Asymptotic efficiency of the OLS estimator with singular limiting sample moment matrices

In the literature on time series analysis, Grenander and Rosenblatt’s theorem is necessary to judge the efficiency of OLS estimators with the requirement of Grenander’s conditions. However, without the conditions, it is not obvious whether the estimator is efficient. In this study, a model with an asymptotically efficient OLS estimator is presented, regardless whether one of the conditions is not satisfied. The regression model is specified with polynomial regressors of a slowly varying function and general stationary disturbances. The regressors are known to display asymptotic singularity in the sample moment matrices, and thereby, Grenander’s condition fails.