Introduction to Asymptotic Theory Lemma 4.2 [WLLN for an IID Random Sample]: Suppose {Z}is IID with E(Z)=u and EZ<.Define Zn=n-1Z. Then Zn马uasn→o. Questions: Why do we need the moment condition EZt<oo? A counter example:Suppose {Zt}is a sequence of IID Cauchy(0,1)ran- dom variables whose moments do not exist.Then nCauchy(0,1)for all n 1,and so it does not converge in probability to some constant as m→00. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 11
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 11 Introduction to Asymptotic Theory Lemma 4.2 Questions: s:
Introduction to Asymptotic Theory Questions: Why do we need the moment condition Et<oo? Convergence in probability with order n: The sequence {Zn,n =1,2,...}is said to be of order smaller than na in probability if Zn/n0 as noo.This is denoted as Zn=op(na). -The sequence{Zm,n=l,2,···}is said to be at most of order na in probability if for any given 6>0,there exist a constant M=M(6)<oo and a finite integer NN(6),such that P(Zn/n>M)<6 for all nN.This is denoted as Zn=Op(n). For Zn=Op(na)with a >0,the order na is the fastest growth rate at which Zn goes to infinity with probability approaching one.When a <0, the order na is the slowest convergence rate at which Zn vanishes to 0 with probability approaching one. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 12
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 12 Introduction to Asymptotic Theory Questions: s: s:
Introduction to Asymptotic Theory Definition 4.3 [Boundedness in Probability]: A sequence of random variables/vectors/matrices {Zn is bounded in prob- ability if for any small constant 6>0,there exists a constant C<oo such that P(Zn‖>C)≤δ asn→o.We denote Zn =Op(1). When Zn=Op(1),the probability thatn exceeds a very large con- stant is small as n→o.Equivalently,.‖Zn‖is smaller than C with a very high probability as n->oo. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 13
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 13 Introduction to Asymptotic Theory Definition 4.3
Introduction to Asymptotic Theory Example 4.3 Suppose Zn N(u,o2)for all n >1.Then Zn =Op(1). Solution:For any 6 0,we always have a sufficiently large constant C=C(6)>0 such that P(Znl>C=1-P(-C≤Zm≤C) =1-P =1-(0,）-(） ≤6. where (z)=P(Z<z)is the CDF of N(0,1).[We can choose C such thatΦ[(C-)/o]≥1-6and④[-(C+)/o]≤6.] ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 14
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 14 Introduction to Asymptotic Theory Example 4.3
Introduction to Asymptotic Theory Lemma 4.3 If Zn 0,then Zn -ZB 0. Proof:By Chebyshev's inequality,we have P(Z,-Z>e)≤ E(Zn -22 0 E2 for any given e>0 as n-oo.This completes the proof. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 15
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 15 Introduction to Asymptotic Theory Lemma 4.3