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LIMIT THEOREMS FOR FACTOR MODELS

Published online by Cambridge University Press:  09 November 2020

Stanislav Anatolyev  and
Anna Mikusheva
Stanislav Anatolyev*
Affiliation:
CERGE-EI New Economic School
Anna Mikusheva
Affiliation:
Massachusetts Institute of Technology
*
Address correspondence to Stanislav Anatolyev, CERGE-EI, Politickych vězňů 7, 11121 Prague 1, Czech Republic; e-mail: stanislav.anatolyev@cerge-ei.cz.
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Abstract

This paper establishes central limit theorems (CLTs) and proposes how to perform valid inference in factor models. We consider a setting where many counties/regions/assets are observed for many time periods, and when estimation of a global parameter includes aggregation of a cross-section of heterogeneous microparameters estimated separately for each entity. The CLT applies for quantities involving both cross-sectional and time series aggregation, as well as for quadratic forms in time-aggregated errors. This paper studies the conditions when one can consistently estimate the asymptotic variance, and proposes a bootstrap scheme for cases when one cannot. A small simulation study illustrates performance of the asymptotic and bootstrap procedures. The results are useful for making inferences in two-step estimation procedures related to factor models, as well as in other related contexts. Our treatment avoids structural modeling of cross-sectional dependence but imposes time-series independence.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 5 , October 2021 , pp. 1034 - 1074
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

We thank the Editor Peter C.B. Phillips, Co-Editor Guido Kuersteiner, and three anonymous referees for guidance and helpful suggestions that have greatly improved this paper. Support for Stanislav Anatolyev by Czech Science Foundation under grants 17-26535S and 20-28055S is gratefully acknowledged. Support for Anna Mikusheva by National Science Foundation under grant 1757199 is gratefully acknowledged.

References

REFERENCES

Anatolyev, S. & Mikusheva, A. (2018) Factor Models with Many Assets: Strong Factors, Weak Factors, and the Two-Pass Procedure. Working paper, CERGE-EI and Massachusetts Institute of Technology. arXiv:1807.04094.Google Scholar
Ando, T. & Bai, J. (2015) Asset pricing with a general multifactor structure. Journal of Financial Econometrics 13, 556–604.CrossRefGoogle Scholar
Bai, J. (2009) Panel data models with interactive fixed effects. Econometrica 77, 1229–1279.Google Scholar
Bai, J. & Ng, S. (2006) Confidence intervals for diffusion index forecast and inference with factor-augmented regressions. Econometrica 74, 1133–1155.CrossRefGoogle Scholar
Bai, J. & Ng, S. (2010) Instrumental variable estimation in a data rich environment. Econometric Theory 26, 1577–1606.CrossRefGoogle Scholar
Bhansali, R.J., Giraitis, L., & Kokoszka, P.S. (2007) Convergence of quadratic forms with nonvanishing diagonal. Statistics & Probability Letters 77, 726–734.CrossRefGoogle Scholar
Cattaneo, M.D., Crump, R.K., & Jansson, M. (2014a) Bootstrapping density-weighted average derivatives. Econometric Theory 30(6), 1135–1164.CrossRefGoogle Scholar
Cattaneo, M.D., Crump, R.K., & Jansson, M. (2014b) Small bandwidth asymptotics for density-weighted average derivatives. Econometric Theory 30(1), 176–200.CrossRefGoogle Scholar
Cattaneo, M.D., Jansson, M., & Newey, W.K. (2018) Alternative asymptotics and the partially linear model with many regressors. Econometric Theory 34(2), 277–301.CrossRefGoogle Scholar
Chao, J.C., Swanson, N.R., Hausman, J.A., Newey, W.K., & Woutersen, T. (2012) Asymptotic distribution of JIVE in a heteroskedastic IV regression with many instruments. Econometric Theory 28, 42–86.CrossRefGoogle Scholar
de Jong, P. (1987) A central limit theorem for generalized quadratic forms. Probability Theory and Related Fields 75, 261–277.CrossRefGoogle Scholar
Fama, E.F. & MacBeth, J. (1973) Risk, return and equilibrium: Empirical tests. Journal of Political Economy 81, 607–636.CrossRefGoogle Scholar
Giraitis, L., Koul, H.L., & Surgailis, D. (2012) Large Sample Inference for Long Memory Processes. Imperial College Press.CrossRefGoogle Scholar
Hagedorn, M., Manovskii, I., & Mitman, K. (2015) The Impact of Unemployment Benefit Extensions on Employment: The 2014 Employment Miracle? Working paper 20884, National Bureau of Economic Research.CrossRefGoogle Scholar
Hahn, J., Kuersteiner, G.M., & Mazzocco, M. (2020) Central Limit Theory for Combined Cross-Section and Time Series. Working paper, University of California, Los Angeles and University of Maryland. arXiv:1610.01697.Google Scholar
Hausman, J.A., Newey, W.K., Woutersen, T., Chao, J.C., & Swanson, N.R. (2012) Instrumental variable estimation with heteroskedasticity and many instruments. Quantitative Economics 3, 211–255.CrossRefGoogle Scholar
Heyde, C. & Brown, B. (1970) On the departure from normality of a certain class of martingales. Annals of Mathematical Statistics 41, 2161–2165.CrossRefGoogle Scholar
Kleibergen, F. & Zhan, Z. (2015) Unexplained factors and their effects on second pass R-squared’s. Journal of Econometrics 189, 101–116.CrossRefGoogle Scholar
Kuersteiner, G.M. & Prucha, I.R. (2013) Limit theory for panel data models with cross sectional dependence and sequential exogeneity. Journal of Econometrics 174, 107–126.CrossRefGoogle ScholarPubMed
Kuersteiner, G.M. & Prucha, I.R. (2020) Dynamic spatial panel models: Networks, common shocks, and sequential exogeneity. Econometrica 88(5), 2109–2146.CrossRefGoogle Scholar
Onatski, A. (2012) Asymptotics of the principal components estimator of large factor models with weakly influential factors. Journal of Econometrics 168, 244–258.CrossRefGoogle Scholar
Pesaran, M.H. (2006) Estimation and inference in large heterogeneous panels with a multifactor error structure. Econometrica 74, 967–1012.CrossRefGoogle Scholar
Pesaran, M.H. & Yamagata, T. (2018) Testing for Alpha in Linear Factor Pricing Models with a Large Number of Securities. Working paper, University of Southern California. http://dx.doi.org/10.2139/ssrn.2943640.CrossRefGoogle Scholar
Rotar, V.I. (1973) Some limit theorems for polynomials of second degree. Theory of Probability and Its Applications 18, 499–507.CrossRefGoogle Scholar
Sarto, A.P. (2018) Recovering Macro Elasticities from Regional Data. Working paper, Massachusetts Institute of Technology.Google Scholar
Serrato, J.C.S. & Wingender, P. (2016) Estimating Local Fiscal Multipliers. NBER Working paper 22425.CrossRefGoogle Scholar
Shanken, J. (1992) On the estimation of beta-pricing models. Review of Financial Studies 5, 1–33.CrossRefGoogle Scholar
Sølvsten, M. (2020) Robust estimation with many instruments. Journal of Econometrics 214, 495–512.CrossRefGoogle Scholar