Goodness-of-fit tests based on series estimators in nonparametric instrumental regression

Abstract

This paper proposes several tests of restricted specification in nonparametric instrumental regression. Based on series estimators, test statistics are established that allow for tests of the general model against a parametric or nonparametric specification as well as a test of exogeneity of the vector of regressors. The tests are asymptotically normally distributed under correct specification and consistent against any alternative model. Under a sequence of local alternative hypotheses, the asymptotic distribution of the tests is derived. Moreover, uniform consistency is established over a class of alternatives whose distance to the null hypothesis shrinks appropriately as the sample size increases.

Nonparametric estimation in case of endogenous selection

“…Our testing procedure builds on Breunig [2015]. But as we consider a constraint estimation procedure we cannot apply the method of Breunig [2015] directly.…”

“…Our testing procedure builds on Breunig [2015]. But as we consider a constraint estimation procedure we cannot apply the method of Breunig [2015] directly. A constraint sieve testing procedure was proposed by Breunig [2013] but for the specific situation of quantile versions of instrumental variable models.…”

Testing Missing at Random Using Instrumental Variables

“…Below we choose (τ j ) j 1 to be a strictly decreasing which implies that we reduce weight to those generalized Fourier coefficients as basis functions are becoming more nonlinear. Additional weighting of the testing procedure was also used by Horowitz [2006], Blundell and Horowitz [2007], and Breunig [2015]. Our test statistic is based on an empirical analog of the left hand side of (2.3) given (∆ 1 , Y 1 , X 1 , W 1 ), .…”

Specification testing in random coefficient models

“…None of these tests is applicable to specification testing in random coefficient models. Moreover, in contrast to nonparametric specification tests in instrumental variable models in Horowitz () and Breunig () who assumed bounded support, we explicitly allow for regressors with large support which is required to ensure identification of random coefficient models in general. This results in a very different setup as densities have to be allowed to be close to zero, which leads to slower rates of convergence and rules out the approach of density weighting considered in Horowitz ().…”