中国科学院数学与系统科学研究院 Academy of Mathematics and Systems Science Chinese Academy of Sciences Linear Regression Models with IID Observations Professor Yongmiao Hong March 2021
Professor Yongmiao Hong March 2021 Linear Regression Models with IID Observations
CONTENTS 4.1 Introduction to Asymptotic Theory 4.2 Framework and Assumptions 4.3 Consistency of OLS 4.4 Asymptotic Normality of OLS 4.5 Asymptotic Variance Estimation 4.6 Hypothesis Testing 4.7 Testing for Conditional Homoskedasticity 4.8 Conclusion ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 2
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 2 4.1 Introduction to Asymptotic Theory 4.2 Framework and Assumptions 4.3 Consistency of OLS 4.4 Asymptotic Normality of OLS 4.5 Asymptotic Variance Estimation 4.6 Hypothesis Testing 4.7 Testing for Conditional Homoskedasticity 4.8 Conclusion CONTENTS
Introduction to Asymptotic Theory The assumptions of classical linear regression models are rather strong. For example,the conditional normality condition on e is crucial to obtain the finite sample distributions of the OLS,GLS and related test statistics, but it has been documented that most economic and financial data have heavy tails. An interesting question is whether the estimators and tests which are based on the same principles as before still make sense in this more general setting. In particular,what happens to the OLS estimator,the t-and F-tests if any of the following assumptions fails? strict exogeneity E(etX)=0; normality (eX~N(0,021); conditional homoskedasticity (var(etX)=g2); auto uncorrelatedness (cov(et,esX)=0 for ts). ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 3
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 3 Introduction to Asymptotic Theory
Introduction to Asymptotic Theory We can pose the question how estimators and test statistics would behave when the number of observations increases without limit.This is called asymptotic analysis.For more systematic introduction of asymptotic the- ory,see,for example,White (1994,2001)and Davidson (1994). To assess how close the OLS estimator B is to the true parameter value Bo and to derive its asymptotic distribution (after suitable normalization),we briefly review some important convergence concepts and limit theorems. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 4
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 4 Introduction to Asymptotic Theory
Introduction to Asymptotic Theory Definition 4.1 [Convergence in Mean Squares (or in Quadratic Mean)]A sequence of random variables/vectors/matrices Zn,n=1,2,...,is said to converge to Z in mean squares as n->oo if ElZm-Z2→0asn→o, where is the sum of the absolute value of each component in Zn -Z. When Zn is a vector or a matrix,convergence can be understood as con- vergence in each element of Zn.When Zn-Z is a l x m matrix,where l and m are fixed positive integers,we can also define the squared norm as Z-Z2=∑∑Zm-☑ t=1s=1 Note that 2n converges to 2 in mean squares if and only if each component of Zn converges to the corresponding component of Z in mean squares. ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31,2021 5
ADVANCED ECONOMETRICS Linear Regression Models with IID Observations March 31, 2021 5 Introduction to Asymptotic Theory Definition 4.1