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# ch08 - Heteroskedasticity Click to edit Master subtitle...

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Chapter 8 Heteroskedasticity Prepared by Vera Tabakova, East Carolina University

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Chapter 8: Heteroskedasticity 8.1 The Nature of Heteroskedasticity 8.2 Using the Least Squares Estimator 8.3 The Generalized Least Squares Estimator 8.4 Detecting Heteroskedasticity Slide 8-2 Principles of Econometrics, 3rd Edition
8.1 The Nature of Heteroskedasticity Slide 8-3 Principles of Econometrics, 3rd Edition (8.1) (8.2) (8.3) 1 2 ( ) E y x = β +β 1 2 ( ) i i i i i e y E y y x = - = -β -β 1 2 i i i y x e = β +β +

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8.1 The Nature of Heteroskedasticity Slide 8-4 Principles of Econometrics, 3rd Edition
8.1 The Nature of Heteroskedasticity Slide 8-5 Principles of Econometrics, 3rd Edition (8.4) 2 ( ) 0 var( ) cov( , ) 0 i i i j E e e e e = = σ = var( ) var( ) ( ) i i i y e h x = = ˆ 83.42 10.21 i i y x = + ˆ 83.42 10.21 i i i e y x = - -

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8.1 The Nature of Heteroskedasticity Slide 8-6 Principles of Econometrics, 3rd Edition
8.2 Using the Least Squares Estimator The existence of heteroskedasticity implies: The least squares estimator is still a linear and unbiased estimator, but it is no longer best. There is another estimator with a smaller variance. The standard errors usually computed for the least squares estimator are incorrect. Confidence intervals and hypothesis tests that use these standard errors may be misleading. Slide 8-7 Principles of Econometrics, 3rd Edition

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8.2 Using the Least Squares Estimator Slide 8-8 Principles of Econometrics, 3rd Edition (8.5) (8.6) (8.7) 2 1 2 var( ) i i i i y x e e = β +β + = σ 2 2 2 1 var( ) ( ) N i i b x x = σ = - 2 1 2 var( ) i i i i i y x e e = β +β + = σ
8.2 Using the Least Squares Estimator Slide 8-9 Principles of Econometrics, 3rd Edition (8.8) (8.9) 2 2 2 2 1 2 2 1 2 1 ( ) var( ) ( ) N i i N i i i N i i i x x b w x x = = = - σ = σ = - · 2 2 2 2 1 2 2 1 2 1 ˆ ( ) ˆ var( ) ( ) N i i N i i i N i i i x x e b w e x x = = = - = = -

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8.2 Using the Least Squares Estimator Slide 8-10 Principles of Econometrics, 3rd Edition ˆ 83.42 10.21 (27.46) (1.81) (White se) (43.41) (2.09) (incorrect se) i i y x = + 2 2 2 2 White: se( ) 10.21 2.024 1.81 [6.55, 13.87] Incorrect: se( ) 10.21 2.024 2.09 [5.97, 14.45] c c b t b b t b ± = ± × = ± = ± × =
8.3 The Generalized Least Squares Estimator Slide 8-11 Principles of Econometrics, 3rd Edition (8.10) 1 2 2 ( ) 0 var( ) cov( , ) 0 i i i i i i i j y x e E e e e e = β +β + = = σ =

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8.3.1 Transforming the Model Slide 8-12 Principles of Econometrics, 3rd Edition (8.11) (8.12) (8.13) ( 29 2 2 var i i i e x = σ = σ 1 2 1 i i i i i i i y x e x x x x = β + ÷ ÷ ÷ ÷ 1 2 1 i i i i i i i i i i i i y x e y x x x e x x x x = = = = =
8.3.1 Transforming the Model Slide 8-13 Principles of Econometrics, 3rd Edition (8.14) (8.15) 1 1 2 2 i i i i y x x e = β + 2 2 1 1 var( ) var var( ) i i i i i i i e e e x x x x = = = σ = σ ÷ ÷

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8.3.1 Transforming the Model To obtain the best linear unbiased estimator for a model with heteroskedasticity of the type specified in equation (8.11): 1.
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