Empirical Likelihood Approach for Non-Gaussian Vector Stationary Processes and Its Application to Minimum Contrast Estimation

A. For a class of vector-valued non-Gaussian stationary processes with unknown parameters, we develop the empirical likelihood approach. In time series analysis it is known that Whittle likelihood is one of the most fundamental tools to get a good estimator of unknown parameters, and that the score functions are asymptotically normal. Motivated by the Whittle likelihood, we apply the empirical likelihood approach to its derivative with respect to unknown parameters. We also consider the empirical likelihood approach to a minimum contrast estimation based on a spectral disparity measure, and apply the approach to the derivative of the spectral disparity. This paper provides rigorous proofs on convergence of our two empirical likelihoods to sums of Gamma distributions. Since the fitted spectral model may be different from the true spectral structure, the results enable us to construct confidence regions for various important time series parameters without Gaussianity. Numerical studies are given, and illuminate some interesting features of the empirical approach.