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 GaussMarkov 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; expfunction 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.
 Two '13
 strachan

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