# Regression for Completely Randomized Experiments

I'm reading Guido W. Imbens and Donald B. Rubin's book on Causal Inference. In chapter 7 they try to justify using a regression model to estimate the mean casual effect in a completely random experiment. To use the same notation let $$W_i \in \{0, 1\}$$ be the assignment of unit $$i$$. The idea is to estimate the mean casual effect by using the $$\tau$$ that solves:

$$\begin{equation} minimize_{\alpha, \tau} \sum_{i=1}^N \left( Y_i^{obs} - \alpha - \tau W_i \right)^2 \end{equation}$$

where $$Y_i^{obs}$$ is the observed potential outcome - that they denote $$Y_i(0)$$ or $$Y_i(1)$$ depending on whether $$W_i = 0$$ or $$1$$.

They argue that under certain regularity conditions the problem above converges, as $$N$$ grows, to the super population limit:

$$\begin{equation} minimize_{\alpha, \tau} \mathbb{E}_{sp} \left[ \frac{1}{N}\sum_{i=1}^N \left( Y_i^{obs} - \alpha - \tau W_i \right)^2\right] \end{equation}$$

where the expected value is taken over the distribution of sampling the potential outcomes from the super population. The super population limit is equal to:

$$\begin{equation} minimize_{\alpha, \tau} \mathbb{E}_{sp} \left[ \left( Y_i^{obs} - \alpha - \tau W_i \right)^2\right] \end{equation}$$

and here is the part I don't understand. They state that this implies that the solution is:

$$\begin{equation} \tau^* := \mathbb{E}_{sp}\left[ Y_i^{obs} | W_i = 1\right] - \mathbb{E}_{sp}\left[ Y_i^{obs} | W_i = 0\right] \end{equation}$$

Where does this last result comes from?

Thank you

• Are you confused about how to show that the regression estimates the conditional expectation function with a binary covariate? Jun 7, 2021 at 19:59
• Hi Dimitriy, I didn't know how to show that $\hat{\tau}_{ols}$ converges to $\tau^*$. But now I know. I was confused by the way it was presented in the book.
– Zee
Jun 7, 2021 at 20:38

Write the OLS estimate for $$\tau$$ that I will denote by $$\hat{\tau}_{ols}$$. You can check that:
$$\begin{equation} \hat{\tau}_{ols} = \overline{Y_1}^{obs} - \overline{Y_0}^{obs} \end{equation}$$
where $$\overline{Y_1}^{obs}$$ is the average of the observed values with $$W_i =1$$. Then,
$$\begin{equation} \hat{\tau}_{ols} = \overline{Y_1}^{obs} - \overline{Y_0}^{obs} \longrightarrow \mathbb{E}\left[ Y_i^{obs} | W_i = 1\right] - \mathbb{E}\left[ Y_i^{obs} | W_i = 0\right] \end{equation}$$
as $$N \rightarrow \infty$$ and that is what they meant in the book.