Dear Fellow POLMETHers:
A doctoral student in my department has encountered the following
methodological problem that she brought to my attention. In a nutshell, the
problem pertains to the proper specification of an instrumental variable
regression containing an endogenous variable that is interacted with
exogenous covariates. Her aim is to arrive at an estimate of X (X_hat) which
will be void of endogeneity bias. The problem is posed below in the
original structural equation to be estimated:
Y_t = b_0 + b_1*X_t + b_2*A + b_3*B + b_4*C + b_5*(A_t*X_t) +
b_6*(B_t*X_t) + b_7*(A_t*B_t) + b_8*(A_t*B_t*X_t) (eq. 1)
+ b_9*D_t + b_10*E_t + å_t
where the dependent variable (Y_t) is a function of an intercept (a), and a
series of additive variables (X, A, B, C, D, E) and corresponding
coefficient terms (b_1, b_2, b_3, b_4, b_9, b_10), three two-way interaction
terms (A_t*X_t, B_t*X_t, A_t*B_t ) and their corresponding coefficients
(b_5, b_6, and b_7), and a single three-way interaction term (A_t*B_t*X_t)
whose relationship with Y_t is captured by the coefficient (b_8) - and å is
the residual disturbance term.
If one presumes that X_t is endogenous (i.e., Y_t causes X_t), then we
need to obtain an instrument for X_t that is truly exogenous. However, the
crux of her problem is handling the instrumentation of X_t in presence of
interaction terms which are partly determined by X_t when estimating the
instrumental variable regression equation (where X_t is the dependent
variable, and its predicted value is utilized for estimating (eq. 1) listed
above. That is, the reduced-form equation is given by:
X_t = c_0 + c_1*Z_t
+ c_2*A + c_3*B + c_4*C + c_5*(A_t*X_t) + c_6*(B_t*X_t) +
c_7*(A_t*B_t) (eq. 2)
+ c_8*(A_t*B_t*X_t) + c_9*D_t + c_10*E_t + í_t
where the endogenous variable (X_t) is a function of an intercept (c), a
(presumably) suitable instrument (denoted by the Z_t variable) and
corresponding coefficient (c_1), plus the exogenous variables from (eq. 1).
The (apparent) problem with (eq. 2) is that the endogenous regressor (i.e.,
X_t) appears on the right hand side via its interaction with A_t (c_5), B_t
(c_6), and A_t & B_t (c_8). This strikes me as a bit odd (and perhaps
tautological) given that we have X_t (in at least some form) appearing on
both the right and left hand sides of (eq. 2) - but maybe I am making much
ado about nothing.
One way to handle this problem [which I can see as being flawed since
it discards relevant information useful and necessary in estimating (eq. 1)]
is to estimate the IV reduced-form equation (eq. 2) without the "offending
interactions" where X_t appears by dropping these variables from the model
[i.e. A_t (c_5), B_t (c_6), and A_t & B_t (c_8)].
Another alternative to dropping the "offending interactions" is to
replace the X_t's used in the interactions with the instrument (Z_t) such
that:
X_t = c_0 + c_1*Z_t
+ c_2*A + c_3*B + c_4*C + c_5*(A_t*Z_t) + c_6*(B_t*Z_t) +
c_7*(A_t*B_t) (eq. 3)
+ c_8*(A_t*B_t*Z_t) + c_9*D_t + c_10*E_t + í_t
This method, however, is problematic since it alters (i.e., is inconsistent
with) the specification of the exogenous portion of the model stated in the
original structural equation model (eq. 1) - plus, the student has found
that estimating (eq. 3) is unsound since it produces nonsensical results
with very large t-statistics and a very high R^squared
statistic near positive unity.
Does anyone have any suggestions for tackling this problem that
circumvents the problems noted above? (unless, of course, they truly are not
problematic?) What would you deem as both an appropriate and feasible way of
estimating the instrumental variable reduced-form equation for using the
predicted values of the endogenous variable (X_t) in a subsequent
structural model given the fact that the components for some of these
(exogenous) interaction terms is a function of the endogenous variable in
question? What would be the logical and/or statistical basis for your
recommendation(s)?
Thank you for your time and consideration regarding my queries to this
problem. Both the student and I will appreciate any advice on how to tackle
this problem that you wish to send along to me.
Best Regards,
George Krause
George A. Krause
Professor
Department of Political Science
University of Pittsburgh
4442 Wesley W. Posvar Hall
230 South Bouquet Street
Pittsburgh, PA 15260 [USA]
412.648.7278 (Office Phone)
412.648.7277 (Department Fax)
412.648.7250 (Department Phone)
[log in to unmask] (E-Mail Address)
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