Abstract
Based on the data-cutoff method, we study quantile regression in linear models, where the noise process is of Ornstein-Uhlenbeck type with possible jumps. In single-level quantile regression, we allow the noise process to be heteroscedastic, while in composite quantile regression, we require that the noise process be homoscedastic so that the slopes are invariant across quantiles. Similar to the independent noise case, the proposed quantile estimators are root-n consistent and asymptotic normal. Furthermore, the adaptive least absolute shrinkage and selection operator (LASSO) is applied for the purpose of variable selection. As a result, the quantile estimators are consistent in variable selection, and the nonzero coefficient estimators enjoy the same asymptotic distribution as their counterparts under the true model. Extensive numerical simulations are conducted to evaluate the performance of the proposed approaches and foreign exchange rate data are analyzed for the illustration purpose.
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References
Barndorff-Nielsen O E. Processes of normal inverse Gaussian type. Finance Stoch, 1998, 2: 41–68
Barndorff-Nielsen O E. Superposition of Ornstein-Uhlenbeck type processes. Theory Probab Appl, 2001, 45: 175–194
Barndorff-Nielsen O E, Shephard N. Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc Ser B Stat Methodol, 2001, 63: 167–241
Barndorff-Nielsen O E, Shephard N. Modelling by Lévy processes for financial econometrics. In: Lévy Processes. Boston: Birkhäuser, 2001, 283–318
Barndorff-Nielsen O E, Shephard N. Integrate OU processes and non-Gaussian OU-based stochastic volatility models. Scand J Statist, 2003, 30: 277–295
Belloni A, Chernozhukov V, Kato K. Valid post-selection inference in high-dimensional approximately sparse quantile regression models. J Amer Statist Assoc, 2019, 114: 749–758
Benth F E, Kiesel R, Nazarova A. A critical empirical study of three electricity spot price models. Energy Econom, 2012, 34: 1589–1616
Bodnarchuk S V, Kulyk O M. Conditions for the existence and smoothness of the distribution density of the Ornstein-Uhlenbeck process with Lévy noise. Theory Probab Math Statist, 2009, 79: 23–38
Bosq D. Nonparametric Statistics for Stochastic Processes. New York: Springer-Verlag, 1998
Cheng Y, Zerom D. A quantile regression model for time series data in the presence of additive components. Comm Statist Theory Methods, 2015, 44: 4354–4379
Davydov Yu A. Mixing conditions for Markov chains. Theory Probab Appl, 1973, 18: 312–328
Halgreen C. Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Probab Theory Related Fields, 1979, 47: 13–17
Jiang L, Wang H, Bondell H. Interquantile shrinkage in regression models. J Comput Graph Statist, 2013, 22: 970–986
Jongbloed G, van der Meulen F H, van der Vaart A W. Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes. Bernoulli, 2005, 11: 759–791
Knight K. Limiting distributions for L1 regression estimators under general conditions. Ann Statist, 1998, 26: 755–770
Koenker R. Quantile Regression. Cambridge: Cambridge University Press, 2005
Koenker R, Basset G. Regression quantiles. Econometrica, 1978, 46: 33–50
Masuda H. On multidimensional Ornstein-Uhlenbeck processes driven by a general Lévy process. Bernoulli, 2004, 10: 97–120
Rosenblatt M. Asymptotic normality, strong mixing and spectral density estimates. Ann Probab, 1984, 12: 1167–1180
Sato K. Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press, 1999
Schoutens W, Cariboni J. Lévy processes in credit risk. Hoboken: John Wiley & Sons, 2009
Schwarz G. Estimating the dimension of a model. Ann Statist, 1978, 6: 461–464
Shiga T. A recurrence criterion for Markov processes of Ornstein-Uhlenbeck type. Probab Theory Related Fields, 1990, 85: 425–447
Tang Y, Wang H. Penalized regression across multiple quantiles under random censoring. J Multivariate Anal, 2015, 141: 132–146
Tang Y L, Song X Y, Zhu Z Y. Variable selection via composite quantile regression with dependent errors. Statist Neerlandica, 2015, 69: 1–20
Tibshirani R J. Regression shrinkage and selection via the LASSO. J R Stat Soc Ser B Stat Methodol, 1996, 58: 267–288
Tweedie R L. Criteria for rates of convergence of Markov chains with application to queueing and storage theory. In: Probability, Statistics and Analysis. London Mathematical Society Lecture Note Series, vol. 79. Cambridge: Cambridge University Press, 1983, 260–276
Wang H, Li G, Tsai C L. Regression coefficients and autoregressive order shrinkage and selection via the LASSO. J R Stat Soc Ser B Stat Methodol, 2007, 69: 63–78
Wu W. M-estimation of linear models with dependent errors. Ann Statist, 2007, 35: 495–521
Wu W, Zhou Z. Nonparametric inference for time-varying coefficient quantile regression. J Bus Econom Statist, 2017, 35: 98–109
Zhang S, Zhang X. Exact simulation of IG-OU processes. Methodol Comput Appl Probab, 2008, 10: 337–355
Zou H. The adaptive lasso and its oracle properties. J R Stat Soc Ser B Stat Methodol, 2006, 101: 1418–1429
Zou H, Yuan M. Composite quantile regression and the oracle model selection theory. Ann Statist, 2008, 36: 1108–1126
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11801355 and 11971116). The authors thank the reviewers for constructive comments and helpful suggestions. The authors also thank Dr. Yanlin Tang from East China Normal University for helpful discussions.
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Wang, Y., Zhang, X. Variable selection via quantile regression with the process of Ornstein-Uhlenbeck type. Sci. China Math. 65, 827–848 (2022). https://doi.org/10.1007/s11425-019-1723-4
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DOI: https://doi.org/10.1007/s11425-019-1723-4
Keywords
- adaptive LASSO
- composite quantile regression
- data-cutoff method
- process of Ornstein-Uhlenbeck type
- quantile regression