Unrestricted, restricted, and regularized models for forecasting multivariate volatility

Stanislav Anatolyev; Filip Staněk

doi:10.1515/snde-2021-0064

Published by De Gruyter April 21, 2022

Unrestricted, restricted, and regularized models for forecasting multivariate volatility

Stanislav Anatolyev and Filip Staněk

From the journal Studies in Nonlinear Dynamics & Econometrics

https://doi.org/10.1515/snde-2021-0064

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Abstract

We perform an extensive investigation of different specifications of the BEKK-type multivariate volatility models for a moderate number of assets, focusing on how the degree of parametrization affects forecasting performance. Because the unrestricted specification may be too generously parameterized, often one imposes restrictions on coefficient matrices constraining them to have a diagonal or even scalar structure. We frame all three model variations (full, diagonal, scalar) as special cases of a ridge-type regularized estimator, where the off-diagonal elements are shrunk towards zero and the diagonal elements are shrunk towards homogeneity. Our forecasting experiments with BEKK-type Conditional Autoregressive Wishart model for realized volatility confirm the superiority of the more parsimonious scalar and diagonal model variations. Even though sometimes a moderate degree of regularization of the diagonal and off-diagonal parameters may be beneficial for forecasting performance, it does not regularly lead to tangible performance improvements irrespective of how precise is tuning of regularization intensity. Additionally, our results highlight the crucial importance of frequent model re-estimation in improving the forecast precision, and, perhaps paradoxically, a slight advantage of shorter estimation windows compared to longer windows.

Keywords: BEKK; CAW; covariance matrix; multivariate volatility; regularization; ridge

Corresponding author: Stanislav Anatolyev, CERGE-EI, Politickych vězňů 7, 11121 Prague 1, Czech Republic, E-mail: stanislav.anatolyev@cerge-ei.cz

Funding source: Grantová Agentura České Republiky http://dx.doi.org/10.13039/501100001824

Award Identifier / Grant number: 20-28055S

Acknowledgment

We thank an anonymous referee for the helpful comments and constructive remarks. Furthermore, we thank the audiences and discussants at the 40th International Symposium on Forecasting, XIt Workshop in Time Series Econometrics, and 5th International Workshop on Financial Markets and Nonlinear Dynamics.

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: Czech Science Foundation support under grant 20-28055S and Grantová agentura UK support under grant 264120 are gratefully acknowledged.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix

Figures 5 –7 and Tables 5 and 6

Figure 5:

Differences of the in-sample (A) and out-of-sample (B) Stein losses of individual models and rCAW (benchmark represented by the horizontal line) plotted for different combinations of a number of assets n and length of the estimation window T.

Figure 6:

Ratios of in-sample (A) and out-of-sample (B) Frobenius losses of individual models and rCAW (benchmark represented by the horizontal line) plotted for different combinations of a number of assets n and length of the estimation window T.

$Figure 7: Distributions of et,i,j for individual models plotted against N ( 0,1 ) $\mathcal{N}(0,1)$ distribution function.$

Figure 7:

Distributions of e_t,i,j for individual models plotted against N ( 0,1 ) distribution function.

Table 5:

Average ranking of individual models in terms of out-of-sample Frobenius loss (1 indicating the best model and 4 indicating the worst model) for combinations of number of assets n and length of the estimation window T.

	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9	n = 10	All
(A) sCAW
2y	2.11	2.24	2.23	2.17	1.96	1.97	2.00	1.94	1.89	2.06
3y	2.21	2.38	2.46	2.26	2.10	2.18	2.11	1.99	2.14	2.20
4y	2.61	2.61	2.88	2.60	2.40	2.54	2.47	2.14	1.78	2.49
5y	2.88	2.77	3.07	3.18	3.18	3.12	3.00	2.94	3.07	3.01
All	2.42	2.47	2.60	2.49	2.35	2.39	2.35	2.19	2.16	2.39
(B) dCAW
2y	2.47	2.72	2.84	2.78	2.88	2.87	2.96	2.92	3.00	2.82
3y	2.36	2.50	2.80	2.89	2.80	2.96	3.04	3.08	3.09	2.83
4y	2.44	2.53	2.86	2.56	2.81	2.94	2.94	3.11	3.22	2.81
5y	2.39	2.32	2.54	2.57	2.57	2.71	2.50	2.68	2.71	2.55
All	2.43	2.59	2.78	2.72	2.81	2.84	2.86	2.97	3.04	2.77
(C) fCAW
2y	2.98	2.84	2.76	2.84	3.04	3.17	3.12	3.36	3.39	3.05
3y	2.81	2.93	2.48	2.77	3.02	2.93	2.91	3.06	3.00	2.88
4y	2.36	2.47	2.08	2.53	2.67	2.50	2.44	2.48	2.89	2.46
5y	2.21	2.36	2.11	2.00	1.82	2.07	2.21	2.34	2.29	2.16
All	2.63	2.64	2.41	2.59	2.69	2.74	2.76	2.83	2.91	2.68
(D) rCAW
2y	2.45	2.20	2.17	2.21	2.12	1.99	1.92	1.78	1.73	2.07
3y	2.62	2.19	2.27	2.08	2.08	1.93	1.93	1.88	1.77	2.10
4y	2.58	2.39	2.18	2.32	2.12	2.01	2.14	2.27	2.11	2.25
5y	2.52	2.55	2.29	2.25	2.43	2.09	2.29	2.04	1.93	2.28
All	2.52	2.30	2.21	2.20	2.15	2.04	2.03	2.00	1.89	2.16

Bold values refer to average rankings across all asset numerosities or all window lengths.

Table 6:

Frequency of rejection of individual models via the MCS for the Frobenius loss at the 5% level for combinations of a number of assets n and length of the estimation window T.

	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9	n = 10	All
(A) sCAW
2y	0.11	0.12	0.14	0.14	0.02	0.14	0.15	0.12	0.08	0.12
3y	0.02	0.20	0.14	0.14	0.09	0.07	0.14	0.06	0.09	0.10
4y	0.11	0.17	0.14	0.06	0.03	0.06	0.03	0.04	0.00	0.07
5y	0.07	0.07	0.11	0.18	0.25	0.21	0.18	0.22	0.29	0.17
All	0.07	0.14	0.15	0.12	0.10	0.11	0.13	0.10	0.10	0.11
(B) dCAW
2y	0.21	0.30	0.26	0.18	0.22	0.36	0.33	0.33	0.46	0.28
3y	0.07	0.20	0.14	0.16	0.16	0.14	0.16	0.20	0.09	0.15
4y	0.17	0.11	0.08	0.06	0.03	0.06	0.08	0.07	0.11	0.08
5y	0.11	0.04	0.07	0.11	0.14	0.18	0.04	0.10	0.14	0.10
All	0.14	0.19	0.16	0.13	0.15	0.19	0.17	0.19	0.20	0.17
(C) fCAW
2y	0.23	0.24	0.28	0.26	0.37	0.31	0.29	0.36	0.38	0.30
3y	0.17	0.23	0.11	0.20	0.23	0.30	0.23	0.30	0.18	0.22
4y	0.17	0.17	0.06	0.08	0.17	0.06	0.08	0.09	0.11	0.11
5y	0.11	0.14	0.07	0.04	0.07	0.11	0.11	0.15	0.00	0.10
All	0.16	0.20	0.13	0.16	0.22	0.19	0.19	0.23	0.23	0.19
(D) rCAW
2y	0.17	0.14	0.18	0.20	0.20	0.19	0.19	0.18	0.23	0.18
3y	0.10	0.16	0.14	0.11	0.09	0.07	0.07	0.08	0.00	0.10
4y	0.14	0.11	0.06	0.00	0.00	0.00	0.00	0.00	0.00	0.04
5y	0.14	0.14	0.11	0.11	0.11	0.14	0.07	0.15	0.14	0.12
All	0.14	0.15	0.13	0.11	0.10	0.11	0.10	0.10	0.10	0.12

Bold values refer to average rankings across all asset numerosities or all window lengths.

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/snde-2021-0064).

Received: 2021-06-29

Revised: 2022-03-19

Accepted: 2022-03-26

Published Online: 2022-04-21

Unrestricted, restricted, and regularized models for forecasting multivariate volatility

Abstract

Acknowledgment

References

Supplementary Material

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