Empirical Likelihood Method for Time Series

Abstract

Empirical likelihood method is one of the nonparametric statistical methods, which is applied to the hypothesis testing or construction of confidence regions for unknown parameters. This method has been developed for the statistical inference for independent and identically distributed random variables. To handle serial correlation, an empirical likelihood method is proposed in the frequency domain for second-order stationary processes. The Whittle likelihood is used as an estimating function in the empirical likelihood. It has been shown that the likelihood ratio test statistic based on the empirical likelihood is asymptotically $\chi ^2$-distributed. We discuss the application of the empirical likelihood method to symmetric $\alpha$-stable linear processes. It is shown that the asymptotic distribution of our test statistic is quite different from the usual one. We illustrate the theoretical result with some numerical simulations.