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

Suppose there are only $T=2$ time periods $t=1982,1988$. This allows us to analyze differences in changes of the fatality rate from year 1982 to 1988. We start by considering the population regression model $FatalityRate_{it} = \beta_0 + \beta_1 BeerTax_{it} + \beta_2 Z_{i} + u_{it}$ where the $Z_i$ are state specific characteristics that differ between states but are constant over time. For $t=1982$ and $t=1988$ we have \begin{align*} FatalityRate_{i1982} =&\, \beta_0 + \beta_1 BeerTax_{i1982} + \beta_2 Z_i + u_{i1982}, \\ FatalityRate_{i1988} =&\, \beta_0 + \beta_1 BeerTax_{i1988} + \beta_2 Z_i + u_{i1988}. \end{align*}

We can eliminate the $Z_i$ by regressing the difference in the fatality rate between 1988 and 1982 on the difference in beer tax between those years: $FatalityRate_{i1988} - FatalityRate_{i1982} = \beta_1 (BeerTax_{i1988} - BeerTax_{i1982}) + u_{i1988} - u_{i1982}.$ This regression model, where the difference in fatality rate between 1988 and 1982 is regressed on the difference in beer tax between those years, yields an estimate for $\beta_1$ that is robust to a possible bias due to omission of $Z_i$, as these influences are eliminated from the model. Next we will use R to estimate a regression based on the differenced data and to plot the estimated regression function.

# compute the differences
diff_fatal_rate <- Fatalities1988$fatal_rate - Fatalities1982$fatal_rate
diff_beertax <- Fatalities1988$beertax - Fatalities1982$beertax

# estimate a regression using differenced data
fatal_diff_mod <- lm(diff_fatal_rate ~ diff_beertax)

coeftest(fatal_diff_mod, vcov = vcovHC, type = "HC1")
#>
#> t test of coefficients:
#>
#>               Estimate Std. Error t value Pr(>|t|)
#> (Intercept)  -0.072037   0.065355 -1.1022 0.276091
#> diff_beertax -1.040973   0.355006 -2.9323 0.005229 **
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Including the intercept allows for a change in the mean fatality rate in the time between 1982 and 1988 in the absence of a change in the beer tax.

We obtain the OLS estimated regression function $\widehat{FatalityRate_{i1988} - FatalityRate_{i1982}} = -\underset{(0.065)}{0.072} -\underset{(0.36)}{1.04} \times (BeerTax_{i1988}-BeerTax_{i1982}).$

# plot the differenced data
plot(x = as.double(diff_beertax),
y = as.double(diff_fatal_rate),
xlab = "Change in beer tax (in 1988 dollars)",
ylab = "Change in fatality rate (fatalities per 10000)",
main = "Changes in Traffic Fatality Rates and Beer Taxes in 1982-1988",
cex.main=1,
xlim = c(-0.6, 0.6),
ylim = c(-1.5, 1),
pch = 20,
col = "steelblue")

# add the regression line to plot
abline(fatal_diff_mod, lwd = 1.5,col="darkred")
legend("topright",lty=1,col="darkred","Estimated Regression Line")
The estimated coefficient on beer tax is now negative and significantly different from zero at $5\%$. Its interpretation is that raising the beer tax by $\1$ causes traffic fatalities to decrease by $1.04$ per $10000$ people. This is rather large as the average fatality rate is approximately $2$ persons per $10000$ people.
# compute mean fatality rate over all states for all time periods
#> [1] 2.040444
The approach presented in this section discards information for years $1983$ to $1987$. The fixed effects method that allows us to use data for more than $T = 2$ time periods and enables us to add control variables to the analysis.