OLS and IV estimation of regression models including endogenous interaction terms

We analyze a class of linear regression models including interactions of endogenous regressors and exogenous covariates. We show that, under typical conditions regarding higher-order dependencies between endogenous and exogenous regressors, the OLS estimator of the coe¢ cient of the interaction term is consistent and asymptotically normally distributed. Applying heteroskedasticity-consistent covariance matrix estimators, we then show that standard inference based on OLS is valid for the coefficient of the interaction term. Furthermore, we analyze several IV estimators, and
show that an implementation assuming exogeneity of the interaction term is valid under fairly weak conditions. In the more general case, we derive that instruments need to be interacted with the exogenous part of the interaction to achieve identification. Finally, we propose several specfication tests to empirically assess the validity of OLS and IV inference for the interaction model. Using our theoretical results we confirm recent empirical findings on the nonlinear causal relation between financial development and economic growth.