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Open AccessOpen Access JUSTC Mathematics 15 January 2024

The fluctuation of eigenvalues in factor models

  • Department of Statistics and Finance, International Institute of Finance, School of Management, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.52396/JUSTC-2023-0016
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  • Author Bio:

    Fanglin Bao is a graduate student under the tutelage of Prof. Bo Zhang at the University of Science and Technology of China. Her research mainly focuses on high-dimensional statistics

    Bo Zhang is an Associate Professor at the University of Science and Technology of China (USTC). He obtained a Bachelor’s degree in Statistics from USTC in 2011, a Master’s degree in Probability Theory and Mathematical Statistics from USTC in 2013, and a Ph.D. degree in Mathematics from Nanyang Technological University in Singapore in 2017. His research mainly focuses on high-dimensional random matrices, high-dimensional time series, and complex networks

  • Corresponding author: E-mail: wbchpmp@ustc.edu.cn
  • Received Date: 07 February 2023
  • Accepted Date: 11 May 2023
    • Available Online: 15 January 2024
    Abstract

  • Abstract

    We consider the fluctuation of eigenvalues in factor models and propose a new method for testing the model. Based on the characteristics of eigenvalues, variables of unknown distribution are transformed into statistics of known distribution through randomization. The test statistic checks for breaks in the structure of factor models, including changes in factor loadings and increases in the number of factors. We give the results of simulation experiments and test the factor structure of the stock return data of China’s and U.S. stock markets from January 1, 2017, to December 31, 2019. Our method performs well in both simulations and real data.

    Graphical abstract

    The randomization algorithm is used to construct the test statistics of the model structure.

    Abstract

    We consider the fluctuation of eigenvalues in factor models and propose a new method for testing the model. Based on the characteristics of eigenvalues, variables of unknown distribution are transformed into statistics of known distribution through randomization. The test statistic checks for breaks in the structure of factor models, including changes in factor loadings and increases in the number of factors. We give the results of simulation experiments and test the factor structure of the stock return data of China’s and U.S. stock markets from January 1, 2017, to December 31, 2019. Our method performs well in both simulations and real data.

    • Public Summary

    • Study the fluctuation characteristics of eigenvalues in factor models.
    • Construct the test statistic using a randomization algorithm.
    • Check for breaks in the structure of factor models, involving changes in factor loadings and increases in the number of factors.

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    • References(24)
  • [1]
    Bai J. Inferential theory for factor models of large dimensions. Econometrica, 2003, 71 (1): 135–171. doi: 10.1111/1468-0262.00392
    [2]
    Bai J, Li K. Statistical analysis of factor models of high dimension. The Annals of Statistics, 2012, 40 (1): 436–465. doi: 10.1214/11-AOS966
    [3]
    Bai J, Ng S. Determining the number of factors in approximate factor models. Econometrica, 2002, 70 (1): 191–221. doi: 10.1111/1468-0262.00273
    [4]
    Onatski A. Determining the number of factors from empirical distribution of eigenvalues. The Review of Economics and Statistics, 2010, 92 (4): 1004–1016. doi: 10.1162/REST_a_00043
    [5]
    El Karoui N. Spectrum estimation for large dimensional covariance matrices using random matrix theory. The Annals of Statistics, 2008, 36 (6): 2757–2790. doi: 10.1214/07-AOS581
    [6]
    Bao Z, Pan G, Zhou W. Universality for the largest eigenvalue of sample covariance matrices with general population. The Annals of Statistics, 2015, 43 (1): 382–421. doi: 10.1214/14-AOS1281
    [7]
    Johnstone I M. On the distribution of the largest eigenvalue in principal components analysis. The Annals of Statistics, 2001, 29 (2): 295–327. doi: 10.1214/aos/1009210544
    [8]
    Baik J, Silverstein J W. Eigenvalues of large sample covariance matrices of spiked population models. Journal of Multivariate Analysis, 2006, 97 (6): 1382–1408. doi: 10.1016/j.jmva.2005.08.003
    [9]
    Paul D. Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statistica Sinica, 2007, 17 (4): 1617–1642.
    [10]
    Bai Z, Yao J. On sample eigenvalues in a generalized spiked population model. Journal of Multivariate Analysis, 2012, 106: 167–177. doi: 10.1016/j.jmva.2011.10.009
    [11]
    Wang W, Fan J. Asymptotics of empirical eigenstructure for high dimensional spiked covariance. The Annals of Statistics, 2017, 45 (3): 1342–1374. doi: 10.1214/16-AOS1487
    [12]
    Cai T T, Han X, Pan G. Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices. The Annals of Statistics, 2020, 48 (3): 1255–1280. doi: 10.1214/18-AOS1798
    [13]
    Trapani L. A randomized sequential procedure to determine the number of factors. Journal of the American Statistical Association, 2018, 113 (523): 1341–1349. doi: 10.1080/01621459.2017.1328359
    [14]
    Pearson E S. On questions raised by the combination of tests based on discontinuous distributions. Biometrika, 1950, 37: 383–398. doi: 10.1093/biomet/37.3-4.383
    [15]
    Corradi V, Swanson N R. The effect of data transformation on common cycle, cointegration, and unit root tests: Monte Carlo results and a simple test. Journal of Econometrics, 2006, 132 (1): 195–229. doi: 10.1016/j.jeconom.2005.01.028
    [16]
    Chen L, Dolado J J, Gonzalo J. Detecting big structural breaks in large factor models. Journal of Econometrics, 2014, 180 (1): 30–48. doi: 10.1016/j.jeconom.2014.01.006
    [17]
    Cheng X, Liao Z, Schorfheide F. Shrinkage estimation of high-dimensional factor models with structural instabilities. The Review of Economic Studies, 2016, 83 (4): 1511–1543. doi: 10.1093/restud/rdw005
    [18]
    Stock J H, Watson M W. Disentangling the channels of the 2007–2009 recession. Cambridge, MA: National Bureau of Economic Research, 2012: 18094.
    [19]
    Breitung J, Eickmeier S. Testing for structural breaks in dynamic factor models. Journal of Econometrics, 2011, 163 (1): 71–84. doi: 10.1214/17-AAP1341
    [20]
    Ding X, Yang F. A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices. The Annals of Applied Probability, 2018, 28 (3): 1679–1738. doi: 10.1214/17-AAP1341
    [21]
    Ding X, Yang F. Tracy–Widom distribution for heterogeneous Gram matrices with applications in signal detection. IEEE Transactions on Information Theory, 2022, 68 (10): 6682–6715. doi: 10.1109/TIT.2022.3176784
    [22]
    Ding X, Yang F. Spiked separable covariance matrices and principal components. The Annals of Statistics, 2021, 49 (2): 1113–1138. doi: 10.1214/20-AOS1995
    [23]
    Knowles A, Yin J. Anisotropic local laws for random matrices. Probability Theory and Related Fields, 2017, 169 (1): 257–352. doi: 10.1007/s00440-016-0730-4
    [24]
    Barigozzi M, Trapani L. Sequential testing for structural stability in approximate factor models. Stochastic Processes and Their Applications, 2020, 130 (8): 5149–5187. doi: 10.1016/j.spa.2020.03.003
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    Figure  1.  The largest eigenvalue $ \lambda_1^{(s)} $ in each sliding window (for $1 \leqslant s\leqslant 509$).

    [1]
    Bai J. Inferential theory for factor models of large dimensions. Econometrica, 2003, 71 (1): 135–171. doi: 10.1111/1468-0262.00392
    [2]
    Bai J, Li K. Statistical analysis of factor models of high dimension. The Annals of Statistics, 2012, 40 (1): 436–465. doi: 10.1214/11-AOS966
    [3]
    Bai J, Ng S. Determining the number of factors in approximate factor models. Econometrica, 2002, 70 (1): 191–221. doi: 10.1111/1468-0262.00273
    [4]
    Onatski A. Determining the number of factors from empirical distribution of eigenvalues. The Review of Economics and Statistics, 2010, 92 (4): 1004–1016. doi: 10.1162/REST_a_00043
    [5]
    El Karoui N. Spectrum estimation for large dimensional covariance matrices using random matrix theory. The Annals of Statistics, 2008, 36 (6): 2757–2790. doi: 10.1214/07-AOS581
    [6]
    Bao Z, Pan G, Zhou W. Universality for the largest eigenvalue of sample covariance matrices with general population. The Annals of Statistics, 2015, 43 (1): 382–421. doi: 10.1214/14-AOS1281
    [7]
    Johnstone I M. On the distribution of the largest eigenvalue in principal components analysis. The Annals of Statistics, 2001, 29 (2): 295–327. doi: 10.1214/aos/1009210544
    [8]
    Baik J, Silverstein J W. Eigenvalues of large sample covariance matrices of spiked population models. Journal of Multivariate Analysis, 2006, 97 (6): 1382–1408. doi: 10.1016/j.jmva.2005.08.003
    [9]
    Paul D. Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statistica Sinica, 2007, 17 (4): 1617–1642.
    [10]
    Bai Z, Yao J. On sample eigenvalues in a generalized spiked population model. Journal of Multivariate Analysis, 2012, 106: 167–177. doi: 10.1016/j.jmva.2011.10.009
    [11]
    Wang W, Fan J. Asymptotics of empirical eigenstructure for high dimensional spiked covariance. The Annals of Statistics, 2017, 45 (3): 1342–1374. doi: 10.1214/16-AOS1487
    [12]
    Cai T T, Han X, Pan G. Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices. The Annals of Statistics, 2020, 48 (3): 1255–1280. doi: 10.1214/18-AOS1798
    [13]
    Trapani L. A randomized sequential procedure to determine the number of factors. Journal of the American Statistical Association, 2018, 113 (523): 1341–1349. doi: 10.1080/01621459.2017.1328359
    [14]
    Pearson E S. On questions raised by the combination of tests based on discontinuous distributions. Biometrika, 1950, 37: 383–398. doi: 10.1093/biomet/37.3-4.383
    [15]
    Corradi V, Swanson N R. The effect of data transformation on common cycle, cointegration, and unit root tests: Monte Carlo results and a simple test. Journal of Econometrics, 2006, 132 (1): 195–229. doi: 10.1016/j.jeconom.2005.01.028
    [16]
    Chen L, Dolado J J, Gonzalo J. Detecting big structural breaks in large factor models. Journal of Econometrics, 2014, 180 (1): 30–48. doi: 10.1016/j.jeconom.2014.01.006
    [17]
    Cheng X, Liao Z, Schorfheide F. Shrinkage estimation of high-dimensional factor models with structural instabilities. The Review of Economic Studies, 2016, 83 (4): 1511–1543. doi: 10.1093/restud/rdw005
    [18]
    Stock J H, Watson M W. Disentangling the channels of the 2007–2009 recession. Cambridge, MA: National Bureau of Economic Research, 2012: 18094.
    [19]
    Breitung J, Eickmeier S. Testing for structural breaks in dynamic factor models. Journal of Econometrics, 2011, 163 (1): 71–84. doi: 10.1214/17-AAP1341
    [20]
    Ding X, Yang F. A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices. The Annals of Applied Probability, 2018, 28 (3): 1679–1738. doi: 10.1214/17-AAP1341
    [21]
    Ding X, Yang F. Tracy–Widom distribution for heterogeneous Gram matrices with applications in signal detection. IEEE Transactions on Information Theory, 2022, 68 (10): 6682–6715. doi: 10.1109/TIT.2022.3176784
    [22]
    Ding X, Yang F. Spiked separable covariance matrices and principal components. The Annals of Statistics, 2021, 49 (2): 1113–1138. doi: 10.1214/20-AOS1995
    [23]
    Knowles A, Yin J. Anisotropic local laws for random matrices. Probability Theory and Related Fields, 2017, 169 (1): 257–352. doi: 10.1007/s00440-016-0730-4
    [24]
    Barigozzi M, Trapani L. Sequential testing for structural stability in approximate factor models. Stochastic Processes and Their Applications, 2020, 130 (8): 5149–5187. doi: 10.1016/j.spa.2020.03.003
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