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Unrestricted, restricted, and regularized models for forecasting multivariate volatility

  • Stanislav Anatolyev ORCID logo EMAIL logo and Filip Staněk ORCID logo
Published/Copyright: April 21, 2022
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Studies in Nonlinear Dynamics & Econometrics
From the journal Studies in Nonlinear Dynamics & Econometrics Volume 27 Issue 2

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:

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.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: Czech Science Foundation support under grant 20-28055S and Grantová agentura UK support under grant 264120 are gratefully acknowledged.

  3. 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 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 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 et,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

  1. 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

  1. 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

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/snde-2021-0064
Keywords for this article
BEKK; CAW; covariance matrix; multivariate volatility; regularization; ridge

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Asymmetry in stochastic volatility models with threshold and time-dependent correlation
  4. Financial crisis spread, economic growth and unemployment: a mathematical model
  5. Article
  6. Augmenting the Realized-GARCH: the role of signed-jumps, attenuation-biases and long-memory effects
  7. Research Articles
  8. Unrestricted, restricted, and regularized models for forecasting multivariate volatility
  9. Controlling chaos in New Keynesian macroeconomics
  10. Conservatorship, quantitative easing, and mortgage spreads: a new multi-equation score-driven model of policy actions
  11. Expected, unexpected, good and bad aggregate uncertainty
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