st: RE: 2nd Step GMM estimation with nonlinear endogenous regressors is biased?

["Schaffer, Mark E" <M.E.Schaffer@hw.ac.uk>]

Jordana,

> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu 
> [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of 
> Jordana Rodrigues Cunha
> Sent: 14 December 2010 15:24
> To: statalist@hsphsun2.harvard.edu
> Subject: st: 2nd Step GMM estimation with nonlinear 
> endogenous regressors is biased?
> 
> Dear all,
> I need to test the endogeneity of two regressors (a 
> dichotomous and an ordinal ranked in five levels ) in a 
> single equation model with a continuous dependent variable. I 
> would like to confirm if I could do this using the amazing 
> GMM option of ivreg2.

Thanks for the kind words.  Or did you mean a new kind of GMM estimator?
2-step efficient GMM, continuously updated GMM, amazing GMM....

> As I am not interested in estimating 
> the full system of equations (fitting in the first stage, in 
> this case a probit and an ordinal probit and in the 2nd stage 
> an OLS), I don't need to give the correct functional form of 
> my regressors in the first stage, do I?

Correct.  ivreg2 estimates single-equation ("limited information")
models only.  Adding functional form assumptions for the first stage is
akin to estimation a system of two equations (a "full information"
setup) where you have to specify both equations correctly.

> I explain:
> 
> All the tests for underidentification (I am using clustered 
> robust errors) and the J Hansen shows that if I need to 
> instrumentalize my regressors I would be able to do it, even 
> if my endog results confirms that my regressors are not 
> endogenous and in this way, seems that my OLS results are 
> more efficient.

I had to read this a couple of times but I think I understand: the
equation isn't underidentified; the overidentifying restrictions are
apparently valid; and an endogeneity test fails to reject the hypothesis
that your endogeneous regressors are actually exogenous.

> My point, than, is: 
> 
> The parameters estimated in the 2nd stage GMM are not 
> significative for the tested regressors, but those in the OLS 
> are...

This is quite possible.  OLS is more efficient than 2-step GMM, so the
standard errors are smaller and the estimates are more precise.

> While the assumption of a linear relationship between 
> the dependents and independents variables in my 2 first stage 
> models is violated,

Not true.  There is no such linearity assumption.  You're using a
single-equation, limited-information setup precisely so that you don't
have to make such assumptions.  The tradeoff is that your results are
more robust but not as efficient vs. using a system estimator.  This is
basically what you yourself say above.

> they recover correct standard errors that 
> will be applied in the second stage, when I need to good 
> estimates. Am I right? The parameters estimated with the 2nd 
> step GMM could be biased after the assumption of linear form 
> in my first stage

"Bias" is the wrong word - you mean "inconsistent".

And no, your second-stage estimates are consistent, because there is no
linear functional form assumption for the first stage - see above.

Best wishes,
Mark

> and I can believe in the results in my OLS, 
> or the parameters estimated with ivreg2 GMM are better?
> Any help will be appreciate, thank you very much,
> 
> jordana 
> 
> 
> Jordana Rodrigues Cunha
> PhD. Candidate
> University of Bologna
> Department of Management
> Via Capo di Lucca, 34, 1st floor
> 40126 - Bologna, ITALY
> Fixed line:  0039 (051) 20 98 073
> Fax: 0039 (051) 20 98 074
> jordana.rodrigues@unibo.it
> www.sa.unibo.it
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