Econ 399 Chapter8b -...

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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 σ = -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 β -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 ) | ( σ =
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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 ) | / ( σ = = = -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 + + + + = β ... 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 
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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 
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This note was uploaded on 03/14/2009 for the course ECON ECON 399 taught by Professor Priemaza during the Spring '09 term at University of Alberta.

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Econ 399 Chapter8b -...

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