lect6_103_2010_Compatibility_Mode_b

lect6_103_2010_Compatibility_Mode_b - 5 Additional Topics...

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4/14/2010 1 5 Additional Topics in Basic Regression Analysis • 1) Heteroskedasticity/BLUE • 2) Measuring Model Fit • 3) Changing Units of Measurement • 4) Dummy Variables • 5) Omitted Variable Bias Heteroskedasticity/BLUE - I • Our regression model is: • Recall that u i represents variables other than X i that affect Y i . u i is a random variable, just like X i and Y i . Let’s think about its variance. • More specifically, let’s think about its conditional variance, i.e. 0 1 = i i i Y X u β + + ( 29 i i Var u X c =
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4/14/2010 2 Heteroskedasticity/BLUE – II • What exactly is the conditional variance: • It tells us the what the variance of u i is given that X i equals some number c, i.e. – Var( u i | X i = 5) is the variance of u i if X i = 5. – Var( u i | X i = 20) is the variance of u i if X i = 20. • Sometimes Var( u i | X i =c) is the same for every possible value of X i (i.e. for every possible c) Other times, Var( u i | X i =c) varies depending on the particular value of X i (i.e c). ( 29 i i Var u X c = Heteroskedasticity/BLUE - III • Defn: • 1) If Var( u i | X i =c) is the same for every value c (i.e. the variance of u i does not depend on X i ), then u i is homoskedastic . • 2) If Var( u i | X i =c) does depend on c, then u i is heteroskedastic . • Example: #WebsiteHits = β 0 + β 1 Advertising$ + u • There are two important implications of this distinction between homoskedasticity and heteroskedasticity.
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4/14/2010 3 Heteroskedasticity/BLUE - IV • If u i is homoskedastic, then: • 1) The OLS estimator β 1 is BLUE, i.e. it is the B est, L inear, conditionally U nbiased E stimator (of β 1 ). – “Best” means that the OLS estimator β 1 is the most efficient of these estimators, i.e. the one with the smallest variance. • 2) We can actually use a slightly simpler formula for Var( β 1 ) and SE( β 1 ) (again, you don’t need to remember this exact formula) ( 29 ( 29 ( 29 ( 29 2 1 1 1 1 2 1 1 ˆ 1 2 ˆ ˆ ˆ Var SE Var 1 n i i n i i u n n X X n β = = - = = - Heteroskedasticity/BLUE - V • If u i is heteroskedastic, then: • 1) The OLS estimator β 1 is not BLUE, i.e. there are potentially more efficient other estimators out there (specifically, something called weighted least squares – but we will not cover that in this class)
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lect6_103_2010_Compatibility_Mode_b - 5 Additional Topics...

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