On forecasting SETAR processes

Abstract

Suppose a time series {Y_t} is generated by a first-order stationary self-exciting threshold autoregressive (SETAR) model with Gaussian innovations. The minimum mean squared error h-step ahead forecast $\text{Y}\text{̂}{}_{\mathrm{t}}\text{(h) = E[Y}{}_{\mathrm{t+h}}\text{|Y}{}_{\mathrm{s}}\text{; s ⩽ t]}$ for h > 2 involves a sequence of complicated numerical integrations and closed-form expressions are very difficult or even impossible to obtain. In this paper we derive explicit approximate expressions for E[Y_t+h|Y_s; s ⩽ t] and Var[Y_t+h|Y_s; s ⩽ t] (h > 2) for various SETAR models. The approximations are reasonably accurate as compared with alternative methods based on numerical integration and Monte Carlo experiments.