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10.2 Panel Data with Two Time Periods: “Before and After” Comparisons 401

Because Zi does not change over time, in the regression model in Equation
(10.4) it will not produce any change in the fatality rate between 1982 and 1988.
Thus, in this regression model, the influence of Zi can be eliminated by analyzing
the change in the fatality rate between the two periods. To see this mathemati-
cally, consider Equation (10.4) for each of the two years 1982 and 1988:

FatalityRatei1982 = b0 + b1BeerTaxi1982 + b2Zi + ui1982, (10.5)

FatalityRatei1988 = b0 + b1BeerTaxi1988 + b2Zi + ui1988. (10.6)

Subtracting Equation (10.5) from Equation (10.6) eliminates the effect of Zi:

FatalityRatei1988 - FatalityRatei1982
= b1(BeerTaxi1988 - BeerTaxi1982) + ui1988 - ui1982. (10.7)

This specification has an intuitive interpretation. Cultural attitudes toward drink-
ing and driving affect the level of drunk driving and thus the traffic fatality rate in
a state. If, however, they did not change between 1982 and 1988, then they did not
produce any change in fatalities in the state. Rather, any changes in traffic fatali-
ties over time must have arisen from other sources. In Equation (10.7), these
other sources are changes in the tax on beer and changes in the error term (which
captures changes in other factors that determine traffic deaths).

Specifying the regression in changes in Equation (10.7) eliminates the effect
of the unobserved variables Zi that are constant over time. In other words, analyzing
changes in Y and X has the effect of controlling for variables that are constant
over time, thereby eliminating this source of omitted variable bias.

Figure 10.2 presents a scatterplot of the change in the fatality rate between
1982 and 1988 against the change in the real beer tax between 1982 and 1988 for
the 48 states in our data set. A point in Figure 10.2 represents the change in the
fatality rate and the change in the real beer tax between 1982 and 1988 for a given
state. The OLS regression line, estimated using these data and plotted in the
figure, is

FatalityRate1988 - FatalityRate1982 = - 0.072 - 1.04(BeerTax1988 - BeerTax1982).
(0.065) (0.36) (10.8)

Including an intercept in Equation (10.8) allows for the possibility that the mean
change in the fatality rate, in the absence of a change in the real beer tax, is non-
zero. For example, the negative intercept ( -0.072) could reflect improvements in
auto safety from 1982 to 1988 that reduced the average fatality rate.