Publication: Testing conditional monotonicity in the absence of smoothness
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2010-03
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Abstract
This article proposes 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. In fact, we only require continuity of the marginal
distributions. 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. The resulting test is able to detect local alternatives converging to the null at the
parametric rate n-1/2; like the classical goodness-of-.t tests. The article also discusses restricted
estimation procedures under monotonicity and extensions of the basic framework to general
conditional moments, estimated parameters and multivariate explanatory variables. The finite
sample performance of the test is examined by means of a Monte Carlo experiment.
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Stochastic monotonicity, Conditional moments, Least concave majorant, Copula process, Distribution-free in finite samples, Tests invariant to monotone transforms