WLS_Heteroskedasticity

# WLS_Heteroskedasticity - OLS Under Heteroskedasticity...

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OLS Under Heteroskedasticity Testing for Heteroskedasticity Heteroskedasticity and Weighted Least Squares Walter Sosa-Escudero Econ 507. Econometric Analysis. Spring 2009 April 14, 2009 Walter Sosa-Escudero Heteroskedasticity and Weighted Least Squares

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OLS Under Heteroskedasticity Testing for Heteroskedasticity The Classical Linear Model: 1 Linearity: Y = + u . 2 Strict exogeneity: E ( u ) = 0 3 No Multicollinearity: ρ ( X ) = K . 4 No heteroskedasticity/ serial correlation: V ( u ) = σ 2 I n . Gauss/Markov Theorem: ˆ β = ( X 0 X ) - 1 X 0 Y is best linear unbiased. ˆ V ( ˆ β ) = S 2 ( X 0 X ) - 1 is an unbiased estimate of V ( ˆ β ) = σ 2 ( X 0 X ) - 1 . Walter Sosa-Escudero Heteroskedasticity and Weighted Least Squares
OLS Under Heteroskedasticity Testing for Heteroskedasticity What happens if we drop the homoskedasticity assumption? ˆ β (the OLS estimator) is still linear and unbiased (Why?). Though linear and unbiased, ˆ β is not the minimum variance estimate (ineﬃcient). ˆ V ( ˆ β ) = S 2 ( X 0 X ) - 1 is biased . This makes standard ‘t’ and ‘F’ tests invalid. Walter Sosa-Escudero Heteroskedasticity and Weighted Least Squares

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OLS Under Heteroskedasticity Testing for Heteroskedasticity Intuition The presence of heteroskedastic errors should not alter the ‘central’ position of the OLS line (unbiasedness). OLS weigths all observations equally, but in this case it makes more sense to pay more attention to observations where the variance is smaller. Walter Sosa-Escudero Heteroskedasticity and Weighted Least Squares
OLS Under Heteroskedasticity Testing for Heteroskedasticity The plan: what to do with heteroskedasticity. 1 Before abandoning OLS we will see how to test for heteroskedasticity. 2 Strategy 1: Propose another more eﬃcient and unbiased estimator for β ( weighted least squares (WLS) ) and a suitable estimator for its variance. 3 Strategy 2: Keep using OLS (it is still unbiased, though ineﬃcient), but ﬁnd a replacement for its variance (the old one is biased under heteroskedasticity). Walter Sosa-Escudero Heteroskedasticity and Weighted Least Squares

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OLS Under Heteroskedasticity Testing for Heteroskedasticity Testing for heteroscedasticity a) The White test H 0 : no heteroscedasticity, H A : there is heterocedasticity of some form. Consider a simple case with K = 3 : Y i = β 1 + β 2 X 2 i + β 3 X 3 i + u i 1 ,...,n Walter Sosa-Escudero Heteroskedasticity and Weighted Least Squares
OLS Under Heteroskedasticity Testing for Heteroskedasticity Steps to implement the test: 1 Estimate by OLS, save squared residuals in e 2 .

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## This note was uploaded on 02/15/2012 for the course GEO 4167 taught by Professor Staff during the Spring '12 term at University of Florida.

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WLS_Heteroskedasticity - OLS Under Heteroskedasticity...

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