Summary: Under a sample selection or non-response problem, where a response variable \(y\) is observed only when a condition \(\delta= 1\) is met, the identified mean \(E(y\mid\delta = 1)\) is not equal to the desired mean \(E(y)\). But the monotonicity condition \(E(y\mid\delta = 1)\leq E(y\mid\delta= 0)\) yields an informative bound \(E(y\mid\delta= 1)\leq E(y)\), which is enough for certain inferences. For example, in a majority voting with \(\delta\) being the vote-turnout, it is enough to know if \(E(y)> 0.5\) or not, for which \(E(y\mid\delta= 1) > 0.5\) is sufficient under the monotonicity. The main question is then whether the monotonicity condition is testable, and if not, when it is plausible.
Answering to these queries, when there is a ‘proxy’ variable \(z\) related to \(y\) but fully observed, we provide a test for the monotonicity; when \(z\) is not available, we provide primitive conditions and plausible models for the monotonicity. Going further, when both \(y\) and \(z\) are binary, bivariate monotonicities of the type \(P(y, z\mid\delta=1)\leq P(y,z\mid\delta=0)\) are considered, which can lead to sharper bounds for \(P(y)\). As an empirical example, a data set on the 1996 U.S. presidential election is analyzed to see if the Republican candidate could have won had everybody voted, i.e., to see if \(P(y)> 0.5\), where \(y=1\) is voting for the Republican candidate.