Homoscedasticity
Regression Analysis
Yusep Suparman
“Tugas hidup kita adalah
memberi manfaat. Mulailah dari
yang paling dekat yaitu diri
sendiri”
—Yusep Suparman
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Let’s
start!
Homoscedastic means
that the errors have the
same or constant
variances.
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Formal formulation
Previously we assumed that
⏷
⏷
yi   0  1 x1i   2 x2i     k xki   i
and
cov   i , x ji   0, i  1, 2, , n; j  1, 2, , k .
Now we are assuming that
⏷
var   i    2 ; i  1, 2, , n.
the homoscedasticity assumption.
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Violation of the assumption
⏷ It is called heteroscedaticity.
var   i    i2 ; i  1, 2, , n
at least one epsilon has a different variance
⏷ The effect on the OLS estimates
var  b j  for j  1, 2, , k
are not the minimum as well as
inefficient → incorrect inferences
see Verbeek (2002) p. 74-75 for the
proof
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What are the
causes?
Misspecified
model
Always reconsider your
model specification if you
detect heteroscedasticity!
Detecting Homoscedasticity
⏷ Scatter plot residual against Xs
⏷ Scatter plot absolute residual against Xs
Narrowing or
widening patterns
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Detecting Homoscedasticity (Breucsh-Pagan test)
⏷ Based on:  i   h  z i α 
e
2
⏷ Hypothesis
H 0 : α  0, homoscedastic
H 0 : α  0, heteroscedastic
⏷ Statistical test
 2  nRe2 ~  2j
⏷ Test criteria
reject H 0 at significance level  ,
if  2  12 , j
Re2 is obtained from the auxilary regression
ei2   0  z i α i  i
ei : residual from the primary regression
z i : vector of observations of size j
explaining error variances
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Detecting Homoscedasticity (Breucsh-Pagan test)
Procedure
⏷ Compute residual from the
primary regression model .
⏷ Compute residual square.
⏷ Determine explanatory
variables in the auxiliary.
regression (often z i  xi )
⏷ Compute the R-square for
the auxiliary regression.
⏷ Compute di chi-square
statistic.
⏷ Determine a significant
level and compute the test
critical value.
⏷ Conclude.
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Detecting Homoscedasticity (White test)
⏷ Hypothesis
H 0 : α  0, homoscedastic
H 0 : α  0, heteroscedastic
⏷ Statistical test
 2  nRe2 ~  2j
⏷ Test criteria
reject H 0 at significance level  ,
if  2  12 , j
Re2 is obtained from the auxilary regression
ei2   0  z i α i  i
ei : residual from the primary regression
z i : vector of observations of size j
explaining error variances. it include
first moment and second moment as well as
cross-product of the original regressor
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