Before revision:  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.
After revision:  This test has power against generic n-alternatives , and failure to reject the null implies no selection , we cannot, 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.