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400 Chapter 10 Regression with Panel Data

In contrast to the regression using the 1982 data, the coefficient on the real
beer tax is statistically significant at the 1% level (the t-statistic is 3.43). Curiously,
the estimated coefficients for the 1982 and the 1988 data are positive: Taken liter-
ally, higher real beer taxes are associated with more, not fewer, traffic fatalities.

Should we conclude that an increase in the tax on beer leads to more traffic
deaths? Not necessarily, because these regressions could have substantial omitted
variable bias. Many factors affect the fatality rate, including the quality of the
automobiles driven in the state, whether the state highways are in good repair,
whether most driving is rural or urban, the density of cars on the road, and
whether it is socially acceptable to drink and drive. Any of these factors may be
correlated with alcohol taxes, and if they are, they will lead to omitted variable
bias. One approach to these potential sources of omitted variable bias would be
to collect data on all these variables and add them to the annual cross-sectional
regressions in Equations (10.2) and (10.3). Unfortunately, some of these variables,
such as the cultural acceptance of drinking and driving, might be very hard or
even impossible to measure.

If these factors remain constant over time in a given state, however, then
another route is available. Because we have panel data, we can in effect hold these
factors constant even though we cannot measure them. To do so, we use OLS
regression with fixed effects.

10.2 Panel Data with Two Time Periods:

“Before and After” Comparisons

When data for each state are obtained for T = 2 time periods, it is possible to
compare values of the dependent variable in the second period to values in the
first period. By focusing on changes in the dependent variable, this “before and
after” or “differences” comparison in effect holds constant the unobserved factors
that differ from one state to the next but do not change over time within the state.

Let Zi be a variable that determines the fatality rate in the ith state but does
not change over time (so the t subscript is omitted). For example, Zi might be the
local cultural attitude toward drinking and driving, which changes slowly and thus
could be considered to be constant between 1982 and 1988. Accordingly, the pop-
ulation linear regression relating Zi and the real beer tax to the fatality rate is

FatalityRateit = b0 + b1BeerTaxit + b2Zi + uit, (10.4)

where uit is the error term and i = 1, c, n and t = 1, c, T.