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UNIFORM INFERENCE IN A GENERALIZED INTERVAL ARITHMETIC CENTER AND RANGE LINEAR MODEL

Published online by Cambridge University Press:  13 August 2021

Yanqin Fan  and
Xuetao Shi
Yanqin Fan*
Affiliation:
University of Washington
Xuetao Shi
Affiliation:
University of Sydney
*
Address correspondence to Yanqin Fan, Department of Economics, University of Washington, Seattle, Washington 98195, USA; e-mail: [email protected].
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Abstract

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Via generalized interval arithmetic, we propose a Generalized Interval Arithmetic Center and Range (GIA-CR) model for random intervals, where parameters in the model satisfy linear inequality constraints. We construct a constrained estimator of the parameter vector and develop asymptotically uniformly valid tests for linear equality constraints on the parameters in the model. We conduct a simulation study to examine the finite sample performance of our estimator and tests. Furthermore, we propose a coefficient of determination for the GIA-CR model. As a separate contribution, we establish the asymptotic distribution of the constrained estimator in Blanco-Fernández (2015, Multiple Set Arithmetic-Based Linear Regression Models for Interval-Valued Variables) in which the parameters satisfy an increasing number of random inequality constraints.

Type
ARTICLES
Information
Econometric Theory , Volume 39 , Issue 1 , February 2023 , pp. 27 - 69
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

We are grateful to Peter C. B. Phillips, Patrik Guggenberger, and two anonymous referees for their insightful comments on the previous version of this paper. We also thank Aman Ullah, participants of the 8th International Symposium on Econometric Analysis and Forecasting at Dongbei University of Finance and Economics, statistics seminar at Northeast Normal University, the 2018 International Symposium of Quantitative Economics at Jilin University, the 2018 Seattle–Vancouver Econometrics Conference at Simon Fraser University, and seminars at University of Alberta, Korea University, Seoul National University, UC Irvine, and UC Riverside for helpful discussions. This work was facilitated by the Hyak supercomputer system at the University of Washington.

References

REFERENCES

Andrews, D. W. (1999) Estimation when a parameter is on a boundary. Econometrica 67(6), 1341–1383.CrossRefGoogle Scholar
Andrews, D. W. (2001) Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica 69(3), 683–734.CrossRefGoogle Scholar
Andrews, D. W. & Cheng, X. (2012) Estimation and inference with weak, semi-strong, and strong identification. Econometrica 80(5), 2153–2211.Google Scholar
Andrews, D. W. & Cheng, X. (2014) GMM estimation and uniform subvector inference with possible identification failure. Econometric Theory 30(2), 287–333.CrossRefGoogle Scholar
Andrews, D. W., Cheng, X. & Guggenberger, P. (2020) Generic results for establishing the asymptotic size of confidence sets and tests. Journal of Econometrics 218(2), 496–531.CrossRefGoogle Scholar
Artstein, Z. & Vitale, R. A. (1975) A strong law of large numbers for random compact sets. The Annals of Probability, 879–882.Google Scholar
Aumann, R. J. (1965) Integrals of set-valued functions. Journal of Mathematical Analysis and Applications 12(1), 1–12.CrossRefGoogle Scholar
Beresteanu, A., Molchanov, I. & Molinari, F. (2011) Sharp identification regions in models with convex moment predictions. Econometrica 79(6), 1785–1821.Google Scholar
Beresteanu, A. & Molinari, F. (2008) Asymptotic properties for a class of partially identified models. Econometrica 76(4), 763–814.CrossRefGoogle Scholar
Berger, R. L. & Boos, D. D. (1994) P values maximized over a confidence set for the nuisance parameter. Journal of the American Statistical Association 89(427), 1012–1016.Google Scholar
Bertoluzza, C., Corral Blanco, N. & Salas, A. (1995) On a new class of distances between fuzzy numbers. Mathware & Soft Computing 2(2), 71–84.Google Scholar
Blanco-Fernández, A., Colubi, A. & González-Rodríguez, G. (2012) Confidence sets in a linear regression model for interval data. Journal of Statistical Planning and Inference 142(6), 1320–1329.CrossRefGoogle Scholar
Blanco-Fernández, A., Corral, N. & González-Rodríguez, G. (2011) Estimation of a flexible simple linear model for interval data based on set arithmetic. Computational Statistics & Data Analysis 55(9), 2568–2578.CrossRefGoogle Scholar
Blanco-Fernández, A., García-Bárzana, M., Colubi, A., & Kontoghiorghes, E.J. (2015) Multiple Set Arithmetic-Based Linear Regression Models for Interval-Valued Variables. Unpublished Working Paper, Department of Statistics and OR and Mathematics Didactics, University of Oviedo.Google Scholar
Bontemps, C., Magnac, T. & Maurin, E. (2012) Set identified linear models. Econometrica 80(3), 1129–1155.Google Scholar
Cheng, X. (2015) Robust inference in nonlinear models with mixed identification strength. Journal of Econometrics 189(1), 207–228.CrossRefGoogle Scholar
Chernozhukov, V. (2005) Extremal quantile regression. Annals of Statistics 33, 806–839.CrossRefGoogle Scholar
Chernozhukov, V. & Fernández-Val, I. (2011) Inference for extremal conditional quantile models, with an application to market and birthweight risks. The Review of Economic Studies 78(2), 559–589.CrossRefGoogle Scholar
Chernozhukov, V., Fernández-Val, I., Hahn, J. & Newey, W. (2013) Average and quantile effects in nonseparable panel models. Econometrica 81(2), 535–580.Google Scholar
Chernozhukov, V. & Hong, H. (2002) Likelihood Inference in a Class of Nonregular Econometric Models. Working Paper No. 02-05, Department of Economics, MIT Press.CrossRefGoogle Scholar
Chernozhukov, V. & Hong, H. (2004) Likelihood estimation and inference in a class of nonregular econometric models. Econometrica 72(5), 1445–1480.CrossRefGoogle Scholar
Cowling, B. J., Fang, V. J., Riley, S., Peiris, J. M. & Leung, G. M. (2009) Estimation of the serial interval of influenza. Epidemiology 20(3), 344.CrossRefGoogle ScholarPubMed
Debreu, G. (1967) Integration of correspondences. In Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 351–372. University of California Press, Berkeley.Google Scholar
Demsetz, H. (1968) The cost of transacting. The Quarterly Journal of Economics 82(1), 33–53.CrossRefGoogle Scholar
Du, Z., Xu, X., Wu, Y., Wang, L., Cowling, B. J. & Meyers, L. A. (2020) Serial interval of COVID-19 among publicly reported confirmed cases. Emerging Infectious Diseases 26(6), 1341.CrossRefGoogle ScholarPubMed
Fréchet, M. (1948) Les éléments aléatoires de nature quelconque dans un espace distancié. Annales de l’institut Henri Poincaré 10(3), 215–310.Google Scholar
Ganyani, T., Kremer, C., Chen, D., Torneri, A., Faes, C., Wallinga, J., & Hens, N. (2020) Estimating the generation interval for coronavirus disease (COVID-19) based on symptom onset data, march 2020. Eurosurveillance 25(17), 2000257.CrossRefGoogle ScholarPubMed
García-Ascanio, C. & Maté, C. (2010) Electric power demand forecasting using interval time series: A comparison between var and imlp. Energy Policy 38(2), 715–725.CrossRefGoogle Scholar
Geyer, C. J. (1994) On the asymptotics of constrained M-estimation. Annals of Statistics 22, No. 4, 1993–2010.CrossRefGoogle Scholar
Geyer, C.J. (1996) On the asymptotics of convex stochastic optimization. Unpublished Manuscript, Department of Statistics, University of Minnesota.Google Scholar
Golan, A. & Ullah, A. (2017) Interval estimation: An information theoretic approach. Econometric Reviews 36(6-9), 781–795.CrossRefGoogle Scholar
González-Rivera, G. & Lin, W. (2013) Constrained regression for interval-valued data. Journal of Business & Economic Statistics 31(4), 473–490.CrossRefGoogle Scholar
Gourieroux, C., Holly, A. & Monfort, A. (1982) Likelihood ratio test, wald test, and kuhn-tucker test in linear models with inequality constraints on the regression parameters. Econometrica 50(1), 63–80.CrossRefGoogle Scholar
Han, A., Hong, Y., & Wang, S. (2012) Autoregressive Conditional Models for Interval-Valued Time Series Data. Unpublished Manuscript, Department of Economics, Cornell University.Google Scholar
Han, A., Yongmiao, H., Lai, K. K. & Shouyang, W. (2008) Interval time series analysis with an application to the sterling-dollar exchange rate. Journal of Systems Science and Complexity 21(4), 558–573.CrossRefGoogle Scholar
Hukuhara, M. (1967) Integration des applications mesurables dont la valeur est un compact convexe. Funkcialaj Ekvacioj 10(3), 205–223.Google Scholar
Judge, G., Yancey, T., Bock, M. & Bohrer, R. (1984) The non-optimality of the inequality restricted estimator under squared error loss. Journal of Econometrics 25(1-2), 165–177.CrossRefGoogle Scholar
Kallenberg, O. (1983) Random Measures. Akademie-Verlag.CrossRefGoogle Scholar
Kaucher, E. (1980) Interval analysis in the extended interval space ir. Computing Supplementum 2, 33–49.CrossRefGoogle Scholar
Kenah, E., Lipsitch, M. & Robins, J. M. (2008) Generation interval contraction and epidemic data analysis. Mathematical Biosciences 213(1), 71–79.CrossRefGoogle ScholarPubMed
Ketz, P. (2018) Subvector inference when the true parameter vector may be near or at the boundary. Journal of Econometrics 207(2), 285–306.CrossRefGoogle Scholar
Knight, K. (1999) Epi-convergence in distribution and stochastic equi-semicontinuity. Unpublished Manuscript, Department of Statistics, University of Toronto.Google Scholar
Knight, K. (2001) Limiting distributions of linear programming estimators. Extremes 4(2), 87–103.CrossRefGoogle Scholar
Knight, K. (2006) Asymptotic theory for M-estimators of boundaries. In Sperlich, S., Härdle, W., & Aydinli, G. (eds.), The Art of Semiparametrics, pp. 1–21. Physica-Verlag.Google Scholar
Körner, R. (1997) On the variance of fuzzy random variables. Fuzzy Sets and Systems 92(1), 83–93.CrossRefGoogle Scholar
Körner, R. & Näther, W. (2002) On the variance of random fuzzy variables. In Bertoluzza, C., Gil, M.Á., & Ralescu, D.A. (eds.), Statistical Modeling, Analysis and Management of Fuzzy Data, pp. 25–42. Physica.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G. & Rootzén, H. (1987) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag.Google Scholar
Lee, C. M. (1993) Market integration and price execution for nyse-listed securities. The Journal of Finance 48(3), 1009–1038.CrossRefGoogle Scholar
Liew, C. K. (1976) Inequality constrained least-squares estimation. Journal of the American Statistical Association 71(355), 746–751.CrossRefGoogle Scholar
Lin, W. & González-Rivera, G. (2016) Interval-valued time series models: Estimation based on order statistics exploring the agriculture marketing service data. Computational Statistics & Data Analysis 100, 694–711.CrossRefGoogle Scholar
Manski, C. F. & Tamer, E. (2002) Inference on regressions with interval data on a regressor or outcome. Econometrica 70(2), 519–546.CrossRefGoogle Scholar
Markov, S. (1996) On directed interval arithmetic and its applications. The Journal of Universal Computer Science 200(1), 514–526.CrossRefGoogle Scholar
McCloskey, A. (2017) Bonferroni-based size-correction for nonstandard testing problems. Journal of Econometrics 200(1), 17–35.CrossRefGoogle Scholar
McShane, E. J. (1969) A Riemann-Type Integral that Includes Lebesgue-Stieltjes, Bochner and Stochastic Integrals. American Mathematical Society.CrossRefGoogle Scholar
Moon, H. R. & Schorfheide, F. (2009) Estimation with overidentifying inequality moment conditions. Journal of Econometrics 153(2), 136–154.CrossRefGoogle Scholar
Neto, E. D. A. L. & de Carvalho, F. D. A. (2010) Constrained linear regression models for symbolic interval-valued variables. Computational Statistics & Data Analysis 54(2), 333–347.CrossRefGoogle Scholar
Nishiura, H., Linton, N. M. & Akhmetzhanov, A. R. (2020) Serial interval of novel coronavirus (COVID-19) infections. International Journal of Infectious Diseases 93, 284–286.CrossRefGoogle ScholarPubMed
Pflug, G.C. (1994) On an argmax-distribution connected to the Poisson process. In Mandl, P., Hušková, M. (eds.), Asymptotic Statistics, pp. 123–129. Physica-Verlag.CrossRefGoogle Scholar
Pflug, G. C. (1995) Asymptotic stochastic programs. Mathematics of Operations Research 20(4), 769–789.CrossRefGoogle Scholar
Resnick, S. I. (1987) Extreme Values, Regular Variation, and Point Processes (Springer).CrossRefGoogle Scholar
Rogers, A. J. (1986) Modified lagrange multiplier tests for problems with one-sided alternatives. Journal of Econometrics 31(3), 341–361.CrossRefGoogle Scholar
Romano, J. P., Shaikh, A. M. & Wolf, M. (2014) A practical two-step method for testing moment inequalities. Econometrica 82(5), 1979–2002.Google Scholar
Romano, J. P. & Wolf, M. (2000) Finite sample nonparametric inference and large sample efficiency. Annals of Statistics 28(3), 756–778.CrossRefGoogle Scholar
Silvapulle, M. & Sen, P. (2005) Constrained Statistical Inference (Wiley).Google Scholar
Silvapulle, M. J. (1996) A test in the presence of nuisance parameters. Journal of the American Statistical Association 91(436), 1690–1693.CrossRefGoogle Scholar
Staiger, D. O. & Stock, J. H. (1997) Instrumental variables regression with weak instruments. Econometrica 65, 557–586.CrossRefGoogle Scholar
Trutschnig, W., González-Rodríguez, G., Colubi, A. & Gil, M. Á. (2009) A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread. Information Sciences 179(23), 3964–3972.CrossRefGoogle Scholar
Van der Vaart, A. W. (2000) Asymptotic Statistics (Cambridge University Press).Google Scholar
Vynnycky, E. & Fine, P. E. (2000) Lifetime risks, incubation period, and serial interval of tuberculosis. American Journal of Epidemiology 152(3), 247–263.CrossRefGoogle ScholarPubMed
Wolak, F. A. (1987) An exact test for multiple inequality and equality constraints in the linear regression model. Journal of the American Statistical Association 82(399), 782–793.CrossRefGoogle Scholar
Wolak, F. A. (1989) Local and global testing of linear and nonlinear inequality constraints in nonlinear econometric models. Econometric Theory 5(1), 1–35.CrossRefGoogle Scholar
Wolak, F. A. (1991) The local nature of hypothesis tests involving inequality constraints in nonlinear models. Econometrica 59, No. 4, 981–995.CrossRefGoogle Scholar