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Re: st: Interaction terms in 2SLS


From   Mark Schaffer <M.E.Schaffer@hw.ac.uk>
To   statalist@hsphsun2.harvard.edu, Glenn Hoetker <ghoetker@uiuc.edu>
Subject   Re: st: Interaction terms in 2SLS
Date   Sun, 13 Mar 2005 18:15:23 +0000 (GMT)

Glenn,

Quoting Glenn Hoetker <ghoetker@uiuc.edu>:

> I am facing the following problem and hope someone can help me with it.
> I have the following model, and do not believe Y1 is exogenous in the
> second equation.
> 
> Y1=X1 + X2
> Y2=X3 + Y1 + Y1*X3
> 
> This seems like a classic application for 2SLS, but I can't use a
> prepackaged 2SLS routine, since estimating the second equation requires
> calculating (Y1_hat*X3). One approach would seem to be doing 2SLS
> manually by modeling
> 
> Y1_hat=X1 + X2 + X3
> 
> and use the results to then estimate the second equation as
> 
> Y2=X3 + Y1_hat + Y1_hat*X3.
> 
> Several questions:
> 
> 1.  Is this approach valid in general?

I'm afraid the answer is "no".  This is an example of what has been dubbed
the "forbidden regression".  Plugging fitted values in place of endogenous
variables into nonlinear functions to try to mimic 2SLS won't give you
consistent estimates of the coefficients.  See Wooldridge's 2002 book, p.
235-36.  FWIW, my own view is that you're better off thinking in terms of
two different endogenous regressors, Y1 and (Y1*X3), and four excluded
instruments, X1, X2, (X1*X3) and (X2*X3).

Hope this helps.

--Mark

> 2.  If so, how would one calculate the standard errors for the
> second equation?
> 3A.  In terms of identification, do I have to count both Y1 and
> Y1*X3 as endogenous variables?
> 3B.  If the answer to 3A is yes, can I follow the approach I outlined
> above, or do I actually need to estimate two first stage equations
> (recognizing that my simplified model has identification
> problems):
> 
> Y1_hat=X1 + X2 + X3
> (Y1*X3)_hat=X1 + X2 + X3
> 
> with a second stage equation of
> 
> Y2=X3 +  Y1_hat + (Y1*X3)_hat ?
> 
> Any suggestions, references, or ideas for alternative modeling
> approaches would be very, very appreciated.  Many thanks in
> advance.
> 
> Glenn
> 
> Glenn Hoetker
> Assistant Professor of Strategy
> College of Business
> University of Illinois at Urbana-Champaign
> 217-265-4081
> ghoetker@uiuc.edu
> 
> 
> 
> 
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> 



Prof. Mark Schaffer
Director, CERT
Department of Economics
School of Management & Languages
Heriot-Watt University, Edinburgh EH14 4AS
tel +44-131-451-3494 / fax +44-131-451-3294
email: m.e.schaffer@hw.ac.uk
web: http://www.sml.hw.ac.uk/ecomes

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