i 0 p the transformed model is homoscedastic 2 e 2

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online