Matching points: Supplementing instruments with covariates in triangular models

Models with a discrete endogenous variable and an instrument that takes on fewer values are common in economics. This paper presents a new method that matches pairs of covariates and instruments to restore the order condition in this scenario and to achieve point-identification of the outcome function. The outcome function must be monotonic in a scalar disturbance, but it can be nonseparable. The first stage for the discrete endogenous variable needs to have a multi-index structure but allows for multidimensional heterogeneity. This paper also provides estimators of the outcome function. Two empirical examples of the return to education and of selection into Head Start illustrate the usefulness and limitations of the method.