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250 Chapter 6 Linear Regression with Multiple Regressors

Like the previous example, the perfect linear relationship among the regressors
involves the constant regressor X0i: For every district, PctESi = 100 - PctELi =
100 * X0i - PctELi, because X0i = 1 for all i.

This example illustrates another point: Perfect multicollinearity is a feature of the
entire set of regressors. If either the intercept (that is, the regressor X0i) or PctELi were
excluded from this regression, the regressors would not be perfectly multicollinear.

The dummy variable trap. Another possible source of perfect multicollinearity arises
when multiple binary, or dummy, variables are used as regressors. For example, sup-
pose you have partitioned the school districts into three categories: rural, suburban,
and urban. Each district falls into one (and only one) category. Let these binary vari-
ables be Rurali, which equals 1 for a rural district and equals 0 otherwise; Suburbani;
and Urbani. If you include all three binary variables in the regression along with a
constant, the regressors will be perfect multicollinearity: Because each district belongs
to one and only one category, Rurali +Suburbani + Urbani = 1 = X0i, where X0i
denotes the constant regressor introduced in Equation (6.6). Thus, to estimate the
regression, you must exclude one of these four variables, either one of the binary
indicators or the constant term. By convention, the constant term is retained, in which
case one of the binary indicators is excluded. For example, if Rurali were excluded,
then the coefficient on Suburbani would be the average difference between test scores
in suburban and rural districts, holding constant the other variables in the regression.

In general, if there are G binary variables, if each observation falls into one
and only one category, if there is an intercept in the regression, and if all G binary
variables are included as regressors, then the regression will fail because of perfect
multicollinearity. This situation is called the dummy variable trap. The usual way
to avoid the dummy variable trap is to exclude one of the binary variables from
the multiple regression, so only G - 1 of the G binary variables are included as
regressors. In this case, the coefficients on the included binary variables represent
the incremental effect of being in that category, relative to the base case of the
omitted category, holding constant the other regressors. Alternatively, all G
binary regressors can be included if the intercept is omitted from the regression.

Solutions to perfect multicollinearity. Perfect multicollinearity typically arises
when a mistake has been made in specifying the regression. Sometimes the mis-
take is easy to spot (as in the first example) but sometimes it is not (as in the
second example). In one way or another, your software will let you know if you
make such a mistake because it cannot compute the OLS estimator if you have.

When your software lets you know that you have perfect multicollinearity, it
is important that you modify your regression to eliminate it. Some software is