I have a question about a simulation set up. Assume there are two groups (Z = 0 and Z = 1). The outcome for Z = 0 and Z = 1 are generated by the following equations:
$Y_0 = \alpha_0 + \beta_{01}*{x_1} + \beta_{02}*{x_2} + \epsilon_{0}$
$Y_1 = \alpha_1 + \beta_{11}*{x_1} + \beta_{12}*{x_2} + \epsilon_{1}$,
where $x_1$ and $\epsilon$ are standard normal $N(0,1)$ and $x_2$ is bernoulli distributed with $\pi = 0.5$. The true values of ($\alpha_0$, $\beta_{01}$, $\beta_{02}$) = (0, 1, 2) and ($\alpha_0$, $\beta_{01}$, $\beta_{02}$) = (1, 2, 3). How can I know the true value of the average treatment effect on the treated (ATT $ = E(Y_1 - Y_0 \mid Z =1)$) based on the above setting? many thanks in advance.
