×

Factor models with local factors – determining the number of relevant factors. (English) Zbl 07538791

Summary: We extend the theory on factor models by incorporating “local” factors into the model. Local factors affect only an unknown subset of the observed variables. This implies a continuum of eigenvalues of the covariance matrix, as is commonly observed in applications. We derive which factors are pervasive enough to be economically important and which factors are pervasive enough to be estimable using the common principal component estimator. We then introduce a new class of estimators to determine the number of those relevant factors. Unlike existing estimators, our estimators use not only the eigenvalues of the covariance matrix, but also its eigenvectors. We find that incorporating partial sums of the eigenvectors into our estimators leads to significant gains in performance in simulations.

MSC:

62-XX Statistics
91-XX Game theory, economics, finance, and other social and behavioral sciences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Acemoglu, D.; Carvalho, V. M.; Ozdaglar, A.; Tahbaz-Salehi, A., The network origins of aggregate fluctuations, Econometrica, 80, 5, 1977-2016 (2012) · Zbl 1274.91317
[2] Ahn, S. C.; Horenstein, A. R., Eigenvalue ratio test for the number of factors, Econometrica, 81, 3, 1203-1227 (2013) · Zbl 1274.62403
[3] Alessi, L.; Barigozzi, M.; Capasso, M., Improved penalization for determining the number of factors in approximate factor models, Statist. Probab. Lett., 80, 23, 1806-1813 (2010) · Zbl 1202.62081
[4] Ando, T.; Bai, J., Clustering huge number of financial time series: A panel data approach with high-dimensional predictors and factor structures, J. Amer. Statist. Assoc., 112, 519, 1182-1198 (2017)
[5] Andrews, D. W.; Cheng, X., Estimation and inference with weak, semi-strong, and strong identification, Econometrica, 80, 5, 2153-2211 (2012) · Zbl 1274.62160
[6] Bai, J., Inferential theory for factor models of large dimensions, Econometrica, 71, 1, 135-171 (2003) · Zbl 1136.62354
[7] Bai, J.; Ng, S., Determining the number of factors in approximate factor models, Econometrica, 70, 1, 191-221 (2002) · Zbl 1103.91399
[8] Bai, J.; Ng, S., Determining the Number of Factors in Approximate Factor Models, ErrataTechnical report (2006), Columbia University
[9] Bai, J.; Ng, S., Principal components estimation and identification of static factors, J. Econometrics, 176, 1, 18-29 (2013) · Zbl 1284.62350
[10] Bai, J.; Ng, S., Rank regularized estimation of approximate factor models, J. Econometrics, 212, 1, 78-96 (2019) · Zbl 1452.62405
[11] Bailey, N.; Kapetanios, G.; Pesaran, M. H., Exponent of cross-sectional dependence: Estimation and inference, J. Appl. Econometrics, 31, 6, 929-960 (2016)
[12] Bailey, N., Kapetanios, G., Pesaran, M.H., 2020. Measurement of Factor Strength: Theory and Practice. Technical report, CESifo Working Paper. · Zbl 1467.62103
[13] Bernanke, B. S., Alternative explanations of the money-income correlation, (Carnegie-Rochester Conference Series on Public Policy, Vol. 25 (1986)), 49-99
[14] Boivin, J.; Ng, S., Are more data always better for factor analysis?, J. Econometrics, 132, 1, 169-194 (2006) · Zbl 1337.62345
[15] Cai, T. T.; Ma, Z.; Wu, Y., Sparse PCA: Optimal rates and adaptive estimation, Ann. Statist., 41, 6, 3074-3110 (2013) · Zbl 1288.62099
[16] Carvalho, C. M.; Chang, J.; Lucas, J. E.; Nevins, J. R.; Wang, Q.; West, M., High-dimensional sparse factor modeling: Applications in gene expression genomics, J. Amer. Statist. Assoc., 103, 484, 1438-1456 (2008) · Zbl 1286.62091
[17] Cattell, R. B., The scree test for the number of factors, Multivariate Behav. Res., 1, 2, 245-276 (1966)
[18] Choi, I.; Kim, D.; Kim, Y. J.; Kwark, N.-S., A multilevel factor model: Identification, asymptotic theory and applications, J. Appl. Econometrics, 33, 3, 355-377 (2018)
[19] Chudik, A.; Pesaran, M. H.; Tosetti, E., Weak and strong cross-section dependence and estimation of large panels, Econom. J., 14, 1, C45-C90 (2011) · Zbl 1218.62097
[20] Cochrane, J. H., Shocks, (Carnegie-Rochester Conference Series on Public Policy, Vol. 41 (1994), Elsevier), 295-364
[21] Connor, G.; Korajczyk, R. A., A test for the number of factors in an approximate factor model, J. Finance, 48, 4, 1263-1291 (1993)
[22] De Mol, C.; Giannone, D.; Reichlin, L., Forecasting using a large number of predictors: Is Bayesian shrinkage a valid alternative to principal components?, J. Econometrics, 146, 2, 318-328 (2008) · Zbl 1429.62659
[23] Dias, F.; Pinheiro, M.; Rua, A., Determining the number of global and country-specific factors in the euro area, Stud. Nonlinear Dyn. Econom., 17, 5, 573-617 (2013) · Zbl 1506.62499
[24] Foerster, A. T.; Sarte, P.-D. G.; Watson, M. W., Sectoral versus aggregate shocks: A structural factor analysis of industrial production, J. Political Econ., 119, 1, 1-38 (2011)
[25] Gabaix, X., The granular origins of aggregate fluctuations, Econometrica, 79, 3, 733-772 (2011) · Zbl 1217.91143
[26] Gagliardini, P.; Ossola, E.; Scaillet, O., A diagnostic criterion for approximate factor structure, J. Econometrics, 212, 2, 503-521 (2019) · Zbl 1452.62909
[27] Gao, C.; Brown, C. D.; Engelhardt, B. E., A latent factor model with a mixture of sparse and dense factors to model gene expression data with confounding effects (2013), arXiv preprint arXiv:1310.4792
[28] Green, R. C.; Hollifield, B., When will mean-variance efficient portfolios be well diversified?, J. Finance, 47, 5, 1785-1809 (1992)
[29] Hallin, M.; Liska, R., The generalized dynamic factor model: Determining the number of factors, J. Amer. Statist. Assoc., 102, 478, 603-617 (2007) · Zbl 1172.62339
[30] Han, X., Shrinkage Estimation of Factor Models with Global and Group-Specific FactorsTechnical report (2017), City University of Hong Kong
[31] Han, X.; Caner, M., Determining the number of factors with potentially strong within-block correlations in error terms, Econometric Rev., 36, 6-9, 946-969 (2017) · Zbl 1524.62285
[32] Horn, R. A.; Johnson, C. R., Matrix Analysis (2012), Cambridge University Press
[33] Horvath, M., Cyclicality and sectoral linkages: Aggregate fluctuations from independent sectoral shocks, Rev. Econ. Dyn., 1, 4, 781-808 (1998)
[34] Kapetanios, G., A New Method for Determining the Number of Factors in Factor Models with Large DatasetsTechnical report (2004), Department of Economics, Queen Mary, University of London
[35] Kapetanios, G., A testing procedure for determining the number of factors in approximate factor models with large datasets, J. Bus. Econom. Statist., 28, 3, 397-409 (2010) · Zbl 1214.62068
[36] Kleibergen, F., Tests of risk premia in linear factor models, J. Econometrics, 149, 2, 149-173 (2009) · Zbl 1429.62680
[37] Long, J. B.; Plosser, C. I., Real business cycles, J. Political Econ., 91, 1, 39-69 (1983)
[38] Moench, E.; Ng, S.; Potter, S., Dynamic hierarchical factor models, Rev. Econ. Stat., 95, 5, 1811-1817 (2013)
[39] Moon, H. R.; Weidner, M., Dynamic linear panel regression models with interactive fixed effects, Econometric Theory, 33, 158-195 (2017) · Zbl 1441.62816
[40] Onatski, A., Testing hypotheses about the number of factors in large factor models, Econometrica, 77, 5, 1447-1479 (2009) · Zbl 1182.62180
[41] Onatski, A., Determining the number of factors from empirical distribution of eigenvalues, Rev. Econ. Stat., 92, 4, 1004-1016 (2010)
[42] Onatski, A., Asymptotics of the principal components estimator of large factor models with weakly influential factors, J. Econometrics, 168, 2, 244-258 (2012) · Zbl 1443.62497
[43] Onatski, A., Asymptotic analysis of the squared estimation error in misspecified factor models, J. Econometrics, 186, 2, 388-406 (2015) · Zbl 1331.62480
[44] Pati, D.; Bhattacharya, A.; Pillai, N. S.; Dunson, D., Posterior contraction in sparse Bayesian factor models for massive covariance matrices, Ann. Statist., 42, 3, 1102-1130 (2014) · Zbl 1305.62124
[45] Paul, D.; Johnstone, I. M., Augmented sparse principal component analysis for high dimensional data (2012), arXiv preprint arXiv:1202.1242
[46] Ross, S. A., The arbitrage theory of capital asset pricing, J. Econom. Theory, 13, 3, 341-360 (1976)
[47] Shukla, R.; Trzcinka, C., Sequential tests of the arbitrage pricing theory: A comparison of principal components and maximum likelihood factors, J. Finance, 45, 5, 1541-1564 (1990)
[48] Stock, J. H.; Watson, M. W., Forecasting using principal components from a large number of predictors, J. Amer. Statist. Assoc., 97, 460, 1167-1179 (2002) · Zbl 1041.62081
[49] Stock, J.H., Watson, M.W., 2005. Implications of Dynamic Factor Models for VAR Analysis. Technical report, NBER Working Paper 114467.
[50] Stock, J. H.; Watson, M. W., Disentangling the channels of the 2007-09 recession, Brook. Pap. Econ. Act., 43, 81-156 (2012)
[51] Stock, J. H.; Watson, M. W., Dynamic factor models, factor-augmented vector autoregressions, and structural vector autoregressions in macroeconomics, (Taylor, J. B.; Uhlig, H., Handbook of Macroeconomics, Vol. 2 (2016), Elsevier), 415-525, Chapter 8
[52] Ten Berge, J. M.; Kiers, H. A., A numerical approach to the approximate and the exact minimum rank of a covariance matrix, Psychometrika, 56, 2, 309-315 (1991) · Zbl 0850.62462
[53] Trzcinka, C., On the number of factors in the arbitrage pricing model, J. Finance, 41, 2, 347-368 (1986)
[54] Uematsu, Y., Yamagata, T., 2020. Estimation of Weak Factor Models. Technical report, ISER Discussion paper 1053.
[55] Wang, P., Large Dimensional Factor Models with a Multi-Level Factor Structure: Identification, Estimation and InferenceWorking paper (2008), New York University
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.