Lagged Unstable Regressor Models and Asymptotic Efficiency of the Ordinary Least Squares Estimator

Lagged regressor models with general stationary errors independent of the regressors are considered. The regressor process is unstable having characteristic roots on the unit circle. If the order of the lag matches the number of roots on the unit circle, the ordinary least squares estimator (OLSE) is asymptotically efficient in that it has the same limiting distribution as the generalized least squares estimator (GLSE) under the same normalization. This result extends the well-known result of Grenander and Rosenblatt (1957) for asymptotic efficiency of the OLSE in deterministic polynomial and/or trigonometric regressor models to a class of models with stochastic regressors.