This paper provides a unified approach for detecting sample selection in nonparametric conditional mean and quantile functions. In fact, as sample selection leads to a loss of point identification in the nonparametric quantile case, our tests are of particular relevance when interest lies in the conditional distribution.  Our testing strategy consists of a two-step procedure: the first test is an omitted predictor test , where the omitted variable is the propensity score . This test has power against generic n-alternatives , and failure to reject the null implies no selection . By contrast, as any omnibus test, we cannot distinguish between a rejection due to genuine selection or to generic mis-specification, when the omitted variable is correlated with the propensity score. Under the maintained assumption of no selection , our second test is therefore designed to detect mis-specification. This is achieved by a localized version of the first test, using only individuals with propensity score close to one . Although the second step requires `identification at infinity' , we can allow for cases of irregular identification. Finally, our testing procedure does not require any parametric assumptions on neither the outcome nor the selection equation(s) , and all our results in the conditional quantile case hold uniformly across quantile ranks in a compact set. We apply our procedure to test for selection in log hourly wages of females and males in the UK using the UK Family Expenditure Survey  .
This paper provides a unified approach for detecting sample selection in nonparametric conditional quantileand mean functions. Our testing strategy consists of a two-step procedure: the first test is an omitted predictor test with the propensity score as omitted variable . This test has power against n-alternatives . While failure to reject the null implies no selection , we cannot, as any omnibus test, distinguish between rejection due to genuine selection or to misspecification. Since differentiation of the latter has implications for nonparametric (point) identification and estimation of the conditional quantile function , our second test is designed to detect misspecification. Using only individuals with propensity score close to one , this test relies on an `identification at infinity' argument, but accommodates cases of irregular identification. Finally, our testing procedure does not require any parametric assumptions on the selection equation , and all our results in the quantile case hold uniformly across quantile ranks in a compact set. We apply our procedure to test for selection in log hourly wages using UK Family Expenditure Survey data .
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