文档介绍：C H A P T E R 1 1 HETERO S K ED ASTI CITY 1
C h a p t e r 1 1 H etero s k ed asti c ity
1 1. 1 W h i t e ’ s test f o r heterosk ed a stic ity
F o r a m od e l w i t h heteros k edastic ity ,
′
y i = X i β+ ε i ,
we hav e
E ( b ) = β an d V a r ( b ) = ( X ′ X ) − 1 X ′
X ( X ′ X ) − 1
where
2 2
= d i ag σ 1 ,
, σ n .
W e may ex p ress
n
′ 2 ′
X
X = σ i x i x i .
i = 1
S u ppose that
− 1 ′− 1 ′
n X X = n x i x i → Q ,
a fi nite and nonsing ular matrix. T hen
− 1 ′− 1 2 ′
n X
X = n σ i x i x i
can b e estimated consistently by
n
ˆ− 1 2 ′
V n = n e i x i x i
i =1
where
′
e i = y i − x i b.
( N ote that b → p β even in the presence of heteroskedasticity.)
2 2 − 1 ′
I f there is no heteroskedasticity ( σ 1 =
= σ n ) , n X
X is consistently estimated
2 − 1 ′ 2 − 1 ′′ˆˆ
either by σˆ( n X X ) where σˆ= n ( y − X b ) ( y − X b ) as V n . Thus, comparing V n and
σ 2 ( n − 1 X ′ X ) provides an indicator of heteroskedasticity. When there is no heter oskedas-
ˆ 2 − 1 ′ p ˆ 2 − 1 ′ p
ticity, V n −σ( n X X ) → 0 . O therwise, V n −σ( n X X ) 0 .
C H A P T E R 1 1 HETERO S K ED ASTI CITY 2
T h e t es t sta ti stic W hite su g gests is
W H = n D b , σˆ 2 B ˆ− 1 D b, σˆ 2 ,
w her e
2 − 1 ′ 2 2
D b, σˆ= n Ψ i e i −σˆ
′
ˆ− 1 2 2 ˆˆ
B = n e i −σˆΨ i −ΨΨ