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THREE-DIMENSIONAL FACTOR MODELS WITH GLOBAL AND LOCAL FACTORS

Published online by Cambridge University Press:  11 August 2025

Xun Lu ,
Sainan Jin  and
Liangjun Su
Xun Lu
Affiliation:
Chinese University of Hong Kong
Sainan Jin
Affiliation:
Tsinghua University
Liangjun Su*
Affiliation:
Tsinghua University
*
Address correspondence to Liangjun Su, School of Economics and Management, Tsinghua University, Beijing, China, e-mail: sulj@sem.tsinghua.edu.cn.
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Abstract

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This article considers a three-dimensional latent factor model in the presence of one set of global factors and two sets of local factors. We show that the numbers of global and local factors can be estimated uniformly and consistently. Given the number of global and local factors, we propose a two-step estimation procedure based on principal component analysis (PCA) and establish the asymptotic properties of the PCA estimators. Monte Carlo simulations demonstrate that they perform well in finite samples. An application to the dataset of international trade reveals the relative importance of different types of factors.

Information

Type
ARTICLE
Information
Econometric Theory , Volume 42 , Issue 4 , August 2026 , pp. 903 - 949
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Footnotes

The authors thank the Editor, Peter C.B. Phillips, a Co-Editor, and two anonymous referees for their constructive comments. Lu acknowledges support from the Hong Kong Research Grants Council (RGC) under grant number 14500121 and Chinese University of Hong Kong for the start-up fund. Su thanks the National Natural Science Foundation of China (NSFC) for financial support under the grant number 72133002. All errors are the authors’ sole responsibilities.

References

REFERENCES

Ahn, S. C., & Horenstein, A. R. (2013). Eigenvalue ratio test for the number of factors. Econometrica , 81(3), 1203–1227.Google Scholar
Anderson, J. E., & van Wincoop, E. (2003). Gravity with gravitas: A solution to the border puzzle. American Economic Review , 93(1), 170–192.CrossRefGoogle Scholar
Ando, T., & Bai, J. (2017). Clustering huge number of financial time series: A panel data approach with high-dimensional predictors and factor structures. Journal of the American Statistical Association , 112(519), 1182–1198.CrossRefGoogle Scholar
Andrade, P., & Zachariadis, M. (2016). Global versus local shocks in micro price dynamics. Journal of International Economics , 98, 78–92.CrossRefGoogle Scholar
Andreou, E., Gagliardini, P., Ghysels, E., & Rubin, M. (2019). Inference in group factor models with an application to mixed-frequency data. Econometrica , 87(4), 1267–1305.CrossRefGoogle Scholar
Babii, A., Ghysels, E., & Pan, J. (2024). Tensor principal component analysis. Working Paper.Google Scholar
Bai, J. (2003). Inferential theory for factor models of large dimensions. Econometrica , 71(1), 135–171.CrossRefGoogle Scholar
Bai, J. (2009). Panel data models with interactive fixed effects. Econometrica , 77(4), 1229–1279.Google Scholar
Bai, J., & Ng, S. (2002). Determining the number of factors in approximate factor models. Econometrica , 70(1), 191–221.CrossRefGoogle Scholar
Bai, J., & Ng, S. (2004). A panic attack on unit roots and cointegration. Econometrica , 72(4), 1127–1177.CrossRefGoogle Scholar
Bai, J., & Ng, S. (2006). Confidence intervals for diffusion index forecasts and inference for factor-augmented regressions. Econometrica , 74(4), 1133–1150.CrossRefGoogle Scholar
Bai, J., & Ng, S. (2013). Principal components estimation and identification of static factors. Journal of Econometrics , 176(1), 18–29.CrossRefGoogle Scholar
Bai, J., & Ng, S. (2023). Approximate factor models with weaker loadings. Journal of Econometrics , 235(2), 1893–1916.CrossRefGoogle Scholar
Bai, J., & Wang, P. (2016). Econometric analysis of large factor models. Annual Review of Economics , 8, 53–80.CrossRefGoogle Scholar
Bai, Z., & Saranadasa, H. (1996). Effect of high dimension: An example of a two sample problem. Statistica Sinica , 6(2), 311–329.Google Scholar
Baier, S. L., & Bergstrand, J. H. (2001). The growth of world trade: Tariffs, transport costs, and income similarity. Journal of International Economics , 53(1), 1–27.CrossRefGoogle Scholar
Bernard, A. B., Dhyne, E., Magerman, G., Manova, K., & Moxnes, A. (2022). The origins of firm heterogeneity: A production network approach. Journal of Political Economy , 130(7), 1765–1804.CrossRefGoogle Scholar
Breitung, J., & Eickmeier, S. (2016). Analyzing international business and financial cycles using multi-level factor models: A comparison of alternative approaches. In E. Hillebrand & S. J. Koopman (Eds.), Dynamic factor models (vol. 35, pp. 177–214). Emerald Group Publishing Limited.Google Scholar
Chen, B., Han, Y., & Yu, Q. (2024). Estimation and inference for CP tensor factor models. arXiv preprint arXiv:2406.17278.Google Scholar
Chen, E. Y., Tsay, R. S., & Chen, R. (2020). Constrained factor models for high-dimensional matrix-variate time series. Journal of the American Statistical Association , 115(530), 775–793.CrossRefGoogle Scholar
Chen, M. (2023). Circularly projected common factors for grouped data. Journal of Business & Economic Statistics , 41(2), 636–649.CrossRefGoogle Scholar
Chen, Q., & Fang, Z. (2019). Improved inference on the rank of a matrix. Quantitative Economics , 10(4), 1787–1824.CrossRefGoogle Scholar
Chen, R., Yang, D., & Zhang, C.-H. (2022). Factor models for high-dimensional tensor time series. Journal of the American Statistical Association , 117(537), 94–116.CrossRefGoogle Scholar
Chen, S. X., & Qin, Y.-L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. The Annals of Statistics , 38(2), 808–835.CrossRefGoogle Scholar
Cheng, X., & Hansen, B. E. (2015). Forecasting with factor-augmented regression: A frequentist model averaging approach. Journal of Econometrics , 186(2), 280–293.CrossRefGoogle Scholar
Cheng, X., Liao, Z., & Schorfheide, F. (2016). Shrinkage estimation of high-dimensional factor models with structural instabilities. The Review of Economic Studies , 83(4), 1511–1543.CrossRefGoogle Scholar
Chiang, H. D., Rodrigue, J., & Sasaki, Y. (2023). Post-selection inference in three-dimensional panel data. Econometric Theory , 39(3), 623–658.CrossRefGoogle Scholar
Choi, I., Kim, D., Kim, Y. J., & Kwark, N.-S. (2018). A multilevel factor model: Identification, asymptotic theory and applications. Journal of Applied Econometrics , 33(3), 355–377.CrossRefGoogle Scholar
Choi, I., Lin, R., & Shin, Y. (2023). Canonical correlation-based model selection for the multilevel factors. Journal of Econometrics , 233(1), 22–44.CrossRefGoogle Scholar
Chudik, A., & Pesaran, M. H. (2015). Common correlated effects estimation of heterogeneous dynamic panel data models with weakly exogenous regressors. Journal of Econometrics , 188(2), 393–420.CrossRefGoogle Scholar
Dhyne, E., Kikkawa, A. K., Mogstad, M., & Tintelnot, F. (2021). Trade and domestic production networks. The Review of Economic Studies , 88(2), 643–668.CrossRefGoogle Scholar
Dias, F., Pinheiro, M., & Rua, A. (2013). Determining the number of global and country-specific factors in the euro area. Studies in Nonlinear Dynamics and Econometrics , 17(5), 573–617.Google Scholar
Fan, J., Ke, Y., & Liao, Y. (2016a). Robust factor models with explanatory proxies. arXiv preprint arXiv:1603.07041.CrossRefGoogle Scholar
Fan, J., Liao, Y., & Wang, W. (2016b). Projected principal component analysis in factor models. The Annals of Statistics , 44(1), 219–254.CrossRefGoogle Scholar
Fan, J., Liao, Y., & Mincheva, M. (2013). Large covariance estimation by thresholding principal orthogonal complements. Journal of the Royal Statistical Society. Series B, Statistical Methodology , 75(4), 603–680.CrossRefGoogle ScholarPubMed
Feenstra, R. C. (2015). Advanced international trade: Theory and evidence . Princeton University Press.Google Scholar
Feng, G., Gao, J., Liu, F., & Peng, B. (2024). Estimation and inference for three-dimensional panel data models. arXiv preprint arXiv:2404.08365.Google Scholar
Freeman, H. (2022). Multidimensional interactive fixed-effects. arXiv preprint arXiv:2209.11691.Google Scholar
Gao, Z., & Tsay, R. S. (2023). Divide-and-conquer: A distributed hierarchical factor approach to modeling large-scale time series data. Journal of the American Statistical Association , 118(544), 2698–2711.CrossRefGoogle Scholar
Giglio, S., & Xiu, D. (2021). Asset pricing with omitted factors. Journal of Political Economy , 129(7), 1947–1990.CrossRefGoogle Scholar
Han, X. (2021). Shrinkage estimation of factor models with global and group-specific factors. Journal of Business & Economic Statistics , 39(1), 1–17.CrossRefGoogle Scholar
Han, Y., Yang, D., Zhang, C.-H., & Chen, R. (2024). CP factor model for dynamic tensors. Journal of the Royal Statistical Society Series B: Statistical Methodology , 86(5), 1383–1413.CrossRefGoogle Scholar
He, Y., Kong, X., Yu, L., Zhang, X., & Zhao, C. (2024). Matrix factor analysis: From least squares to iterative projection. Journal of Business & Economic Statistics , 42(1), 322–334.CrossRefGoogle Scholar
Head, K., & Mayer, T. (2014). Gravity equations: Workhorse, toolkit, and cookbook. In G. Gopinath, E. Helpman, & K. Rogoff (Eds.), Handbook of international economics (vol. 4, pp. 131–195). Elsevier.Google Scholar
Jin, S., Lu, X., & Su, L. (2025). Three-dimensional heterogeneous panel data models with multi-level interactive fixed effects. Journal of Econometrics , 249(2), 105957.CrossRefGoogle Scholar
Kapetanios, G., Serlenga, L., & Shin, Y. (2021). Estimation and inference for multi-dimensional heterogeneous panel datasets with hierarchical multi-factor error structure. Journal of Econometrics , 220(2), 504–531.CrossRefGoogle Scholar
Kelly, B. T., Pruitt, S., & Su, Y. (2019). Characteristics are covariances: A unified model of risk and return. Journal of Financial Economics , 134(3), 501–524.CrossRefGoogle Scholar
Kleibergen, F., & Paap, R. (2006). Generalized reduced rank tests using the singular value decomposition. Journal of Econometrics , 133(1), 97–126.CrossRefGoogle Scholar
Koren, M., & Tenreyro, S. (2007). Volatility and development. The Quarterly Journal of Economics , 122(1), 243–287.CrossRefGoogle Scholar
Latała, R. (2005). Some estimates of norms of random matrices. Proceedings of the American Mathematical Society , 133(5), 1273–1282.CrossRefGoogle Scholar
Lettau, M. (2022). High-dimensional factor models with an application to mutual fund characteristics. Technical report, National Bureau of Economic Research.CrossRefGoogle Scholar
Lu, X., Miao, K., & Su, L. (2021). Determination of different types of fixed effects in three-dimensional panels. Econometric Reviews , 40(9), 867–898.CrossRefGoogle Scholar
Lu, X., & Su, L. (2016). Shrinkage estimation of dynamic panel data models with interactive fixed effects. Journal of Econometrics , 190(1), 148–175.CrossRefGoogle Scholar
Ma, S., Lan, W., Su, L., & Tsai, C.-L. (2020). Testing alphas in conditional time-varying factor models with high-dimensional assets. Journal of Business & Economic Statistics , 38(1), 214–227.CrossRefGoogle Scholar
Matyas, L. (2017). The econometrics of multi-dimensional panels . Springer.CrossRefGoogle Scholar
Moench, E., & Ng, S. (2011). A hierarchical factor analysis of us housing market dynamics. Econometrics Journal , 14(1), 1–24.CrossRefGoogle Scholar
Moench, E., Ng, S., & Potter, S. (2013). Dynamic hierarchical factor models. The Review of Economics and Statistics , 95(5), 1811–1817.CrossRefGoogle Scholar
Moon, H. R., & Weidner, M. (2015). Linear regression for panel with unknown number of factors as interactive fixed effects. Econometrica , 83(4), 1543–1579.CrossRefGoogle Scholar
Moon, H. R., & Weidner, M. (2017). Dynamic linear panel regression models with interactive fixed effects. Econometric Theory , 33(1), 158–195.CrossRefGoogle Scholar
Onatski, A. (2010). Determining the number of factors from empirical distribution of eigenvalues. The Review of Economics and Statistics , 92(4), 1004–1016.CrossRefGoogle Scholar
Onatski, A. (2012). Asymptotics of the principal components estimator of large factor models with weakly influential factors. Journal of Econometrics , 168(2), 244–258.CrossRefGoogle Scholar
Pesaran, M. H. (2006). Estimation and inference in large heterogeneous panels with a multifactor error structure. Econometrica , 74(4), 967–1012.CrossRefGoogle Scholar
Santos Silva, J., & Tenreyro, S. (2006). The log of gravity. The Review of Economics and Statistics , 88(4), 641–658.CrossRefGoogle Scholar
Stock, J. H., & Watson, M. W. (2002). Macroeconomic forecasting using diffusion indexes. Journal of Business & Economic Statistics , 20(2), 147–162.CrossRefGoogle Scholar
Su, L., & Ju, G. (2018). Identifying latent grouped patterns in panel data models with interactive fixed effects. Journal of Econometrics , 206(2), 554–573.CrossRefGoogle Scholar
Vershynin, R. (2011). Spectral norm of products of random and deterministic matrices. Probability Theory and Related Fields , 150(3–4), 471–509.CrossRefGoogle Scholar
Vershynin, R. (2012). Introduction to the non-asymptotic analysis of random matrices. In Y. C. Eldar & G. Kutyniok (Eds.), Compressed sensing, theory and applications (pp. 210–268). Cambridge University Press.CrossRefGoogle Scholar
Wang, D., Liu, X., & Chen, R. (2019). Factor models for matrix-valued high-dimensional time series. Journal of Econometrics , 208(1), 231–248.CrossRefGoogle Scholar
Wang, P. (2014). Large dimensional factor models with a multi-level factor structure: identification, estimation and inference. Technical report, Hong Kong University of Science and Technology.Google Scholar
Yang, Y., & Schmidt, P. (2021). An econometric approach to the estimation of multi-level models. Journal of Econometrics , 220(2), 532–543.CrossRefGoogle Scholar
Yu, L., He, Y., Kong, X., & Zhang, X. (2022). Projected estimation for large-dimensional matrix factor models. Journal of Econometrics , 229(1), 201–217.CrossRefGoogle Scholar
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