Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-13T04:23:21.930Z Has data issue: false hasContentIssue false
  • English
  • Français

NONPARAMETRIC ESTIMATION OF LARGE SPOT VOLATILITY MATRICES FOR HIGH-FREQUENCY FINANCIAL DATA

Published online by Cambridge University Press:  07 April 2025

Ruijun Bu,
Degui Li,
Oliver Linton and
Hanchao Wang
Ruijun Bu
Affiliation:
University of Liverpool
Degui Li
Affiliation:
University of Macau
Oliver Linton*
Affiliation:
University of Cambridge
Hanchao Wang
Affiliation:
Shandong University
*
Address correspondence to Oliver Linton, Faculty of Economics, University of Cambridge, Cambridge, UK; e-mail: obl20@cam.ac.uk.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper, we consider estimating spot/instantaneous volatility matrices of high-frequency data collected for a large number of assets. We first combine classic nonparametric kernel-based smoothing with a generalized shrinkage technique in the matrix estimation for noise-free data under a uniform sparsity assumption, a natural extension of the approximate sparsity commonly used in the literature. The uniform consistency property is derived for the proposed spot volatility matrix estimator with convergence rates comparable to the optimal minimax one. For high-frequency data contaminated by microstructure noise, we introduce a localized pre-averaging estimation method that reduces the effective magnitude of the noise. We then use the estimation tool developed in the noise-free scenario and derive the uniform convergence rates for the developed spot volatility matrix estimator. We further combine kernel smoothing with the shrinkage technique to estimate the time-varying volatility matrix of the high-dimensional noise vector. In addition, we consider large spot volatility matrix estimation in time-varying factor models with observable risk factors and derive the uniform convergence property. We provide numerical studies including simulation and empirical application to examine the performance of the proposed estimation methods in finite samples.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 42 , Issue 1 , February 2026, pp. 63 - 100
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Footnotes

The authors would like to thank the Co-Editor and two reviewers for the constructive comments, which helped to substantially improve the article. The first author’s research was partly supported by the BA Talent Development Award (Grant No. TDA21∖210027). The second author’s research was partly supported by the Leverhulme Research Fellowship (Grant No. RF-2023-396∖9), the BA/Leverhulme Small Research Grant (Grant No. SRG1920/100603), and the National Natural Science Foundation of China (Grant No. 72033002).

References

REFERENCES

Aït-Sahalia, Y., & Jacod, J. (2014). High-frequency financial econometrics . Princeton University Press.Google Scholar
Aït-Sahalia, Y., Jacod, J., & Li, J. (2012). Testing for jumps in noisy high frequency data. Journal of Econometrics , 168, 207–222.CrossRefGoogle Scholar
Aït-Sahalia, Y., Kalnina, I., & Xiu, D. (2020). High-frequency factor models and regressions. Journal of Econometrics , 216, 86–105.CrossRefGoogle Scholar
Aït-Sahalia, Y., & Xiu, D. (2017). Using principal component analysis to estimate a high dimensional factor model with high-frequency data. Journal of Econometrics , 201, 384–399.CrossRefGoogle Scholar
Aït-Sahalia, Y., & Xiu, D. (2019). Principal component analysis of high-frequency data. Journal of the American Statistical Association , 114, 287–303.CrossRefGoogle Scholar
Andersen, T. G., & Bollerslev, T. (1998). Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review , 39, 885–905.CrossRefGoogle Scholar
Andersen, T. G., Bollerslev, T., & Diebold, F. X. (2010). Parametric and nonparametric volatility measurement. In Aït-Sahalia, Y., & Hansen, L. P. (Eds.), Handbook of financial econometrics: Tools and techniques (pp. 67–137). Elsevier.CrossRefGoogle Scholar
Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (2003). Modeling and forecasting realized volatility. Econometrica , 71, 579–625.CrossRefGoogle Scholar
Andersen, T. G., Su, T., Todorov, V., & Zhang, Z. (2024). Intraday periodic volatility curves. Journal of the American Statistical Association , 119(546), 1181–1191 CrossRefGoogle Scholar
Andersen, T. G., Thyrsgaard, M., & Todorov, V. (2021). Recalcitrant betas: Intraday variation in the cross-sectional dispersion of systematic risk. Quantitative Economics , 12, 647–682.CrossRefGoogle Scholar
Bai, Z., & Silverstein, J. W. (2010). Spectral analysis of large dimensional random matrices . Springer Series in Statistics. Springer.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., & Shephard, N. (2008). Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise. Econometrica , 76, 1481–1536.Google Scholar
Barndorff-Nielsen, O. E., & Shephard, N. (2002). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society Series B , 64, 253–280.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E., & Shephard, N. (2004). Econometric analysis of realized covariation: High frequency based covariance, regression and correlation in financial economics. Econometrica , 72, 885–925.CrossRefGoogle Scholar
Bibinger, M., Hautsch, N., Malec, P., & Reiss, M. (2019). Estimating the spot covariation of asset prices—statistical theory and empirical evidence. Journal of Business and Economic Statistics , 37(3), 419–435.CrossRefGoogle Scholar
Bibinger, M., Jirak, M., & Vetter, M. (2017). Nonparametric change-point analysis of volatility. Annals of Statistics , 45, 1542–1578.CrossRefGoogle Scholar
Bickel, P., & Levina, E. (2008). Covariance regularization by thresholding. Annals of Statistics , 36, 2577–2604.CrossRefGoogle Scholar
Cai, T. T., Hu, J., Li, Y., & Zheng, X. (2020). High-dimensional minimum variance portfolio estimation based on high-frequency data. Journal of Econometrics , 214, 482–494.CrossRefGoogle Scholar
Cai, T. T., & Zhou, H. H. (2012). Optimal rates of convergence for sparse covariance matrix estimation. Annals of Statistics , 40, 2389–2420.CrossRefGoogle Scholar
Chang, J., Hu, Q., Liu, C., & Tang, C. (2024). Optimal covariance matrix estimation for high-dimensional noise in high-frequency data. Journal of Econometrics , 239, Article 105329.CrossRefGoogle Scholar
Chen, J., Li, D., & Linton, O. (2019). A new semiparametric estimation approach of large dynamic covariance matrices with multiple conditioning variables. Journal of Econometrics , 212, 155–176.CrossRefGoogle Scholar
Chen, X., Xu, M., & Wu, W. (2013). Covariance and precision matrix estimation for high-dimensional time series. Annals of Statistics , 41, 2994–3021.CrossRefGoogle Scholar
Chen, Z., & Leng, C. (2016). Dynamic covariance models. Journal of the American Statistical Association , 111, 1196–1207.CrossRefGoogle Scholar
Christensen, K., Kinnebrock, S., & Podolskij, M. (2010). Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data. Journal of Econometrics , 159, 116–133.CrossRefGoogle Scholar
Dai, C., Lu, K., & Xiu, D. (2019). Knowing factors or factor loadings, or neither? Evaluating estimators for large covariance matrices with noisy and asynchronous data. Journal of Econometrics , 208, 43–79.CrossRefGoogle Scholar
Fan, J., Fan, Y., & Lv, J. (2007). Aggregation of nonparametric estimators for volatility matrix. Journal of Financial Econometrics , 5, 321–357.CrossRefGoogle Scholar
Fan, J., Furger, A., & Xiu, D. (2016). Incorporating global industrial classification standard into portfolio allocation: A simple factor-based large covariance matrix estimator with high frequency data. Journal of Business and Economic Statistics , 34, 489–503.CrossRefGoogle Scholar
Fan, J., & Gijbels, I. (1996). Local polynomial modelling and its applications . Chapman and Hall.Google Scholar
Fan, J., Liao, Y., & Mincheva, M. (2011). High-dimensional covariance matrix estimation in approximate factor models. Annals of Statistics , 39, 3320–3356.CrossRefGoogle ScholarPubMed
Fan, J., Liao, Y., & Mincheva, M. (2013). Large covariance estimation by thresholding principal orthogonal complements (with discussion). Journal of the Royal Statistical Society, Series B , 75, 603–680.CrossRefGoogle Scholar
Fan, J., & Wang, Y. (2008). Spot volatility estimation for high-frequency data. Statistics and Its Interface , 1, 279–288.CrossRefGoogle Scholar
Figueroa-López, J. E., & Li, C. (2020). Optimal kernel estimation of spot volatility of stochastic differential equations. Stochastic Processes and Their Applications , 130, 4693–4720.CrossRefGoogle Scholar
Jacod, J., Li, Y., Mykland, P. A., Podolskij, M., & Vetter, M. (2009). Microstructure noise in the continuous case: The pre-averaging approach. Stochastic Processes and Their Applications , 119, 2249–2276.CrossRefGoogle Scholar
Jacod, J., & Protter, P. (2012). Discretization of processes . Springer.CrossRefGoogle Scholar
Jacod, J., & Todorov, V. (2010). Do price and volatility jump together? Annals of Applied Probability , 20, 1425–1469.CrossRefGoogle Scholar
Kalnina, I., & Linton, O. (2008). Estimating quadratic variation consistently in the presence of endogenous and diurnal measurement error. Journal of Econometrics , 147, 47–59.CrossRefGoogle Scholar
Kanaya, S., & Kristensen, D. (2016). Estimation of stochastic volatility models by nonparametric filtering. Econometric Theory , 32, 861–916.CrossRefGoogle Scholar
Kim, D., Wang, Y., & Zou, J. (2016). Asymptotic theory for large volatility matrix estimation based on high-frequency financial data. Stochastic Processes and Their Applications , 126, 3527–3577.CrossRefGoogle Scholar
Kong, X. (2018). On the systematic and idiosyncratic volatility with large panel high-frequency data. Annals of Statistics , 46, 1077–1108.CrossRefGoogle Scholar
Kristensen, D. (2010). Nonparametric filtering of the realized spot volatility: A kernel-based approach. Econometric Theory , 26, 60–93.CrossRefGoogle Scholar
Lam, C., & Feng, P. (2018). A nonparametric eigenvalue-regularized integrated covariance matrix estimator for asset return data. Journal of Econometrics , 206, 226–257.CrossRefGoogle Scholar
Lee, S., & Mykland, P. (2008). Jumps in financial markets: A new nonparametric test and jump dynamics. Review of Financial Studies , 21, 2535–2563.CrossRefGoogle Scholar
Li, J., Todorov, V., & Tauchen, G. (2017). Jump regression. Econometrica , 85, 173–195.CrossRefGoogle Scholar
Li, J., Todorov, V., & Tauchen, G. (2019). Jump factor models in large cross-sections. Quantitative Economics , 10, 419–456.CrossRefGoogle Scholar
Li, Q., & Racine, J. (2007). Nonparametric econometrics . Princeton University Press.Google Scholar
Li, Z. M., & Linton, O. (2022). A ReMeDI for microstructure noise. Econometrica , 90, 367–389.CrossRefGoogle Scholar
Liao, Y., & Todorov, V. (2024). Changes in the span of systematic risk exposures. Quantitative Economics , 15(3), 817–847.CrossRefGoogle Scholar
Mancini, C. (2009). Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps. Scandinavian Journal of Statistics , 36, 270–296.CrossRefGoogle Scholar
Mykland, P. A., & Zhang, L. (2006). ANOVA for diffusions and Itô processes. Annals of Statistics , 34, 1931–1963.CrossRefGoogle Scholar
Park, S., Hong, S. Y., & Linton, O. (2016). Estimating the quadratic covariation matrix for asynchronously observed high frequency stock returns corrupted by additive measurement error. Journal of Econometrics , 191, 325–347.CrossRefGoogle Scholar
Podolskij, M., & Vetter, M. (2009). Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps. Bernoulli , 15, 634–658.CrossRefGoogle Scholar
Reiß, M., Todorov, V., & Tauchen, G. (2015). Nonparametric test for a constant beta between Itô semi-martingales based on high-frequency data. Stochastic Processes and Their Applications , 125, 2955–2988.CrossRefGoogle Scholar
Revuz, D., & Yor, M. (1999). Continuous martingales and Brownian motion . Grundlehren der Mathematischen Wissenschaften (vol. 293). Springer.CrossRefGoogle Scholar
Shephard, N. (2005). Stochastic volatility: Selected readings . Oxford University Press.CrossRefGoogle Scholar
Tao, M., Wang, Y., & Zhou, H. H. (2013). Optimal sparse volatility matrix estimation for high-dimensional Itô processes with measurement errors. Annals of Statistics , 41, 1816–1864.CrossRefGoogle Scholar
Wang, Y., & Zou, J. (2010). Vast volatility matrix estimation for high-frequency financial data. Annals of Statistics , 38, 943–978.CrossRefGoogle Scholar
Xia, N., & Zheng, X. (2018). On the inference about the spectral distribution of high-dimensional covariance matrix based on high-frequency noisy observations. Annals of Statistics , 46, 500–525.CrossRefGoogle Scholar
Xiao, Z., & Linton, O. (2002). A nonparametric prewhitened covariance estimator. Journal of Time Series Analysis , 23, 215–250.CrossRefGoogle Scholar
Zhang, L. (2011). Estimating covariation: Epps effect, microstructure noise. Journal of Econometrics , 160, 33–47.CrossRefGoogle Scholar
Zhang, L., Mykland, P. A., & Aït-Sahalia, Y. (2005). A tale of two time scales: Determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association , 100, 1394–1411.CrossRefGoogle Scholar
Zheng, X., & Li, Y. (2011). On the estimation of integrated covariance matrices of high dimensional diffusion processes. Annals of Statistics , 39, 3121–3151.CrossRefGoogle Scholar
Zu, Y., & Boswijk, H. P. (2014). Estimating spot volatility with high-frequency financial data. Journal of Econometrics , 181, 117–135.CrossRefGoogle Scholar
Supplementary material: File

Bu et al. supplementary material

Bu et al. supplementary material
Download Bu et al. supplementary material(File)
File 300.1 KB