Bootstrap tests for an autoregressive unit root in the presence of weakly dependent errors (Q2759339)

Let \(\{x_{t}\}_{t\in {\mathbb N}}\) be a real-valued time series given by \(x_{t}=d_{t}+v_{t}\), \(v_{t}=\alpha v_{t-1}+u_{t}\), where \(\{d_{t}\}_{t\in {\mathbb N}}\) is a nonstochastic sequence, \(\{u_{t}\}_{t\in {\mathbb Z}}\) is an unobservable stationary stochastic process with zero mean, and \(\alpha\in (-1,1]\). Let \(\{u_{t}\}_{t\in {\mathbb Z}}\) satisfy the following conditions:NEWLINENEWLINENEWLINE1) \(\{u_{t}\}_{t\in {\mathbb Z}}\) is the linear process \(u_{t}=\sum_{j=0}^{\infty}\psi_{j}\epsilon_{t-j},\;\psi_0=1\), where \(\{\epsilon_{t}\}_{t\in {\mathbb Z}}\) are i.i.d. random variables with \(E[\epsilon_1]=0,\;E[\epsilon_1^2]=\sigma^2_{\epsilon}>0\) and \(E[\epsilon^4_1]<\infty\); 2) The sequence of constants \(\{\psi_{t}\}_{t\in {\mathbb N}_0}\) is such that \(\sum_{j=0}^{\infty}j|\psi_{j}|<\infty\), \(\sum_{j=0}^{\infty}\psi_{j}\neq 0\) and \(\sum_{j=0}^{\infty}\psi_{j}z^{j}\neq 0\) in \(\{z\in {\mathbb C}: |z|\leq 1\}\).NEWLINENEWLINENEWLINEThe paper deals with testing the null hypothesis \(H_0:\;\alpha=1\) against the alternative \(H_1:\;\alpha<1\) given a sample \(\{x_1,\ldots,x_{n}\}\) of \(n\) observations of \(x_{t}\). The author describes the sieve bootstrap procedure and establishes its asymptotic validity for testing the unit-root hypothesis. The presented Monte Carlo study demonstrates for the case of negatively correlated moving-average errors that sieve bootstrap tests yield impressive improvements in Type 1 error probability accuracy over conventional asymptotic unit-root tests.