effective is the bootstrap correction in small samples.