Lecture 18 The Asymptotic variance of OLS and heteroskedasticity

# Lecture 18 The Asymptotic variance of OLS and heteroskedasticity

This preview shows pages 1–5. Sign up to view the full content.

Economics 326 Methods of Empirical Research in Economics Lecture 18: The asymptotic variance of OLS and heteroskedasticity Vadim Marmer University of British Columbia March 24, 2009

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Asymptotic normality I In the previous lecture, we showed that when the data are iid and the regressors are exogenous: Y i = β 0 + β 1 X i + U i , EU i = E ( X i U i ) = 0 , the OLS estimator of β 1 is asymptotically normal: p n ° ˆ β 1 , n ° β 1 ± ! d N ( 0 , V ) , V = E ² ( X i ° EX i ) 2 U 2 i ³ ( Var ( X i )) 2 . I For the purpose of hypothesis testing, we need to obtain a consistent estimator of the asymptotic variance V : ˆ V n ! p V . 1/14
Homoskedastic errors I Let°s assume that the errors are homoskedastic: E ° U 2 i j X i ± = σ 2 for all X i °s. I In this case, the asymptotic variance can be simpli±ed using the Law of Iterated Expectation: E ² ( X i ° EX i ) 2 U 2 i ³ = EE h ( X i ° EX i ) 2 U 2 i j X i i = E ² ( X i ° EX i ) 2 E ´ U 2 i j X i µ ³ = E ² ( X i ° EX i ) 2 σ 2 ³ = σ 2 E ( X i ° EX i ) 2 = σ 2 Var ( X i ) . 2/14

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Homoskedastic errors I Thus, when the errors are homoskedastic with EU 2 i = σ 2 , V = E ² ( X i ° EX i ) 2 U 2 i ³ ( Var ( X i )) 2 = σ 2 Var ( X i ) ( Var ( X i )) 2 = σ 2 Var ( X i ) . I Let ˆ U i = Y i ° ˆ β 0 , n ° ˆ β 1 , n X i , where ˆ β 0 , n and ˆ β 1 , n are the OLS estimators of β 0 and β 1 . I A consistent estimator for the asymptotic variance can be constructed by using the Method of Moments . ˆ σ 2 n = 1 n n i = 1 ˆ U 2 i , d Var ( X i ) = 1 n n i = 1 ( X i ° ¯ X n ) 2 , and ˆ V n = ˆ σ 2 n 1 n n i = 1 ( X i ° ¯ X n ) 2 .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern