A Likelihood Approximation for Locally Stationary Processes

A new approximation to the Gaussian likelihood of a multivariate locally stationary process is introduced. It is based on an approximation of the inverse of the covariance matrix of such processes. The new quasi-likelihood is a generalisation of the classical Whittle-likelihood for stationary processes. For parametric models asymptotic normality and efficiency of the resulting estimator are proved. Since the likelihood has a special local structure it can be used for nonparametric inference as well. This is briefly sketched for different estimates.