Introduction to Asymptotic Theory Lemma 4.6 [Continuity]: (1)Suppose an Ba and bn b,and g()and h()are continuous functions. Then g(an)+h(on)g(a)+h(b),and g(an)h(bn）g(a)h(b). (2)Similar results hold for almost sure convergence. Proof:Left as an exercise. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 26
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 26 Introduction to Asymptotic Theory Lemma 4.6
Introduction to Asymptotic Theory Concept of convergence in distribution:a sequence fm converges in distribution to random variable Z if the CDF F (z)of random variable Zn converges to the CDF F(z)of random variable Z at all continuity points (where F(z)is continuous)when n->oo. ● Convergence in distribution implies that one can obtain an asymptotic approximation to the exact distribution of n that depends on the posi- tive integer n and the underlying population distribution.In practice,the distribution F()of Zn is often rather complicated and even unknown for finite n.However,if we know the unknown distribution Fn()converges to a known distribution F()as n->oo is sufficiently large,we can use F()to approximate Fn()in finite samples,and the resulting approxima- tion errors will be arbitrarily small for n sufficiently large.This provides convenient statistical inferences in practice. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 27
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 27 Introduction to Asymptotic Theory
Introduction to Asymptotic Theory We emphasize that convergence in mean squares,convergence in proba- bility and almost sure convergence all measure the closeness between the random variable Zn and the random variable Z as n->oo.This differs from the concept of convergence in distribution,which is defined in terms of the closeness of the CDF Fn(z)of Zt to the CDF F(z)of Z,not between the closeness of the random variable Zn to the random variable 2. For convergence in mean squares,convergence in probability and almost sure convergence,Zn converges to Z if and only if convergence of Zn to Z occurs element by element (that is,each element of n converges to the corresponding element of Z). For the convergence in distribution of Zn to Z,however,element by el- ement convergence does not imply convergence in distribution of Zn to 2,because element-wise convergence in distribution ignores the relation- ships among the components of Zn.Nevertheless,Zn does imply element by element convergence in distribution.That is,convergence in joint distribution implies convergence in marginal distribution. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 28
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 28 Introduction to Asymptotic Theory
Introduction to Asymptotic Theory [CLT for an IID Random Sample]: Suppose {Z}is IID(u,2),and Zn =n-1Z.Then as n Zm-E(⑦n）_Zm-4 Vvar(Zn) Vo2/n =n(Z。-四 0 4N(0,1). ●Proof:Put Yi= Zt-业 and Yn=n-l∑t-lY.Then √(Zn- ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 29
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 29 Introduction to Asymptotic Theory [CLT for an IID Random Sample]:
Introduction to Asymptotic Theory [CLT for an IID Random Sample]: The characteristic function of ny 中n(u)=E[exp(iunYn)小， i=W-1 =(君 =Ⅱ()] by independence -r()] by identical distribution. =0+@后+0o+ =(1- +o(1) →exp 2 asn-→o, where the third equality follows from independence,the fourth equality follows from identical distribution,the fifth equality follows from the Taylor series ex- pansion,andp(0)=1,'(0)=0,Φ"(0)=-1. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 30
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 30 Introduction to Asymptotic Theory [CLT for an IID Random Sample]: