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FACTORISABLE MULTITASK QUANTILE REGRESSION

Published online by Cambridge University Press:  22 September 2020

Shih-Kang Chao ,
Wolfgang K. Härdle  and
Ming Yuan
Shih-Kang Chao*
Affiliation:
University of Missouri
Wolfgang K. Härdle
Affiliation:
Humboldt-Universität zu Berlin Xiamen University Charles University
Ming Yuan
Affiliation:
Columbia University
*
Address correspondence to Shih-Kang Chao, Department of Statistics, University of Missouri, Columbia, MO 65211, USA; e-mail: chaosh@missouri.edu.
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Abstract

A multivariate quantile regression model with a factor structure is proposed to study data with multivariate responses with covariates. The factor structure is allowed to vary with the quantile levels, which is more flexible than the classical factor models. Assuming the number of factors is small, and the number of responses and the input variables are growing with the sample size, the model is estimated with the nuclear norm regularization. The incurred optimization problem can only be efficiently solved in an approximate manner by off-the-shelf optimization methods. Such a scenario is often seen when the empirical loss is nonsmooth or the numerical procedure involves expensive subroutines, for example, singular value decomposition. To show that the approximate estimator is still statistically accurate, we establish a nonasymptotic bound on the Frobenius risk and prediction risk. For implementation, a numerical procedure that provably marginalizes the approximation error is proposed. The merits of our model and the proposed numerical procedures are demonstrated through the Monte Carlo simulation and an application to finance involving a large pool of asset returns.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 4 , August 2021 , pp. 794 - 816
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

We would like to thank the Editor, Peter Phillips, the Co-Editor, Dennis Kristensen, and two anonymous referees for their very helpful and constructive comments. We also thank Scott Holan for providing many helpful comments.

References

REFERENCES

Adrian, T. & Brunnermeier, M.K. (2016) CoVaR. American Economic Review 106(7), 1705–1741.CrossRefGoogle Scholar
Ando, T. & Tsay, R.S. (2011) Quantile regression models with factor-augmented predictors and information criterion. Econometrics Journal 14, 1–24.CrossRefGoogle Scholar
Bai, J. & Ng, S. (2010) Instrumental variable estimation in a data rich environment. Econometric Theory 26(6), 1577–1606.CrossRefGoogle Scholar
Beck, A. & Teboulle, M. (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM Journal on Imaging Sciences 2(1), 183–202.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
Bisias, D., Flood, M., Lo, A., & Valavanis, S. (2012) A Survey of Systemic Risk Analytics, Office of Financial Research Working Paper.CrossRefGoogle Scholar
Brunnermeier, M.K. & Oehmke, M. (2013) Chapter 18—Bubbles, financial crises, and systemic risk. Handbook of the Economics of Finance, vol. 2, pp. 1221–1288. Elsevier.http://www.sciencedirect.com/science/article/pii/B9780444594068000184 .Google Scholar
Buldygin, V.V. & Moskvichova, K.K. (2013 The sub-Gaussian norm of a binary random variable. Theory of Probability and Mathematical Statistics 86, 33–49.CrossRefGoogle Scholar
Bunea, F., She, Y., & Wegkamp, M.H. (2011) Optimal selection of reduced rank estimators of high-dimensional matrices. Annals of Statistics 39(2), 1282–1309.CrossRefGoogle Scholar
Chao, S.-K., Härdle, W.K., & Huang, C. (2018) Multivariate factorizable expectile regression with application to fMRI data. Computational Statistics & Data Analysis 121, 1 – 19.CrossRefGoogle Scholar
Chao, S.-K., Härdle, W.K., & Yuan, M. (2015) Factorisable sparse tail event curves. Preprint, arXiv:1507.03833 Google Scholar
Chen, L., Dolado, J.J., & Gonzalo, J. (2019) Quantile factor models. Preprint arXiv:1911.02173.Google Scholar
Chen, X., Lin, Q., Kim, S., Carbonell, J.G., & Xing, E.P. (2012) Smoothing proximal gradient method for general structured sparse regression. The Annals of Applied Statistics 6(2), 719–752.CrossRefGoogle Scholar
Chernozhukov, V. (1999) Conditional extremes and near-extremes: Estimation, inference, and economic applications, PhD thesis, Department of Economics, Stanford University.Google Scholar
Chernozhukov, V. (2005) Extremal quantile regression. Annals of Statistics 33(2), 806–839.CrossRefGoogle Scholar
Chernozhukov, V., Fernández-Val, I., & Kaji, T. (2017) Extremal quantile regression: An overview. Preprint, arXiv:1612.06850.CrossRefGoogle Scholar
Ciliberto, C., Stamos, D., & Pontil, M. (2017) Reexamining low rank matrix factorization for trace norm regularization. Preprint, arXiv:1706.08934 Google Scholar
Davis, R.A., Zang, P., & Zheng, T. (2016) Sparse vector autoregressive modeling. Journal of Computational and Graphical Statistics 25(4), 1077–1096.CrossRefGoogle Scholar
Engle, R. & Manganelli, S. (2004) CAViaR: Conditional autoregressive value at risk by regression quantiles. Journal of Business & Economic Statistics 22, 367–381.CrossRefGoogle Scholar
Falk, M. (1999) A simple approach to the generation of uniformly distributed random variables with prescribed correlation. Communications in Statistics—Simulation and Computation 28(3), 785–791.CrossRefGoogle Scholar
Fazel, M., Hindi, H., & Boyd, S.P. (2001) A rank minimization heuristic with application to minimum order system approximation. Proceedings of the American Control Conference, pp. 4734–4739.CrossRefGoogle Scholar
Koenker, R. & Bassett, G.S. (1978) Regression quantiles. Econometrica 46(1), 33–50.CrossRefGoogle Scholar
Koenker, R. & Hallock, K.F. (2001) Quantile regression. Journal of Economic Perspectives 15(4), 143–156.CrossRefGoogle Scholar
Koenker, R. & Portnoy, S. (1990) M estimation of multivariate regressions. Journal of American Statistical Association 85(412), 1060–1068.Google Scholar
Koltchinskii, V. (2011) Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems. Springer.CrossRefGoogle Scholar
Koltchinskii, V., Lounici, K., & Tsybakov, A.B. (2011) Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion. Annals of Statistics 39(5), 2243–2794.CrossRefGoogle Scholar
Maurer, A. & Pontil, M. (2013) Excess risk bounds for multitask learning with trace norm regularization. JMLR: Workshop and Conference Proceedings 30, 1–22.Google Scholar
Maurer, A., Pontil, M., & Romera-Paredes, B. (2016) The benefit of multitask representation learning. Journal of Machine Learning Research 17, 1–32.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
Negahban, S.N. & 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
Nesterov, Y. (2005) Smooth minimization of non-smooth functions. Mathematical Programming 103(1), 127–152.CrossRefGoogle Scholar
Nicholson, W.B., Matteson, D.S., & Bien, J. (2017) VARX-L: Structured regularization for large vector autoregressions with exogenous variables. International Journal of Forecasting 33(3), 627–651.CrossRefGoogle Scholar
Nze, P.A. & Doukhan, P. (2004) Weak dependence: Models and applications to econometrics. Econometric Theory 20(6), 995–1045.CrossRefGoogle Scholar
Reinsel, G.C. & Velu, R.P. (1998) Multivariate Reduced-Rank Regression. Springer.CrossRefGoogle Scholar
Rennie, J.D.M. & Srebro, N. (2005) Fast maximum margin matrix factorization for collaborative prediction. Proceedings of the 22nd International Conference on Machine Learning, ICML ’05. ACM, pp. 713–719.CrossRefGoogle Scholar
Srebro, N., Rennie, J.D.M., & Jaakkola, T.S. (2005) Maximum-margin matrix factorization. Proceedings of the Conference on Neural Information Processing Systems, pp. 1329–1336.Google Scholar
Stewart, G.W. & Sun, J.-G. (1990) Matrix Perturbation Theory. Academic Press.Google Scholar
Stock, J.H. & Watson, M.W. (2002) Macroeconomic forecasting using diffusion indexes. Journal of Business & Economic Statistics 20(2), 147–162.CrossRefGoogle Scholar
Tropp, J.A. (2011) User-friendly tail bounds for sums of random matrices. Foundations of Computational Mathematics 12(4), 389–434.CrossRefGoogle Scholar
Vershynin, R. (2012) How close is the sample covariance matrix to the actual covariance matrix? Journal of Theoretical Probability 25(3), 655–686.CrossRefGoogle Scholar
Wallis, K.F. (1999) Asymmetric density forecasts of inflation and the Bank of England’s fan chart. National Institute Economic Review 167, 106–112.CrossRefGoogle Scholar
Wallis, K.F. (2014) The two-piece normal, binormal, or double Gaussian distribution: Its origin and rediscoveries, Statistical Science 29(1), 106–112.CrossRefGoogle Scholar
Weimer, M., Karatzoglou, A., Le, Q.V., & Smola, A.J. (2008) COFI RANK—Maximum margin matrix factorization for collaborative ranking. In Platt, J.C., Koller, D., Singer, Y., & Roweis, S.T. (eds.), Advances in Neural Information Processing Systems 20. Curran Associates, Inc., pp. 1593–1600. http://papers.nips.cc/paper/3359-cofi-rank-maximum-margin-matrix-factorization-for-collaborative-ranking.pdf Google Scholar
Weimer, M., Karatzoglou, A., & Smola, A. (2008) Improving maximum margin matrix factorization. Machine Learning 72(3), 263–276.CrossRefGoogle Scholar
White, H., Kim, T.-H., & Manganelli, S. (2015) VAR for VaR: Measuring systemic risk using multivariate regression quantiles. Journal of Econometrics 187, 169–188.CrossRefGoogle Scholar
Yousefi, N., Lei, Y., Kloft, M., Mollaghasemi, M., & Anagnastapolous, G. (2018) Local Rademacher complexity-based learning guarantees for multi-task learning. Journal of Machine Learning Research 19(1), 1385–1431.Google Scholar
Yu, Y., Wang, T., & Samworth, R.J. (2015) A useful variant of the Davis–Kahan theorem for statisticians. Biometrika 102(2), 315–323.CrossRefGoogle Scholar
Yuan, M., Ekici, A., Lu, Z., & Monteiro, R. (2007) Dimension reduction and coefficient estimation in multivariate linear regression. Journal of the Royal Statistical Society: Series B 69(3), 329–346.CrossRefGoogle Scholar
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