Monotonicity in a scalar unobservable is a now common assumption in economic theory and applications. Among other things, it allows one to recover the underlying structural function from certain conditional quantiles of observables. Nevertheless, monotonicity is a strong assumption, and its failure can have substantive adverse consequences for structural inference. So far, there are no generally applicable nonparametric specification tests designed to detect monotonicity failure. This paper provides such a test for cross-section data. We show how to exploit an exclusion restriction together with a conditional independence assumption, plausible in a variety of applications, to construct a test. Our statistic is asymptotically normal under local alternatives and consistent against nonparametric alternatives violating the conditional quantile representation. Monte Carlo experiments show that a suitable bootstrap procedure yields tests with reasonable level behavior and useful power. We apply our test to study the role of unobserved ability in determining Black-White wage differences and to study whether Engel curves are monotonically driven by a scalar unobservable
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