 # A Discontinuity Test of the Validity of an IV

The idea is related to that of my endogeneity paper, but the problem has several new complications. Here I tell you the gist of it very informally. Suppose that the structural equation is

$Y=g(X,U)$
But X is endogenous. Now, suppose that you got a variable Z which you think is an IV, and you plan to use a control function approach. If the approach is correct and all the conditions are met, then

$U\perp X|V,$
where V is the control function. Therefore, provided

A1) g is continuous in X,

$E(Y|X,V)=\int g(X,U) dF(U|V)$
is also continuous. This seems like the same setup as in my paper, where instead of controlling for the covariates, we control for the control function. The test would have power if

A2) F(U|X,V) is discontinuous in X

where F is the distribution of U. This happens, for example, when X has bunching points. We pursue the following case:

$X=\max\{0,h(Z,V)\},$
where the bunching, and therefore the discontinuity, is generated by a corner solution type of restriction. The test could be based on a quantity such as

$\Delta(V)=E(Y|X=0,V)-\lim_{x\downarrow 0} E(Y|X=x,V).$
Here the complications begin. The first problem is that we need to estimate V, but V is identified only when X>0. The way around it is to look at the following quantity instead:

$\theta=E(E(Y|X=0,V)-\lim_{x\downarrow 0} E(Y|X=x,V)|X=0)$
$=E(Y|X=0)-E(\lim_{x\downarrow 0} E(Y|X=x,V)|X=0)$
which eliminates the need to estimate V when X=0 thanks to the law of iterated expectations. However, the second term,

$\int \lim_{x\downarrow 0} E(Y|X=x,V)dF(V|X=0)$
requires the estimation of F(V|X=0). It turns out that although it is impossible to estimate V when X=0, it is possible to estimate F(V|X=0). The trick is to observe that without loss of generality, V is uniformly distributed, and thus F(V|X=0)=V-F(V|X>0). Since V is identified when X>0, we can estimate F(V|X>0).  