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 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 L₂ 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.