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Heteroscedasticity—Page 1
Heteroscedasticity
[NOTE: These notes draw heavily from Berry and Feldman, and, to a lesser extent, Allison, and
Pindyck and Rubinfeld.]
What heteroscedasticity is.
Recall that OLS makes the assumption that
V
j
(
)
2
for all j. That is, the variance of the error term is constant. (Homoscedasticity). If the
error terms do not have constant variance, they are said to be heteroscedastic.
When heteroscedasticity might occur.
Errors may increase as the value of an IV increases. For example, consider a model in which
annual family income is the IV and annual family expenditures on vacations is the DV.
Families with low incomes will spend relatively little on vacations, and the variations in
expenditures across such families will be small. But for families with large incomes, the
amount of discretionary income will be higher. The mean amount spent on vacations will be
higher, and there will also be greater variability among such families, resulting in
heteroscedasticity. Note that, in this example, a high family income is a necessary but not
sufficient condition for large vacation expenditures. Any time a high value for an IV is a
necessary but not sufficient condition for an observation to have a high value on a DV,
heteroscedasticity is likely.
Similar examples: Error terms associated with very large firms might have larger variances
than error terms associated with smaller firms. Sales of larger firms might be more volatile
than sales of smaller firms.
Errors may also increase as the values of an IV become more extreme in either direction, e.g.
with attitudes that range from extremely negative to extremely positive. This will produce
something that looks like an hourglass shape:
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View Full Document Heteroscedasticity—Page 2
Plot of X versus residuals
4
3
2
1
0
1
2
3
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3
2
1
0
1
2
3
X variable
Residuals
Measurement error can cause heteroscedasticity. Some respondents might provide more
accurate responses than others. (Note that this problem arises from the violation of another
assumption, that variables are measured without error.)
Heteroscedasticity can also occur if there are subpopulation differences or other interaction
effects (e.g. the effect of income on expenditures differs for whites and blacks). (Again, the
problem arises from violation of the assumption that no such differences exist or have already
been incorporated into the model.) For example, in the following diagram suppose that Z
stands for three different populations. At low values of X, the regression lines for each
population are very close to each other. As X gets bigger, the regression lines get further and
further apart. This means that the residual values will also get further and further apart.
Other model misspecifications can produce heteroskedasticity. For example, it may be that
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This note was uploaded on 02/29/2012 for the course SOC 63993 taught by Professor Richardwilliams during the Spring '11 term at Notre Dame.
 Spring '11
 RichardWilliams

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