Assessing prediction error in autoregressive models

Abstract:

Assessing prediction error is a problem which arises in time series analysis. The distinction between the conditional prediction error $e$ and the unconditional prediction error $E(e)$ has not received much attention in the literature. Although one can argue that the conditional version is more practical, we show in this article that assessing $e$ is nearly impossible. In particular, we use the correlation coefficient $\operatorname {corr} (\hat e,e)$, where $\hat e$ is an estimate of $e$, as a measure of performance and show that ${\lim _{T \to \infty }}\sqrt T \operatorname {corr} (\hat e,e) = C$ where $T$ is the sample size and $C > 0$ is some constant. Furthermore, the value of $C$ is large only when the process is extremely non-Gaussian or nearly nonstationary.