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NUCLEAR NORM REGULARIZED QUANTILE REGRESSION WITH INTERACTIVE FIXED EFFECTS

Published online by Cambridge University Press:  24 April 2023

Junlong Feng
Junlong Feng*
Affiliation:
Hong Kong University of Science and Technology
*
Address correspondence to Junlong Feng, Department of Economics, Hong Kong University of Science and Technology, Kowloon, Hong Kong; e-mail: [email protected].
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Abstract

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This paper studies large N and large T conditional quantile panel data models with interactive fixed effects. We propose a nuclear norm penalized estimator of the coefficients on the covariates and the low-rank matrix formed by the interactive fixed effects. The estimator solves a convex minimization problem, not requiring pre-estimation of the (number of) interactive fixed effects. It also allows the number of covariates to grow slowly with N and T. We derive an error bound on the estimator that holds uniformly in the quantile level. The order of the bound implies uniform consistency of the estimator and is nearly optimal for the low-rank component. Given the error bound, we also propose a consistent estimator of the number of interactive fixed effects at any quantile level. We demonstrate the performance of the estimator via Monte Carlo simulations.

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 6 , December 2024 , pp. 1391 - 1421
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Footnotes

This paper is based on the third chapter of my doctoral dissertation at Columbia. An earlier working paper version of it was circulated under the title “Regularized Quantile Regression with Interactive Fixed Effects.” I am very grateful for helpful comments provided by the Editor, the Co-Editor, and three referees. I thank Jushan Bai, Sokbae (Simon) Lee, and Bernard Salanié, who were gracious with their advice, support, and feedback. I have also greatly benefited from comments and discussions with Songnian Chen, Roger Koenker, José Luis Montiel Olea, Roger Moon, Serena Ng, Jörg Stoye, Peng Wang, Martin Weidner, and participants at the Columbia Econometrics Colloquium, 2021 Econometric Society North American, Asian, and China meetings, and 2021 IAAE. All errors are my own.

References

REFERENCES

Abrevaya, J. & Dahl, C.M. (2008) The effects of birth inputs on birthweight: Evidence from quantile estimation on panel data. Journal of Business & Economic Statistics 26(4), 379–397.CrossRefGoogle Scholar
Agarwal, A., Negahban, S., & Wainwright, M.J. (2012) Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions. Annals of Statistics 40(2), 1171–1197.CrossRefGoogle Scholar
Ahn, S.C., Lee, Y.H., & Schmidt, P. (2001) GMM estimation of linear panel data models with time-varying individual effects. Journal of Econometrics 101(2), 219–255.CrossRefGoogle Scholar
Ando, T. & Bai, J. (2020) Quantile co-movement in financial markets: A panel quantile model with unobserved heterogeneity. Journal of the American Statistical Association 115(529), 266–279.CrossRefGoogle Scholar
Athey, S., Bayati, M., Doudchenko, N., Imbens, G., & Khosravi, K. (2021) Matrix completion methods for causal panel data models. Journal of the American Statistical Association 116(536), 1716–1730.CrossRefGoogle Scholar
Bai, J. (2003) Inferential theory for factor models of large dimensions. Econometrica 71(1), 135–171.CrossRefGoogle Scholar
Bai, J. (2009) Panel data models with interactive fixed effects. Econometrica 77(4), 1229–1279.Google Scholar
Bai, J. & Feng, J. (2019) Robust principal component analysis with non-sparse errors. Preprint, arXiv:1902.08735.Google Scholar
Bai, J. & Ng, S. (2019) Rank regularized estimation of approximate factor models. Journal of Econometrics 212(1), 78–96.CrossRefGoogle Scholar
Bai, J. & Ng, S. (2021) Matrix completion, counterfactuals, and factor analysis of missing data. Journal of the American Statistical Association 116(536), 1746–1763.CrossRefGoogle Scholar
Belloni, A., Chen, M., Padilla, O.H.M., & Wang, Z.K. (2023) High-dimensional latent panel quantile regression with an application to asset pricing. Annals of Statistics 51(1), 96–121.CrossRefGoogle Scholar
Belloni, A. & Chernozhukov, V. (2011) $\ell 1$ -penalized quantile regression in high-dimensional sparse models. Annals of Statistics 39(1), 82–130.CrossRefGoogle Scholar
Beyhum, J. & Gautier, E. (2019) Square-root nuclear norm penalized estimator for panel data models with approximately low-rank unobserved heterogeneity. Preprint, arXiv:1904.09192.Google Scholar
Canay, I.A. (2011) A simple approach to quantile regression for panel data. Econometrics Journal 14(3), 368–386.CrossRefGoogle Scholar
Candès, E.J., Li, X., Ma, Y., & Wright, J. (2011) Robust principal component analysis? Journal of the ACM 58(3), 1–37.CrossRefGoogle Scholar
Candès, E.J. & Recht, B. (2009) Exact matrix completion via convex optimization. Foundations of Computational Mathematics 9(6), 717–772.CrossRefGoogle Scholar
Chao, S.-K., Härdle, W.K., & Yuan, M. (2021) Factorisable multitask quantile regression. Econometric Theory 37(4), 794–816.CrossRefGoogle Scholar
Chen, L. (2022) Two-step estimation of quantile panel data models with interactive fixed effects. Econometric Theory, 1–28, first view 18 August 2022. doi:10.1017/S0266466622000366.Google Scholar
Chen, L., Dolado, J.J., & Gonzalo, J. (2021) Quantile factor models. Econometrica 89(2), 875–910.CrossRefGoogle Scholar
Chernozhukov, V., Hansen, C., Liao, Y., & Zhu, Y. (2019) Inference for heterogeneous effects using low-rank estimations. Preprint, arXiv:1812.08089.Google Scholar
Galvao, A.F. & Kato, K. (2016) Smoothed quantile regression for panel data. Journal of Econometrics 193(1), 92–112.CrossRefGoogle Scholar
Galvao, A.F., Lamarche, C., & Lima, L.R. (2013) Estimation of censored quantile regression for panel data with fixed effects. Journal of the American Statistical Association 108(503), 1075–1089.CrossRefGoogle Scholar
Ganesh, A., Wright, J., Li, X., Candès, E. J., & Ma, Y. (2010) Dense error correction for low-rank matrices via principal component pursuit. In 2010 IEEE International Symposium on Information Theory , pp. 1513–1517. IEEE.CrossRefGoogle Scholar
Harding, M. & Lamarche, C. (2014) Estimating and testing a quantile regression model with interactive effects. Journal of Econometrics 178, 101–113.CrossRefGoogle Scholar
Hsu, D., Kakade, S.M., & Zhang, T. (2011) Robust matrix decomposition with sparse corruptions. IEEE Transactions on Information Theory 57(11), 7221–7234.CrossRefGoogle Scholar
Kato, K., Galvao, A.F. Jr., & Montes-Rojas, G.V. (2012) Asymptotics for panel quantile regression models with individual effects. Journal of Econometrics 170(1), 76–91.CrossRefGoogle Scholar
Knight, K. (1998) Limiting distributions for ${l}_1$ regression estimators under general conditions. Annals of Statistics 26(2), 755–770.CrossRefGoogle Scholar
Koenker, R. (2004) Quantile regression for longitudinal data. Journal of Multivariate Analysis 91(1), 74–89.CrossRefGoogle Scholar
Lamarche, C. (2010) Robust penalized quantile regression estimation for panel data. Journal of Econometrics 157(2), 396–408.CrossRefGoogle Scholar
Lin, Z., Chen, M., and Ma, Y. (2010). The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices. Preprint, arXiv:1009.5055.Google Scholar
Ma, S., Su, L., & Zhang, Y. (2022) Detecting latent communities in network formation models. Journal of Machine Learning Research 23(1), 13971–14031.Google Scholar
Moon, H.R. & Weidner, M. (2015) Linear regression for panel with unknown number of factors as interactive fixed effects. Econometrica 83(4), 1543–1579.CrossRefGoogle Scholar
Moon, H. R. and Weidner, M. (2019) Nuclear norm regularized estimation of panel regression models. Preprint, arXiv:1810.10987.Google Scholar
Negahban, S. & Wainwright, M.J. (2011) Estimation of (near) low-rank matrices with noise and high-dimensional scaling. Annals of Statistics 39(2), 1069–1097.CrossRefGoogle Scholar
Negahban, S. & Wainwright, M.J. (2012) Restricted strong convexity and weighted matrix completion: Optimal bounds with noise. Journal of Machine Learning Research 13(May), 1665–1697.Google Scholar
Negahban, S.N., Ravikumar, P., Wainwright, M.J., & Yu, B. (2012) A unified framework for high-dimensional analysis of $m$ -estimators with decomposable regularizers. Statistical Science 27(4), 538–557.CrossRefGoogle Scholar
Pesaran, M.H. (2006) Estimation and inference in large heterogeneous panels with a multifactor error structure. Econometrica 74(4), 967–1012.CrossRefGoogle Scholar
Yuan, X. & Yang, J. (2013) Sparse and low-rank matrix decomposition via alternating direction method. Pacific Journal of Optimization 9(1), 167.Google Scholar
Zhou, Z., Li, X., Wright, J., Candès, E., & Ma, Y. (2010) Stable principal component pursuit. In 2010 IEEE International Symposium on Information Theory , pp. 1518–1522. IEEE.CrossRefGoogle Scholar
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