### 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|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 the 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 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.

Original language | English |
---|---|

Pages (from-to) | 175-194 |

Number of pages | 20 |

Journal | Econometric Reviews |

Volume | 24 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2005 Aug 8 |

Externally published | Yes |

### Fingerprint

### Keywords

- Imputation
- Monotonicity
- Non-response
- Orthant dependence
- Sample selection

### ASJC Scopus subject areas

- Economics, Econometrics and Finance(all)

### Cite this

**Monotonicity conditions and inequality imputation for sample-selection and non-response problems.** / Lee, Myoung-jae.

Research output: Contribution to journal › Article

*Econometric Reviews*, vol. 24, no. 2, pp. 175-194. https://doi.org/10.1081/ETC-200067910

}

TY - JOUR

T1 - Monotonicity conditions and inequality imputation for sample-selection and non-response problems

AU - Lee, Myoung-jae

PY - 2005/8/8

Y1 - 2005/8/8

N2 - 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 the 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 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.

AB - 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 the 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 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.

KW - Imputation

KW - Monotonicity

KW - Non-response

KW - Orthant dependence

KW - Sample selection

UR - http://www.scopus.com/inward/record.url?scp=22944472683&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=22944472683&partnerID=8YFLogxK

U2 - 10.1081/ETC-200067910

DO - 10.1081/ETC-200067910

M3 - Article

AN - SCOPUS:22944472683

VL - 24

SP - 175

EP - 194

JO - Econometric Reviews

JF - Econometric Reviews

SN - 0747-4938

IS - 2

ER -