6.14 Problem with TWFE in Multiple Group with Treatment Timing Variation

To understand what TWFE is estimating, write the potential outcome as:

\[\begin{equation} Y_{it}(0) = \theta_{t} + \eta_{i} + v_{it}..... (A) \end{equation}\]

Then write the TWFE using the potential outcome As

\[\begin{equation} Y_{it} = Y_{it}(0) + 1\{t \geq G_{i}\}(Y_{it}(G_i)-Y_{it}(0))..... (B) \end{equation}\]

Note that substituting \((A)\) into \((B)\) gives the TWFE. Let’s look at the terms in the above equation closely. $ 1{t G_{i}}$ represents \(D_{it}\) and
\((Y_{it}(G_i)-Y_{it}(0))\) is the \(\alpha\) parameter in equation TWFE. The equality \((Y_{it}(G_i)-Y_{it}(0))=\alpha\) indicates that the effects of being treated is the same for each group in \(\zeta\) and the effects do not vary over time. In other words, the effects of treatment are homogeneous across units and over time.