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Matrix Quantile Factor Model

Author

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  • Xin-Bing Kong
  • Yong-Xin Liu
  • Long Yu
  • Peng Zhao

Abstract

This paper introduces a matrix quantile factor model for matrix-valued data with a low-rank structure. We estimate the row and column factor spaces via minimizing the empirical check loss function over all panels. We show the estimates converge at rate $1/\min\{\sqrt{p_1p_2}, \sqrt{p_2T},$ $\sqrt{p_1T}\}$ in average Frobenius norm, where $p_1$, $p_2$ and $T$ are the row dimensionality, column dimensionality and length of the matrix sequence. This rate is faster than that of the quantile estimates via ``flattening" the matrix model into a large vector model. Smoothed estimates are given and their central limit theorems are derived under some mild condition. We provide three consistent criteria to determine the pair of row and column factor numbers. Extensive simulation studies and an empirical study justify our theory.

Suggested Citation

  • Xin-Bing Kong & Yong-Xin Liu & Long Yu & Peng Zhao, 2022. "Matrix Quantile Factor Model," Papers 2208.08693, arXiv.org, revised May 2023.
    • Handle: RePEc:arx:papers:2208.08693
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    References listed on IDEAS

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    1. Xin-Bing Kong, 2017. "On the number of common factors with high-frequency data," Biometrika, Biometrika Trust, vol. 104(2), pages 397-410.
    2. Lorenzo Trapani, 2018. "A Randomized Sequential Procedure to Determine the Number of Factors," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(523), pages 1341-1349, July.
    3. Jianqing Fan & Yuan Liao & Martina Mincheva, 2013. "Large covariance estimation by thresholding principal orthogonal complements," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(4), pages 603-680, September.
    4. Jushan Bai & Serena Ng, 2002. "Determining the Number of Factors in Approximate Factor Models," Econometrica, Econometric Society, vol. 70(1), pages 191-221, January.
    5. Liang Chen & Juan J. Dolado & Jesús Gonzalo, 2021. "Quantile Factor Models," Econometrica, Econometric Society, vol. 89(2), pages 875-910, March.
    6. Pelger, Markus, 2019. "Large-dimensional factor modeling based on high-frequency observations," Journal of Econometrics, Elsevier, vol. 208(1), pages 23-42.
    7. Barigozzi, Matteo & Trapani, Lorenzo, 2020. "Sequential testing for structural stability in approximate factor models," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5149-5187.
    8. Stock, James H & Watson, Mark W, 2002. "Macroeconomic Forecasting Using Diffusion Indexes," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(2), pages 147-162, April.
    9. Alexei Onatski, 2010. "Determining the Number of Factors from Empirical Distribution of Eigenvalues," The Review of Economics and Statistics, MIT Press, vol. 92(4), pages 1004-1016, November.
    10. Wang, Dong & Liu, Xialu & Chen, Rong, 2019. "Factor models for matrix-valued high-dimensional time series," Journal of Econometrics, Elsevier, vol. 208(1), pages 231-248.
    11. Seung C. Ahn & Alex R. Horenstein, 2013. "Eigenvalue Ratio Test for the Number of Factors," Econometrica, Econometric Society, vol. 81(3), pages 1203-1227, May.
    12. Yu, Long & He, Yong & Kong, Xinbing & Zhang, Xinsheng, 2022. "Projected estimation for large-dimensional matrix factor models," Journal of Econometrics, Elsevier, vol. 229(1), pages 201-217.
    13. Elynn Y. Chen & Ruey S. Tsay & Rong Chen, 2020. "Constrained Factor Models for High-Dimensional Matrix-Variate Time Series," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(530), pages 775-793, April.
    14. Aït-Sahalia, Yacine & Xiu, Dacheng, 2017. "Using principal component analysis to estimate a high dimensional factor model with high-frequency data," Journal of Econometrics, Elsevier, vol. 201(2), pages 384-399.
    15. Xinbing Kong & Jiangyan Wang & Jinbao Xing & Chao Xu & Chao Ying, 2019. "Factor and Idiosyncratic Empirical Processes," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(527), pages 1138-1146, July.
    16. Yong He & Xinbing Kong & Long Yu & Xinsheng Zhang, 2022. "Large-Dimensional Factor Analysis Without Moment Constraints," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 40(1), pages 302-312, January.
    17. Stock J.H. & Watson M.W., 2002. "Forecasting Using Principal Components From a Large Number of Predictors," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 1167-1179, December.
    18. Jushan Bai, 2003. "Inferential Theory for Factor Models of Large Dimensions," Econometrica, Econometric Society, vol. 71(1), pages 135-171, January.
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    Cited by:

    1. Yang, Shuquan & Ling, Nengxiang, 2023. "Robust projected principal component analysis for large-dimensional semiparametric factor modeling," Journal of Multivariate Analysis, Elsevier, vol. 195(C).

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