Relevant sample quantities such as the sample autocorrelation function and extremes contain useful information about autoregressive time series with heteroskedastic errors. As these quantities usually depend on the tail index of the underlying heteroskedastic time series, estimating the tail index becomes an important task. Since the tail index of such a model is determined by a moment equation, one can estimate the underlying tail index by solving the sample moment equation with the unknown parameters being replaced by their quasi-maximum likelihood estimates. To construct a confidence interval for the tail index, one needs to estimate the complicated asymptotic variance of the tail index estimator, however. In this paper the asymptotic normality of the tail index estimator is first derived, and a profile empirical likelihood method to construct a confidence interval for the tail index is then proposed. A simulation study shows that the proposed empirical likelihood method works better than the bootstrap method in terms of coverage accuracy, especially when the process is nearly nonstationary.
A penalized empirical likelihood method in high dimensions