Bootstrap Probability Errors of the Whittle MLE for Linear Regression Processes with Strongly Dependent Disturbances

Abstract

This paper determines bounds on the asymptotic orders of the coverage probability errors of parametric bootstrap confidence intervals (CIs) and tests for the covariance parameters of a time series generated by a regression model with Gaussian, stationary, and strongly dependent errors. The CIs and tests are based on the plug-in Whittle maximum likelihood (PWML) estimators. It is shown that, under some sets of conditions on the regression coefficients, the spectral density function, and the parameter values, the bounds on the coverage probability errors of symmetric two-sided and one-sided parametric bootstrap confidence intervals on the plug-in Whittle log-likelihood function are shown to be O(n^{-3/2}\ln{n}) and O(n^{-1}\ln{n}), respectively. Apart from the \ln{n} term, the magnitudes of the coverage probability errors of the one-sided bootstrap confidence intervals for our model is shown to be essentially the same as that of the independent and identically distributed (iid) data. The error for the two-sided confidence intervals is not as small as the error O(n^{-2}) that has been established for many confidence intervals in the literature, see Hall (1992), pp 102-108.