Higher order approximations for Wald statistics in time series regressions with integrated processes

Asymptotic expansions are developed for Wald test statistics in time series regressions with integrated processes. These expansions provide an opportunity to reduce size distortion in testing by suitable bandwidth selection, and automated rules for doing so are calculated. A band spectral regression and the associated Wald test are also considered. Both the first order and second order properties of the estimator are studied.