Estimation and Prediction for Positive Autoregressive Processes

Abstract

Two issues on estimation and prediction for positive AR(1) processes are addressed. We start by considering the nearly unstable case in which the AR coefficient converges to 1 at rate 1/n, where n is the sample size. Based on moment bounds for the extreme-value (EV) and least squares (LS) estimates, asymptotic expressions for the mean squared prediction errors (MSPE) of the EV and LS predictors are obtained. These asymptotic expressions are further extended to a general class of nearly unstable models, thereby allowing one to understand to what degree such general models can be used to establish a link between stationary and unstable models from a prediction perspective. As applications, we illustrate the usefulness of these results in conducting finite sample approximations. The second issue is concerned with the estimation of the AR coefficient in stationary AR(1) models whose positive error has a density function regularly varying at the origin and infinity with index α and β, respectively. We first derive the limiting distribution of the EV estimate using point process techniques. This result reveals that the rate of convergence of the EV estimate varies with the ordering of α and β. We then construct consistent estimates of α and β, leading to an asymptotically valid confidence interval of the AR coefficient without knowing α and β.