Asymptotics Of Ml Estimator For Regression Models With A Stochastic Trend Component

This paper investigates the asymptotic properties of the maximum marginal likelihood estimator for a regression model with a stochastic trend component when the signal-to-noise ratio is near zero. In particular, the local level model in Harvey (1989, Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge: Cambridge University Press) and its variants where a time trend or an intercept is included are considered. A local-to-zero parameterization is adopted. Two sets of asymptotic properties are presented for the local maximizer: consistency and the limiting distribution. The estimator is found to be super-consistent. The limit distribution is derived and found to possess a long tail and a mass point at zero. It yields a good approximation for samples of moderate size. Simulation also documents that the empirical distribution converges less rapidly to the limit distribution as number of regression parameters increases. The results could be viewed as a transition step toward establishing new likelihood ratio–type or Wald-type tests for the stationarity null.