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st: Interaction with an instrumented variable - IV


From   "Ngo,PT (pgr)" <P.T.Ngo@lse.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: Interaction with an instrumented variable - IV
Date   Thu, 26 Aug 2004 14:28:32 +0100

Dear Statalisters,

I have a question re. IVs.
There was an earlier posting in the statalist archive, but it was not conclusive. After some correspondence and a discussion today with a colleague in my department, I am still unsure as to what is to be done. So I am putting the question to you.

Y=b0+b1*X1+b2*X2+b3*X3+b4*X2*X3+U

X2 is the endogenous variable, which is instrumented with Z.
X2 is itself interacted with X3.
How does one model the interaction between an instrumented variable (X2 instrumented with Z) and an exogenous variable (X3)?

1. How does one model the first stage of the instrumentation? 
Someone suggested to interact all the instruments with the interaction variables of interest. 
Using ivreg, that would be:
ivreg y (X2 X2_X3= Z Z*X3) X1 X3

2. Alternatively, someone suggested the following solution:
run the first stage regression of all exogenous variables, generate the predicted values and interact these with the other second stage variables of interest. In this case, one has to bootstrap the whole process.

This would give:
program bootit
version 8.0
//stage 1
reg X2 X1 Z X3
predict double X2hat, xb
//stage 2
reg Y X1 X2hat X3 X3*X2hat
end
  
set seed 1001
forvalues i=7/7 {
bootstrap "bootit`i'" _b, reps (5) dots
}

3- a third opinion says that these approaches are not valid as the first-stage equations would be as follows (I omit the coefficients):
X2hat= Z + X1 + X3 
(X2_X3)hat=Z + X1 + X3 + W (that is, I need to find another instrument W

I have tried to find a good reference on this in textbooks but could not find anything specific.
Any help would be most welcome.

Thanks a lot in advance,

Thi Minh

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