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252 Chapter 6 Linear Regression with Multiple Regressors

but rather just a feature of OLS, your data, and the question you are trying to
answer. If the variables in your regression are the ones you meant to include—the
ones you chose to address the potential for omitted variable bias—then imperfect
multicollinearity implies that it will be difficult to estimate precisely one or more
of the partial effects using the data at hand.

6.8 Conclusion

Regression with a single regressor is vulnerable to omitted variable bias: If an
omitted variable is a determinant of the dependent variable and is correlated with
the regressor, then the OLS estimator of the slope coefficient will be biased and
will reflect both the effect of the regressor and the effect of the omitted variable.
Multiple regression makes it possible to mitigate omitted variable bias by includ-
ing the omitted variable in the regression. The coefficient on a regressor, X1, in
multiple regression is the partial effect of a change in X1, holding constant the
other included regressors. In the test score example, including the percentage of
English learners as a regressor made it possible to estimate the effect on test
scores of a change in the student–teacher ratio, holding constant the percentage
of English learners. Doing so reduced by half the estimated effect on test scores
of a change in the student–teacher ratio.

The statistical theory of multiple regression builds on the statistical theory of
regression with a single regressor. The least squares assumptions for multiple regres-
sion are extensions of the three least squares assumptions for regression with a single
regressor, plus a fourth assumption ruling out perfect multicollinearity. Because the
regression coefficients are estimated using a single sample, the OLS estimators have
a joint sampling distribution and therefore have sampling uncertainty. This sampling
uncertainty must be quantified as part of an empirical study, and the ways to do so
in the multiple regression model are the topic of the next chapter.

Summary

1. Omitted variable bias occurs when an omitted variable (1) is correlated with
an included regressor and (2) is a determinant of Y.

2. The multiple regression model is a linear regression model that includes
multiple regressors, X1, X2, c, Xk. Associated with each regressor is a
regression coefficient, b1, b2, c, bk. The coefficient b1 is the expected
change in Y associated with a one-unit change in X1, holding the other
regressors constant. The other regression coefficients have an analogous
interpretation.