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

## Author

Listed:
• 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.

## Suggested Citation

• Myoung-Jae Lee, 2005. "Monotonicity Conditions and Inequality Imputation for Sample-Selection and Non-Response Problems," Econometric Reviews, Taylor & Francis Journals, vol. 24(2), pages 175-194.
• Handle: RePEc:taf:emetrv:v:24:y:2005:i:2:p:175-194 DOI: 10.1081/ETC-200067910
as

File URL: http://www.tandfonline.com/doi/abs/10.1081/ETC-200067910

As the access to this document is restricted, you may want to search for a different version of it.

## References listed on IDEAS

as
1. Pierre Perron & Serena Ng, 1996. "Useful Modifications to some Unit Root Tests with Dependent Errors and their Local Asymptotic Properties," Review of Economic Studies, Oxford University Press, pages 435-463.
2. Harris, David, 1997. "Principal Components Analysis of Cointegrated Time Series," Econometric Theory, Cambridge University Press, pages 529-557.
3. Peter C. B. Phillips & Hyungsik R. Moon, 1999. "Linear Regression Limit Theory for Nonstationary Panel Data," Econometrica, Econometric Society, pages 1057-1112.
4. Phillips, P C B, 1991. "Optimal Inference in Cointegrated Systems," Econometrica, Econometric Society, pages 283-306.
5. Peter C. B. Phillips & Bruce E. Hansen, 1990. "Statistical Inference in Instrumental Variables Regression with I(1) Processes," Review of Economic Studies, Oxford University Press, vol. 57(1), pages 99-125.
6. Phillips, Peter C B, 1995. "Fully Modified Least Squares and Vector Autoregression," Econometrica, Econometric Society, pages 1023-1078.
7. Yongcheol Shin & Ron P Smith & Mohammad Hashem Pesaran, 1998. "Pooled Mean Group Estimation of Dynamic Heterogeneous Panels," ESE Discussion Papers 16, Edinburgh School of Economics, University of Edinburgh.
8. Schwert, G William, 2002. "Tests for Unit Roots: A Monte Carlo Investigation," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 5-17, January.
9. Arellano, M, 1987. "Computing Robust Standard Errors for Within-Groups Estimators," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 49(4), pages 431-434, November.
10. Chang, Yoosoon, 2004. "Bootstrap unit root tests in panels with cross-sectional dependency," Journal of Econometrics, Elsevier, pages 263-293.
11. Newey, Whitney & West, Kenneth, 2014. "A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix," Applied Econometrics, Publishing House "SINERGIA PRESS", pages 125-132.
12. Wu, Jyh-Lin & Wu, Shaowen, 2001. "Is Purchasing Power Parity Overvalued?," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 33(3), pages 804-812, August.
13. O'Connell, Paul G. J., 1998. "The overvaluation of purchasing power parity," Journal of International Economics, Elsevier, pages 1-19.
14. Engle, R.F. & Yoo, B.S., 1989. "Cointegrated Economic Time Series: A Survey With New Results," Papers 8-89-13, Pennsylvania State - Department of Economics.
15. repec:cup:etheor:v:13:y:1997:i:4:p:529-57 is not listed on IDEAS
16. Johansen, Soren, 1991. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models," Econometrica, Econometric Society, vol. 59(6), pages 1551-1580, November.
17. Peter Pedroni, 2000. "Fully Modified OLS for Heterogeneous Cointegrated Panels," Department of Economics Working Papers 2000-03, Department of Economics, Williams College.
18. Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254.
19. Johansen, Soren & Juselius, Katarina, 1994. "Identification of the long-run and the short-run structure an application to the ISLM model," Journal of Econometrics, Elsevier, pages 7-36.
20. Rolf Larsson & Johan Lyhagen & Mickael Lothgren, 2001. "Likelihood-based cointegration tests in heterogeneous panels," Econometrics Journal, Royal Economic Society, vol. 4(1), pages 1-41.
21. Johansen, Soren, 1995. "Likelihood-Based Inference in Cointegrated Vector Autoregressive Models," OUP Catalogue, Oxford University Press, number 9780198774501.
22. Pentti Saikkonen, 1999. "Testing normalization and overidentification of cointegrating vectors in vector autoregressive processes," Econometric Reviews, Taylor & Francis Journals, vol. 18(3), pages 235-257.
23. Levin, Andrew & Lin, Chien-Fu & James Chu, Chia-Shang, 2002. "Unit root tests in panel data: asymptotic and finite-sample properties," Journal of Econometrics, Elsevier, vol. 108(1), pages 1-24, May.
24. Boswijk, H Peter, 1996. "Testing Identifiability of Cointegrating Vectors," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(2), pages 153-160, April.
Full references (including those not matched with items on IDEAS)

### Keywords

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

## Corrections

All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:taf:emetrv:v:24:y:2005:i:2:p:175-194. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (). General contact details of provider: http://www.tandfonline.com/LECR20 .

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

We have no references for this item. You can help adding them by using this form .

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.