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10.4 Regression with Time Fixed Effects 407

Application to Traffic Deaths

The OLS estimate of the fixed effects regression line relating the real beer tax to
the fatality rate, based on all 7 years of data (336 observations), is

FatalityRate = -0.66 BeerTax + StateFixedEffects, (10.15)
(0.29)

where, as is conventional, the estimated state fixed intercepts are not listed to save
space and because they are not of primary interest in this application.

Like the “differences” specification in Equation (10.8), the estimated coeffi-
cient in the fixed effects regression in Equation (10.15) is negative, so, as pre-
dicted by economic theory, higher real beer taxes are associated with fewer traffic
deaths, which is the opposite of what we found in the initial cross-sectional regres-
sions of Equations (10.2) and (10.3). The two regressions are not identical because
the “differences” regression in Equation (10.8) uses only the data for 1982 and
1988 (specifically, the difference between those two years), whereas the fixed
effects regression in Equation (10.15) uses the data for all 7 years. Because of the
additional observations, the standard error is smaller in Equation (10.15) than in
Equation (10.8).

Including state fixed effects in the fatality rate regression lets us avoid omitted
variables bias arising from omitted factors, such as cultural attitudes toward drink-
ing and driving, that vary across states but are constant over time within a state. Still,
a skeptic might suspect that other factors could lead to omitted variables bias. For
example, over this period cars were getting safer and occupants were increasingly
wearing seat belts; if the real tax on beer rose on average during the mid-1980s, then
BeerTax could be picking up the effect of overall automobile safety improvements.
If, however, safety improvements evolved over time but were the same for all states,
then we can eliminate their influence by including time fixed effects.

10.4 Regression with Time Fixed Effects

Just as fixed effects for each entity can control for variables that are constant over
time but differ across entities, so can time fixed effects control for variables that
are constant across entities but evolve over time.

Because safety improvements in new cars are introduced nationally, they
serve to reduce traffic fatalities in all states. So, it is plausible to think of automo-
bile safety as an omitted variable that changes over time but has the same value