**Due:** At the start of class on Tuesday, April 18.
Please turn in a hard copy.

**Textbook Questions:** Hansen 17.2

**Additional Question 1:** For this problem, we will be
interested in computing an estimate of the \(ATT\) of a job training program. You can download
the data here and download
a description here. For this problem, the outcome of interest is
`re78`

, the treatment is `train`

, and suppose that
unconfoundedness holds after conditioning on `age`

,
`educ`

, `black`

, `hisp`

,
`married`

, `re75`

, and `unem75`

.

Given our discussion in class, we know that if we believe unconfoundedness and that untreated potential outcomes are linear in covariates, then we have that

\[\begin{align*} ATT = \textrm{E}[Y|D=1] - \textrm{E}[X'|D=1]\beta \end{align*}\]

where \(\beta\) can be estimated from the regression of \(Y\) on \(X\) using untreated observations. Estimate \(ATT\) based on the above expression for it and report your result.

Show that \(\sqrt{n}(\widehat{ATT} - ATT) \xrightarrow{d} N(0,V)\) and provide an expression for \(V\). Based on this result, provide standard errors for your estimate of \(ATT\).

Use the bootstrap to compute standard errors for your estimate of \(ATT\). How do these compare to the standard errors that you reported previously?

Run a regression of \(Y\) on \(D\) and \(X\) (where \(X\) includes the same additional variables as above). Compare the coefficient on \(D\) to the estimate of \(ATT\). How similar are they? What about their standard errors? Do you have any comment on these results?

**Additional Question 2:** This question uses the same
data/setup as for the job training problem from the previous question.
We will be interested in computing an estimate of the \(ATT\) of a job training program.
For this problem, the outcome of interest is `re78`

, the
treatment is `train`

, and suppose that unconfoundedness holds
after conditioning on `age`

, `educ`

,
`black`

, `hisp`

, `married`

,
`re75`

, and `unem75`

.

For this problem, use the algorithm we discussed in class to estimate
the \(ATT\) using machine learning. In
particular, I’d like for you to use the `randomForest`

package to estimate the propensity score and the outcome regression
model. How does this estimate compare to the one from the previous
problem? [You do not need to report a standard error for the estimate,
just the point estimate.]

**Additional Question 3:** Suppose you are interested in
estimating the \(ATT\). You also have
access to two periods of panel data where no units were treated in the
first period and some units became treated in the second period.
Further, suppose you are willing to assume that \(\textrm{E}[\Delta Y_t(0) | D=1] =
\textrm{E}[\Delta Y_t(0) | D=0]\). Your friend suggests running
the following regression \[\begin{align*}
Y_{it} = \theta_t + \eta_i + \alpha D_{it} + e_{it}
\end{align*}\] and interpreting \(\alpha\) as the \(ATT\). This friend also suggests that this
approach is robust to treatment effect heterogeneity. Are they correct?
Explain.