Consider an observational study with binary treatment. I denote treatment variable as $z_i$, denote observed outcome as $y_i$, denote potential outcome as $y_i(1),y_i(0)$, and denote covariates as $x_i$. Assume the unconfoundedness $P(Z | Y,X) = P(Z|X)$ holds in this problem.
I want to inference the sample average treatment effect on treatment group, i.e. $\frac{1}{\left| \{i|z_i = 1\} \right|} \sum_{i,z_i = 1} \left[Y_i(1) - Y_i(0)\right]$. A way to do this is to regress $Y(0) \sim X$ using data from control group, and directly apply the regression function $\hat{f}(X)$ to treatment group to estimate $Y_i(0)$ , finally, use $\frac{1}{\left| \{i|z_i = 1\} \right|} \sum_{i,z_i = 1} \left[Y_i(1) - \hat{f}(x_i)\right]$ to estimate $\frac{1}{\left| \{i|z_i = 1\} \right|} \sum_{i,z_i = 1} \left[Y_i(1) - Y_i(0)\right]$.
The method sounds weird, I actually want to estimate $Y(0)$ on the treatment group, but I only use data from the control group in regression. How can I improve it?