Testing for Monotonicity in Unobservables under

Monotonicity in a scalar unobservable is a now common assumption when modeling het-erogeneity in structural models. Among other things, it allows one to recover the underlying structural function from certain conditional quantiles of observables. Nevertheless, monotonic-ity is a strong assumption and in some economic applications unlikely to hold. Its failure can have substantive adverse consequences; having a test for this hypothesis is hence desirable. This paper provides such a test for cross-section data. We show how to exploit an exclusion restriction together with a conditional independence assumption, which in the binary treat-ment literature is commonly called unconfoundedness, to construct a test. Our statistic is asymptotically normal under local alternatives and consistent against nonparametric alterna-tives. 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 di¤erences and to study whether Engel curves are monotonically driven by a scalar unobservable