ALMOST SURE BOUNDS ON THE ESTIMATION ERROR FOR OLS ESTIMATORS WHEN THE REGRESSORS INCLUDE CERTAIN MFI(1) PROCESSES

Abstract

Lai and Wei (1983, Annals of Statistics 10, 154–166) state in their Theorem 1 that the estimators of the regression coefficients in the regression $y_t = x_t^' \beta + \varepsilon _{\rm{t}} $, t ∈ ℕ are almost surely (a.s.) consistent under the assumption that the minimum eigenvalue λmin(T) of $\sum\nolimits_{t = 1}^T {x_t } 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 $O(\root \of {\log (\lambda _{max} (T))/\lambda _{min} (T)})$. 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.