Quantitative Methods
Heteroskedasticity
Heterskedasticity
OLS assumes homoskedastic error terms. In
OLS, the data are homoskedastic if the
error term does not have constant
variance.
If there is non-constant variance of the error
terms, the error terms are related to some
variable (or set of variables), or to case #.
The data is then heteroskedastic.
Heteroskedasticity
Example (from wikipedia, I confess—it has relevant graphs which are easily pasted!)
 Note: as X increases, the variance of the
error term increases (the “goodness of fit”
gets worse)
Heteroskedasticity
 As you can see from the graph, the “b”
(parameter estimate – estimated slope or
effect of x on y) will not necessarily change.
 However, heteroskedasticity changes the
standard errors of the b’s—making us more
or less confident in our slope estimates
than we would be otherwise.
Heteroskedasticity
 Note that whether one is more confident or less
confident depends in large part on the
distribution of the data—if there is relatively
poor goodness of fit near the mean of X, where
most of the data points tend to be, then it is
likely that you will be less confident in your
slope estimates than you would b otherwise. If
the data fit the line relatively well near the mean
of X, then it is likely that you will be more
confident in your slope estimates than you
would be otherwise.
Heteroskedasticity: why?
 Learning?—either your coders learn
(in which case you have
measurement error), or your cases
actually learn. For example, if you
are predicting wages with experience,
it is likely that variance is reduced
among those with more experience.
Heteroskedasticity: why?
 Scope of choice: some subsets of
your data may have more discretion.
So, if you want to predict saving
behavior with wealthwealthier
individuals might show greater
variance in their behavior.
Heteroskedasticity
 Heteroskedasticity is very common in
pooled data, which makes sense—for
example, some phenomenon (i.e., voting)
may be more predictable in some states
than in others.
Heteroskedasticity
But note that what looks like
heteroskedasticity could actually be
measurement error (improving or
deteriorating, thus causing differences
in goodness of fit), or specification
issues (you have failed to control for
something which might account for
how predictable your dependent
variable is across different subsets of
data).
Heteroskedasticity Tests
 The tests for heteroskedasticity tend to
incorporate the same basic idea of
figuring out – through an auxiliary
regression analysis – whether the
independent variables (or case #, or some
combination of independent variables)
have a significant relationship to the
goodness of fit of the model.
Heteroskedasticity Tests
In other words, all of the tests seek to answer
the question: Does my model fit the data
better in some places than in others? Is
the goodness of fit significantly better at
low values of some independent variable
X? Or at high values? Or in the mid-range
of X? Or in some subsets of data?
Heteroskedasticity Tests
Also note that no single test is definitive—in
part because, as observed in class, there
could be problems with the auxiliary
regressions themselves.
We’ll examine just a few tests, to give you the
basic idea.
Heteroskedasticity Tests
The first thing you could do is just examine
your data in a scatterplot.
Of course, it is time consuming to examine all
the possible ways in which your data could
be heteroskedastic (that is, relative to
each X, to combinations of X, to case #, to
other variables that aren’t in the model
such as pooling unit, etc.)
Heteroskedasticity Tests

Another test is the Goldfeld-Quandt. The Goldfeld Quandt essentially asks
you to compare the goodness of fit of two areas of your data.

Disadvantagesyou need to have pre-selected an X that you think is
correlated with the variance of the error term.

G-Q assumes a monotonic relations between X and the variance of the error
term.

That is, is will only work to diagnose heteroskedasticity where the goodness
of fit at the low levels of X is different than the goodness of fit of high levels
of X (as in the graph above). But it won’t work to diagnose
heteroskedasticity where the goodness of fit in the mid-range of X is
different from the goodness of fit at both the low end of X and the high end
of X.
Heteroskedasticity Tests
 Goldfeld-Quandt test--steps
 First, order the n cases by the X that you think is
correlated with ei2.
 Then, drop a section of c cases out of the middle
(one-fifth is a reasonable number).
 Then, run separate regressions on both upper and
lower samples. You will then be able to compare the
“goodness of fit” between the two subsets of your
data.
Heteroskedasticity Tests
 Obtain the residual sum of squares from each
regression (ESS-1 and ESS-2).
 Then, calculate GQ, which has an F distribution.
 12
GQ  2  F[( N  K ),( N
2
1
1
2  K 2 )]
Heteroskedasticity Tests
 The numerator
represents the residual
“mean square” from the
first regression—that is,
ESS-1 / df. The df
(degrees of freedom) are
n-k-1. “n” is the number
of cases in that first
subset of data, and k is
the # of independent
variables (and then, 1 is
for the intercept
estimate).
 12
GQ  2  F[( N  K ),( N  K )]
2
1
1
2
2
Heteroskedasticity Tests
 The denominator
represents the residual
“mean square” from the
first regression—that is,
ESS-2 / df. The df
(degrees of freedom) are
n-k-1. “n” is the number
of cases in that second
subset of data, and k is
the # of independent
variables (and then, 1 is
for the intercept
estimate).
 12
GQ  2  F[( N  K ),( N  K )]
2
1
1
2
2
Heteroskedasticity Tests

Note that the F test is useful in comparing the goodness of
fit of two sets of data.

How would we know if the goodness of fit was significantly
different across the two subsets of data?

By comparing them (as in the ratio above), we can see if
one goodness of fit is significantly better than the other
(accounting for degrees of freedomsample size, number
of variables, etc.)

In other words, if GQ is significantly greater or less than 1,
that means that the “ESS-1 / df” is significantly greater or
less than the “ESS-2 / df”in other words, we have
evidence of heteroskedasticity.
Heteroskedasticity Tests
A second test is the Glejser test
 Perform the regression analysis and save the
residuals.
 Regress the absolute value of the residuals on
possible sources of heteroskedasticity
 A significant coefficient indicates
heteroskedasticity
u i = b0 + b1Xi + ei
Heteroskedasticity Tests
Glejser test
 This makes sense conceptually—you are
testing to see if one of your independent
variables is significantly related to the
variance of your residuals.
u i = b0 + b1Xi + ei
Heteroskedasticity Tests
 White’s Test
 Regress the squared residuals (as the dependent
variables) on...
 All the X variables, all the cross products (i.e.,
possible interactions) of the X variables, and all
squared values of the X variables.
 Calculate an “LM test statistics”, which is = n *
R2
 The LM test statistic has a chi-squared
distribution, with the degrees of freedom = #
independent variables.
Heteroskedasticity Tests
 White’s Test
 The advantage of White’s test is that it does not
assume that there is a monotonic relationship
between any one X and the variance of the error
terms—the inclusion of the interactions allows some
non-linearity in that relationship.
 And, it tests for heteroskedasticity in the entire
model—you do not have to choose a particular X to
examine.
 However, if you have many variables, the number of
possible interactions plus the squared variables plus
the original variables can be quite high!
Heteroskedasticity Solutions

GLS / Weighted Least Squares

In a perfect world, we would actually know what heteroskedasticity
we could expect—and we would then use ‘weighted least squares’.

WLS essentially transforms the entire equation by dividing through
every part of the equation with the square root of whatever it is
that one thinks the variance is related to.

In other words, if one thinks one’s variance of the error terms is related
to X1 2, then one divides through every element of the equation
(intercept, each bx, residual) by X1.
Heteroskedasticity Solutions
 GLS / Weighted Least Squares
 In this way, one creates a transformed equation,
where the variance of the error term is now constant
(because you’ve “weighted” it appropriately).
 Note, however, that since the equation has been
“transformed”, the parameter esimates are different
than in the non-transformed version—in the example
above, for b2, you have the effect of X2/X1 on Y, not
the effect of X2 on Y. So, you need to think about
that when you are interpreting your results.
Heteroskedasticity Solutions
 However...
 We almost never know the precise form that we
expect heteroskedasticity to take.
 So, in general, we ask the software package to give
us White’s Heteroskedastic-Constant Variances and
Standard Errors (White’s robust standard errors).
(alternatively, less commonly, Newey-West is similar.)
(For those of you who have dealt with clustering—the basic idea here is
somewhat similar, except that in clustering, you identify an X that you
believe your data are “clustered on”. When I have repeated states in a
database—that is, multiple cases from California, etc.—I might want to
cluster on state (or, if I have repeated legislators, I could cluster on
legislator. Etc.) In general, it’s a recognition that the error terms will be
related to those repeated observations—the goodness of fit within the
observations from California will be better than the goodness of fit across
the observations from all states.)