I would like to understand how one gets the same coefficient estimate from 2 different model specifications.
Consider single difference estimation model:
$y_{\{Time=1\}}=\alpha+\beta_3 \textbf{Treatment}+\beta_4 y_{\{Time=0\}}+\epsilon, $
where $time:\{0,1\}$ or simply before/after and $\textbf{Treatment}:\{0,1\}$ or simply control/treatment groups.
Now consider double difference estimation model:
$y=\alpha+\beta_1 \textbf{Treatment}+\beta_2 \textbf{Time}+ \beta_3 (\textbf{Treatment}*\textbf{Time}) +\epsilon. $
The source, which I am questioning, claims that one can estimate $ \beta_3$ coefficient using either of above-mentioned models. However when I do simple rearrangement of terms and writing the model while changing group or time I find the following:
Well-known double difference estimator using DID model is the following (suppressing expected values):
$\beta_3=\Delta y_{\{Time=1\}}-\Delta y_{\{Time=0\}}$,
where $\Delta$ is the difference in treatment and control groups.
When I use the single difference model, I get the following for $\beta_3$:
$\beta_3=\Delta y_{\{Time=1\}}-\beta_4 \Delta y_{\{Time=0\}}$,
which shows that unless we put contstraint that $\beta_4=1$, I can not estimate the treatment effect using single difference estimator.
Question
Do I calculate wrongly or miss something? Could someone confirm that both models can result in the same estimate of $\beta_3$ ?
Best Answer
You are correct that the ANCOVA estimator and the DID do not estimate the same parameter. ANCOVA estimates $$(\bar Y^T_{POST}−\bar Y^C_{POST}) − \hat \theta \cdot (\bar Y^T_{PRE} - \bar Y^C_{PRE}),$$ where $\hat \theta$ is the coefficient on the lagged outcome, while DID is $$(\bar Y^T_{POST}−\bar Y^T_{PRE}) − (\bar Y^C_{POST} - \bar Y^C_{PRE})$$
These formulas are given in McKenzie (2012).
You can verify this with yourself with a regression: