On forecasting SETAR processes

Suppose a time series {Yt} 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 ~(h)=E[Yt+h]g; 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[Yt+h ] g; s ~< t] and Var[ Yt+h ] g; 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.