Estimation for Non-Gaussian Locally Stationary Processes with Empirical Likelihood Method

An application of the empirical likelihood method to non-Gaussian locally stationary processes is presented. Based on the central limit theorem for locally stationary processes, we give the asymptotic distributions of the maximum empirical likelihood estimator and the empirical likelihood ratio statistics, respectively. It is shown that the empirical likelihood method enables us to make inferences on various important indices in a time series analysis. Furthermore, we give a numerical study and investigate a finite sample property. 1. Introduction The empirical likelihood is one of the nonparametric methods for a statistical inference proposed by Owen [1, 2]. It is used for constructing confidence regions for a mean, for a class of M-estimates that includes quantile, and for differentiable statistical functionals. The empirical likelihood method has been applied to various problems because of its good properties: generality of the nonparametric method and effectiveness of the likelihood method. For example, we can name applications to the general estimating equations, [3] the regression models [4–6], the biased sample models [7], and so forth. Applications are also extended to dependent observations. Kitamura [8] developed the blockwise empirical likelihood for estimating equations and for smooth functions of means. Monti [9] applied the empirical likelihood method to linear processes, essentially under the circular Gaussian assumption, using a spectral method. For short- and long-range dependence, Nordman and Lahiri [10] gave the asymptotic properties of the frequency domain empirical likelihood. As we named above, some applications to time series analysis can be found but it seems that they were mainly for stationary processes. Although stationarity is the most fundamental assumption when we are engaged in a time series analysis, it is also known that real time series data are generally nonstationary (e.g., economics analysis). Therefore we need to use nonstationary models in order to describe the real world. Recently Dahlhaus [11–13] proposed an important class of nonstationary processes, called locally stationary processes. They have so-called time-varying spectral densities whose spectral structures smoothly change in time. In this paper we extend the empirical likelihood method to non-Gaussian locally stationary processes with time-varying spectra. First, We derive the asymptotic normality of the maximum empirical likelihood estimator based on the central limit theorem for locally stationary processes, which is stated in Dahlhaus [13, Theorem