Lecture 18 The Asymptotic variance of OLS and heteroskedasticity

Lecture 18 The Asymptotic variance of OLS and heteroskedasticity

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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
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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
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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
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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 .
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