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?

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?