The potential outcomes framework in Section 3 - 7 e can be extended to more
than two potential
outcomes. In fact, we can think of the policy variable, \(w\), as taking on many
different values, and then \(y(w)\) denotes the outcome for policy level \(w\).
For concreteness, suppose \(w\) is the dollar
amount of a grant that can be used for purchasing books and electronics in
college, \(y(w)\) is a measure of college performance, such as grade point
average. For example, \(y(0)\) is the resulting GPA if the student receives no
grant and \(y(500)\) is the resulting GPA if the grant amount is \(\$ 500\).
For a random draw \(i\), we observe the grant level, \(w_{i} \geq 0\) and
\(y_{i}=y\left(w_{i}\right)\). As in the binary program evaluation case, we
observe the policy level, \(w_{i}\), and then only the outcome associated with
that level.
i. Suppose a linear relationship is assumed:
$$
y(w)=\alpha+\beta w+v(0)
$$
where \(y(0)=\alpha+v .\) Further, assume that for all \(i, w_{i}\) is independent
of \(v_{i}\). Show that for each \(i\)
we can write
$$
\begin{aligned}
y_{i} &=\alpha+\beta w_{i}+v_{i} \\
\mathrm{E}\left(v_{i} | w_{i}\right) &=0
\end{aligned}
$$
ii. In the setting of part (i), how would you estimate \(\beta\) (and \(\alpha\) )
given a random sample? Justify
your answer:
iii. Now suppose that \(w_{i}\) is possibly correlated with \(v_{i},\) but for a
set of observed variables \(x_{y,}\)
$$
\mathbf{E}\left(v_{i} | w_{i}, x_{i 1}, \ldots, x_{i
k}\right)=\mathrm{E}\left(v_{i} | x_{i 1}, \ldots, x_{i
k}\right)=\eta+\gamma_{1} x_{i 1}+\cdots+\gamma_{k} x_{i k}
$$
The first equality holds if \(w_{i}\) is independent of \(v_{i}\) conditional on
\(\left(x_{i}, \ldots, x_{i k}\right)\) and the second equality assumes a linear
relationship. Show that we can write
$$
\begin{aligned}
& y_{i}=\psi+\beta w_{i}+\gamma_{1} x_{i 1}+\cdots+\gamma_{k} x_{i k}+u_{i}
\\\
\mathrm{E}\left(u_{i} | w_{i}, x_{i 1}, \ldots, x_{i k}\right) &=0
\end{aligned}
$$
What is the intercept \(\psi ?\)
iv. How would you estimate \(\beta\) (along with \(\psi\) and the \(\gamma_{j}\) )
in part (iii)? Explain.
