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Re: st: Another method of squaring an instrumental variable

From   Stas Kolenikov <>
Subject   Re: st: Another method of squaring an instrumental variable
Date   Sat, 5 Jun 2010 09:27:42 -0500

I don't think this is the right approach. Recall that E[x^2] is not
equal to ( E[x] )^2 (Jensen's inequality). You need to have E[
regressor | instrument ] in the RHS, so you should've run two
regressions of x and x^2 on your instrument(s). But since you have
only one, that would create perfect collinearity between your
instrumented variables in the main regression -- in other words, you
don't have enough instruments for your endogenous variables x and x^2.

On Sat, Jun 5, 2010 at 7:47 AM, Nir Regev <> wrote:
> I am trying to estimate an equation of     y = x + x^2+u
> x is endogenous (which makes x^2 endogenous too) and I have only one
> instrumental variable z, which is a dummy.
> On a regular case, I would square z and let z and z^2 be the
> instrumentals, however, since z is a dummy this is irrelevant.
> I believe it is possible to run the first stage of 2SLS where:
> x=z+u,
> predict x_hat,
> square x_hat
> and run the second stage both on x_hat and x_hat^2.
> I think this is a consistent estimate but my standard errors and
> hypothesis tests will no doubt be miscalculated.
> Has anyone ever heard of such a method and knows how to fix the standart errors?

Stas Kolenikov, also found at
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