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

## Author Info

• Myoung-Jae Lee

## Abstract

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&7Cδ = 1) is not equal to the desired mean E(y). But the monotonicity condition E(y&7Cδ = 1) ≤ E(y&7Cδ =  0) yields an informative bound E(y&7Cδ = 1) ≤ E(y), which is enough for certain inferences. For example, in a majority voting with δ being the vote-turnout, it is enough to know if E(y) > 0.5 or not, for which E(y&7Cδ = 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&7Cδ = 1) ≤ P(y, z&7Cδ = 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.

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## Bibliographic Info

Article provided by Taylor & Francis Journals in its journal Econometric Reviews.

Volume (Year): 24 (2005)
Issue (Month): 2 ()
Pages: 175-194

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Handle: RePEc:taf:emetrv:v:24:y:2005:i:2:p:175-194

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## Related research

Keywords: Imputation; Monotonicity; Non-response; Orthant dependence; Sample selection;

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