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10.3 Fixed Effects Regression 403

10.3 Fixed Effects Regression

Fixed effects regression is a method for controlling for omitted variables in panel
data when the omitted variables vary across entities (states) but do not change over
time. Unlike the “before and after” comparisons of Section 10.2, fixed effects regres-
sion can be used when there are two or more time observations for each entity.

The fixed effects regression model has n different intercepts, one for each
entity. These intercepts can be represented by a set of binary (or indicator) vari-
ables. These binary variables absorb the influences of all omitted variables that
differ from one entity to the next but are constant over time.

The Fixed Effects Regression Model

Consider the regression model in Equation (10.4) with the dependent variable
(FatalityRate) and observed regressor (BeerTax) denoted as Yit and Xit, respectively:

Yit = b0 + b1Xit + b2Zi + uit, (10.9)

where Zi is an unobserved variable that varies from one state to the next but does
not change over time (for example, Zi represents cultural attitudes toward drink-
ing and driving). We want to estimate b1, the effect on Y of X holding constant
the unobserved state characteristics Z.

Because Zi varies from one state to the next but is constant over time, the popu-
lation regression model in Equation (10.9) can be interpreted as having n intercepts,
one for each state. Specifically, let ai = b0 + b2Zi. Then Equation (10.9) becomes

Yit = b1Xit + ai + uit. (10.10)

Equation (10.10) is the fixed effects regression model, in which a1, c, an are
treated as unknown intercepts to be estimated, one for each state. The interpretation
of ai as a state-specific intercept in Equation (10.10) comes from considering the popu­
lation regression line for the ith state; this population regression line is ai + b1Xit.
The slope coefficient of the population regression line, b1, is the same for all states,
but the intercept of the population regression line varies from one state to the next.

Because the intercept ai in Equation (10.10) can be thought of as the “effect”
of being in entity i (in the current application, entities are states), the terms
a1, c, an are known as entity fixed effects. The variation in the entity fixed
effects comes from omitted variables that, like Zi in Equation (10.9), vary across
entities but not over time.