st: xtivreg: Interaction of endogenuous treatment and (exogenuous) weather variable as instrument?

[Henry Arnfeld <harnfeld@gmail.com>]

Dear all,

I have been puzzled by whether I can use an interaction term as an
instrumental variable.

The reduced form relationship I like to study is the following:

Y_dt = a_d + b_t + g*Z_dt + b*T_dt + c*Z_dt*T_dt +e_dt

I am interested in the coffecicient on the interaction term Z_dt*T_dt
, which tells me how the relationship between the weather variable
Z_dt and the left hand side changed, after the treatment. I want to
argue that the reason for this change is due to an effect of Z_dt*T_dt
on a variable w_dt.

The variable w_dt is likely to be endogenuous to the left-hand side;
also the treatment indicator T is endogenuous to the left hand side as
well.
The variable Z_dt (some weather variable) may have a direct effect on
Y_dt other than through w_dt, so it is not a valid instrument in
itself.

The process for w_dt is:

w_dt = a_d + b_t + b*T_dt + c*Z_dt + d*T_dt*Z_dt +e_dt

The product of Z_dt and T_dt is likely to be exogenuous to Y_dt as
treatment assignment was not a function of the Z; reverse causality is
also not an issue as Z is a weather variable.

Can I use solely the interaction term Z_dt * T_dt as an instrument for w_dt?

I.e. in stata I would try the following:

xtivreg2 Y Z T (w = ZT)  , fe  cluster(dd)

This means that I still allow for a direct effect of Z and T on the
left-hand side, and explain the variable of interest w only through
the variation I have in the product Z*T.

I am not sure whether this makes any sense -- does anybody have some
thoughts on this approach, whether it is valid? I have not come across
an approach of this form...

Thanks a lot for your help and thoughts,

Henry
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