Testing Conditional Monotonicity in the Absence of Smoothness

• Activity Number: 210
• Type: Invited
• Date/Time: Monday, August 2, 2010 : 2:00 PM to 3:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Miguel A. Delgado*+ and Juan Carlos Escanciano
• Companies: Universidad Carlos III de Madrid and Indiana University
• Keywords: Testing Monotonicity ; Conditional Distribution ; Nonparametric Tests ; Empirical Copula Process ; Least Concave Majorant ; Kiefer Process

Abstract

We propose an omnibus test for monotonicity of nonparametric conditional distributions and its moments. Unlike previous proposals, our method does not require smooth estimation of the derivatives of nonparametric curves and it can be implemented even when the probability densities do not exist. Distinguishing features of our approach are that the test statistic is pivotal under the null and invariant to any monotonic continuous transformation of the explanatory variable in finite samples. The test statistic is the sup-norm of the difference between the empirical copula function and its least concave majorant with respect to the explanatory variable coordinate. We also discuss restricted estimation procedures under monotonicity and extensions of the basic framework to general conditional moments, estimated parameters and multivariate explanatory variables.