EMET2007 Lecture 7

# i 0 p the transformed model is homoscedastic 2 e 2

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Unformatted text preview: el has no intercept) sav pi inci savi 1 inci εi + β1 p +p inci inci inci = β0 x0,i + β1 inci + εi = β0 p The transformed model is homoscedastic ! 2 E ε2 jxi εi i p E εi 2 jxi = E j xi = hi inci = σ2 hi = σ2 hi If the other Gauss-Markov assumptions hold as well, OLS applied to the transformed model is the best linear unbiased estimator! Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 23 / 34 OLS in the transformed model is weighted least squares (WLS) n min ∑ i =1 n = min ∑ yi i =1 n , min ∑ i =1 1 bp β0 hi y pi hi yi b x0,i β0 b β0 x1 b p ,i β1 hi b x1,i β1 b x1,i β1 hi xk b p ,i βk hi b xk ,i βk b xk ,i βk 2 2 2 This implies that observations with a large variance get a smaller weight in the optimization problem Why is WLS more e¢ cient than OLS in the original model? Observations with a large variance are less informative than observations with small variance and therefore should get less weight WLS is a special case of generalized least squares (GLS) Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 24 / 34 Unknown heteroscedasticity function (Feasible GLS) Assumed general form of heteroscedasticity; exp-function is used to ensure positivity Var (εjx ) = σ2 exp fδ0 + δ1 x1 + δ2 x2 + + δk xk g = σ2 h (xi ) Multiplicative error ν (assumption: independent of the explanatory variables) ε2 log b2 ε = σ2 exp fδ0 + δ1 x1 + δ2 x2 + + δk xk g ν2 ) log ε2 = α0 + δ1 x1 + δ2 x2 + + δk xk + e = b0 + b1 x1 + b2 x2 + α δ δ + bk xk + b δ e Use inverse values of the estimated heteroscedas...
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## This note was uploaded on 06/15/2013 for the course EMET 2007 taught by Professor Strachan during the Two '13 term at Australian National University.

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