Control Function steps when the Explanatory Endogenous Variable is squared

[Daipayan Dhar]

@Wooldridge
My main model is
y1 = b0 + b1*y2 +b2*(y2)^2 + b3*(Other_Controls) +ui

I assume that y2 is endogenous, and I have z1 as an instrument for y2. As 2SLS cannot be used in this model because of its non-linearity, I am using the Control Function Approach.
From the ppt and the paper written by Wooldridge(2015), I could infer that the steps are as follows-

xtreg y2 z1 z1^2 Other_Controls, fe cluster(i)
From this model I predict the Residuals.
Then I run the main model
xtreg y1 y2 y2^2 Residuals Other_controls, fe cluster(i)

If the residuals come out as significant, then there is a problem of endogeneity.
I am a little confused about steps as there is no other blog specifically checking for the endogeneity of a squared Explanatory Endogenous Variable. Am I doing it correctly? Any other suggestions are welcome.

[Jeff Wooldridge]

This is correct. A benefit of the CF approach is that it recognizes y2^2 is a function of y2. So, y2 is endogenous or it isn't. If y2 is endogenous, so is y2^2. If y2 is exogenous, so is y2^2. So the test is one degree-of-freedom on the residuals in the second estimation.

[Daipayan Dhar]

Thanks for your reply sir.

[Daipayan Dhar]

Jeff Wooldridge If I have two endogenous variables suppose. Hence the model is
y1 = b0 + b1*y2 +b2*(y2)^2 + b3*y3+ b4*(y3)^2+ b5(Other_Controls) +ui

I assume that y2 and y3 are endogenous, and I have z1 and z2 as the instruments for y2 and y3, respectively. As 2SLS cannot be used in this model because of its non-linearity, I am using the Control Function Approach.
Please let me know if the steps are correct-

xtreg y2 z1 z1^2 z2 z2^2 Other_Controls, fe cluster(i)
From this model I predict the Residuals for y2, say ui

xtreg y3 z1 z1^2 z2 z2^2 Other_Controls, fe cluster(i)
From this model I predict the residuals for y3, say vi

Then I run the main model
xtreg y1 y2 y2^2 ui vi Other_controls, fe cluster(i)

If the residuals come out as significant, then there is a problem of endogeneity.
Am I doing it correctly? Any other suggestions are welcome.
Also, if you can tell the steps in case of 2SLS when I have such variables, it will be helpful.