st: Multiple endogenous variables IV. Estimating the first stage regression with only a subset of the instruments

[Nicolai Borgen <nicolai.borgen@sosgeo.uio.no>]

I have a question regarding the use of instrumental variables when
having multiple endogenous variables. I am estimating the economic
return of attending different educational institutions X1, X2 and X3.
For simplicity, the models are presented without control variables and
constant.

[A1]		Y= δ1*X1 + δ2*X2 + δ3*X3 + ε

Since I have strong reasons to suspect that X1, X2 and X3 is correlated
with ε, it is necessary to use an instrumental variable approach to
estimate the return. I therefore use proximity (in km) between
municipality of adolescence and these institutions as instruments (Z1,
Z2 and Z3). Using regression commands such as ivregress and ivreg2 in
STATA, all instruments are included in the first stage for each
endogenous variable X1, X2 and X3:

[B1]		X1 = B1*Z1 + B2*Z2 + B3*Z3 + ε
[C1]		X2 = B1*Z1 + B2*Z2 + B3*Z3 + ε
[D1]		X3 = B1*Z1 + B2*Z2 + B3*Z3 + ε

My problem occurs because X1 and X2 are located in the same city, and
Z1 and Z2 are therefore perfectly collinear. Thus, in [B1] and [C1]
either Z1 or Z2 is dropped. My model is therefore basically
under-identified.  Based on this, I have the following two questions:

(1) Is it possible to estimate the first stage regression using a
subset of the instruments?

[B2]		X1 = B1*Z1 + ε
[C2]		X2 = B2*Z2 + ε
[D2]		X3 = B3*Z3 + ε


(2) This STATA page http://www.stata.com/support/faqs/stat/ivr_faq.html
shows an example of how to perform the two-step computations for the
instrumental variable estimator without using ivregress or ivreg2. Is
this a feasible solution? Are there any STATA commands I can use that do
this?

Many thanks,
Nicolai Borgen
University of Oslo
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