Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T18:28:52.988Z Has data issue: false hasContentIssue false

INFERENCE ON TWO-COMPONENT MIXTURES UNDER TAIL RESTRICTIONS

Published online by Cambridge University Press:  04 April 2016

Koen Jochmans*
Affiliation:
Sciences Po
Marc Henry
Affiliation:
The Pennsylvania State University
Bernard Salanié
Affiliation:
Columbia University
*
*Address correspondence to Koen Jochmans, Sciences Po, Department of Economics, 28 rue des Saints Pères 75007, Paris, France, e-mail. [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Many econometric models can be analyzed as finite mixtures. We focus on two-component mixtures, and we show that they are nonparametrically point identified by a combination of an exclusion restriction and tail restrictions. Our identification analysis suggests simple closed-form estimators of the component distributions and mixing proportions, as well as a specification test. We derive their asymptotic properties using results on tail empirical processes and we present a simulation study that documents their finite-sample performance.

Type
ARTICLES
Copyright
Copyright © Cambridge University Press 2016 

Footnotes

We are grateful to Peter Phillips, Arthur Lewbel, and three referees for comments and suggestions, and to Victor Chernozhukov and Yuichi Kitamura for fruitful discussions. Parts of this paper were written while Henry was visiting the University of Tokyo Graduate School of Economics and while Salanié was visiting the Toulouse School of Economics. The hospitality of both institutions is gratefully acknowledged. Jochmans’ research has received funding from the SAB grant “Nonparametric estimation of finite mixtures”. Henry’s research has received funding from the SSHRC Grants 410-2010-242 and 435-2013-0292, and NSERC Grant 356491-2013. Salanié thanks the Georges Meyer endowment. Some of the results presented here previously circulated as part of Henry, Kitamura, and Salanié (2010), whose published version (Henry et al., 2014) only contains results on partial identification.

References

REFERENCES

Acemoglu, D., Carvalho, V., Ozdaglar, A., & Tabaz-Salehi, A. (2012) The network origins of aggregate fluctuations. Econometrica 80(5), 19772016.Google Scholar
Allman, E.S., Matias, C., & Rhodes, J.A. (2009) Identifiability of parameters in latent structure models with many observed variables. Annals of Statistics 37, 30993132.CrossRefGoogle Scholar
Andrews, D.W.K. & Schafgans, M.M.A. (1998) Semiparametric estimation of the intercept of a sample selection model. Review of Economic Studies 65, 497517.CrossRefGoogle Scholar
Arkolakis, C., Costinot, A., & Rodriguez-Clare, A. (2012) New trade models, same old gains? American Economic Review 102, 94130.Google Scholar
Atkinson, A.B., Piketty, T., & Saez, E. (2011) Top incomes in the long run of history. Journal of Economic Literature 49, 371.CrossRefGoogle Scholar
Azzalini, A. (1985) A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12, 171178.Google Scholar
Bollinger, C.R. (1996) Bounding mean regressions when a binary regressor is mismeasured. Journal of Econometrics 73, 387399.CrossRefGoogle Scholar
Bonhomme, S., Jochmans, K., & Robin, J.-M. (2016) Estimating multivariate latent-structure models. Annals of Statistics, 44, 540563.CrossRefGoogle Scholar
Bonhomme, S., Jochmans, K., & Robin, J.-M. (2016) Nonparametric estimation of finite mixtures from repeated measurements. Journal of the Royal Statistical Society, Series B 78, 211229.CrossRefGoogle Scholar
Bordes, L., Mottelet, S., & Vandekerkhove, P. (2006) Semiparametric estimation of a two-component mixture model. Annals of Statistics 34, 12041232.CrossRefGoogle Scholar
Capitanio, A. (2010) On the approximation of the tail probability of the scalar skew-normal distribution. METRON 68, 299308.CrossRefGoogle Scholar
Carroll, R.J., Ruppert, D., Stefanski, L.A., & Crainiceanu, C. (2006) Measurement Error in Nonlinear Models: A Modern Perspective. Chapman and Hall, CRC Press.Google Scholar
D’Haultfœuille, X. & Février, P. (2015) Identification of mixture models using support variations. Journal of Econometrics 189, 7082.Google Scholar
D’Haultfœuille, X. & Maurel, A. (2013) Another look at identification at infinity of sample selection models. Econometric Theory 29, 213224.CrossRefGoogle Scholar
Einmahl, J. (1992) Limit theorems for tail processes with application to intermediate quantile estimation. Journal of Statistical Planning and Inference 32, 137145.CrossRefGoogle Scholar
Einmahl, U. & Mason, D. (1997) Gaussian approximation of local empirical processes indexed by functions. Probability Theory and Related Fields 107, 283311.CrossRefGoogle Scholar
Frisch, R. (1934) Statistical confluence analysis by means of complete regression systems. Technical Report 5, University of Oslo, Economics Institute, Oslo, Norway.Google Scholar
Gabaix, X. (2009) Power laws in economics and finance. Annual Review of Economics 1, 255294.CrossRefGoogle Scholar
Gassiat, E. & Rousseau, J. (2016) Nonparametric finite translation hidden Markov models and extensions. Bernoulli 22, 193212.CrossRefGoogle Scholar
Ghysels, E., Harvey, A., & Renault, E. (1996) Stochastic volatility. In Maddala, G.S. & Rao, C.R. (eds.), Handbook of Statistics Volume 14: Statistical Methods in Finance. Elsevier.Google Scholar
Hall, P. & Zhou, X.-H. (2003) Nonparametric identification of component distributions in a multivariate mixture. Annals of Statistics 31, 201224.Google Scholar
Hamilton, J.D. (1989) A new approach to the analysis of nonstationary times series and the business cycle. Econometrica 57, 357384.CrossRefGoogle Scholar
Heckman, J.J. (1974) Shadow prices, market wages, and labor supply. Econometrica 42, 679694.CrossRefGoogle Scholar
Heckman, J.J. (1990) Varieties of selection bias. American Economic Review 80, 313318.Google Scholar
Henry, M., Kitamura, Y., & Salanié, B. (2010) Identifying Finite Mixtures in Econometric Models. Cowles Foundation Discussion paper 1767.CrossRefGoogle Scholar
Henry, M., Kitamura, Y., & Salanié, B. (2014) Partial identification of finite mixtures in econometric models. Quantitative Economics 5, 123144.CrossRefGoogle Scholar
Hu, Y. & Schennach, S.M. (2008) Instrumental variable treatment of nonclassical measurement error models. Econometrica 76, 195216.CrossRefGoogle Scholar
Jochmans, K., Henry, M., & Salanié, B. (2014) Inference on mixtures under tail restrictions. Discussion paper No. 2014-01, Department of Economics, Sciences Po.Google Scholar
Kasahara, H. & Shimotsu, K. (2009) Nonparametric identification of finite mixture models of dynamic discrete choices. Econometrica 77, 135175.Google Scholar
Khan, S. & Tamer, E. (2010) Irregular identification, support conditions and inverse weight estimation. Econometrica 78, 20212042.Google Scholar
Lewbel, A. (2007) Estimation of average treatment effects with misclassification. Econometrica 75, 537551.CrossRefGoogle Scholar
Mahajan, A. (2006) Identification and estimation of regression models with misclassification. Econometrica 74, 631665.Google Scholar
Schwarz, M. & Van Bellegem, S. (2010) Consistent density deconvolution under partially known error distribution. Statistics and Probability Letters 80, 236241.CrossRefGoogle Scholar
Shimer, R. L. Smith (2000) Assortative matching and search. Econometrica 68, 343369.Google Scholar