# 2SLS with a discretized endogenous variable

Imagine you would like to estimate by OLS the following: $$y=\beta_0+\beta_1 med + \beta_2 high + u$$ $med$ and $high$ are dummy values with respect to some underlying variable $x \in [0,\infty)$. $$med = \begin{cases} 1, & \text{if x>a} \\ 0, & \text{else} \end{cases}$$ $$high= \begin{cases} 1, & \text{if x>b} \\ 0, & \text{else} \end{cases}$$

We also have that $b>a>0$. $x$ is endogenous, so $med$ and $high$ should also be endogenous. Fortunately, we have some potential instrument $z$. To estimate the model, I would however need two instruments. One for $med$, one for $high$. My question is:

What are potential ways to construct such a vector of instruments?

I have the following idea:

$$z_{med} = \begin{cases} 1, & \text{if z>c} \\ 0, & \text{else} \end{cases}$$

$$z_{high} = \begin{cases} 1, & \text{if z>d} \\ 0, & \text{else} \end{cases}$$ Where $c>d$, and $c$ and $d$ are chosen as follows: Let $F(x)$ be the cumulative density function (cdf) of $x$. Let $\alpha=F(a)$ and $\beta=F(b)$. Let $G(z)$ be the cdf of $z$. Then $c=G^{-1}(\alpha)$ and $d=G^{-1}(\beta)$. Do you think this is a reasonable approach? In gerenal I would probably like to have a strong first stage.

• Sorry I might be reading it wrong, but won't the sum of med and high always be 1? – Repmat May 9 '16 at 16:35
• No, if $x \in [0,a]$, $med$ and $high$ are both $0$. – Felix H May 9 '16 at 16:38
• Aha, didnt see the assumption of $a > 0$. – Repmat May 9 '16 at 16:49