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C h a p t e r Instrumental Variables

12 Regression

C hapter 9 discussed several problems, including omitted variables, errors in
variables, and simultaneous causality, that make the error term correlated with
the regressor. Omitted variable bias can be addressed directly by including the
omitted variable in a multiple regression, but this is only feasible if you have data
on the omitted variable. And sometimes, such as when causality runs both from X to
Y and from Y to X so that there is simultaneous causality bias, multiple regression
simply cannot eliminate the bias. If a direct solution to these problems is either
infeasible or unavailable, a new method is required.

Instrumental variables (IV) regression is a general way to obtain a consistent
estimator of the unknown coefficients of the population regression function when
the regressor, X, is correlated with the error term, u. To understand how IV regres-
sion works, think of the variation in X as having two parts: one part that, for what-
ever reason, is correlated with u (this is the part that causes the problems) and a
second part that is uncorrelated with u. If you had information that allowed you to
isolate the second part, you could focus on those variations in X that are uncorre-
lated with u and disregard the variations in X that bias the OLS estimates. This is, in
fact, what IV regression does. The information about the movements in X that are
uncorrelated with u is gleaned from one or more additional variables, called instru-
mental variables or simply instruments. Instrumental variables regression uses
these additional variables as tools or “instruments” to isolate the movements in X
that are uncorrelated with u, which in turn permit consistent estimation of the
regression coefficients.

The first two sections of this chapter describe the mechanics and assumptions
of IV regression: why IV regression works, what is a valid instrument, and how to
implement and to interpret the most common IV regression method, two stage
least squares. The key to successful empirical analysis using instrumental variables
is finding valid instruments, and Section 12.3 takes up the question of how to
assess whether a set of instruments is valid. As an illustration, Section 12.4 uses IV
regression to estimate the elasticity of demand for cigarettes. Finally, Section 12.5
turns to the difficult question of where valid instruments come from in the first
place.

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