It is obvious that from these assumptions we have GT=TXe(O, Var(T-1PX'e)=N(O,VT) and XtX t=1 t Mr Therefore T(A-B)2M71N(0,V ≡N(0,M7VrM7) MVIM)-/VT(B-B)N(O,I From results above, the asymptotic normality of OLS estimator depend cru cial on the existence of at least second moments of the regressors Xti and from that we have lln such that E(=1x As we ive seen from last chapter that a I(1) variables does not have a finite sec- ond moments, therefore when the regressor is a unit root process, then tradi- tional asymptotic results for OLS estimator would not apply. However, there is a case that the regressor is not stochastic, but it violate the condition that Xx s E(Zixexi)=Mr=O(1), as we will see in the following that the asymptotic normality still valid though the rate convergence to the normality changesIt is obvious that from these assumptions we have √ T  X0ε T  = T −1/2X0 ε L−→ N(0, V ar(T −1/2X0 ε) ≡ N(0, VT ) and  X0X T  = PT t=1 xtx 0 t T ! a.s −→ E PT t=1 xtx 0 t T ! = MT . Therefore √ T(βˆ − β) L−→ M−1 T N(0, VT ) ≡ N(0,M−1 T VTM−1 T ), or (M−1 T VTM−1 T ) −1/2 √ T(βˆ − β) L−→ N(0, I). From results above, the asymptotic normality of OLS estimator depend cru￾cial on the existence of at least second moments of the regressors Xti and from that we have LLN such that X0X T a.s −→ E PT t=1 xtx 0 t T  = MT = O(1). As we have seen from last chapter that a I(1) variables does not have a finite sec￾ond moments, therefore when the regressor is a unit root process, then tradi￾tional asymptotic results for OLS estimator would not apply. However, there is a case that the regressor is not stochastic, but it violate the condition that X0X T a.s −→ E PT t=1 xtx 0 t T  = MT = O(1), as we will see in the following that the asymptotic normality still valid though the rate convergence to the normality changes. 7