  In unit root testing, a piecewise locally stationary process is adopted to
accommodate nonstationary errors that can have both smooth and abrupt changes
in second- or higher-order properties. Under this framework, the limiting null
distributions of the conventional unit root test statistics are derived and
shown to contain a number of unknown parameters. To circumvent the difficulty
of direct consistent estimation, we propose to use the dependent wild bootstrap
to approximate the non-pivotal limiting null distributions and provide a
rigorous theoretical justification for bootstrap consistency. The proposed
method is compared through finite sample simulations with the recolored wild
bootstrap procedure, which was developed for errors that follow a
heteroscedastic linear process. Further, a combination of autoregressive sieve
recoloring with the dependent wild bootstrap is shown to perform well. The
validity of the dependent wild bootstrap in a nonstationary setting is
demonstrated for the first time, showing the possibility of extensions to other
inference problems associated with locally stationary processes.
