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ESTIMATION OF TIME-VARYING COVARIANCE MATRICES FOR LARGE DATASETS

Published online by Cambridge University Press:  08 February 2021

Yiannis Dendramis ,
Liudas Giraitis  and
George Kapetanios
Yiannis Dendramis
Affiliation:
Athens University of Economics and Business
Liudas Giraitis*
Affiliation:
Queen Mary University of London
George Kapetanios
Affiliation:
King’s College London
*
Corresponding address: School of Economics and Finance, Queen Mary University of London, LondonE1 4NS, UK: e-mail: L.Giraitis@qmul.ac.uk.
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Abstract

Time variation is a fundamental problem in statistical and econometric analysis of macroeconomic and financial data. Recently, there has been considerable focus on developing econometric modelling that enables stochastic structural change in model parameters and on model estimation by Bayesian or nonparametric kernel methods. In the context of the estimation of covariance matrices of large dimensional panels, such data requires taking into account time variation, possible dependence and heavy-tailed distributions. In this paper, we introduce a nonparametric version of regularization techniques for sparse large covariance matrices, developed by Bickel and Levina (2008) and others. We focus on the robustness of such a procedure to time variation, dependence and heavy-tailedness of distributions. The paper includes a set of results on Bernstein type inequalities for dependent unbounded variables which are expected to be applicable in econometric analysis beyond estimation of large covariance matrices. We discuss the utility of the robust thresholding method, comparing it with other estimators in simulations and an empirical application on the design of minimum variance portfolios.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 6 , December 2021 , pp. 1100 - 1134
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

We thank the Editor, the Co-Editor and the Referees for constructive comments. Yiannis Dendramis acknowledges support from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 708501.

References

REFERENCES

Abadir, K. M., Distaso, W., & Zikes, F. (2014) Design-free estimation of variance matrices. Journal of Econometrics 181, 165–180.CrossRefGoogle Scholar
Bai, J. & Ng, S. (2002) Determining the number of factors in approximate factor models. Econometrica 70(1), 191–221.CrossRefGoogle Scholar
Bailey, N., Kapetanios, G., & Pesaran, M. H. (2016) Exponent of cross-sectional dependence: estimation and inference. Journal of Applied Econometrics 31, 929—960.CrossRefGoogle Scholar
Bickel, M., Wang, Y., & Zhou, H. (2013) Optimal sparse volatility matrix estimation for high-dimensional Itô processes with measurement errors. The Annals of Statistics 41, 1816–1864.Google Scholar
Bickel, P. & Levina, E. (2008) Covariance regularization by thresholding. The Annals of Statistics 36, 2577–2604.CrossRefGoogle Scholar
Cai, T. & Liu, W. (2011) Adaptive thresholding for sparse covariance matrix estimation. Journal of the American Statistical Association 106, 672–684.CrossRefGoogle Scholar
Chen, X., Xu, M., & Wu, W. B. (2013) Covariance and precision matrix estimation for high-dimensional time series. The Annals of Statistics 41, 2994–3021.CrossRefGoogle Scholar
Dahlhaus, R. (1997) Fitting time series models to nonstationary processes. Annals of Statistics 25, 1–37.CrossRefGoogle Scholar
Engle, R. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation. Econometrica 50, 987–1008.CrossRefGoogle Scholar
Engle, R. F. & Kroner, K. F. (1995) Multivariate simultaneous generalized Arch. Econometric Theory 11(1), 122–150.CrossRefGoogle Scholar
Fan, J., Liao, Y., & Liu, H. (2016) An overview of the estimation of large covariance and precision matrices. Econometrics Journal 19, C1–C32.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 75, 603–680.CrossRefGoogle ScholarPubMed
Fan, J., Wang, W., & Zhong, Y. (2016) Robust covariance estimation for approximate factor models. arXiv:1602.00719v1, pp. 1–31.Google Scholar
Giraitis, L., Kapetanios, G., & Price, S. (2013) Adaptive forecasting in the presence of recent and ongoing structural change. Journal of Econometrics 177(2), 153–170.CrossRefGoogle Scholar
Giraitis, L., Kapetanios, G., & Yates, T. (2014) Inference on stochastic time-varying coefficient models. Journal of Econometrics 179, 46–65.CrossRefGoogle Scholar
Giraitis, L., Kapetanios, G., & Yates, T. (2018) Inference on multivariate heteroscedastic time varying random coefficient models. Journal of Time Series Analysis 39, 129–149.CrossRefGoogle Scholar
Han, F., & Liu, H. (2017) Statistical analysis of latent generalized correlation matrix estimation in transelliptical distribution. Bernoulli 23, 23–57.CrossRefGoogle Scholar
Jagannathan, R. & Ma, T. (2003) Risk reduction in large portfolios: why imposing the wrong constraints helps. Journal of Finance 58, 1651–1684.CrossRefGoogle Scholar
Kapetanios, G. (2010) Testing for strict stationarity in financial variables. Journal of Banking and Finance 33, 2346–2362.CrossRefGoogle Scholar
Kolar, M. & Xing, E. (2011) On time varying undirected graphs. Proceedings of the 14th International Conference on Artificial Intelligence and Statistics (AISTATS) 2011 (JMLR) 15, 407–415.Google Scholar
Ledoit, O. & Wolf, M. (2003) Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance 10, 603–621.CrossRefGoogle Scholar
Ledoit, O. & Wolf, M. (2004) A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis 88, 365–411.CrossRefGoogle Scholar
Ledoit, O. & Wolf, M. (2015) Spectrum estimation: A unified framework for covariance matrix estimation and PCA in large dimensions. Journal of Multivariate Analysis 139, 360–384.CrossRefGoogle Scholar
Marchenko, V. A. & Pastur, L. A. (1968) Distribution of eigenvalues for some sets of random matrices. Mathematics of the USSR, Sbornik 1, 457–483.CrossRefGoogle Scholar
Markowitz, H. M. (1952) Portfolio selection. Journal of Finance 7, 77–91.Google Scholar
Noureldin, D., Shephard, N., & Sheppard, K. (2014) Multivariate rotated ARCH models. Journal of Econometrics 179(1), 16–30.CrossRefGoogle Scholar
Pourahmadi, M. (2013) High-Dimensional Covariance Estimation. Hoboken: John Wiley.CrossRefGoogle Scholar
Stock, J. H. & Watson, M. W. (2002a) Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association 97, 1167–1179.CrossRefGoogle Scholar
Stock, J. H. & Watson, M. W. (2002b) Macroeconomic forecasting using diffusion indexes. Journal of Business & Economic Statistics 20, 147–162.CrossRefGoogle Scholar
Wegkamp, M. & Zhao, Y. (2016) Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas. Bernoulli 22, 1184–1226.CrossRefGoogle Scholar
White, H. & Wooldridge, J. M. (1991) Some results on sieve estimation with dependent observations. In Barnett, W., Powell, J., & Tauchen, G. (eds.), Semiparametric and Nonparametric Methods in Econometrics and Statistics, pp. 459–493. Cambridge University Press.Google Scholar
Zhou, S., Lafferty, J., & Wasserman, L. (2010) Time varying undirected graphs. Machine Learning 80, 295–319.CrossRefGoogle Scholar
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