Local Whittle likelihood estimators and tests for non-Gaussian stationary processes

In this paper, we propose a local Whittle likelihood estimator for spectral densities of non-Gaussian processes and a local Whittle likelihood ratio test statistic for the problem of testing whether the spectral density of a non-Gaussian stationary process belongs to a parametric family or not. Introducing a local Whittle likelihood of a spectral density f _θ (λ) around λ, we propose a local estimator [^(q)] = [^(q)] (l){hat{theta } = hat{theta } (lambda ) } of θ which maximizes the local Whittle likelihood around λ, and use f_{[^(q)] (l)} (l){f_{hat{theta } (lambda )} (lambda )} as an estimator of the true spectral density. For the testing problem, we use a local Whittle likelihood ratio test statistic based on the local Whittle likelihood estimator. The asymptotics of these statistics are elucidated. It is shown that their asymptotic distributions do not depend on non-Gaussianity of the processes. Because our models include nonlinear stationary time series models, we can apply the results to stationary GARCH processes. Advantage of the proposed estimator is demonstrated by a few simulated numerical examples.