Econ 399 Chapter8b - statistics, heteroskedasticity...

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8.4 Weighted Least Squares Estimation Before the existence of heteroskedasticity-robust statistics, one needed to know the form of heteroskedasticity -Het was then corrected using WEIGHTED LEAST SQUARES (WLS) -This method is still useful today, as if heteroskedasticity can be correctly modeled, WLS becomes more efficient than OLS -ie: WLS becomes BLUE
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8.4 Known Heteroskedasticity -Assume first that the form of heteroskedasticity is known and expressed as: ) ( ) | ( 2 X h X u Var -Where h(X) is some function of the independent variables -since variance must be positive, h(X)>0 for all valid combinations of X -given a random sample, we can write: i i i i h X u Var 2 2 ) | (
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8.4 Known Het Example -Assume that sanity is a function of econometrics knowledge and other factors: u rs otherfacto econ crazy 1 0 -However, by studying econometrics two things happen: either one becomes more sane as one understands the world, or one becomes more crazy as one is pulled into a never-ending vortex of causal relationships. Therefore: i i i econ X u Var 2 ) | (
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8.4 Known Heteroskedasticity -Since h is a function of x, we know that: i i i i h X u E X X h u E 2 2 i ) | ( ) | Var(u and 0 ) | / ( -Therefore h | ) ( ] ) | / [( 2 i 2 2 2 i i i i i h h u E X h u E -So inclusion of the h term in our model can solve heteroskedasticity
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8.4 Fixing Het – And Stay Down! -We therefore have the modified equation: i i i ik k i i i i i h u h x h x h h y ... 1 1 0 -Or alternately: (8.26) ... * * * 1 1 0 * i ik k i i u x x y -Note that although our estimates for B J will change (and their standard errors become valid), their interpretation is the same as the straightforward OLS model (don’t try to bring h into your interpretation)
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8.4 Het Fixing – “I am the law” -(8.26) is linear and satisfied MLR.1 -if the original sample was random, nothing chances so MLR.2 is satisfied -If no perfect collinearity existed before, MLR.3 is still satisfied now -E(u i *|X i *)=0, so MLR.4 is satisfied -Var(u i *|X i *)=σ 2 , so MLR.5 is satisfied -if u i has a normal distribution, so does u i *, so MLR. 6 is satisfied -Thus if the original model satisfies everything but het, the new model satisfies MLR. 1 to 6
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8.4 Het Fix – Control the Het Pop -These B J * estimates are different from typical OLS estimates and are examples of GENERALIZED LEAST SQUARES (GLS) ESTIMATORS
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