1.2.2 Asymptotic Normality To prove asymptotic normality of B, we use revised Markov's LLN and Liapounov and Lindeberg- Feller's central limit theorem of Ch 4. Rewrite VT(B-B) we have the following result Theorem In addition to(1), suppose (i).I(x/, Et) is an independent sequences (a)E(x Et=0; (b)E|X+6<△<∞, for some>0,i=1,2,…,k; (c)Vr=Var(T-1/X'e)is positive definite (a)M=e(X'X/T) is positive definite (b)E|X+<△<∞, for some6>0,i=1,2,…,k; Then D= VT(B-B)N(O, I), where DT=M-]= 1. Assumption (ii. a)is talking about of the mean of this independent sequences (XtiEt, i=1, 2,. ),(ii. b)is about its(2+8)moment exist which is needed for the application of Liapounov's central limit theorem(see p 23 of Ch. 4 )and(ii. c) is to standardize the random vector T-1/(X'e)so that the asymptotic distribu- tion is unit multivariate normal 2. Assumption (ii. a) is talking about of the limits of almost sure convergence of X7x and(ii. b) guarantee its(1+8)moment exist of (Xt Xti, i= 1, 2, ,k;j= 1, 2,. by Cauchy-Schwarz inequality. An existence of the(1+8) moment is what is need for lln of independent sequence. See p. 15 of Ch 4 Proof:
1.2.2 Asymptotic Normality To prove asymptotic normality of βˆ, we use revised Markov’s LLN and Liapounov and Lindeberg-Feller’s central limit theorem of Ch 4. Rewrite √ T(βˆ − β) = X0X T −1 √ T X0ε T = PT t=1 xtx 0 t T !−1 √ T PT t=1 xtεt T ! , we have the following result. Theorem: In addition to (1), suppose (i). {(x 0 t , εt) 0} is an independent sequences; (ii). (a) E(xtεt) = 0; (b) E|Xtiεt | 2+δ < ∆ < ∞, for some δ > 0, i = 1, 2, ..., k; (c) VT ≡ V ar(T −1/2X0ε) is positive definite; (iii). (a) M ≡ E(X0X/T) is positive definite; (b) E|X2 ti| 1+δ < ∆ < ∞, for some δ > 0, i = 1, 2, ..., k; Then D −1/2 T √ T(βˆ − β) L−→ N(0, I), where DT ≡ M−1 T VTM−1 T . Remark: 1. Assumption (ii.a) is talking about of the mean of this independent sequences (Xtiεt , i = 1, 2, ..., k), (ii.b) is about its (2 + δ) moment exist which is needed for the application of Liapounov’s central limit theorem (see p.23 of Ch. 4) and (ii.c) is to standardize the random vector T −1/2 (X0ε) so that the asymptotic distribution is unit multivariate normal. 2. Assumption (iii.a) is talking about of the limits of almost sure convergence of X0X T and (iii.b) guarantee its (1 + δ) moment exist of (XtiXtj , i = 1, 2, .., k; j = 1, 2, ..., k) by Cauchy-Schwarz inequality. An existence of the (1 + δ) moment is what is need for LLN of independent sequence. See p.15 of Ch.4. Proof: 6
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 changes
It 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 crucial 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 second moments, therefore when the regressor is a unit root process, then traditional 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