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IDENTIFICATION OF JOINT DISTRIBUTIONS IN DEPENDENT FACTOR MODELS

Published online by Cambridge University Press:  21 February 2017

Dan Ben-Moshe
Dan Ben-Moshe*
Affiliation:
The Hebrew University of Jerusalem
*
*Address correspondence to Dan Ben-Moshe, Department of Economics, The Hebrew University of Jerusalem, Mt. Scopus, Jerusalem 91905, Israel; e-mail: [email protected].
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Abstract

This paper studies linear factor models that have arbitrarily dependent factors. Assuming that the coefficients are known and that their matrix representation satisfies rank conditions, we identify the nonparametric joint distribution of the unobserved factors using first and then second-order partial derivatives of the log characteristic function of the observed variables. In conjunction with these identification strategies the mean and variance of the vector of factors are identified. The main result provides necessary and sufficient conditions for identification of the joint distribution of the factors. In an illustrative example, we show identification of an earnings dynamics model with a subset of arbitrarily dependent income shocks. Closed-form formulas lead to estimators that converge uniformly and despite being based on inverse Fourier transforms have tight confidence bands around their theoretical counterparts in Monte Carlo simulations.

Type
ARTICLES
Information
Econometric Theory , Volume 34 , Issue 1 , February 2018 , pp. 134 - 165
Copyright
Copyright © Cambridge University Press 2017 

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Footnotes

This paper is based on the first chapter of my PhD Thesis and is a revised version of my job market paper. I am grateful to Rosa Matzkin and Jinyong Hahn for their generous support, advice, and guidance. I benefited greatly from the feedback of two anonymous referees and the editor Liangjun Su, comments at seminars at Haifa University, Hebrew University of Jerusalem, Tel Aviv University, UCL and UCLA and discussions with Stefan Hoderlein, Max Kasy, Arthur Lewbel, Áureo de Paula, Daniel Wilhelm and Martin Weidner. The support of a grant from the Maurice Falk Institute for Economic Research is gratefully acknowledged.

References

REFERENCES

Abadir, K. & Magnus, J. (2005) Matrix Algebra (Econometric Exercises). Cambridge University Press, ISBN 9780521822893.CrossRefGoogle Scholar
Aigner, D.J., Hsiao, C., Kapteyn, A., & Wansbeek, T. (1984) Latent variable models in econometrics. Handbook of Econometrics 2, 1321–1393.CrossRefGoogle Scholar
Anderson, T.W. & Rubin, H. (1956) Statistical inference in factor analysis. In Neyman, J. (ed.), Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 5, p. 1. Berkeley: University of California Press.Google Scholar
Anselin, L. (2013) Spatial Econometrics: Methods and Models, vol. 4. Springer Science & Business Media.Google Scholar
Arellano, M. & Bonhomme, S. (2012) Identifying distributional characteristics in random coefficients panel data models. The Review of Economic Studies 79(3), 987–1020.CrossRefGoogle Scholar
Ben-Moshe, D. (2016) Identification and Estimation of Coefficients in Dependent Factor Models. Working paper, The Hebrew University of Jerusalem.Google Scholar
Beran, R., Feuerverger, A., & Hall, P. (1996) On nonparametric estimation of intercept and slope distributions in random coefficient regression. The Annals of Statistics 24(6), 2569–2592.Google Scholar
Bondesson, L. (1974) Characterizations of probability laws through constant regression. Probability Theory and Related Fields 30(2), 93–115.Google Scholar
Bonhomme, S. & Robin, J.M. (2009) Consistent noisy independent component analysis. Journal of Econometrics 149(1), 12–25.CrossRefGoogle Scholar
Bonhomme, S. & Robin, J.M. (2010) Generalized non-parametric deconvolution with an application to earnings dynamics. The Review of Economic Studies 77(2), 491–533.CrossRefGoogle Scholar
Browning, M., Ejrnaes, M., & Alvarez, J. (2010) Modelling income processes with lots of heterogeneity. The Review of Economic Studies 77(4), 1353–1381.CrossRefGoogle Scholar
Carneiro, P., Hansen, K., & Heckman, J.J. (2003) Estimating distributions of treatment effects with an application to the returns to schooling and measurement of the effects of uncertainty on college choice. International Economic Review 44(2), 361–422.CrossRefGoogle Scholar
Chamberlain, G. & Griliches, Z. (1975) Unobservables with a variance-components structure: Ability, schooling, and the economic success of brothers. International Economic Review, vol. 16, 422–449.Google Scholar
Comon, P. & Jutten, C. (2010) Handbook of Blind Source Separation: Independent Component Analysis and Applications. Academic press.Google Scholar
Comte, F. & Lacour, C. (2013) Anisotropic adaptive kernel deconvolution. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, vol. 49, pp. 569–609. Institut Henri Poincaré.Google Scholar
Conti, G. & Heckman, J.J. (2010) Understanding the early origins of the education–health gradient a framework that can also be applied to analyze gene–environment interactions. Perspectives on Psychological Science 5(5), 585–605.Google ScholarPubMed
Conti, G., Heckman, J., & Urzua, S. (2010) The education-health gradient. The American Economic Review 100(2), 234.CrossRefGoogle ScholarPubMed
Cunha, F., Heckman, J.J., & Schennach, S.M. (2010) Estimating the technology of cognitive and noncognitive skill formation. Econometrica 78(3), 883–931.Google ScholarPubMed
Delaigle, A. (2008) An alternative view of the deconvolution problem. Statistica Sinica, vol. 18, 1025–1045.Google Scholar
Delaigle, A. & Gijbels, I. (2004) Practical bandwidth selection in deconvolution kernel density estimation. Computational Statistics & Data Analysis 45(2), 249–267.CrossRefGoogle Scholar
Delaigle, A., Hall, P., & Meister, A. (2008) On deconvolution with repeated measurements. The Annals of Statistics, vol. 36, 665–685.Google Scholar
Ejrnæs, M. & Browning, M. (2014) The persistent–transitory representation for earnings processes. Quantitative Economics 5(3), 555–581.CrossRefGoogle Scholar
Evdokimov, K. & White, H. (2012) Some extensions of a lemma of kotlarski. Econometric Theory 28(04), 925–932.CrossRefGoogle Scholar
Fama, E.F. & French, K.R. (1993) Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33(1), 3–56.Google Scholar
Fan, J. (1991) Global behavior of deconvolution kernel estimates. Statistica Sinica 1(2), 541–551.Google Scholar
Forni, M. & Reichlin, L. (1998) Let’s get real: A factor analytical approach to disaggregated business cycle dynamics. The Review of Economic Studies 65(3), 453–473.CrossRefGoogle Scholar
Geer, S.A. (2000) Empirical Processes in M-estimation, vol. 6. Cambridge University Press.Google Scholar
Goldberger, A.S. (1972) Structural equation methods in the social sciences. Econometrica, vol. 40, 979–1001.CrossRefGoogle Scholar
Gorman, W.M. (1981) Some engel curves. In Deaton, A. (ed.), Essays in the Theory and Measurement of Consumer Behaviour in Honor of Sir Richard Stone, 7–29. New York: Cambridge University Press.CrossRefGoogle Scholar
Graham, B.S. & Powell, J.L. (2012) Identification and estimation of average partial effects in “irregular” correlated random coefficient panel data models. Econometrica, vol. 80, 2105–2152.Google Scholar
Hamedani, G., Volkmer, H., & Behboodian, J. (2011) A note on sub-independent random variables and a class of bivariate mixtures. Studia Scientiarum Mathematicarum Hungarica 49(1), 19–25.Google Scholar
Hoderlein, S., Holzmann, H., & Meister, A. (2014) The Triangular Model with Random Coefficients. Working paper, Boston College.Google Scholar
Hoderlein, S., Klemelä, J., & Mammen, E. (2010) Analyzing the random coefficient model nonparametrically. Econometric Theory 26(03), 804–837.CrossRefGoogle Scholar
Horowitz, J.L. & Markatou, M. (1996) Semiparametric estimation of regression models for panel data. The Review of Economic Studies 63(1), 145–168.Google Scholar
Hyvärinen, A., Karhunen, J., & Oja, E. (2004) Independent Component Analysis, vol. 46. John Wiley & Sons.Google Scholar
Jolliffe, I. (2002) Principal Component Analysis. Wiley Online Library.Google Scholar
Jöreskog, K.G. (1970) A general method for analysis of covariance structures. Biometrika 57(2), 239–251.CrossRefGoogle Scholar
Khatri, C. & Rao, C.R. (1968) Solutions to some functional equations and their applications to characterization of probability distributions. Sankhyā: The Indian Journal of Statistics, Series A, vol. 30, 167–180.Google Scholar
Kotlarski, I. (1967) On characterizing the gamma and the normal distribution. Pacific Journal of Mathematics 20(1), 69–76.Google Scholar
Lewbel, A. (1991) The rank of demand systems: Theory and nonparametric estimation. Econometrica, vol. 59, 711–730.Google Scholar
Li, T. & Vuong, Q. (1998) Nonparametric estimation of the measurement error model using multiple indicators. Journal of Multivariate Analysis 65(2), 139–165.CrossRefGoogle Scholar
Masten, M. (2014) Random Coefficients on Endogenous Variables in Simultaneous Equations Models. Tech. rep. Cemmap Working paper, Centre for Microdata Methods and Practice.Google Scholar
Meghir, C. & Pistaferri, L. (2011) Earnings, consumption and life cycle choices. Handbook of Labor Economics 4, 773–854.CrossRefGoogle Scholar
Pollard, D. (1984) Convergence of Stochastic Processes. Springer.CrossRefGoogle Scholar
Ross, S.A. (1976) The arbitrage theory of capital asset pricing. Journal of Economic Theory 13(3), 341–360.CrossRefGoogle Scholar
Schennach, S.M. (2004) Estimation of nonlinear models with measurement error. Econometrica 72(1), 33–75.CrossRefGoogle Scholar
Schennach, S. (2007) Instrumental variables estimation of nonlinear errors-in-variables models. Econometrica 75, 201–239.CrossRefGoogle Scholar
Schennach, S.M. (2013) Convolution Without Independence. Tech. rep. Cemmap Working paper,Centre for Microdata Methods and Practice.Google Scholar
Schennach, S.M. & Hu, Y. (2013) Nonparametric identification and semiparametric estimation of classical measurement error models without side information. Journal of the American Statistical Association 108, 177–186.Google Scholar
Stock, J.H. & Watson, M.W. (1999) Forecasting inflation. Journal of Monetary Economics 44(2), 293–335.CrossRefGoogle Scholar
Székely, G. & Rao, C. (2000) Identifiability of distributions of independent random variables by linear combinations and moments. Sankhyā: The Indian Journal of Statistics, Series A, vol. 62, 193–202.Google Scholar
Williams, B. (2015) Identification of the Linear Factor Model. Working paper, George Washington University.Google Scholar
Zinde-Walsh, V. (2014) Measurement error and deconvolution in spaces of generalized functions. Econometric Theory 30, 1207–1246.CrossRefGoogle Scholar