`Weak` trends for inference and forecasting in finite samples. - Chevillon, G (Department of Economics (University of Oxford), 2004)
This paper studies the small sample properties of processes which exhibit both a stochastic and a deterministic trend. Whereas for estimation, inference and forecasting purposes the latter asymptotically dominates the former, it is not so when only a finite number of observations is available and large non-linearities in the parameters of the process result. To analyze this dependence, we resort to local-asymptotics and present the concept of a ‘weak’ trend whose coefficient is of order O(T−1/2), so that the deterministic trend is O(T1/2) and the process Op(T1/2). In this framework, parameter estimates, unit-root test statistics and forecast errors are functions of ‘drifting’ Ornstein-Uhlenbeck processes. We derive a comparison of direct and iterated multi-step estimation and forecasting of a—potentially misspecified—random walk with drift, and show that we explain well the non-linearities exhibited in finite samples. Another main benefit of direct multi-step estimation stems from some different behaviors of the ‘multi-step’ unit-root and slope tests under the weak and strong (constant coefficient) trend frameworks which could lead to testing which framework is more relevant. A Monte Carlo analysis validates the local-asymptotics approximation to the distributions of finite sample biases and test statistics.