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10.4 Regression with Time Fixed Effects 409
in this version of the time effects model the intercept is included, and the first binary
variable (B1t) is omitted to prevent perfect multicollinearity.
When there are additional observed “X” regressors, then these regressors
appear in Equations (10.17) and (10.18) as well.
In the traffic fatalities regression, the time fixed effects specification
allows us to eliminate bias arising from omitted variables like nationally intro-
duced safety standards that change over time but are the same across states in
a given year.
Both Entity and Time Fixed Effects
If some omitted variables are constant over time but vary across states (such as
cultural norms) while others are constant across states but vary over time (such as
national safety standards), then it is appropriate to include both entity (state) and
time effects.
The combined entity and time fixed effects regression model is
Yit = b1Xit + ai + lt + uit, (10.19)
where ai is the entity fixed effect and lt is the time fixed effect. This model can
equivalently be represented using n - 1 entity binary indicators and T - 1 time
binary indicators, along with an intercept:
Yit = b0 + b1Xit + g2D2i + g + gnDni
+ d2B2t + g + dTBTt + uit, (10.20)
where b0, b1, g2, c, gn, and d2, c, dT are unknown coefficients.
When there are additional observed “X” regressors, then these appear in
Equations (10.19) and (10.20) as well.
The combined state and time fixed effects regression model eliminates omit-
ted variables bias arising both from unobserved variables that are constant over
time and from unobserved variables that are constant across states.
Estimation. The time fixed effects model and the entity and time fixed effects
model are both variants of the multiple regression model. Thus their coefficients
can be estimated by OLS by including the additional time binary variables. Alter-
natively, in a balanced panel the coefficients on the X’s can be computed by first
deviating Y and the X’s from their entity and time-period means and then by
