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.