2
$\begingroup$

I am estimating a model using the control function approach (also "2SRI").

My model includes an endogenous variable $y_2$, an instrument $z_2$ and an interaction of $y_2$ with an exogenous variable $w$: $y_2w$ (a similar question has been asked here Endogenous interaction term in a triangular system using control function (CF) approach). Regardless of the estimation approach, $y_2$ and $y_2w$ are endogenous variables, so we need at least two instruments.

The way I understand Wooldridge 2010 §6.2 pp128-129 and §9.5, especially §9.5.3, a second instrument can be $z_2w$ (the original instrument interacted with $w$).

The model would then be:

  1. Regress $y_2$ on $z, w, z_2w$ and obtain $\hat{u}_2$.
  2. Regress $y_1$ on $y_2, w, y_2w, \hat{u}_2, z_1$.

with $z$ the vector of all exogenous variables $z_1$ and the instrument $z_2$.

So is this correct?

$\endgroup$
1
$\begingroup$

It is only correct with an additional assumption. Using the notation of the post you referenced, you need to assume that $E[u_1|w,z,x_1,x_2,u_2] = \lambda u_2$. This is analogous to (9.72) on page 270 of Wooldridge. With this assumption, it's actually okay to just use one instrument.

The assumption that $E[u_1|w,z,x_1,x_2,u_2] = \lambda u_2$ is rather strong and arbitrary. It's very difficult to justify. Conventional practice is to just use two stage least squares, or the equivalent control function version of it that I described in my previous answer.

$\endgroup$
  • $\begingroup$ I see. So the conditional expectation of $u_1$ given the exogenous variables has to be some multiplicative of $u_2$. Do you have a reference that discusses the implications of this assumption in more detail? $\endgroup$ – Phil C Apr 18 '16 at 11:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.