Nonparametric testing for exogeneity with discrete regressors and instruments

This paper presents new approaches to testing for exogeneity in non-parametric models with discrete regressors and instruments. Our interest is in learning about an unknown structural (conditional mean) function. An interesting feature of these models is that under endogeneity the identifying power of a discrete instrument depends on the number of support points of the instruments relative to that of the regressors, a result driven by the discreteness of the variables. Observing that the simple nonparametric additive error model can be interpreted as a linear regression, we present two test-statistics. For the point identifying model, the test is an adapted version of the standard Wu-Hausman approach. This extends the work of Blundell and Horowitz (2007) to the case of discrete regressors and instruments. For the set identifying model, the Wu-Hausman approach is not available. In this case the test-statistic is derived from a constrained minimization problem. The asymptotic distributions of the test-statistics are derived under the null and fixed and local alternatives. The tests are shown to be consistent, and a simulation study reveals that the proposed tests have satisfactory finite-sample properties.