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103lect6hand - 5 Additional Topics in Basic Regression Analysis 1 Heteroskedasticity/BLUE 2 Measuring Model Fit 3 Changing Units of Measurement 4

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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. 01 = ii i YX u β ++ ( ) Var u X c = 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. –Va r ( u i | X i = 5) is the variance of u i if X i = 5. r ( 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). ( ) Var u X c =
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Heteroskedasticity/BLUE - III •D e f n : • 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. 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. – “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) () 2 1 11 1 2 1 1 ˆ 1 2 ˆˆ ˆ Var SE Var 1 n i i n i i u n n XX n β ββ = = == 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
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This note was uploaded on 05/26/2008 for the course ECON 103 taught by Professor Sandrablack during the Spring '07 term at UCLA.

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103lect6hand - 5 Additional Topics in Basic Regression Analysis 1 Heteroskedasticity/BLUE 2 Measuring Model Fit 3 Changing Units of Measurement 4

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