Weak IV problem with interaction term

[Hanx Cui]

Hi statalist,

I'm trying to estimate the following equation with one endogenous variable x and the interaction of the endogenous variable with an exogenous variable w.

y=b₁ x + b₂ x*w + b₃ w + u

I have z as an IV for x. It seems natural to use z*w as an instrument for x*w. So I ran the following code:

 ivreghdfe y (x c.x#c.w = z c.z#c.w) w, a(CZ#year industry#year) cluster(CZ) // CZ stands for commuting zone

The problem is, if I regress x on z alone using the code below, the t-stat on z is around 8 (sorry I'm not able to post actual results here because I'm currently using administrative data), which seems like a pretty strong instrument.

 reghdfe x z w, a(CZ#year industry#year)) cluster(CZ)

but when considering the interaction term, the Kleibergen-Paap F-stat from the first regression is around 7. The reason seems to be that conditional on z, z*w is irrelevant to x, and similarly, conditional on z*w, z is irrelevant to x*w. Therefore coefficients a₂ and c₁ in the following 2 regressions are both insignificant, potentially causing a small first-stage F-stat.

x=a₁ z + a₂ z*w + a₃ w + e₁

x*w=c₁ z + c₂ z*w + c₃ w + e₂

Would my reasoning be correct in this case? If so, is there any way I could address this situation apart from getting a stronger instrument for x? Any thoughts from you are greatly appreciated. Thanks in advance!