Linear prediction of ARMA processes with infinite variance

Abstract

In order to predict unobserved values of a linear process with infinite variance, we introduce a linear predictor which minimizes the dispersion (suitably defined) of the error distribution. When the linear process is driven by symmetric stable white noise this predictor minimizes the scale parameter of the error distribution. In the more general case when the driving white noise process has regularly varying tails with index α, the predictor minimizes the size of the error tail probabilities. The procedure can be interpreted also as minimizing an appropriately defined lα-distance between the predictor and the random variable to be predicted. We derive explicitly the best linear predictor of Xn+1 in terms of X1,..., Xn for the process ARMA(1, 1) and for the process AR(p). For higher order processes general analytic expressions are cumbersome, but we indicate how predictors can be determined numerically. © 1985.