st: re:Several endogenous dummies, one instrument for each, in a binary model

[Kit Baum <baum@bc.edu>]

<>
Nick said

Just to get in before Kit Baum: it's a misunderstanding to think that
instruments map on to other variables one to one. That's not how
instrumental variables work.

I don't have an answer the question.

Nick
n.j.cox@durham.ac.uk

LachZak@web.de

I'm estimating a series of probit models with several possibly
endogenous dummy variables. Following the suggestions by Angrist (2001):
Estimation of Limited Dependent Variable Models with Dummy Endogenous
Regressors: Simple Strategies for Empirical Practice, Journal of
Business and Economic Statistics, 19 (1) I'm using "standard" 2SLS.
However, there is a problem with my instruments...because I just have
one instrument for each of the endogenous dummies. The instruments are
not dummies (but regional-level percentages).
I was thinking about running regressions of the following type:
y1 = b0 + b1z1 + b2z2 + b3x3+....+u
where z1 and z2 are possibly endogenous dummies, to instrument just one
of the dummies (for instance, z1) and keep the other dummy (for
instance, z2) as a control without instrumenting it. As I said, the
instruments can just be used for 1 of the endogenous dummies. Are the
estimates on the instrumented dummy consistent in this case? Does anyone
know of any other procedure that I could use in this case?



Nick is quite correct, as a Stata FAQ (and papers by Baum, Schaffer,  
Stillman) point out. But I don't think that is the question here.  
You're talking about estimating

ivreg2  y (z1 = R1 R2)  z2

where R1, R2 are instruments for z1. In this case you will be  
essentially estimating the first-stage regression

reg z1 R1 R2 z2

forming the 2SLS instrument z1-hat as a linear combination of those  
three factors: including the questionable regressor z2. Now if you  
doubt the exogeneity (or statistical independence) of z2 in the  
original 2SLS regression, it makes no sense to include it here among  
variables which you assert are exogenous. Given that you only have one  
potential instrument per potential endogenous regressor, you cannot  
test anything with, e.g., -ivreg2- -orthog()- or -endog()- options.

In general terms, though, if in reality both z1 and z2 are correlated  
with the 2SLS error term, and you take care of Z1 and treat Z2 as  
exogenous as above, the estimates of all coefficients in that equation  
are likely to be biased and inconsistent. You are treating Z2 as an  
exogenous variable that can instrument itself when running 2SLS. The  
consistency of 2SLS is based on the assumption that the matrix of  
instruments contains nothing correlated with the equation error, which  
you doubt is true. So in this case I would just estimate the exactly- 
identified equation

ivreg2 y (z1 z2 = R1 R2)

You cannot conduct any overid tests on that equation, but you don't  
"have a problem with the instruments". It would be better if you had  
more, but if these are the only ones you can conjure up, so be it.  
Keep in mind Wooldridge's advice, though: if R1 and R2 are valid  
instruments, then why not R1^2, R2^2, R1*R2? That would allow you to  
conduct overid tests.


Kit Baum, Boston College Economics and DIW Berlin
http://ideas.repec.org/e/pba1.html
An Introduction to Modern Econometrics Using Stata:
http://www.stata-press.com/books/imeus.html


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