My question is similar to the one posted here ( I can't comment in there yet so that is why I opened this thread).
Given a the regression posted in the question:
$Y_{it}= \alpha_i + \rho_t + \beta_1 X_{it}+ \beta_2C_i \times D_t + \epsilon_{it} \quad \forall t=1,2,...,T ; i=1,2,...,N$
Where $D_t=1 \quad \forall t\geq \tau $
According to the answer posted there, we can interpret $\beta_2$ as the effect of $C_i$ on $Y_{it}$ from period $t \geq \tau$ on compared with the base period.
Is there a way to show this more formally?
Here is my try:
$E(Y_{it}|X_{it},C_i=1,D_t=1)=\beta_1X_{it}+ \beta_2$
$E(Y_{it}|X_{it},C_i=0,D_t=1)=\beta_1X_{it}$
$E(Y_{it}|X_{it},C_i=1,D_t=0)=\beta_1X_{it}$
$E(Y_{it}|X_{it},C_i=0,D_t=0)=\beta_1X_{it}$
Hence
$\Big[E(Y_{it}|X_{it},C_i=1,D_t=1)-E(Y_{it}|X_{it},C_i=0,D_t=1)\Big]-\Big[E(Y_{it}|X_{it},C_i=1,D_t=0)-E(Y_{it}|X_{it},C_i=0,D_t=0)\Big]=\beta_2$
How can we make sure that this is indeed the base category and not any other combination of 0 and 1's?
For instance, in Stata the previous method can be achieved using:
xtreg Y X i.date 1.D#1.C, fe vce(robust)
Does the interpretation of $\beta_2$ apply to the last interaction term?
