UC San Diego

We consider the problem of using an autoregressive (AR) approximation to estimate the spectral density function and the $n\times n$ autocovariance matrixbased on stationary data $X_1,\dots,X_n$. The consistency of the autoregressive spectral density estimator has been proven since the 1970s under a linearity assumption. We extend these ideas to the non-linear setting, and give an application to estimating the $n \times n$ 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.

Cleveland (1972) introduced the inverse autocovariance function (iacf) for weaklystationary time series. He proposed two ways to estimate the inverse autocovariances: one way is to fit an autoregressive (AR) model to the data and use the fitted model's inverse autocovariance as the iacf estimator, and the other method is to employ a kernel-smoothed spectral density estimator to construct iacf estimator. Bhansali (1980) then proved consistency of the iacf estimator at a fixed lag based on a linear time series condition. In the paper at hand, we relax the linearity assumption and provide sufficient conditions for the consistency of the iacf estimator. We further consider the problem of estimating the vector consisting of the iacf at lags up to $n$, based on a sample of size $n$. We propose several competing estimators of the iacf vector, and study their convergence. Lastly, we consider the inverse autocovariance matrix, i.e., the $n$ by $n$ Toeplitz matrix with $i,j$ element given by the iacf at lag $i-j$; we propose an estimator and investigate its consistency properties.