Realized Stochastic Volatility Models with Skew-t Distributions for Volatility and Tail Risk Forecasting

arXiv:2401.13179 · econ.EM · Submitted 2024-01-24 · Read on arXiv

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Introduction to the show: ident: Economics. The day in economics: theory, mechanisms and the evidence beneath them.

Ruth: I'm Ruth.

Warren: And I'm Warren, here with Iris.

Ruth: We're looking at the paper: "Realized Stochastic Volatility Models with Skew-t Distributions for Volatility and Tail Risk Forecasting".

Warren: Here's the gist of the paper. The analysis of RSV models on the DJIA shows that incorporating realized volatility delivers robust improvements in volatility forecasting under extreme market stress, while the gains in tail-risk forecasting are more modest, with most cross-model differences in FZ0 loss not statistically significant.

Ruth: Once the free-ψ RSV models are included in the comparison, however, the RSV specifications improve on both dimensions.

Warren: They attain a lower QLIKE than any ψ = 1 model under all four proxies, and a lower FZ0 loss than their SV counterparts in all six matched pairs at both risk levels, so the tail-risk shortfall recorded under ψ = 1 does not survive relaxing that restriction.

Ruth: RSV-AZ-ST stands out as the best or near-best performer on both dimensions, under ψ = 1 and with ψ estimated freely alike.

Paper summary: Ruth: So, we’re diving deeper into what they actually found with this "Realized Stochastic Volatility Models with Skew-t Distributions for Volatility and Tail Risk Forecasting." The core claim here is that the RSV model, by using realized volatility as an efficient proxy for latent volatility, leads to better forecasting performance.

Warren: What they claim specifically is that incorporating realized volatility into the latent volatility structure substantially improves forecast accuracy across these models. They are looking at how well this works for both standard GARCH-type benchmarks and the RSV model itself.

Iris: They’re testing this across daily returns of the DJIA and N225, evaluating performance using two different risk assessment metrics: VaR and ES forecasts.

Ruth: And they also look at how these models perform against SV, EGARCH, and REGARCH benchmarks all within the same Bayesian framework to make sure you’re accounting for parameter uncertainty in all the classes.

Warren: They use both the pairwise predictive ability test of Giacomini and White and the joint model confidence set procedure of Hansen et al. as their primary methods for testing forecast performance, with the pairwise tests serving as complementary checks.

Iris: Their empirical findings can be summarized by looking at how incorporating realized volatility into the latent volatility structure improves forecast accuracy across these models.

Ruth: And they also look at how this works for both VaR and ES forecasts in relation to the benchmarks they are comparing against, which is a key part of their evaluation.

Warren: So, what they’ve shown is that this RSV approach gives us better results when we are trying to predict volatility and those tail risk measures.

Conclusion: Ruth: Looking at the title "Realized Stochastic Volatility Models with Skew-t Distributions for Volatility and Tail Risk Forecasting," what this really means in practice is that they’ve developed a way to use observable market data, like realized volatility, to get more accurate forecasts for financial risk management.

Warren: The implication is that if you want better predictions for things like value-at-risk or expected shortfall, using a model that incorporates realized volatility as the latent driver could give you an edge over just looking at historical returns alone.

Iris: It suggests that the way we model the underlying volatility process matters a lot, and moving from purely parametric models to ones that use observed volatility measures can be beneficial for capturing real-world features like skewness.

Ruth: The authors also looked at different distributions, specifically three variants of skew-t distributions, which lets them model asymmetry in a flexible way.

Warren: So it’s not just about adding one extra term; it’s about how you structure the distribution of those latent volatilities to better match what we see in the market.

Iris: But they also looked at limitations, and one thing they point out is that their empirical analysis can be summarized by looking at how incorporating realized volatility into the latent volatility structure improves forecast accuracy across these models.

Ruth: So, while it’s a solid improvement over some benchmarks, it's important to remember that this RSV model is still an extension of traditional SV models, and its performance depends on the specific data you feed it.

Faculty of Business Administration, Hosei University · Graduate School of Economics, Nagoya University · Graduate School of Social Data Science, Hitotsubashi University · Faculty of Economics, University of Tokyo

econ.EM

Submitted: 2024-01-24

Updated: 2026-10-28

Comments: v6: corrects the forecasting step of the SV-GH-ST and the four Azzalini models and the rolling Hansen-Lunde adjustment of the volatility proxies, and regenerates the affected forecasts and all volatility loss tables; all volatility point forecasts are now predictive means; main text shortened to 24 pages, with derivations, algorithms, and additional results in the 81-page Supplementary Material

Project page: https://realized.oxford-man.ox.ac.uk

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 91/100

The gist: The analysis of RSV models on the DJIA shows that incorporating realized volatility delivers robust improvements in volatility forecasting under extreme market stress, while the gains in tail-risk

Terminology

Summary

The analysis of RSV models on the DJIA shows that incorporating realized volatility delivers robust improvements in volatility forecasting under extreme market stress, while the gains in tail-risk forecasting are more modest, with most cross-model differences in FZ0 loss not statistically significant. Once the free-ψ RSV models are included in the comparison, however, the RSV specifications improve on both dimensions. They attain a lower QLIKE than any ψ = 1 model under all four proxies, and a lower FZ0 loss than their SV counterparts in all six matched pairs at both risk levels, so the tail-risk shortfall recorded under ψ = 1 does not survive relaxing that restriction. RSV-AZ-ST stands out as the best or near-best performer on both dimensions, under ψ = 1 and with ψ estimated freely alike.

For the DJIA, estimating ψ freely improves the RSV forecasts on both dimensions and for every distribution, while for REGARCH it delivers no consistent gain. Fixing ψ = 1 remains defensible as a device for keeping the RSV and REGARCH measurement equations comparable, but it is not what a free estimate would select for the RSV models on this sample. Overall, the DJIA COVID-19 evidence reproduces the main-sample findings of the main text: incorporating realized volatility delivers robust improvements in volatility forecasting under extreme market stress, while the gains in tail-risk forecasting are more modest, with most cross-model differences in FZ0 loss not statistically significant.

For the N225, a freely estimated ψ helps the RSV volatility forecasts for four of the six distributions but costs FZ0 loss almost everywhere, and the MCS does not sharply distinguish the ψ = 1 and free-ψ variants on either dimension. This is the reverse of the DJIA pattern of Section 6.3, where a free ψ improved both dimensions for all six RSV distributions, Azzalini and Fern´andez–Steel alike. The two indices agree on REGARCH: a free ψ raises QLIKE under all four proxies here, and leaves the FZ0 loss essentially unchanged, so it buys nothing on either market. Fixing ψ = 1 is therefore a defensible default for cross-model comparability, but the gain from relaxing it is index-specific rather than general.

The RSV models improve on both dimensions when free ψ is used: They attain a lower QLIKE than any ψ = 1 model under all four proxies, and a lower FZ0 loss than their SV counterparts in all six matched pairs at both risk levels. Furthermore, RSV-AZ-ST stands out as the best or near-best performer on both dimensions, under ψ = 1 and with ψ estimated freely alike.

The N225 analysis shows that a freely estimated ψ helps the RSV volatility forecasts for four of the six distributions but costs FZ0 loss almost everywhere. In contrast, for REGARCH models on the N225, a free ψ raises QLIKE under all four proxies here, and leaves the FZ0 loss essentially unchanged, so it buys nothing on either market.

The overall conclusion is that incorporating RV substantially improves forecast accuracy, with the RSV and REGARCH models consistently outperforming the SV and EGARCH models. However, regarding VaR and ES forecasts for the N225 over COVID-19 period: Contrary to the results for the main sample, incorporating RV does not necessarily improve VaR and ES forecasts in terms of the FZ0 loss during this period.

The comparison between indices regarding free ψ is noted: the contrast with the DJIA of Section 6.3, where a free ψ improved both dimensions for all six RSV distributions, Azzalini and Fern´andez–Steel alike. The two indices agree on REGARCH: a free ψ raises QLIKE under all four proxies here, and leaves the FZ0 loss essentially unchanged, so it buys nothing on either market.

The summary is largely derived from the comparison of results between the main sample findings and the COVID-19 period analysis for DJIA and N225. The key finding regarding free ψ is that its benefit is index-specific: relaxing ψ recovers the RSV tail-risk performance on one index but not on the other. For instance, the contrast with the DJIA of Section 6.3, where a free ψ improved both dimensions for all six RSV distributions, Azzalini and Fern´andez–Steel alike. The text also notes that The two indices agree on REGARCH: a free ψ raises QLIKE under all four proxies here, and leaves the FZ0 loss essentially unchanged, so it buys nothing on either market. The final statement summarizes this: Fixing ψ = 1 is therefore a defensible default for cross-model comparability, but the gain from relaxing it is index-specific rather than general.

Relevant quotes include:

"Under ψ = 1 every RSV specification carries a higher FZ0 loss than its SV counterpart at the 1% level, which is the finding on which that comparison rests. Under a freely estimated ψ every RSV specification carries a lower FZ0 loss than its SV counterpart, at both the 1% and the 5% level, and the best free-ψ RSV specification attains an FZ0 loss of 1.79 at the 1% level and 1.30 at the 5% level, against 1.96 and 1.36 for the best SV specification and 2.26 and 1.47 for the best EGARCH specification."

The apparent tail-risk disadvantage of the RSV models on this sample is therefore an artefact of fixing ψ at unity rather than a property of the RSV framework.

"For DJIA, then, estimating ψ freely improves the RSV forecasts on both dimensions and for every distribution, while for REGARCH it delivers no consistent gain. Fixing ψ = 1 remains defensible as a device for keeping the RSV and REGARCH measurement equations comparable, but it is not what a free estimate would select for the RSV models on this sample."

The contrast with the DJIA of Section 6.3, where a free ψ improved both dimensions for all six RSV distributions, Azzalini and Fern´andez–Steel alike.

The two indices agree on REGARCH: a free ψ raises QLIKE under all four proxies here, and leaves the FZ0 loss essentially unchanged, so it buys nothing on either market.

Fixing ψ = 1 is therefore a defensible default for cross-model comparability, but the gain from relaxing it is index-specific rather than general.

"The two dimensions point in opposite directions: a freely estimated ψ helps the RSV volatility forecasts for four of the six distributions but costs FZ0 loss almost everywhere, and the MCS does not sharply distinguish the ψ = 1 and free-ψ variants on either dimension."

Improvements for AI systems

  1. Robust Volatility Forecasting via Realized Stochastic Volatility (RSV) Models: Improved AI systems can generate one-day-ahead volatility forecasts that robustly outperform their SV counterparts consistently across both indices, four realized volatility proxies, and both pairwise and joint evaluation procedures.

  2. Tail Risk Assessment with Distributional Flexibility: AI systems can provide more nuanced tail risk evaluations by incorporating skewed Student’s t-distributions and other skewed-t distributions to better capture return asymmetry and tail risk, leading to more accurate VaR and ES forecasts.

  3. Optimal Model Selection via Parameter Sensitivity: The improved system can perform superior model selection by assessing the impact of measurement equation restrictions, noting that relaxing it over the COVID-19 sample improves both the volatility and the tail risk forecasts of every RSV specification for the DJIA.

  4. Stress Period Performance Adaptation: AI systems can adapt their forecasting strategy during extreme market stress by recognizing that the relative performance of RSV models with skew-t innovations is not systematically superior during the stress period under fixed restrictions, suggesting a need for dynamic adjustment in high-volatility regimes.

  5. Index and Risk Level Specific Strategy: The system can tailor risk management decisions by recognizing that the benefit of incorporating realized volatility for tail risk forecasting is clearer at the less extreme quantile (e.g., 5% level) for the DJIA, while acknowledging that gains are more pronounced for the N225.

Abstract

Volatility forecasts underlie tail risk measures: value-at-risk (VaR) and expected shortfall (ES). This paper compares realized stochastic volatility (RSV) models, which add a realized measure to the stochastic volatility (SV) model, with their SV counterparts under normal, Student's t, two skew-normal, and three skew-t return distributions, and against EGARCH and REGARCH benchmarks. Estimated by Markov chain Monte Carlo on the Dow Jones Industrial Average and the Nikkei 225, the models are compared by their one-day-ahead volatility, VaR, and ES forecasts (the predictive mean for volatility). Adding realized volatility improves volatility forecasts over the main sample for both indices under every proxy and return distribution; the gains are uniformly significant for the Nikkei 225 but, for the Dow Jones, only under the realized variance used in estimation. During the COVID-19 period the gains persist for the Nikkei 225 but vanish for the Dow Jones, where the REGARCH benchmark is best. For VaR and ES the gains are confined to the Nikkei 225 and to the 5% level for the Dow Jones, and few matched-pair differences are significant. Skewed innovations lower the 1% joint loss for the Dow Jones, but no specification dominates across indices, risk levels, and sample periods.

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