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_t the stochastic term which follows an AR(1) process; u_t=θu_t−1+ϵ_t, with normal innovations ϵ_t. Various assumptions about the start‐up will be made. Our main interest lies in the behaviour of the l‐period‐ahead forecast y_n+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 λ_l^(s) of the forecast y_n+1 near θ=1. This measures the sensitivity of the forecast at the unit root. In this study we consider two deterministic trends: d_t=β_t and d_t=β_t+β_tt. The forecast will be the Best Linear Unbiased forecast. We show that, when d_t=β_t, 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.