Abstract

This paper introduces a novel two-way factor model designed to analyze high-dimensional panel

time series. The proposed model assumes that the low-dimensional hidden factors are separably influenced

by rows and columns. We conduct likelihood inference via a matrix decomposition technique. To obtain

maximum likelihood estimates (MLE) of factor loadings and other parameters, we apply the traditional

delta method, which relies on the score function and the Hessian matrix. Additionally, we develop fast

computational algorithms based on diagonal block matrices to estimate the model’s parameters. Under

regularity conditions, we establish the theoretical properties of the proposed estimators, including consistency and asymptotic normality. Notably, the proposed approach achieves

√

T-consistency, representing

a substantial improvement over the convergence rates established in prior literature. The effectiveness of

the methodology is further validated through simulation experiments and real data analysis.

Key words and phrases: Two-way factor model, panel data, high dimensionality, cross-section, time series, likelihood inference Corresponding author (Rubing Liang): liang ru bing@scau.edu.cn, South China Agricultural University

Information

Preprint No.SS-2025-0292
Manuscript IDSS-2025-0292
Complete AuthorsQiang Xia, Zhigen Gao, Gaorong Li, Rubing Liang
Corresponding AuthorsQiang Xia
Emailsxiaqiang@scau.edu.cn

References

  1. Ahn, S. C. and Horenstein, A. R. (2013). Eigenvalue ratio test for the number of factors, Econometrica 81: 1203–1227.
  2. Bai, J. (2003). Inferential theory for factor models of large dimensions, Econometrica 71: 135– 171.
  3. Bai, J. and Li, K. (2012). Statistical analysis of factor models of high dimension, The Annals of Statistics 40: 436–465.
  4. Bernanke, B., Bovian, J. and Eliasz, P. (2005). Measuring the effects of monetary policy: a factor-augmented vector autoregressive (favar) approach, Quarterly Journal of Economics 120: 387–422.
  5. Chang, J. Y., Guo, B. and Yao, Q. (2015). High dimensional stochastic regression with latent factors, endogeneity and nonlinearity, Journal of Econometrics 189: 297–312.
  6. Chen, E. Y. and Fan, J. (2023). Statistical inference for high-dimensional matrix-variate factor model, Journal of the American Statistical Association 118: 1038–1055.
  7. Chen, E. Y., Tsay, R. S. and Chen, R. (2020). Constrained factor models for high-dimensional matrix-variate time series, Journal of the American Statistical Association 115: 775–793.
  8. Doz, C., Giannone, D. and Reichlin, L. (2012). A quasi-maximum likelihood approach for large, approximate dynamic factor models, The Review of Economics and Statistics 94: 1014–1024.
  9. Fan, J., Fan, Y. and Lv, J. (2008). High dimensional covariance matrix estimation using a factor model, Journal of Econometrics 147: 186–197.
  10. Fan, J., Guo, J. and Zheng, S. (2022). Estimating number of factors by adjusted eigenvalues thresholding, Journal of the American Statistical Association 117(538): 852–861.
  11. Forni, M., Hallin, M., Lippi, M. and Reichlin, L. (2005). The generalized dynamic-factor model: one-sided estimation and forecasting, Journal of the American Statistical Association 100: 830–840.
  12. Gao, Z., Yuan, C., Jing, B., Wei, H. and Guo, J. (2021). A two-way factor model for highdimensional matrix data, arXiv.org .
  13. He, Y., Kong, X., Yu, L., Zhang, X. and Zhao, C. (2024). Matrix factor analysis: From least squares to iterative projection, Journal of Business and Economic Statistics 42(1): 322–334.
  14. He, Y., Trapani, L., Kong, X. and Yu, L. (2023). One-way or two-way factor model for matrix sequences?, Journal of Econometrics 235: 1981–2004.
  15. Hornstein, M., Fan, R., Shedden, K. and Zhou, S. (2019). Joint mean and covariance estimation with unreplicated matrix-variate data, Journal of the American Statistical Association 114(526): 682–696.
  16. Lam, C., Yao, Q. and Bathia, N. (2011). Estimation for latent factor models for highdimensional time series, Biometrika 98: 901–918.
  17. Lee, S. (1978). The gauss-newton algorithm for the weighted least squares factor analysis, The Statistician 27: 103114.
  18. McCracken, M. and Ng, S. (2015). Fred-md: a monthly database for macroeconomic research, Journal of Business and Economic Statistics 34: 574–589.
  19. Ng, C. T., Yau, C. Y. and Chan, N. H. (2015). Likelihood inferences for high-dimensional factor analysis of time series with applications in finance, Journal of Computational and Graphical Statistics 24: 866–884.
  20. Pan, J. and Yao, Q. (2008). Modelling multiple time series via common factors, Biometrika 95: 356–379.
  21. Stock, J. and Watson, M. (2002). Forecasting using principal components from a large number of predictors, Journal of the American Statistical Association 97: 1167–1179.
  22. Wang, D., Liu, X. and Chen, R. (2019). Factor models for matrix-valued high-dimensional time series, Journal of Econometrics 208: 231–248.
  23. Xia, Q., Xu, W. and Zhu, L. X. (2015). Consistently determining the number of factors in multivariate volatility modelling, Statistica Sinica 25: 1025–1044.
  24. Yu, L., He, Y., Kong, X. and Zhang, X. (2022). Projected estimation for large-dimensional matrixfactor models, Journal of Econometrics 229: 201–217.
  25. Yuan, C., Gao, Z., He, X., Huang, W. and Guo, J. (2023). Two-way dynamic factor models for high-dimensional matrix-valued time series, Journal of the Royal Statistical Society: Series B 85: 1517–1537.
  26. Zhang, X., Liu, C. C., Guo, J., Yuen, K. C. and Welsh, A. H. (2025). Modeling and learning on high-dimensional matrix-variate sequences, Journal of the American Statistical Association 120: 419–434.

Acknowledgments

We sincerely thank Co-Editor Prof. John Stufken, the Associate Editor, and two referees for

their thorough review and perceptive comments that greatly strengthened this manuscript. We

are also grateful to Prof. Jianhua Guo for his helpful suggestions. This work was supported

in part by the National Statistical Scientific Research Projects (No. 2025LZ029), the National

Natural Science Foundation of China (Nos. 12171161 and No. 12271046), and the Technology

Development Plan Project of Jilin Province, China (No. 20240101022JJ).