Control Function (CF) Approach with Nonlinear Functions of Endogenous Variables

Question

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?

Answer 1

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.