Testing against smooth stochastic trends



A trend estimated from an unobserved components model tends to be smoother when it is modelled as an integrated random walk rather than a random walk with drift. This article derives a test of the null hypothesis that the trend is deterministic against the alternative that it is an integrated random walk. It is assumed that the other component in the model is normally distributed white noise. Critical values are tabulated, the asymptotic distribution is derived and the performance of the test is compared with the test against a trend specified as a random walk with drift. The test is extended to allow for serially correlated and evolving seasonal components. When there is a stationary process containing a single autoregressive unit root close to one, a bounds test can be applied. In the case of a first-order autoregressive disturbance, it is shown that a consistent test can still be obtained by carrying out estimation of the nuisance parameters under the null hypothesis. The overall conclusion is that the most effective test against an integrated random walk is a parametric one based on the random walk plus drift test statistic, constructed from innovations, with the nuisance parameters estimated in the unrestricted model. Copyright © 2001 John Wiley & Sons, Ltd.