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# Parameter Set Inference in a Class of Econometric Models

## Author

Listed:
• E. Tamer
• V. Chernozhukov
• H. Hong

## Abstract

We provide new methods for inference in econometric models where the parameter of interest is a set. These models arise in many situations where point identification requires strong (and sometimes untestable) assumptions. Every parameter vector in the set of interest represents a feasible economic model that generated the data. Our point of departure is {\it set} $\Theta_I$ that minimizes a given population criterion function $Q(\theta)$. To obtain valid inferences on $\Theta_I$, we characterize first the large sample properties of the sample criterion function $Q_n(\theta)$. These are then used to construct confidence sets for $\Theta_I$ that contain this set with a given prespecified probability. The method we use picks an appropriate level set of the objective function by cutting off'' this function at a level that corresponds to an appropriately chosen percentile of a key coverage statistic." This is a likelihood ratio type quantity (and reduces to the usual likelihood ratio when the set $\Theta_I$ is a singleton). Our confidence set then is an appropriately chosen level set of the objective function. This general framework is illustrated in two important examples: regressions where data on outcomes is observed in intervals, and method of moments where the criterion function is minimized on a set. A subsampling procedure that is used to obtain the asymptotic critical values for the coverage statistic is provided and shown to be consistent. For the two examples, we also provide indirect bootstrap procedures that are based on resampling a dual statistic that has the same asymptotic distribution as the coverage statistic. We illustrate our methods in an empirical example of returns to schooling using data from the CPS

## Suggested Citation

• E. Tamer & V. Chernozhukov & H. Hong, 2004. "Parameter Set Inference in a Class of Econometric Models," Econometric Society 2004 North American Winter Meetings 382, Econometric Society.
• Handle: RePEc:ecm:nawm04:382
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## Citations

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Cited by:

1. Aguirregabiria, Victor & Mira, Pedro, 2010. "Dynamic discrete choice structural models: A survey," Journal of Econometrics, Elsevier, vol. 156(1), pages 38-67, May.
2. repec:pit:wpaper:428 is not listed on IDEAS
3. Kreider, Brent & Pepper, John V., 2011. "Identification of Expected Outcomes in a Data Error Mixing Model With Multiplicative Mean Independence," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(1), pages 49-60.
4. Andrew Chesher, 2005. "Nonparametric Identification under Discrete Variation," Econometrica, Econometric Society, vol. 73(5), pages 1525-1550, September.
5. Molinari, Francesca, 2010. "Missing Treatments," Journal of Business & Economic Statistics, American Statistical Association, vol. 28(1), pages 82-95.
6. Arie Beresteanu & Francesca Molinari, 2008. "Asymptotic Properties for a Class of Partially Identified Models," Econometrica, Econometric Society, vol. 76(4), pages 763-814, July.
7. Guido W. Imbens & Jeffrey M. Wooldridge, 2009. "Recent Developments in the Econometrics of Program Evaluation," Journal of Economic Literature, American Economic Association, vol. 47(1), pages 5-86, March.
8. Rosen, Adam M., 2008. "Confidence sets for partially identified parameters that satisfy a finite number of moment inequalities," Journal of Econometrics, Elsevier, vol. 146(1), pages 107-117, September.
9. Molinari, Francesca, 2008. "Partial identification of probability distributions with misclassified data," Journal of Econometrics, Elsevier, vol. 144(1), pages 81-117, May.
10. Andrews, Donald W.K. & Guggenberger, Patrik, 2009. "Validity Of Subsampling And “Plug-In Asymptotic” Inference For Parameters Defined By Moment Inequalities," Econometric Theory, Cambridge University Press, vol. 25(3), pages 669-709, June.
11. Hyungsik Roger Moon & Frank Schorfheide, 2006. "Boosting Your Instruments: Estimation with Overidentifying Inequality Moment Conditions," IEPR Working Papers 06.56, Institute of Economic Policy Research (IEPR).
12. Patrick Bajari & C. Lanier Benkard & Jonathan Levin, 2007. "Estimating Dynamic Models of Imperfect Competition," Econometrica, Econometric Society, vol. 75(5), pages 1331-1370, September.
13. Guido W. Imbens & Charles F. Manski, 2004. "Confidence Intervals for Partially Identified Parameters," Econometrica, Econometric Society, vol. 72(6), pages 1845-1857, November.
14. Patrik Guggenberger, 2006. "The limit of finite sample size and a problem with subsampling (joint with D.W.K. Andrews), June 2005, this version March 2007," UCLA Economics Online Papers 372, UCLA Department of Economics.
15. Adam Rosen, 2007. "Identification and estimation of firms' marginal cost functions with incomplete knowledge of strategic behavior," CeMMAP working papers CWP03/07, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
16. Fan, Yanqin & Park, Sang Soo, 2009. "Partial identification of the distribution of treatment effects and its confidence sets," MPRA Paper 37148, University Library of Munich, Germany.
17. Fan, Yanqin & Park, Sang Soo, 2010. "Confidence sets for some partially identified parameters," MPRA Paper 37149, University Library of Munich, Germany.
18. J. Stoye, 2009. "Charles F. Manski, Identification for Prediction and Decision (Harvard University Press 2007)," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 24(5), pages 857-862.
19. Andrew M. Cohen & Beth A. Freeborn & Brian McManus, 2007. "Competition and Crowding-Out among Public, Non-Profit and For-Profit Organizations: Evidence from Outpatient Substance Abuse Treatment," Working Papers 52, Department of Economics, College of William and Mary.
20. Esteban-Bravo, Mercedes & Vidal-Sanz, Jose M., 2007. "Worst-case estimation for econometric models with unobservable components," Computational Statistics & Data Analysis, Elsevier, vol. 51(7), pages 3330-3354, April.

### Keywords

Set Inference; level sets; Subsampling;
All these keywords.

### JEL classification:

• C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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