Almost sure bounds on the Estimation Error for Ols Estimators when the Regressors include certain MFI(1) Processes

Lai and Wei (1982, Annals of Statistics 10, 154-166) state in their Theorem 1 that the estimators of the regression coefficients in the regression yt = x tβ +εt , t ∈ N are almost surely (a.s.) consistent under the assumption that the minimum eigenvalue λmin(T ) of ΣT t=1 xt x t tends to infinity (a.s.) and log(λmax(T ))/λmin(T )→0 (a.s.) where λmax(T ) denotes the maximal eigenvalue. Moreover the rate of convergence in this case equals O( √ log(λmax(T ))/λmin(T )). In this note xt is taken to be a particular multivariate multifrequency I(1) processes, and almost sure rates of convergence for least squares estimators are established.