1.(10 points) The conditional average treatment
effect on the treated (CATT) given \(X\) and the conditional average treatment
effect (CATE) given \(X\) are defined
as \[\begin{eqnarray*}
CATE(X) &\equiv& E[Y(1)- Y(0)|X], \\
CATT(X) &\equiv& E[Y(1)- Y(0)|D=1,X]. \\
\end{eqnarray*}\]
Assume that \(Y(1), Y(0) \perp
D|X.\) and that, for some \(0<\varepsilon<0.5\), \(\varepsilon < P(D=1|X) <
1-\varepsilon\).
Under these assumptions, is the \(CATE(X)\) different from the \(CATT(X)\)? If so, how? If not, why not?
Prove it mathematically.
2.(10 points) Let all the assumptions in question 1
hold. Prove that, under these assumptions, the ATT can be written as
\[\begin{eqnarray*}
ATT &\equiv& E[Y(1)- Y(0)|D=1] = E[Y|D=1] - E[E[Y|D=0,X]|D=1].
\end{eqnarray*}\] You must justify all the steps in your proof.
3.(30 points) Let all the assumptions in question 1
hold. Let
\(X_s\) be a subset of all
\(X\).
- (10 points) Under the above assumptions, is the \(CATE(X_s)\) and \(CATT(X_s)\) identified? If so, prove it,
and state the identified target parameter. If not, explain why not.
- (10 points) If the \(CATE(X_s)\)
and \(CATT(X_s)\) are identified, how
would you could estimate them? Explain the steps in detail.
- (10 points) Is the \(CATE(X_s)\)
and \(CATT(X_s)\) different from each
other? Justify your answer.