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Unformatted text preview: Chapter 8 Heteroskedasticity Prepared by Vera Tabakova, East Carolina University 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 82 Principles of Econometrics, 3rd Edition 8.1 The Nature of Heteroskedasticity Slide 83 Principles of Econometrics, 3rd Edition (8.1) (8.2) (8.3) 1 2 ( ) E y x = + 1 2 ( ) i i i i i e y E y y x = =  1 2 i i i y x e = + + 8.1 The Nature of Heteroskedasticity Slide 84 Principles of Econometrics, 3rd Edition 8.1 The Nature of Heteroskedasticity Slide 85 Principles of Econometrics, 3rd Edition (8.4) 2 ( ) var( ) cov( , ) i i i j E e e e e = = = var( ) var( ) ( ) i i i y e h x = = 83.42 10.21 i i y x = + 83.42 10.21 i i i e y x = 8.1 The Nature of Heteroskedasticity Slide 86 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 87 Principles of Econometrics, 3rd Edition 8.2 Using the Least Squares Estimator Slide 88 Principles of Econometrics, 3rd Edition (8.5) (8.6) (8.7) 2 1 2 var( ) i i i i y x e e = + + = 2 2 2 1 var( ) ( ) N i i b x x = = 2 1 2 var( ) i i i i i y x e e = + + = 8.2 Using the Least Squares Estimator Slide 89 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 = = =  = =  2 2 2 2 1 2 2 1 2 1 ( ) var( ) ( ) N i i N i i i N i i i x x e b w e x x = = =  = =  8.2 Using the Least Squares Estimator Slide 810 Principles of Econometrics, 3rd Edition 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 811 Principles of Econometrics, 3rd Edition (8.10) 1 2 2 ( ) var( ) cov( , ) i i i i i i i j y x e E e e e e = + + = = = 8.3.1 Transforming the Model Slide 812 Principles of Econometrics, 3rd Edition (8.11) (8.12) (8.13) ( 29 2 2 var i i i e x = = 1 2 1 i i i i i i i y x e x x x x = + + 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 813 Principles of Econometrics, 3rd Edition (8.14) (8.15) 1 1 2 2 i i i i y x x e = + + 2 2 1 1 var( ) var var( ) i i i i i i i e e e x x x x = = = = 8.3.1 Transforming the Model8....
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