If X is endogenous, then all of the interaction terms involving X are also endogenous. However, if Z is an appropriate instrument for X, then A*Z is an appropriate instrument for A*X, B*Z is an appropriate instrument for B*X, and A*B*Z is an appropriate instrument for A*B*X (assuming that A and B are genuinely exogenous). That is true regardless of whether or not these interaction terms appear as exogenous variables in their own right elsewhere in the equation system. You can simply add them (and additional interaction terms if you happen to have more than one Z) to the list of available instruments. The "reduced-form equation" below is confusing because it is not in reduced form; if you do some tedious algebra you can reduce it to an equation in which the right-hand side consists entirely of As, Bs, Zs, stochastic disturbances, and interactions among these, plus constant parameters--and the configuration of interaction terms will imply the strategy above.
-----Original Message-----
From: Political Methodology Society [mailto:[log in to unmask]] On Behalf Of George Krause
Sent: Tuesday, March 21, 2006 4:48 PM
To: [log in to unmask]
Subject: Re: [POLMETH] Instrumental Variables with Interactions (when the Interactions are Partly Comprised as an Endogenous Variable)
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
**********************************************************
Political Methodology E-Mail List
Editor: Karen Long Jusko <[log in to unmask]>
**********************************************************
Send messages to [log in to unmask]
To join the list, cancel your subscription, or modify
your subscription settings visit:
http://polmeth.wustl.edu/polmeth.php
**********************************************************
|
