Consistent autoregressive spectral estimates: Nonlinear time series and large autocovariance matrices

Abstract

We consider the problem of using an autoregressive (AR) approximation to estimate the spectral density function and the n × n autocovariance matrix based on stationary data X1, … , Xn. The consistency of the autoregressive spectral density estimator has been proven since the 1970s under a linearity assumption. We extend these ideas to the nonlinear setting, and give an application to estimating the n × n autocovariance matrix. Under mild assumptions on the underlying dependence structure and the order p of the fitted AR(p) model, we are able to show that the autoregressive spectral estimate and the associated AR‐based autocovariance matrix estimator are consistent. We are also able to establish an explicit bound on the rate of convergence of the proposed estimators.