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.

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

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