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Re: st: Another method of squaring an instrumental variable

 From Nir Regev To statalist@hsphsun2.harvard.edu Subject Re: st: Another method of squaring an instrumental variable Date Sat, 5 Jun 2010 17:50:35 +0300

```Hi Stas, Thank you for your answer.

I know I have a case of under-identification. I wish I had another
instrument but I don't.

The question is, maybe there is someway I can fix for the standard
errors of my forced "second stage".
What I am thinking is using the Variance of x,
meaning, Var(x) = E(x^2) - (E(x))^2, since I have two of the three,
maybe I can derive the right SE for x_hat^2 (This is just a hunch)
Alas, Econometrics is not my strongest point and I am looking for a
reference on this matter.

By the way, I ran the procedure I thought of and technically, it works.
It is not a case of perfect collinearity since squaring is non-linear.

Nir

On  Sat, Jun 5, 2010 at 5:27 PM, Stas Kolenikov <skolenik@gmail.com> wrote:
> 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 <regev.nir@gmail.com> 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 http://stas.kolenikov.name
> Small print: I use this email account for mailing lists only.
>
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```