**You do not need to turn these in. These are just some additional questions mainly related to the material since the previous homework that will be covered on the first midterm.**

**Textbook Questions** 2.21, 3.2, 3.3, 3.5, 3.7, 3.22

**Extra Question**

Consider the case with a binary treatment and suppose that unconfoundedness holds. Show that \(ATE\) is identified in this case.

How does this expression compare to the one that we derived for \(ATT\) in class?

Now, additionally suppose a linear model for untreated potential outcomes: \(Y_i(0) = X_i'\beta + e_i\) with \(\E[e|X]=0\) and treatment effect homogeneity: \(Y_i(1) - Y_i(0) = \alpha\) for all units. Explain how to use a linear regression to estimate the causal effect of participating in the treatment in this case (and explain where you use each condition above to provide this result).

How does this regression compare to the one that we derived in class after we had identified the ATT? Is it the same or different? Any comments/explanations?