Transcript HETEROSCEDASTICITY

HETEROSCEDASTICITY
The assumption of equal variance
Var(ui) = σ2, for all i, is called
homoscedasticity, which means
“equal scatter” (of the error terms
ui around their mean 0)
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 Equivalently, this means that
the dispersion of the observed
values of Y around the
regression line is the same
across all observations
If the above assumption of
homoscedasticity does not hold,
we have heteroscedasticity
(unequal scatter)
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Consequences of ignoring
heteroscedasticity during the OLS
procedure
 The estimates and forecasts based
on them will still be unbiased and
consistent
 However, the OLS estimates are no
longer the best (B in BLUE) and
thus will be inefficient. Forecasts
will also be inefficient
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 The estimated variances and
covariances of the regression
coefficients will be biased and
inconsistent, and hence the t- and
F-tests will be invalid
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Testing for heteroscedasticity
1. Before any formal tests,
visually examine the model’s
residuals ûi
 Graph the ûi or ûi2 separately
against each explanatory
variable Xj, or against Ŷi, the
fitted values of the dependent
variable
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Residuals
Example of heteroscedasticity
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0
-1 0
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Series1
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Income (X) ordered by size
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2. The Goldfeld-Quandt test
Step 1. Arrange the data from small
to large values of the indp variable Xj
Step 2. Run two separate regressions,
one for small values of Xj and one for
large values of Xj, omitting d middle
observations (app. 20%), and record
the residual sum of squares RSS for
each regression: RSS1 for small
values of Xj and RSS2 for large Xj’s.
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Step 3. Calculate the ratio
F = RSS2/RSS1, which has an F
distribution with
d.f. = [n – d – 2(k+1)]/2 both in the
numerator and the denominator,
where n is the total # of
observations, d is the # of omitted
observations, and k is the # of
explanatory variables.
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• Step4. Reject H0: All the variances
σi2 are equal (i.e., homoscedastic) if
F > Fcr, where Fcr is found in the
table of the F distribution for
[n-d-2(k+1)]/2 d.f. and for a
predetermined level of significance
α, typically 5%.
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 Drawbacks of the the
Goldfeld-Quandt test
 It cannot accommodate
situations where several
variables jointly cause
heteroscedasticity
The middle d
observations are lost
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3. Lagrange Multiplier (LM) tests
(for large n>30)
 The Breusch-Pagan test
Step 1. Run the regression of ûi2
on all the explanatory variables. In
our example (CN p. 37), there is
only one explanatory variable, X1,
therefore the model for the OLS
estimation has the form:
ûi2 = α0 + α1X1i + vi
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Step 2. Keep the R2 from this
regression. Let’s call it Rû22
Calculate either
 (a) F = (Rû22/k)/[(1-Rû22)/(n-(k+1)],
where k is the # of explanatory
variables; the F statistic has an F
distribution with d.f. = [k, n-(k+1)]
Reject H0: All the variances σi2 are
equal (i.e., homoscedastic) if F >Fcr
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or
 (b) LM = n Rû22, where LM is
called the Lagrangian Multiplier
(LM) statistic and has an
asymptotic chi-square (χ2)
distribution with d.f. = k
Reject H0: All the variances σi2 are
equal (i.e., homoscedastic)
if LM> χcr2
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 Drawbacks of the Breusch-
Pagan test
 It has been shown to be
sensitive to any violation of the
normality assumption
Three other popular LM tests: the
Glejser test; the Harvey-Godfrey
test, and the Park test, are also
sensitive to such violations (won’t
be covered in this course)
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One LM test, the White
test, does not depend
on the normality
assumption; therefore
it is recommended
over all the other tests
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 The White test
Step 1.The test is based on the regr.
of û2 on all the explanatory variables (Xj), their squares (Xj2), and
all their cross products. E.g., when
the model contains k = 2 explanat.
variables, the test is based on an
estim. of the model: û2 =β0+ β1X1
+β2X2+β3X12+β4X22 + β5X1X2 + v
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Step 2. Compute the statistic
χ2 = nRû22, where n is the sample size
and Rû22 is the unadjusted R-squared
from the OLS regression in Step 1. The
statistic χ2 = nRû22, has an asymptotic
chi-square (χ2) distrib. with d.f. = k,
where k is the # of ALL explanatory
variables in the AUXILIARY model.
Reject H0: All the variances σi2 are
equal (i.e., homoscedastic) if χ2 > χcr2
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Estimation Procedures when H0 is
rejected
• 1. Heteroscedasticity with a known
proportional factor
 If it can be assumed that the error variance is
proportional to the square of the indep. variable
Xj2, we can correct for heteroscedasticity by
dividing every term of the regression by X1i and
then reestimating the model using the transformed
variables. In the two-variable case, we will have to
reestimate the following model (CN, p. 39):
Yi/X1i = β0/X1i + β1 + ui/X1i
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• 2. Heteroscedasticity consistent
covariance matrix (HCCM)
 As we know, the usual OLS inference is
faulty in the presence of heteroscedasticity because in this case the
estimators of variances Var(bj) are
biased. Therefore, new ways have been
developed for estimation of heteroscedasticity-robust variances.
 The most popular is the HCCM
procedure proposed by White.
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The heteroscedasticity consistent
covariance matrix (HCCM)
procedure.
 Let’s consider the model: Yi = β0 +
β1X1i + β2X2i + ... + βkXki + ui
• Step 1. Estimate the initial model
by the OLS method. Let ûi denote
the OLS residuals from the initial
regression of Y on X1, X2, .., Xk
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• Step 2. Run the OLS regression of
Xj (each time for a different j) on
all other independent variables. Let
ŵij denotes the ith residual from
regressing Xj on all other
independent variables.
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• Step 3. Let RSSj be the residual
sum of squares from this
regression: RSSj = SXjXj(1-R2).
 RSSj can also be calculated as
RSSj = [n-(k+1)]SER2, where SER
is the standard error of regression
and can easily be found in the
Excel’s OLS solution.
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• Step 4. The heteroscedasticityrobust variance Var(bj) can be
calculated as follows:
Var(bj) = Σŵij2ûi2/RSSj2.
 The square root of Var(bj) is called
the heteroscedasticity-robust
standard error for bj.
 Example: CN, p. 44.
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• 3. Feasible Generalized Least
Squares (FGLS) method
Step 1. Compute the residuals ûi
from the OLS of the initial
regression model
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• Step 2. Regress ûi2 against a constant
term and all the explanatory variables
from either the Breusch-Pagan test for
heteroscedasticity (e.g., when k =2:
 ûi2 = α0 + α1X1i + α2X2i + vi )
or the White test for heteroscedasticity:
 ûi2 = α0 + α1X1i + α2X2i + α3X1i2 + α4X2i2
+ α5X1i X2i + vi
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• Step 3. Estimate the original model
by OLS using the weights zi = 1/σi,
where σi2 are the predicted values
of the dependent variable (the ûi2)
in the Breusch-Pagan (or White)
model. Note: the model must be
estimated without a constant term.
 Such OLS procedure is called
WLS (weighted least squares).
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 It may happen that the predicted
values σi2 of the dependent variable
may not be positive, so we cannot
calculate the corresponding weights
zi = 1/σi. If this situation arises for
some observations, then we can use
the original ûi2 and take their
positive square roots.
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