instrumental variable for quadratic term

[Johnny YANG]

In a regression model with an explanatory variable (x) we have also the squared term of the variable x (sq_x) to capture possible nonlinearities in the relationship.
In order to check for endogeneity of x we use an instrumental variable (z) that is strongly correlated with the explanatory variable x. What kind of instrumental variable could be used for the squared term of x (sq_x) in the two stage least squares regression (TSLS/IV)?
I'm aware that in Wooldridge (2000), there is a nice discussion about this issue.
There are two approaches you can adopt. The first is to use the squares of the other exogenous variables as additional instruments. The second approach requires the use of (xhat)² as instrument in the second-stage regression. Therefore, after obtaining (xhat) from first-stage regression and squaring this value, you must estimate:
ivreg2 y (x x² = z (xhat)² ) c
y-dependent
c- controls
Note that this approach adds a nonlinear function of your exogenous variables to your instrument set. The (xhat)² should not be used as "regressor" in the second-stage model instead as an "instrument".
But I don't know how to implement this theory in Stata.
Thank you very much in advance for suggestions!

[Joro Kolev]

You just generate the variables, e.g.,

gen z2 = z^2

reg x z other_controls

predict xhat

gen xhat2 = xhat^2

ivregress 2sls y (x x2 = z z2) other_contorls

ivregress 2sls y (x x2 = xhat xhat2) other_contorls

[Johnny YANG]

You just generate the variables, e.g.,

gen z2 = z^2

reg x z other_controls

predict xhat

gen xhat2 = xhat^2

ivregress 2sls y (x x2 = z z2) other_contorls

ivregress 2sls y (x x2 = xhat xhat2) other_contorls

Thank you Joro! It's really helpful!

[Rune Schmidt Qvist]

You just generate the variables, e.g.,

gen z2 = z^2

reg x z other_controls

predict xhat

gen xhat2 = xhat^2

ivregress 2sls y (x x2 = z z2) other_contorls

ivregress 2sls y (x x2 = xhat xhat2) other_contorls

Sorry to butt into an old thread. Are these two methods meant to give the same results? My instrument is a dummy, so I can only perform one of the two following (as z^2 = z):

[CODE]
ivregress 2sls y (x x2 = z xhat2) other_controls

ivregress 2sls y (x x2 = xhat xhat2) other_controls
[CODE]

However, these two give me somewhat different results. What's the reason for that?

Also, when is inference valid? I use cluster-robust standard errors on my ID variable. Is it correct to cluster on ID in my first stage (where I find xhat) and in my second stage?

Sincerely,
Rune