# Re: st: Reduced form specification for endogenous interaction variable

 From "Mark Schaffer" To statalist@hsphsun2.harvard.edu Subject Re: st: Reduced form specification for endogenous interaction variable Date Thu, 14 Oct 2004 13:38:59 +0100

```Yvonne,

Date sent:      	Thu, 14 Oct 2004 11:23:49 +0200
From:           	"" <mhnyvo001@mail.uct.ac.za>
To:             	"" <statalist@hsphsun2.harvard.edu>
Subject:        	st: Reduced form specification for endogenous interaction variable

> Hi,
> This is a follow up to a response you gave on your website (on 26 August,
> 2004) to someone querying  'RE: Interaction with an instrumented variable -
> IV'.
>
> The person with the query has the following specification and query,
> "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)?"
>
> And this was your response,
> "You've got two endogenous variables, just as if you had another endogenous
> variable X4 and you did not realize that X4=X2*X3.  But if Z is a valid
> instrument, and X3 is truly exogenous, then Z*X3 is a valid instrument, too."
>
> However, its not clear from this how you go about estimating the reduced
> form.  What would you use as your LHS variable in the reduced form.
>
> Would the specification for the reduced form for the endogenous interaction
> variable look as follows,
> X2*X3=b0 + Z*X3 + b1*X1 + b2*X2 + b3*X3 + u?

You forgot Z, the reduced form coefficient on Z*X3 is missing, and
you should use different symbols for your reduced form coefficients.
Say the b's are the structural coefficients, and a's are RF coeffs.
Then the RF is:

X2*X3=a0 + a1*Z + a2*Z*X3 + a3*X1 + a4*X2 + a5*X3 + u

--Mark
>
> Regards,
> Yvonne Mhango
> University of Cape Town
> South Africa
>
>
>
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Prof. Mark E. Schaffer
Director
Centre for Economic Reform and Transformation
Department of Economics
School of Management & Languages
Heriot-Watt University, Edinburgh EH14 4AS  UK
44-131-451-3494 direct
44-131-451-3008 fax