Regression for Completely Randomized Experiments

Question

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

Answer 1

In the meantime I managed to find an answer:

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.