Monotonicity Conditions and Inequality Imputation for Sample Selection and Non-Response Problems

Under a sample selection or non-response problem where a response variable y is observed only when a condition δ=1 is met, the identified mean E(y|δ=1) is not equal to the desired mean E(y). But the monotonicity condition E(y|δ=1)≤E(y|δ=0) yields an informative bound E(y|δ=1)≤E(y), which is enough for certain inferences. For example, in a majority voting with δ being vote-turnout, it is enough to know if E(y)>0.5 or not, for which E(y|δ=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|δ=1)≤P(y,z|δ=0) are considered, which can lead to sharper bounds for P(y). As an empirical example, a data set on the 1996 US 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