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 distribu￾tion 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