Before revision:  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.
After revision:  By contrast, as any omnibus test, we cannot distinguish between a 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 therefore designed to detect mis-specification.