I am running regressions of the sort:
$$ y_{i}= \alpha + \beta T_{i} + \gamma G_{i} + \delta( T_{i} * G_{i}) + \rho X_{i} + \epsilon_{i} $$
where $T_{i}$ is binary treatment variable, $G_{i}$ is binary variable that indicates whether or not observation $i$ belongs to a group of interest and, $X_{i}$ is a set of covariates.
I am interested in estimating $\beta + \delta$, the average treatment effect of the group of interest. Then, I cut my full-sample in two sub-samples and I estimate:
- $y_{i}= (\alpha+\gamma) + (\beta + \delta) T_{i} + \epsilon_{i}$ for all $i$ such that $G_{i}=1$ and,
- $y_{i}= \alpha + \beta T_{i} + \epsilon_{i}$ for all $i$ such that $G_{i}=0$
which provide me with estimates of $(\beta + \delta)$ and $\beta$. Next, to estimate $\delta$ alone, I use:
$$ y_{i}= \alpha + \beta T_{i} + \gamma G_{i} + \delta( T_{i} * G_{i}) + \epsilon_{i} $$
These regressions are totally equivalent and I have $\hat{(\beta + \delta)}=\hat{\beta} + \hat{\delta}$, which is fine.
Now, when I introduce the set of covariates into these regressions, they are no longer equivalent. That is, when using:
$y_{i}= (\alpha+\gamma) + (\beta + \delta) T_{i} + \rho X_{i} + \epsilon_{i}$ for all $i$ such that $G_{i}=1$
and
$y_{i}= \alpha + \beta T_{i} + \gamma G_{i} + \delta( T_{i} * G_{i}) + \rho X_{i} + \epsilon_{i}$,
I no longer get $\hat{(\beta + \delta)}=\hat{\beta} + \hat{\delta}$.
Is that normal? I can't understand why. Then, how can I estimate $\beta$, $\delta$ and $(\beta + \delta)$ with control variables (covariates)?