Causal Inference with unconfoundedness

Question

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?

Answer 1

The units in the control group provide information about the relationship between the $X$ and $Y(0)$, because for those units, $Y$ (the observed outcome) is equal to $Y(0)$. If you want $Y(0)$ in the treated group, you can use that information to estimate it from $X$ in the treated group, just as you describe.

For the treated units, you have $Y_i(1)$ because $Y_i = Y_i(1)$ for them. It's not an estimate of $Y_i(1)$; it is $Y_i(1)$. From the treated group alone, we have half of the information required to estimate $\frac{1}{|{i|z_i=1}|} \sum_{i,z_i=1} [Y_i(1)-Y_i(0)]$, i.e., the SATT, because we have $Y_i(1)$ for $i$ where $z_i=1$ without doing any modeling. However, for these units, $Y_i(0)$ is missing. We don't know what would have happened had the treated units received control.

How could we get $Y_i(0)$ for the treated units? For those units, it's unobserved, so no modeling of just the treated units could tell you the relationship between $X$ and $Y_i(0)$. That information only exists in the control group. So even though we're interested in a quality of the treated group (i.e., the treated group SATT), we have to consult the control group because only the control group contains information that can help us fill in the other half of the estimate of the SATT.

There are, of course, other methods of estimating the SATT, including weighting and matching, that don't require you to estimate $Y_i(0)$ for each treated unit. But, because $Y_i(0)$ is only observed for the control units, we have to include them in the analysis.