Interval Regression Models with Endogenous Explanatory Variables

We consider the estimation of linear models where the dependent variable is observed by intervals and some continuous regressors may be endogenous. Our approach, an IV version of the technique devised by Stewart (Rev Econ Stud 50(3):737–753, 1983), is fully parametric and two estimators are proposed: a two-step estimator and a limited-information maximum-likelihood estimator. The results can be summarized as follows: the two-step estimator has an intuitive appeal, and a Monte Carlo experiment suggests that its relative efficiency is rather satisfactory. The limited-information maximum-likelihood estimator, however, is probably simpler to implement and has the advantage of providing a framework in which several testing procedures are more straightforward to perform. The application of two-stage least squares to a proxy of the dependent variable built by taking midpoints, on the other hand, leads to inconsistent estimates; Monte Carlo evidence suggests that the bias arising from the “midpoint” technique is much worse than the effect of distributional misspecification. An example application is also included, which uses Australian data on migrants’ remittances; endogeneity effects are substantial and using conventional estimation methods leads to substantially misleading inference.