Linear Prediction by Autoregressive Model Fitting in the Time Domain

Abstract

Let $\{x_t\}$ be a purely nondeterministic stationary process satisfying all the assumptions made by Berk (1974), and $\{y_t\}$ be another purely nondeterministic stationary process. Assume that $y_t$ is independent of $x_t$ but has exactly the same statistical properties as that of $x_t$. Consider the linear prediction of future values of $y_t$ on the basis of past values, using prediction constants estimated from a realisation of $T$ observations of $x_t$ by least-squares fitting of an autoregression of order $k$. By assuming that $k \rightarrow \infty, k^3/T \rightarrow 0$ as $T \rightarrow \infty$, the effect on the mean square error of prediction of estimating the autoregressive coefficients is determined. This effect is the same as for the case when the prediction constants are estimated by factorising a "windowed" estimate of the spectral density function of $x_t$.