Introduction to Asymptotic Theory Example 4.1 Suppose {is IID(u,2),and Zn=n-1Z.Then 7n9y4. Solution:Because E(Zn)=u,we have 以原-w以-（空4-京店 1 var(Zt) t=1 n -→0asn→o. It follows that E2,-P2=g2→0asn→c m ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 6
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 6 Introduction to Asymptotic Theory Example 4.1
Introduction to Asymptotic Theory Definition 4.2 [Convergence in Probability]Zn converges to Z in probability if for any given constant e>0, Pr[Zm-Z>e→0asn→oor Pr[lZn-Z≤d→1asn→oo. For convergence in probability,we can write Zn-Z0 or Zn -Z=op(1), where op(1)means that Zn-Z vanishes to zero in probability.When Z=b is a constant,we can write Zn2b and b=plim Zn is called the probability limit of Zn. Convergence in probability is also called weak convergence or convergence with probability approaching one.When Zn2,the probability that the differencen-exceeds any given small constant e is rather small for all n sufficiently large.In other words,Zn will be arbitrarily close to Z with very high probability when the sample size n is sufficiently large. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 7
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 7 Introduction to Asymptotic Theory Definition 4.2
Introduction to Asymptotic Theory Definition 4.2 [Convergence in Probability]Zn converges to Z in probability if for any given constant e>0, Pr[Zm-Z>e→0asn→oor Pr[lZm-Z≤e→1asn→o. To gain more intuition of the convergence in probability,we define the event An(e)={w∈2：|Zn(w)-Z(w)川>e}, where w is a basic outcome in sample space n.Then convergence in prob- ability says that the probability of event An(e)may be nonzero for any finite n,but such a probability will eventually vanish to zero as n->oo. e can be viewed as a pre-specified tolerance level such that one can view that the difference between Zn and Zn is small when Zn-Z<e,and the difference is large when Zn-ZI>e.Thus,when ZnBZ,,it becomes less and less likely that the difference Zn-is larger than a prespecified constant e>0. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 8
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 8 Introduction to Asymptotic Theory Definition 4.2
Introduction to Asymptotic Theory Lemma 4.1 [Weak Law of Large Numbers (WLLN)for an IID Random Sample]: Suppose (Z}is IID(u,2),and define Zn =nZ,n =1,2..... Then ZmB4asn→o. Proof:For any given constant e>0,we have by Chebyshev's inequality Pr1Z-川>)≤E2-四2 E2 03 ne2→0asno. Hence, Zn B u as n→o. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 9
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 9 Introduction to Asymptotic Theory Lemma 4.1
Introduction to Asymptotic Theory Example 4.2 Buy and Hold Trading Strategy and Economic Interpretation of WLLN In finance,there is a popular trading strategy called buy-and-hold trading strategy.An investor buys a stock at some day and then hold it for a long time period before he sells it out.This is called a buy-and-hold trading strategy. How is the average return of this trading strategy? Suppose Zt is the return of the stock on period t,and the returns over different time periods are IID(u,o2).Also assume the investor holds the stock for a total of n period.Then the average return over each time period is the sample mean t=1 When the number n of holding periods is large,we have =E(Z) as n->oo.Thus,the average return of the buy-and-hold trading strategy is approximately equal to u when n is sufficiently large. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 10
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 10 Introduction to Asymptotic Theory Example 4.2 Buy and Hold Trading Strategy and Economic Interpretation of WLLN