Inverse Autocovariance Estimates

Abstract

The notion of the inverse autocovariance function (iacf) for stationary time series was introduced by W. Cleveland in the 1970s who proposed two ways to estimate it: 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 via a kernel-smoothed spectral density estimator. Consistency of the iacf estimator at a fixed lag was subsequently proved by R.J. Bhansali in the 1980s based on a linear time series condition. In this article, 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. In addition, we discuss the difficult problem of choosing the order $p$ of a fitted AR model, and provide some alternative ways to approach it. Finally, 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. Numerical simulations illustrate the finite sample performance of all iacf estimators, including the estimators of the order $p$.