Estimation of Conditional Ranks and Tests of Exogeneity in Nonparametric Nonseparable Models

Abstract

Consider a nonparametric nonseparable regression model Y = ϕ(Z, U), where ϕ(Z, U) is strictly increasing in U and U ∼ U[0, 1]. We suppose that there exists an instrument W that is independent of U. The observable random variables are Y, Z and W, all one-dimensional. We construct test statistics for the hypothesis that Z is exogenous, i.e. that U is independent of Z. The test statistics are based on the observation that Z is exogenous if and only if V = F Y|Z (Y|Z) is independent of W, and hence they do not require the estimation of the function ϕ. The asymptotic properties of the proposed tests are proved, and a bootstrap approximation of the critical values of the tests is shown to be consistent and to work for finite samples via simulations. An empirical example using the U.K. Family Expenditure Survey is also given. As a by-product of our results we obtain the asymptotic properties of a kernel estimator of the distribution of V, which equals U when Z is exogenous. We show that this estimator converges to the uniform distribution at faster rate than the parametric n −1/2 -rate.

Nonparametric Instrumental Regression With Right Censored Duration Outcomes

“…Assumption (G) amounts to strong conditional completeness of Z given U and W as introduced in Chernozhukov and Hansen (2005) and studied in Fève et al (2018). Under this hypothesis we can now show the global identification of our model.…”

“…The aim is to find conditions under which the system (2.2) has a unique solution. Our presentation is borrowed from Appendix A of Fève et al (2018). Consider the density of (U, Z) conditional on W .…”

Nonparametric estimation of accelerated failure-time models with unobservable confounders and random censoring

“…We focus here on the latter that are easier to derive and are more easily interpretable. Interested readers can refer to Centorrino et al [14], Chernozhukov and Hansen [21], and Fève et al [30], for a discussion of global identification conditions in this context.…”

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“…A test of monotonicity in unobservables of ϕ has been proposed by Hoderlein et al [2016] but requires conditional exogeneity of Z and hence, is not related to instrumental variables methodology. Recently and independently of this paper, Fève et al [2018] developed a test of whether Z is independent of the nonseparable disturbance V in the model (1.2). Our test statistic is based on the L 2 -norm of the empirical conditional quantile restriction and involves sieve methodology.…”