Page 411 -

410 Chapter 10 Regression with Panel Data

estimating the multiple regression equation of deviated Y on the deviated X’s.
This algorithm, which is commonly implemented in regression software, eliminates
the need to construct the full set of binary indicators that appear in Equation
(10.20). An equivalent approach is to deviate Y, the X’s, and the time indicators
from their entity (but not time) means and to estimate k + T coefficients by mul-
tiple regression of the deviated Y on the deviated X’s and the deviated time indi-
cators. Finally, if T = 2, the entity and time fixed effects regression can be
estimated using the “before and after” approach of Section 10.2, including the
intercept in the regression. Thus the “before and after” regression reported in
Equation (10.8), in which the change in FatalityRate from 1982 to 1988 is regressed
on the change in BeerTax from 1982 to 1988 including an intercept, provides the
same estimate of the slope coefficient as the OLS regression of FatalityRate on
BeerTax, including entity and time fixed effects, estimated using data for the two
years 1982 and 1988.

Application to traffic deaths. Adding time effects to the state fixed effects regres-
sion results in the OLS estimate of the regression line:

FatalityRate = -0.64 BeerTax + StateFixedEffects + TimeFixedEffects. (10.21)
(0.36)

This specification includes the beer tax, 47 state binary variables (state fixed
effects), 6 single-year binary variables (time fixed effects), and an intercept, so this
regression actually has 1 + 47 + 6 + 1 = 55 right-hand variables! The coeffi-
cients on the time and state binary variables and the intercept are not reported
because they are not of primary interest.

Including time effects has little impact on the coefficient on the real beer tax
[compare Equations (10.15) and (10.21)]. Although this coefficient is less pre-
cisely estimated when time effects are included, it is still significant at the 10%,
but not 5%, significance level (t = -0.64>0.36 = -1.78).

This estimated relationship between the real beer tax and traffic fatalities is
immune to omitted variable bias from variables that are constant either over time
or across states. However, many important determinants of traffic deaths do not
fall into this category, so this specification could still be subject to omitted variable
bias. Section 10.6 therefore undertakes a more complete empirical examination
of the effect of the beer tax and of laws aimed directly at eliminating drunk driv-
ing, controlling for a variety of factors. Before turning to that study, we first dis-
cuss the assumptions underlying panel data regression and the construction of
standard errors for fixed effects estimators.