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

We consider the linear time-series model y(t) = d(t) + u(t) (t = 1,..., n), where d(t) is the deterministic trend and u, the stochastic term which fol lows an AR(1) process; u(t) = thetau(t-1) + epsilon (t), with normal innova tions epsilon (t). Various assumptions about the start-up will be made. Our main interest lies in the behaviour of the l-period-ahead forecast (y) ove r cap (n + l) near theta = 1. Unlike in other studies of the AR(1) unit roo t process, we do not wish to ask the question whether theta = 1 but are con cerned with the behaviour of the forecast estimate near and at theta = 1. F or this purpose we define the sth (s = 1, 2) order sensitivity measure lamb da ((s))(t) of the forecast (y) over cap (n + l) near theta = 1. This measu res the sensitivity of the forecast at the unit root. In this study we cons ider two deterministic trends: d(t) = beta (1) and d(t) = beta (1) + beta ( 2)t. The forecast will be the Best Linear Unbiased forecast. We show that, when d(t) = beta (1), the number of observations has no effect on forecast sensitivity. When the deterministic trend is linear, the sensitivity is zer o. We also develop a large-sample procedure to measure the forecast sensiti vity when we are uncertain whether to include the linear trend. Our analysi s suggests that, depending on the initial conditions, it is better to inclu de a linear trend for reduced sensitivity of the medium-term forecast. Copy right (C) 2001 John Wiley & Sons, Ltd.