Nonlinear Wavelet Estimation of Time-Varying Autoregressive Processes

Abstract

We consider nonparametric estimation of the parameter functions a(i)(.), i = 1, ..., p, of a time-varying autoregressive process. Choosing an orthonormal wavelet basis representation of the Functions a(i), the empirical wavelet coefficients are derived from the time series data as the solution of a least-squares minimization problem. In order to allow the a(i) to be functions of inhomogeneous regularity, we apply nonlinear thresholding to the empirical coefficients and obtain locally smoothed estimates of the a(i). We show that the resulting estimators attain the usual minimax L-2 rates up to a logarithmic factor, simultaneously in a large scale of Besov classes. The finite-sample behaviour of our procedure is demonstrated by application to two typical simulated examples.