6.12 Various ways of estimation
- Typical Estimation
reg1 <- lm(Y ~ treat:period + treat + period, data)
summary(reg1)##
## Call:
## lm(formula = Y ~ treat:period + treat + period, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -18.830 -3.455 -0.093 3.467 16.072
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.95751 0.22379 13.216 <2e-16 ***
## treat 4.26159 0.31648 13.466 <2e-16 ***
## period -0.07377 0.31648 -0.233 0.816
## treat:period 19.95384 0.44757 44.583 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.004 on 1996 degrees of freedom
## Multiple R-squared: 0.8002, Adjusted R-squared: 0.7999
## F-statistic: 2665 on 3 and 1996 DF, p-value: < 2.2e-16- Within Estimator
reg2 <- lm(Ytrans ~ D + period, data)
summary(reg2)##
## Call:
## lm(formula = Ytrans ~ D + period, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -18.830 -3.455 -0.093 3.467 16.072
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.03689 0.19376 0.190 0.849
## D 19.95384 0.44746 44.594 <2e-16 ***
## period -0.07377 0.31640 -0.233 0.816
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.003 on 1997 degrees of freedom
## Multiple R-squared: 0.6641, Adjusted R-squared: 0.6637
## F-statistic: 1974 on 2 and 1997 DF, p-value: < 2.2e-16- First Difference
reg3 <- lm(Y_FD ~ D, FDdata)
summary(reg3)##
## Call:
## lm(formula = Y_FD ~ D, data = FDdata)
##
## Residuals:
## Min 1Q Median 3Q Max
## -32.074 -4.372 0.223 4.794 21.751
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.07377 0.31216 -0.236 0.813
## D 19.95384 0.44146 45.200 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.98 on 998 degrees of freedom
## Multiple R-squared: 0.6718, Adjusted R-squared: 0.6715
## F-statistic: 2043 on 1 and 998 DF, p-value: < 2.2e-16- Imputation method
This traces the counterfactual (untreated potential outcome) of the treated group using paths of the untreated group. Using the first difference (or within transformation): \[\begin{equation} Y_{i2}(0) - Y_{i1}(0) = \theta_{2} - \theta_{1} + v_{i2}-v_{i1} \nonumber \\ \Delta Y_{it}(0) = \theta_t + \Delta v_{it} \end{equation}\]
where, \(\theta_{t-1}\) is normalized to 0. Estimate the above equation using only the untreated group and estimate \(\hat{\theta_t}\). This is the time trend in the untreated group. The parallel trend assumption states that the outcome in treated group would have moved in a similar way to the untreated group in absence of the treatment. So, let’s use \(\hat{\theta_t}\) to adjust for the pathway in the treated group and find the potential outcome in the treated group in absence of the treatment. Note that \(\hat{\theta_t} = \frac{1}{n_0}\sum_1^n (1-D_i) \Delta Y_{it}\).
\[\begin{equation} \hat{Y_{it}(0)} = Y_{it-1} + \hat{\theta_t} \end{equation}\]
Then write \(ATT_{imp}\) as:
\[\begin{equation} ATT_{imp} = \frac{1}{n_1}\sum_{1}^{n}D_{i}(Y_{it}-\hat{Y_{it}(0)}) \\ = \frac{1}{n_1}\sum_{1}^{n}D_{i}(Y_{it}-({Y_{it-1}+\hat{\theta_{t}}}) ) \\ = \frac{1}{n_1}\sum_{1}^{n}D_{i}(Y_{it}-({Y_{it-1}}) - \frac{1}{n_0}\sum_{1}^{n}(1-D_{i})(Y_{it}-({Y_{it-1}}) \\ = \frac{1}{n_1}\sum_{1}^{n}D_{i}\Delta Y_{it} - \frac{1}{n_0}\sum_{1}^{n}(1-D_{i}) \Delta Y_{it} \end{equation}\]
reg <- lm(Y_FD ~ period2, subset(FDdata, treat1 == 0))
FDdata$yhattreat = FDdata$Y1 + reg[[1]][[1]]
FDdata$imp = FDdata$Y2 - FDdata$yhattreat
mean(FDdata$imp[FDdata$treat1 == 1])## [1] 19.95384- Frisch-Waugh Theorem