Whether interested in the differential impact of a particular factor in various institutional settings or in the heterogeneous effect of policy or random experiment, the empirical researcher confronts a problem if the factor of interest is correlated with an omitted variable. This paper presents the circumstances under which it is possible to arrive at a consistent estimate of the mentioned effect. We find that if the source of heterogeneity and omitted variable are jointly independent of policy or treatment, then the OLS estimate on the interaction term between the treatment and endogenous factor turns out to be consistent.
This paper has benefited from valuable comments and suggestions of Jeffrey Wooldridge. We are also thankful to Tom Coupe, Soiliou Namoro, Jean-Francois Richard, Peter Schmidt for helpful comments and discussions. The authors acknowledge financial support from a UCSUR Steven Manners Research Development Award (University of Pittsburgh).
The popular econometric textbook by Greene (2007) derives the following general result. Suppose the correct specification of the regression model for all observations stacked together is
where i is a n×1 vector of ones. Premultiplying equation (7) by matrix M=I–i(i′i)–1i′, where I is an n×n identity matrix, yields a demeaned version of the original model:
where denotes mean-differenced Z for any Z.  Further, suppose we do not include W into our regression (7) and, therefore, estimate where We make a standard assumption that E(ε*)=0. Then, we can modify the omitted variable formula from Greene (2007) to report the probability limit of
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