3
$\begingroup$

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

$\endgroup$
2
  • $\begingroup$ Are you confused about how to show that the regression estimates the conditional expectation function with a binary covariate? $\endgroup$ – Dimitriy V. Masterov Jun 7 at 19:59
  • $\begingroup$ 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. $\endgroup$ – Zee Jun 7 at 20:38
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.