Sensitivity of univariate AR(1) time-series forecasts near the unit root

Abstract

We consider the linear time-series model yt=dt+ut(t=1,...,n), where dt is the deterministic trend and ut the stochastic term which follows an AR(1) process; ut=θut−1+ϵt, with normal innovations ϵt. Various assumptions about the start-up will be made. Our main interest lies in the behaviour of the l-period-ahead forecast yn+1 near θ=1. Unlike in other studies of the AR(1) unit root process, we do not wish to ask the question whether θ=1 but are concerned with the behaviour of the forecast estimate near and at θ=1. For this purpose we define the sth (s=1,2) order sensitivity measure λl(s) of the forecast yn+1 near θ=1. This measures the sensitivity of the forecast at the unit root. In this study we consider two deterministic trends: dt=βt and dt=βt+βtt. The forecast will be the Best Linear Unbiased forecast. We show that, when dt=βt, the number of observations has no effect on forecast sensitivity. When the deterministic trend is linear, the sensitivity is zero. We also develop a large-sample procedure to measure the forecast sensitivity when we are uncertain whether to include the linear trend. Our analysis suggests that, depending on the initial conditions, it is better to include a linear trend for reduced sensitivity of the medium-term forecast. Copyright © 2001 John Wiley & Sons, Ltd.