st: Interaction Variables as Instruments

[Jessie C <jessiecoh@gmail.com>]

Dear statalist,

I am having some programming trouble with 2SLS and would appreciate any help :)

I want to estimate a 2 stage least squares model.  The goal is to
estimate the relationship between x and y.  x is not exogenous.

Consider 2 groups (e.g. men and women), represented by z_1.  Consider
a policy that affects one group relative to the other, represented by
z_2.  The first stage is therefore:
x = constant + a*z_1 + b*z_2 + c*z_1*z_2 + e

I am not sure what the second-stage should be and how to implement it
in Stata. Would the second stage be
y = constant + k*x + u, instrument x with the z_1, z_2, and z_1*z_2
y = constant + k*x + l*z_1 + u, instrument x with z_2 and z_1*z_2
y = constant + k*x + l*z_1 + m*x*z_1 + u, instrument x and x*z_1 with
z_2 and z_1*z_2

I am having difficulty because z_1 is not the exogenous source of variation.

The goal is to say that the gap between group 1 (e.g. men) and group 2
(e.g. women) with regards to y has narrowed because of a reduction in
the corresponding gap in x, and the reduction in the gap in x is the
result of exogenous policy z_2.

Any help would be very much appreciated :)

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