Unformatted text preview: ares and interactions leads to a large
number of estimated parameters (e.g. k = 6 leads to 27 parameters
to be estimated)
Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 19 / 34 Alternative form of the White test
b2 = δ0 + δ1 y + δ2 y 2 + ν
ε
b
b save 2
Rb2
ε This regression indirectly tests the dependence of the squared residuals on
the explanatory variables, their squares, and interactions, because the
predicted value of y and its square implicitly contain all of these terms.
H0 : Var (εi jx1 , x2 ,
becomes H0
LM : , xk ) = Var (εi jxi ) = σ2 δ1 = δ2 = 0 2
= nRb2 s χ2
2
ε Example: Heteroscedasticity in (log) housing price equations
2
Rb2 = 0.0392 LM = 88 (0.0392) t 3.45 p
ε
Lecture 7 (heteroscedasticity) EMET2007/6007 value = 0.178 24 th April 2013 20 / 34 What can I do about it (I)?
Adjust the data to remove it  Weighted least squares (WLS)
estimation
To use this form of WLS, we …rst assume that the form of the
heteroscedasticity is known up to a multiplicative constant
Var (εi jx1 , x2 , , xk ) = σ2 h (xi ) , h (xi ) = hi > 0 That is, The functional form of the heteroscedasticity, hi , is known
The transformation we apply will result in an error with a constant
variance, σ2 Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 21 / 34 With this function we transform the variables as follows.
yi
where εi
yi
p
hi
yi where εi = β0 + β1 x1,i + β2 x2,i +
+ βk xk ,i + εi
2
s N 0, σ h (xi )
1
x1,i
x2,i
xk
εi
= β0 p + β1 p + β2 p +
+ β k p ,i + p
hi
hi
hi
hi
hi
= β0 x0,i + β1 x1,i + β2 x2,i +
+ βk xk ,i + εi
2
s N 0, σ The Transformed model (Note that this regression model has no intercept)
yi = β0 x0,i + β1 x1,i + β2 x2,i + + βk xk ,i + εi has the new regressand yi and the regressors xj ,i
Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 22 / 34 Example: Savings and income
savi = β0 + β1 inci + εi
Var (εi jinci ) = σ2 inci
p
p
) hi = inci ) hi = inci Transform (Note that this regression mod...
View
Full Document
 Two '13
 strachan
 Normal Distribution, Regression Analysis, Variance, Heteroscedasticity

Click to edit the document details