Introduction to Asymptotic Theory Example 4.4 Suppose Assumptions 3.1-3.4 hold.Does the OLS estimator B converges in probability to B0? Solution:From Theorem 3.5,we have 7E[(3-3)(3-3)Y1X]r=2T(X'X)-1 ≤o2λmin(X'X) →0 for any TERK,T'T=1 as noo with probability one.It follows that EB-B2=EE[lB-B2]}0 as n->oo.Therefore,we have 3BB°. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 16
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 16 Introduction to Asymptotic Theory Example 4.4
Introduction to Asymptotic Theory Example 4.5 Suppose Assumptions 3.1,3.3 and 3.5 hold.Does s2 converge in probability t0g2? Solution:Under the given assumptions and conditional on X. -K)2X2-K 02 and so we have(s)-o2 and var(s2)It follows that E(s2-2)2=2σ4/(n-K)→0. Therefore,s22 and s2Bo2 because convergence in quadratic mean implies convergence in probability. While convergence in mean squares implies convergence in probability,the converse is not true. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 17
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 17 Introduction to Asymptotic Theory Example 4.5
Introduction to Asymptotic Theory Example 4.6 Suppose ={ with prob1-月 with prob 1. Then Zn B0 as noo but E(Zn-0)2=noo.Please verify it. Solution: (i)For any given e>0,we have P(Zm-0>e)=P(Z=m)=1→0. () E(Zm-0)2=∑(2n-0)2f(an) zn∈{0，n} =(0-0)2.(1-n-1)+(m-0)2.n1 =n→0. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 18
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 18 Introduction to Asymptotic Theory Example 4.6
Introduction to Asymptotic Theory Definition 4.4 [Almost Sure Convergence): {Zn}converges to Z almost surely if Prlin llZn-l =0-1. We denote Zn-Z0. To gain intuition for almost sure convergence,recall the definition of a random variable:any random variable is a mapping from the sample space n to the real line,namely 2:-R.Let w be a basic outcome in the sample space Define a subset in Ac={w∈2：lim Zn(w)=Z(w)}. That is,Ac is the set of basic outcomes on which the sequence of [n()} converges to Z(·)asn→oo. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 19
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 19 Introduction to Asymptotic Theory Definition 4.4
Introduction to Asymptotic Theory Definition 4.4 [Almost Sure Convergence: {Zn}converges to Z almost surely if Pr[z-Z=0=1 We denote Zn-Z0. Then almost sure convergence can be stated as P(Ac)=1. In other words,the convergent set Ac has probability one to occur. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 20
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 20 Introduction to Asymptotic Theory Definition 4.4