Can I instrument an endogenous variable and its quadratic form with the instrument and its quadratic form?

Question

I have the following problem, where I would like to have an advise or a reference:

I'm estimating a model which usually (in other papers) only includes the explanatory variable in its linear form, but, in my case, I have seen that the model would be correctly specified if it also includes the quadratic form of my explanatory variable. Despite this, usually the literature only makes an instrumental variable for the linear version of my variable, but not for the quadratic term. Given this, would it be incorrect if I also instrument this last term with the quadratic form of my instrumental variable? Is there an "universal" answer to this question, or it depends on the type of instrumental variable of my model?

Thank you for your time.

Answer 1

The reference given in a comment under the question by @Bertrand, of Cameron and Trivedi (2005, Section 6.5.4, page 198), is very useful, showing how linear 2SLS can be inconsistent. I just stress two things based on the authors' example:

1. The basic IV estimation involves using as an instrument $z$ (which is correlated with the endogenous regressor $x$), and not $z^2$ (even if the regressor $x$ enters squared in the regression equation of their example).
2. For this IV-estimator to perform it must be the case then that $$E(x^2\cdot z) \neq 0.$$ This is the familiar "relevance" criterion for an instrument, and I am mentioning it because, it is often the case that existing correlation in levels may disappear or become very weak if we square one of the two variables, especially if $x$ takes both negative and positive values, but even when it is of uniform sign throughout.

The good thing is that we have available both $x$ and $z$ so we can at least assess what happens to their association of we square $x$ alone.

It would be instructive to use the Cameron-Trivedi setup and try to include $z^2$ as an instrument instead, to see what happens.