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Unformatted text preview: his test is much more general.
It looks for evidence of an association between variance of disturbance term
and regressors.
H0 : Homoscedasticity (σui =σu )
H1 : Heteroscedasticity (σui changes with X’s)
Mechanics:
1 Run OLS and save residuals. 2 Regress squared residual on explanatory variables, their squares and
crossproducts → get R 2 from this second regression. 3 4 Under H0 , nR 2 ∼ χ2k −1) where k is the number of parameters estimated
(
in above step 2 regression.
Intuition for White Test:
Squared residual proxies for disturbance variance.
2
If σui related to regressors, R 2 will tend to be large.
Test statistic will tend to be large and lead to rejecting H0 . Introduction Heteroscedasticity Past Exam Practice Question Moving Forward: Model B Detection White Test Important point:
White test much more general.
Generality comes with the cost of:
1
2
3 Large sample test, so credibility questionable when sample small.
Test tends to have low power (fail to reject H0 wrongly)
Signiﬁcant losses in degrees of freedom when many explanatory variables in
original regression. (Due to including cross products in 2nd regression) Introduction Heteroscedasticity Past Exam Practice Question Moving Forward: Model B Remedy Weighted Least Squares Intuition: weight observations by how reliably they convey information
about the true line.
Assumption: we have Zi where σui = λZi
2
True Model: Yi = β1 + β2 Xi + ui , with Var(ui )= σui = λ2 Zi2
Rewrite: 1
Xi
ui
Yi
= β1 + β2 +
Zi
Zi
Zi
Zi
Yi = β1 Hi + β2 Xi + vi
Transformed model:
u
Homoscedastic: Var(vi )=Var( Zii )= Zi2 λ2
Zi2 OLS on transformed model is BLUE.
Special case: Zi = Xi = λ2 (1)
(2) Introduction Heteroscedasticity Past Exam Practice Question Moving Forward: Model B Remedy Nonlinear Model True Model: Y = β1 X β2 u
You (wrongly) use regression model: Y = c1 + c2 X + v
Here disturbance term equals: v = β1 X β2 u − c1 − c2 X
Naturally Var(v ) varies with size of X .
(Apparent) Heteroscedasticity due to deeper model misspeciﬁcation.
→ If you can identify deeper model misspeciﬁcation, then ﬁx problem at root.
→ Here estimate model using logarithmic speciﬁcation. Introduction Heteroscedasticity Past Exam Practice Question Moving Forward: Model B Remedy White’s Heteroscedasticityconsistent s.e.’s White (1980) shows how to get heteroscedasticityconsistent s.e.’s based on
residuals.
Pros:
Robust: don’t need to specify source of heteroscedasticity
Easy: press button in STATA
Cons:
Large sample result: but how well does it perform in small samples?
Coefﬁcient estimators still inefﬁcient! Introduction Heteroscedasticity Past Exam Practice Question Moving Forward: Model B 2007 Question 4 Setup of the Question Cost Function: EXP = β1 + β2 N + β3 NSQ + u RSS of the same regression using 19 smallest (N ) schools is 8.0 million.
On the other hand RSS of the same regression using 19 largest (N )
schools is 64.0 million. Introduction Heteroscedasticity Past Exam Practice Question Moving Forward: Model B 2007 Question 4 Setup of the Question, continued
Next deﬁne EXPN = EXP /N , and NREC = 1/N .
EXP
β1 + β2 N + β3 NSQ + u
=
N
N
u
1
EXPN = β1 + β2 + β3 N +
N
N
EX...
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This document was uploaded on 03/12/2014 for the course ECON 202 at University of London University of London International Programmes (Distance Learning).
 Spring '13
 ChristopherDougherty
 Econometrics

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