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

We consider the problem of using an autoregressive (AR) approximation to estimate the spectral density function and the n x n autocovariance matrix based on stationary data X-1,...,X-n. 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 x 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.