Page 409 -

408 Chapter 10 Regression with Panel Data

for all states. The population regression in Equation (10.9) can be modified to
make explicit the effect of automobile safety, which we will denote St:

Yit = b0 + b1Xit + b2Zi + b3St + uit, (10.16)

where St is unobserved and where the single t subscript emphasizes that safety
changes over time but is constant across states. Because b3St represents variables
that determine Yit, if St is correlated with Xit, then omitting St from the regression
leads to omitted variable bias.

Time Effects Only

For the moment, suppose that the variables Zi are not present so that the term
b2Zi can be dropped from Equation (10.16), although the term b3St remains. Our
objective is to estimate b1, controlling for St.

Although St is unobserved, its influence can be eliminated because it varies
over time but not across states, just as it is possible to eliminate the effect of Zi,
which varies across states but not over time. In the entity fixed effects model, the
presence of Zi leads to the fixed effects regression model in Equation (10.10), in
which each state has its own intercept (or fixed effect). Similarly, because St varies
over time but not over states, the presence of St leads to a regression model in
which each time period has its own intercept.

The time fixed effects regression model with a single X regressor is

Yit = b1Xit + lt + uit. (10.17)

This model has a different intercept, lt, for each time period. The intercept lt in
Equation (10.17) can be thought of as the “effect” on Y of year t (or, more gener-
ally, time period t), so the terms l1, c, lT are known as time fixed effects. The
variation in the time fixed effects comes from omitted variables that, like St in
Equation (10.16), vary over time but not across entities.

Just as the entity fixed effects regression model can be represented using
n - 1 binary indicators, so, too, can the time fixed effects regression model be
represented using T - 1 binary indicators:

Yit = b0 + b1Xit + d2B2t + g + dTBTt + uit, (10.18)

where d2, c, dT are unknown coefficients and where B2t = 1 if t = 2 and B2t = 0
otherwise, and so forth. As in the fixed effects regression model in Equation (10.11),