Deterministic Sensitivity Analysis of the Factor Score Estimation in the Approximate Factor Model

Abstract

In this paper, by employing the matrix perturbation theory we present a deterministic sensitivity analysis of the factor score estimation in the approximate factor model with respect to two different but popular methods. The derived results precisely describe the mechanics of how the measurement and estimation errors bound the forward error of the estimated factor scores in first-order sense, and can also provide useful suggestions for designing efficient estimation procedures of the approximate factor models. Numerical experiments are also given to illustrate our theoretical results.

Keywords

Matrix perturbation / GLS method / Regression method / Condition number / 65F35 / 15A60 / 62H25

References

[1] AbouraS, van RoyeB. Financial stress and economic dynamics: the case of France. Int. Econ., 2017, 149: 57-73

[2] AndersonTWAn Introduction to Multivariate Statistical Analysis, 20033Hoboken. Wiley.

[3] BaiJ, LiK. Maximum likelihood estimation and inference for approximate factor models of high dimension. Rev. Econ. Stat., 2016, 98(2): 298-309

[4] BaiJ, LiaoY. Efficient estimation of approximate factor models via penalized maximum likelihood. J. Econom., 2016, 191(1): 1-18

[5] BaiJ, NgS. Determining the number of factors in approximate factor models. Econometrica, 2002, 70(1): 191-221

[6] ChamberlainG, RothschildM. Arbitrage, factor structure, and mean-variance analysis on large asset markets. Econometrica, 1983, 51(5): 1281-1304

[7] ChoiI. Efficient estimation of factor models. Econom. Theory, 2012, 28(2): 274-308

[8] DiaoHA, LiangL, QiaoS. A condition analysis of the weighted linear least squares problem using dual norms. Linear Multilinear Algebra, 2018, 66(6): 1085-1103

[9] DozC, GiannoneD, ReichlinL. A two-step estimator for large approximate dynamic factor models based on Kalman filtering. J. Econom., 2011, 164(1): 188-205

[10] LedoitO, WolfM. A well-conditioned estimator for large-dimensional covariance matrices. J. Multivar. Anal., 2004, 88(2): 365-411

[11] LiuH, WangL, ZhaoT. Sparse covariance matrix estimation with eigenvalue constraints. J. Comput. Graph. Stat., 2014, 23(2): 439-459

[12] MeyerCDMatrix Analysis and Applied Linear Algebra, 2000, Philadelphia. SIAM. 71

[13] RossSA. The arbitrage theory of capital asset pricing. J. Econom. Theory, 1976, 13(3): 341-360

[14] van RoyeB. Financial stress and economic activity in Germany. Empirica, 2014, 41(1): 101-126

[15] WangS. An efficient numerical method for condition number constrained covariance matrix approximation. Appl. Math. Comput., 2021, 397125925

[16] WangY, GuoX. Simultaneous inference for high-dimensional approximate factor model. Entropy, 2020, 22111258

[17] WangS, YangH. Conditioning theory of the equality constrained quadratic programming and its applications. Linear Multilinear Algebra, 2021, 69(6): 1161-1183

[18] WangS, YangH, YaoC. On the penalized maximum likelihood estimation of high-dimensional approximate factor model. Comput. Stat., 2019, 34(2): 819-846

[19] WeiMSThe Theory and Computation of the Generalized Least Squares Problems, 2006, Beijing. Science Press. (in Chinese)

Funding

Natural Science Foundation of Shandong Province(ZR2020QA034)

National Natural Science Foundation of China(11671059)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature