Semiparametric estimation by model selection for locally stationary processes

Over recent decades increasingly more attention has been paid to the problem of how to fit a parametric model of time series with time-varying parameters. A typical example is given by autoregressive models with time-varying parameters. We propose a procedure to fit such time-varying models to general non-stationary processes. The estimator is a maximum Whittle likelihood estimator on sieves. The results do not assume that the observed process belongs to a specific class of time-varying parametric models. We discuss in more detail the fitting of time-varying AR("p") processes for which we treat the problem of the selection of the order "p", and we propose an iterative algorithm for the computation of the estimator. A comparison with model selection by Akaike's information criterion is provided through simulations. Copyright 2006 Royal Statistical Society.