A Class of Structured High-Dimensional Dynamic Covariance Matrices

Abstract

High-dimensional covariance matrices have attracted much attention of statisticians and econometricians during the past decades. Vast literature is devoted to the research in high-dimensional covariance matrices. However, most of them are for constant covariance matrices. In many applications, constant covariance matrices are not appropriate, e.g., in portfolio allocation, dynamic covariance matrices would make much more sense. Simply assuming each entry of a covariance matrix is a function of time to introduce a dynamic structure would not work. In this paper, we are going to introduce a class of high-dimensional dynamic covariance matrices in which a kind of additive structure is embedded. We will show the proposed high-dimensional dynamic covariance matrices have many advantages in applications. An estimation procedure is also proposed to estimate the proposed high-dimensional dynamic covariance matrices. Asymptotic properties are built to justify the proposed estimation procedure. Intensive simulation studies show the proposed estimation procedure works very well when sample size is finite. Finally, we apply the proposed high-dimensional dynamic covariance matrices, together with the proposed estimation procedure, to portfolio allocation. The results look very interesting.

Keywords

Additive structure / B-spline / Factor models / High-dimensional dynamic covariance matrices / Portfolio allocation

References

[1] Avella-MadinaM, BatteyHS, FanJ, LiQ. Robust estimation of high-dimensional covariance and precision matrices. Biometrika, 2018, 1052271-284

[2] BerthetQ, RigolletP. Optimal detection of sparse principal components in high dimension. Ann. Stat., 2013, 4141780-1815

[3] BickelP, LevinaE. Covariance regularization by thresholding. Ann. Stat., 2008, 3662577-2604

[4] BickelP, LevinaE. Regularized estimation of large covariance matrices. Ann. Stat., 2008, 361199-227

[5] BirnbaumA, JohnstoneIM, NadlerB, PaulD. Minimax bounds for sparse PCA with noisy high-dimensional data. Ann. Stat., 2013, 4131055-1084

[6] ChenZ, FanJ, LiR. Error variance estimation in ultrahigh dimensional additive models. J. Am. Stat. Assoc., 2018, 113521315-327

[7] EL KarouiN. Operator norm consistent estimation of large-dimensional sparse covariance matrices. Ann. Stat., 2008, 3662712-2756

[8] FamaEF, FrenchKR. The cross-section of expected stock returns. J Finance, 1992, 472427-465

[9] FamaEF, FrenchKR. Common risk factors in the returns on stocks and bonds. J. Financ. Econ., 1993, 3313-56

[10] FanJ, ZhangW. Statistical estimation in varying coefficient models. Ann. Stat., 1999, 2751491-1518

[11] FanJ, ZhangW. Simultaneous confidence bands and hypothesis testing in varying-coefficient models. Scand. J. Stat., 2000, 274715-731

[12] FanJ, YaoQ, CaiZ. Adaptive varying-coefficient linear models. J. R. Stat. Soc. Ser. B, 2003, 65157-80

[13] FanJ, FanY, LvJ. High dimensional covariance matrix estimation using a factor model. J. Econometr., 2008, 1471186-197

[14] FanJ, LiaoY, MinchevaM. High-dimensional covariance matrix estimation in approximate factor models. Ann. Stat., 2011, 3963320-3356

[15] FanJ, ZhangJ, YuK. Vast portfolio selection with gross-exposure constraints. J. Am. Stat. Assoc., 2012, 107498592-606

[16] FanJ, LiuH, WangW. Large covariance estimation through elliptical factor models. Ann. Stat., 2018, 4641383-1414

[17] FangY, WangB, FengY. Tuning-parameter selection in regularized estimations of large covariance matrices. J. Stat. Comput. Simul., 2016, 86: 494-509

[18] Francq, J., Zakoïan, J.M.: GARCH Models: Structure, Statistical Inference and Financial Applications. Wiley (2010)

[19] GuoS, BoxJL, ZhangW. A dynamic structure for high dimensional covariance matrices and its application in portfolio allocation. J. Am. Stat. Assoc., 2017, 112517235-253

[20] Hastie, H., Tibshirani, R.: Generalized Additive Models. CRC Press (1990)

[21] HastieH, TibshiraniR. Varying-coefficient models. Ann. Stat., 1993, 554757-796

[22] James, W., Stein, C.M.: Estimation with quadratic loss. Prco. Fourth Berkeley Symp. Math. Stat. Probab. 1, 361–379 (1961)

[23] KaiB, LiR, ZouH. New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. Ann. Stat., 2011, 391305-332

[24] LiJ, ZhangW. A semiparametric threshold model for censored longitudinal data analysis. J. Am. Stat. Assoc., 2011, 106494685-696

[25] LintonOB. Miscellanea efficient estimation of additive nonparametric regression models. Biometrika, 1997, 842469-473

[26] RothmanA, LevinaE, ZhuJ. Generalized thresholding of large covariance matrices. J. Am Stat. Assoc., 2009, 104485177-186

[27] SunY, ZhangW, TongH. Estimation of covariance matrix of random effects in longitudinal studies. Ann. Stat., 2007, 3562795-2814

[28] SunY, YanH, ZhangW, LuZ. A semiparametric spatial dynamic model. Ann. Stat., 2014, 422700-727

[29] WuWB, PourahmadiM. Nonparametric estimation of large covariance matrices of longitudinal data. Biometrika, 2003, 904831-884

[30] YuanM. High dimensional inverse covariance matrix estimation via linear programming. J. Mach. Learn. Res., 2010, 11: 2261-2286

[31] ZhangW, FanJ, SunY. A semiparametric model for cluster data. Ann. Stat., 2009, 375A2377-2408

Funding

national natural science foundation of china(11901315)

Eunice Kennedy Shriver National Institute of Child Health and Human Development(Intramural Research Program)

RIGHTS & PERMISSIONS

This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply