Econometrics (EM2008/EM2Q05) Heteroskedasticity: Irene Mammi
Econometrics (EM2008/EM2Q05) Heteroskedasticity: Irene Mammi
Econometrics (EM2008/EM2Q05) Heteroskedasticity: Irene Mammi
Lecture 5
Heteroskedasticity
Irene Mammi
irene.mammi@unive.it
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outline
I heteroskedasticity
I properties of OLS estimators
I tests for heteroskedasticity
I estimation under heteroskedasticity
I References:
I Johnston, J. and J. DiNardo (1997), Econometrics Methods, 4th
Edition, McGraw-Hill, New York, Chapter 6.
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introduction
y = X β + u with u ∼ N (0, σ2 Ω)
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introduction (cont.)
I when heteroskedasticity alone occurs, the variance for the error vector
is 2
···
σ1 0 0
0 σ22 ··· 0
var(u ) = E(uu 0 ) = . .. = V
.. ..
.. . . .
0 0 · · · σn2
I there are n + k unknown parameters; n unknown variances; and k
elements in the β vector
I additional assumptions are needed
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properties of OLS estimators
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properties of OLS estimators (cont.)
which may also be expressed as
−1 −1
σ2
1 0 1 0 1 0
var(b ) = (X X ) (X ΩX ) (X X )
n n n n
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tests for heteroskedasticity
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tests for heteroskedasticity (cont.)
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tests for heteroskedasticity (cont.)
I this is an LM test
I assume
yt = x t0 β + ut i = 1, 2, . . . , n
where xt0 = 1 x2t x3t x2t 2 2
x3t x2t x3t , and that
heteroskedasticity takes the form
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tests for heteroskedasticity (cont.)
I the null hypothesis of homoskedasticity is then
H0 : α2 = α3 = · · · = αp = 0
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tests for heteroskedasticity (cont.)
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estimation under heteroskedasticity
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estimation under heteroskedasticity (cont.)
σ12
··· 0 z1 ··· 0
.. .. .. = α .. .. .
V = . . . 1 . . .. = α1 Ω
0 · · · σn 2 0 · · · zn
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estimation under heteroskedasticity (cont.)
I we have
.. .. 1
···
. . z1 0 y1 n
. .. .. = 1
X 0 Ω −1 y = ..
∑ zt x t yt
x
1 ··· xn .. . . .
.. .. 1 t =1
. . 0 ··· zn yn
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