Valid Edgeworth Expansion of the Bootstrap t-statistic of the Whittle MLE for Linear Regression Models with Long-Memory Residuals

In this paper we provide a valid Edgeworth expansion of the parametric bootstrap t-statistic for the Whittle maximum likelihood estimator of a linear regression time series model whose residuals are stationary, Gaussian, and long-memory. Under some sets of conditions on the spectral density function and the parametric values, an Edgeworth expansion of the bootstrap t-statistic of arbitrarily large order of the model is established to have an error of \(o(n^{1-s/2})\) $o(n^{1-s/2})$ , where \(s \ge 3\) $s\ge 3$ is a positive integer. The result is obtained by extending the Edgeworth expansion obtained by Andrew et al. (2006), which was established for the parametric bootstrap t-statistic of the same model without the linear regression component.