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REVIEW 4 major objections 6 minor 35 references

Deep Learning Enhanced Multivariate GARCH

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read An LSTM-driven covariance term improves multivariate volatility forecasts beyond DCC and Scalar BEKK.

desk verdict Genuine LSTM-BEKK hybrid with credible N=50 evidence, but the high-dimensional and tail-risk claims outrun the reported data. read the letter →

arxiv 2506.02796 v1 pith:OIB2HJ2T submitted 2025-06-03 q-fin.CP cs.AIecon.EM

Haoyuan Wang , Chen Liu , Minh-Ngoc Tran , Chao Wang This is my paper · ORCID
classification q-fin.CPcs.AIecon.EM MSC 62M1091G1062P0568T07
keywords multivariatevolatilitymodelingLongShort-TermMemoryLSTM-BEKKBEKKmodelDCCportfoliooptimizationhigh-dimensionalfinanceminimumvariance
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that a multivariate GARCH model can be made more accurate by letting a long short-term memory (LSTM) network generate part of the conditional covariance matrix. The proposed LSTM-BEKK recursion keeps the interpretable BEKK structure—a static intercept, a shock term, and a persistence term—and adds a time-varying lower-triangular matrix produced by an LSTM. On daily returns from the largest U.S., U.K., and Japanese stocks, the paper reports that LSTM-BEKK produces lower out-of-sample negative log-likelihood than DCC and Scalar BEKK, and that minimum-variance portfolios built from its forecasts carry less annualized volatility. If the result holds, it would give risk managers a volatility model that scales to 250 assets while remaining closer to the data than standard econometric benchmarks.

What carries the argument

The load-bearing object is the LSTM-BEKK recursion $H_t = CC' + C_t C_t' + a r_{t-1} r_{t-1}' + b H_{t-1}$, in which the LSTM generates the dynamic lower-triangular matrix $C_t$ from recent returns while $C$, $a$, and $b$ stay fixed. The lower-triangular construction keeps $H_t$ positive definite by design, and the BEKK terms $a r_{t-1} r_{t-1}' + b H_{t-1}$ preserve the familiar interpretation of shock response and volatility persistence. A stated theorem, relying on a uniform bound on $\|C_t C_t'\|$, gives a sufficient condition for the expected covariance matrix to stay finite; the paper imposes $a+b<1$ during estimation but reports that bounding the LSTM output norm was unnecessary in practice.

What would settle it

One concrete check would be to train LSTM-BEKK on a new dataset, record $\|C_t C_t'\|$ for every $t$, and see whether the maximum stays finite; if it grows without bound, the assumption behind Theorem 1 fails and the stability result does not cover the fitted model. A second check would be to rerun the estimation with an explicit norm constraint on $C_t$ and observe whether the reported out-of-sample gains persist or shrink.

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Extended reading notes

Core claim

The paper's central claim is that adding an LSTM-generated component to the Scalar BEKK covariance recursion improves multivariate volatility forecasts without sacrificing the model's interpretability. Specifically, the conditional covariance is $H_t = CC' + C_t C_t' + a r_{t-1} r_{t-1}' + b H_{t-1}$, where $C$ is a static lower-triangular matrix, $C_t$ is a lower-triangular matrix reshaped from the LSTM output, and $a,b\ge0$ with $a+b<1$. The paper argues that the static term captures long-run covariance structure while the LSTM term adapts to short-run, nonlinear, and stress-period dynamics. Empirically, the model is reported to beat DCC and Scalar BEKK on out-of-sample negative log-likelihood in all three markets and across portfolio sizes from 50 to 250 assets, and to produce global minimum-variance portfolios with the lowest annualized volatility in most configurations. The authors read these results as evidence that the hybrid is both statistically superior and practically useful for high-dimensional risk management.

Load-bearing premise

The paper's stability guarantee rests on the assumption that the LSTM-generated matrix $C_t$ never grows too large over time, but the training procedure does not enforce that bound, so the theoretical guarantee depends on an unverified property of the trained network.

Editorial extensions

If this is right

  • LSTM-BEKK's lower out-of-sample negative log-likelihood implies that its covariance forecasts assign higher probability to realized returns than DCC or Scalar BEKK, not only in one market but across U.S., U.K., and Japanese equities.
  • Global minimum-variance portfolios built from LSTM-BEKK forecasts have lower annualized volatility in most 100-, 175-, and 250-asset configurations, which would translate into less risky portfolios for an investor who trusts the forecasts.
  • The model is retained in the 90% Model Confidence Set in all nine market-and-size settings, whereas Scalar BEKK is never retained, so the reported advantage is not a single lucky draw.
  • At the 5% tail-risk level, LSTM-BEKK generally improves joint Value-at-Risk and Expected Shortfall loss, suggesting the extra flexibility helps in moderate tail events.
  • Because the model keeps the BEKK form, the fitted $a$ and $b$ still give interpretable measures of shock sensitivity and persistence, so the improved forecasts do not require abandoning econometric structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same nesting of a neural network inside a structured covariance recursion could be applied to other MGARCH families, such as DCC or factor BEKK, with the LSTM generating correlation or factor-loading increments instead of raw covariance contributions.
  • If the unenforced norm bound is the real weakness, adding a soft penalty or spectral normalization on $C_t C_t'$ would make the stability guarantee honest and could be tested for whether it weakens the empirical gains.
  • The reported widening gap at 175 and 250 assets suggests the LSTM contribution matters most when the covariance matrix is large and the sample covariance is least reliable; this could be checked by comparing against shrinkage estimators on the same GMV backtests.
  • Since the paper uses multivariate normality for likelihood, a natural extension is to replace it with a Student-$t$ or copula tail specification; the 1% tail-loss results hint that the gain may concentrate at moderate rather than extreme quantiles.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes LSTM-BEKK, a hybrid multivariate volatility model that augments Scalar BEKK with an LSTM-generated lower-triangular component C_t, giving H_t = CC' + C_t C_t' + a r_{t-1} r'_{t-1} + b H_{t-1}. Parameters are estimated by Gaussian negative log-likelihood with RMSprop, subject to a+b<1. The empirical work compares LSTM-BEKK with DCC and Scalar BEKK on US, UK, and Japanese equity universes. At N=50, 500 random portfolios per market are used with paired t-tests; at N=100, 175, and 250, a single representative portfolio is used. The paper also reports GMV portfolio backtests, an MCS analysis, and an appendix with VaR/ES tail-risk losses. The central claim is that LSTM-BEKK achieves consistently superior out-of-sample portfolio risk forecasts from 50 to 250 assets while retaining BEKK interpretability.

Significance. The architecture is a natural and potentially useful hybrid: it preserves positive definiteness by construction, keeps the interpretable BEKK parameters a and b, and adds data-driven nonlinear dynamics through an LSTM. The out-of-sample NLL protocol is standard, and the 500-portfolio repeated experiment with paired t-tests at N=50 is a genuine strength. The theorem on boundedness of E(H_k) is a useful first step, and the appendix gives elicitable-loss tail-risk comparisons. If the high-dimensional claims were supported by repeated sampling, this could be a solid contribution to the growing literature on deep-learning-augmented GARCH models. As it stands, the evidence for the scalability and universal-superiority claims is incomplete, and the tail-risk appendix partially contradicts the abstract.

major comments (4)
  1. [§4.5, Tables 7–10] The high-dimensional results are based on a single representative portfolio at each size (N=100, 175, 250), as stated in §4.5; no random draws, seeds, or confidence intervals are reported. The MCS analysis in Table 10 is computed on these same single paths, so the repeated p=1.000 for LSTM-BEKK does not quantify sampling uncertainty. Since the NLL gaps at high dimension are small relative to total NLL (e.g., US N=100: 166.090 vs 166.325), a single favorable asset draw or optimization run could produce the reported ordering. The claim that LSTM-BEKK is 'consistently better ... from 50 to 250 assets' is therefore only rigorously supported at N=50, where 500 portfolios are used. Please add repeated random portfolios, multiple optimization seeds, or a block-bootstrap over test days for all high-dimensional comparisons.
  2. [Appendix C, Tables 14–16] The tail-risk results in Appendix C contradict the unqualified 'superior performance' claim in the abstract and in the Appendix's own cross-market discussion. For Japan N=100, LSTM-BEKK has the worst QLoss1% (12.328) and JointLoss1% (3.333) of the three models; for UK N=100, DCC has the lowest QLoss5% (23.222) and JointLoss5% (1.595). The text in Appendix C says LSTM-BEKK 'consistently outperforms' and 'achieves the lowest QLoss5% and JointLoss5% across nearly all settings,' which is not what the tables show. The central claim should be qualified to the metrics and settings where the evidence actually holds, and the Appendix discussion should be rewritten to match its tables.
  3. [§5.3.3, Table 13] The Japan GMV subsection reports that LSTM-BEKK achieves the lowest AV in '29 out of 50 portfolio combinations' and refers to 'N=25 and N=30 configurations,' but Table 13 only reports N=100, 175, and 250. The 50-combination experiment is not described anywhere, so the majority claim cannot be checked. Either report the full set of portfolio sizes and the number of wins/losses, or remove the 29/50 statement and the N=25/N=30 references.
  4. [§2.2, §3.1, Theorem 1] Theorem 1's stability bound assumes sup_t ||C_t C_t'|| < infinity almost surely, but §3.1 states that this bound was not enforced during estimation. The assumption is plausibly satisfied automatically because the LSTM output in (12)–(13) is produced from bounded tanh activations, but the paper nowhere makes this argument; as written, the theorem is not connected to the estimation procedure. In addition, the theorem only bounds E(H_k) for fixed initial H0; it does not establish a.s. convergence or ergodicity of H_t, so its label 'stability' is stronger than what is proven. Please state explicitly why the norm bound holds for the trained model, or weaken the theoretical claims accordingly.
minor comments (6)
  1. [§4.1, Tables 1–3] The data-section dates disagree with the table notes: Table 2 says the U.K. sample runs from January 2014 but §4.1 says July 2014, and Table 3 says Japan runs from March 2014 but §4.1 says January 2014.
  2. [Table 9] For Japan N=100, LSTM-BEKK reports a=0.002 and b=0.998, which sum to 1.000, inconsistent with the stated constraint a+b<1 and with Theorem 1. Please report more decimal places or clarify that the displayed values round to a sum below one.
  3. [§4.5.4] The MCS description does not specify how the daily loss series is constructed or which resampling scheme (e.g., stationary bootstrap) is used; identical p-values for DCC and Scalar BEKK in several rows (for example, US N=100 both 0.031) suggest a degenerate loss ranking that should be explained.
  4. [Figures 1 and 2] The figures are referenced and described in the text but are not visible in the manuscript; please ensure they are included with axes, legends, and model labels.
  5. [Notation] The name 'LSTM BEKK' appears without a hyphen in several tables and appendix headings (e.g., Tables 11–13); standardize to 'LSTM-BEKK' throughout.
  6. [§4.1] The text describes 'de-meaned returns' but the descriptive statistics report small nonzero means; clarify whether returns are centered before estimation or whether the mean is estimated jointly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: LSTM-BEKK is a newly defined hybrid recursion evaluated out-of-sample; the cited prior work is motivational, not load-bearing.

full rationale

The central claim is that LSTM-BEKK delivers lower out-of-sample NLL and better GMV portfolio risk than DCC and Scalar BEKK. The evaluation is conducted on a held-out test set (Sections 4.4.1 and 4.5), so the performance comparison is not forced by the estimation objective. The model equation (11), H_t = CC' + C_t C_t' + a r_{t-1} r'_{t-1} + b H_{t-1}, is a novel specification whose LSTM-generated C_t is an explanatory input produced from past returns; the target quantities are future covariance matrices and test-set likelihoods, not the model's own fitted components. Theorem 1 (Appendix B) proves a conditional-expectation contraction bound under the stated assumption that ||C_t C_t'|| is bounded almost surely; this assumption is an input to the theorem, not the theorem's conclusion, so the proof is not circular. The paper's later admission in Section 3.1 that it was 'unnecessary to bound the norm of C_t C_t'' is a gap between the theorem's assumptions and the estimation procedure, which is a correctness/robustness concern rather than a circular-derivation concern. The citation to Nguyen, Tran, and Kohn (2022) includes a coauthor of the present paper, but it is used only as motivation for combining deep learning with GARCH and is not load-bearing for the LSTM-BEKK derivation or for the empirical ranking. Other manuscript passages raise evidence-quality issues: Section 4.5 reports only 'a single representative portfolio' for N=100, 175, and 250; the MCS p-values of 1.000 in Table 10 are computed on those same single paths; Appendix C shows LSTM-BEKK is worst at Japan N=100 for QLoss1% and JointLoss1%; and Section 5.3.3 refers to '29 out of 50 portfolio combinations' without reporting that 50-combination experiment. These issues affect whether the broad 'consistently superior' claim is fully supported, but none of them amounts to a prediction that is equivalent to its inputs by construction. The derivation chain is therefore self-contained with respect to circularity.

Assumptions & free parameters 6 free parameters · 5 assumptions · 1 invented entities

The model introduces a large set of free parameters (static C, a, b, LSTM weights, Swish beta, and hand-chosen hyperparameters). The only genuinely new entity is the LSTM-generated C_t. The most consequential assumption is the boundedness of C_t C_t', which is required by Theorem 1 but not enforced in estimation.

free parameters (6)
  • a = 0.002 to 0.038 across markets and portfolios (Tables 5 to 9)
    Scalar shock coefficient in the BEKK recursion; estimated by maximum likelihood.
  • b = 0.669 to 0.998
    Scalar persistence coefficient; estimated by maximum likelihood. In several Japan and 250-asset configurations b is close to 0.99 to 1.00, near the nonstationarity boundary.
  • Static lower-triangular matrix C = n(n+1)/2 parameters per dataset, values not reported
    Constant covariance factor CC' estimated from data.
  • LSTM network parameters (gate weights and biases) = not reported; 3 to 5 layers with hidden size equal to input size
    All LSTM parameters are estimated by NLL minimization; the model's flexibility is concentrated here.
  • Swish temperature beta = not reported
    Learnable parameter in the diagonal activation of C_t (Eq. 13).
  • Hyperparameters: learning rate, batch size, epochs, dropout, layer count = dropout 0.1 to 0.2, layers 3 to 5, no learning rate or batch size reported
    Chosen by hand or validation; not part of the likelihood but affect results.
assumptions (5)
  • domain assumption Returns follow a conditional multivariate normal distribution (Eq. 1)
    Underlies the NLL objective; the paper explicitly sets aside heavier-tailed alternatives.
  • domain assumption Scalar BEKK stationarity constraints a, b >= 0 and a + b < 1 are sufficient for volatility decay
    Imposed in estimation following Theorem 1; standard for MGARCH.
  • ad hoc to paper Norm bound sup_t ||C_t C_t'|| < infinity (Theorem 1) holds
    The theorem's conclusion depends on it, but Section 3.1 states the bound is not enforced.
  • ad hoc to paper LSTM hidden size equal to input size and 3 to 5 layers give enough capacity
    Architecture choice stated without sensitivity analysis.
  • domain assumption The 500 random 50-asset portfolios are independent draws from each market's asset universe
    Paired t-tests and MCS treat each portfolio as an independent replication.
invented entities (1)
  • Dynamic lower-triangular matrix C_t
    purpose: Acts as a time-varying covariance factor generated by an LSTM, added as C_t C_t' to the BEKK recursion.
    This is a new latent, endogenous model component with no external observable counterpart; its existence and behavior are only inferred from the model fit.

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Cite this review

Pith. "Pith review of Deep Learning Enhanced Multivariate GARCH." pith.science (2026). https://pith.science/paper/OIB2HJ2T

@misc{pith2026250602796,
  author       = {Pith},
  title        = {Pith review of: Deep Learning Enhanced Multivariate GARCH},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OIB2HJ2T}},
  note         = {Machine review of arXiv:2506.02796}
}
read the original abstract

This paper introduces a novel multivariate volatility modeling framework, named Long Short-Term Memory enhanced BEKK (LSTM-BEKK), that integrates deep learning into multivariate GARCH processes. By combining the flexibility of recurrent neural networks with the econometric structure of BEKK models, our approach is designed to better capture nonlinear, dynamic, and high-dimensional dependence structures in financial return data. The proposed model addresses key limitations of traditional multivariate GARCH-based methods, particularly in capturing persistent volatility clustering and asymmetric co-movement across assets. Leveraging the data-driven nature of LSTMs, the framework adapts effectively to time-varying market conditions, offering improved robustness and forecasting performance. Empirical results across multiple equity markets confirm that the LSTM-BEKK model achieves superior performance in terms of out-of-sample portfolio risk forecast, while maintaining the interpretability from the BEKK models. These findings highlight the potential of hybrid econometric-deep learning models in advancing financial risk management and multivariate volatility forecasting.

Figures

Figures reproduced from arXiv: 2506.02796 by the authors.

Figure 1
Figure 1. Volatility dynamics (i.e, the diagonal elements of the covariance matrix) across [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Covariance dynamics comparison across models for asset pairs with differing [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

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Reference graph

Works this paper leans on

35 extracted references · 34 canonical work pages

  1. [1]

    Aielli, G. P. (2013). Dynamic conditional correlation: On properties and estimation. Journal of Business & Economic Statistics, 31(3):282–299

    work page 2013

  2. [2]

    Asai, M., McAleer, M., and Yu, J. (2006). Multivariate stochastic volatility: A review. Econometric Reviews, 25(2-3):145–175

    work page 2006

  3. [3]

    Bauwens, L., Laurent, S., and Rombouts, J. V. K. (2006). Multivariate GARCH models: A survey.Journal of Applied Econometrics, 21(1):79–109

    work page 2006

  4. [4]

    and Otranto, E

    Bauwens, L. and Otranto, E. (2020). Nonlinearities and regimes in conditional correlations with different dynamics.Journal of Econometrics, 217(2):496–522

    work page 2020

  5. [5]

    Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity.Journal of Econometrics, 31(3):307–327

    1986

  6. [6]

    F., and Wooldridge, J

    Bollerslev, T., Engle, R. F., and Wooldridge, J. M. (1988). A capital asset pricing model with time-varying covariances.Journal of Political Economy, 96(1):116–131

    work page 1988

  7. [7]

    and McAleer, M

    Caporin, M. and McAleer, M. (2008). Scalar BEKK and indirect DCC.Journal of Forecasting, 27(6):537–549

    work page 2008

  8. [8]

    DeMiguel, V., Garlappi, L., and Uppal, R. (2009). Optimal versus naive diversifica- tion: How inefficient is the 1/n portfolio strategy?The Review of Financial Studies, 22(5):1915–1953

    work page 2009

Show all 35 references
  1. [9]

    Engle, R. F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.Econometrica, 50(4):987–1007

    1982

  2. [10]

    Engle, R. F. (2002). Dynamic conditional correlation: A simple class of multivariate GARCH models.Journal of Business & Economic Statistics, 20(3):339–350

    2002

  3. [11]

    Engle, R. F. and Kelly, B. (2012). Dynamic equicorrelation.Journal of Business & Economic Statistics, 30(4):384–397. 43

    2012

  4. [12]

    Engle, R. F. and Kroner, K. F. (1995). Multivariate simultaneous Generalized ARCH. Econometric Theory, 11(1):122–150

    1995

  5. [13]

    Fang, Y., Liu, L., and Liu, J. (2015). A dynamic double asymmetric copula generalized autoregressive conditional heteroskedasticity model: application to China’s and US stock market.Journal of Applied Statistics, 42(2):327–346

    2015

  6. [14]

    and Ziegel, J

    Fissler, T. and Ziegel, J. F. (2016). Higher order elicitability and Osband’s principle.The Annals of Statistics, 44(4):1680–1707

    2016

  7. [15]

    and Zakoian, J.-M

    Francq, C. and Zakoian, J.-M. (2019).GARCH Models: Structure, Statistical Inference, and Financial Applications. John Wiley & Sons, Hoboken, NJ, 2nd edition

    2019

  8. [16]

    and Zakoïan, J.-M

    Francq, C. and Zakoïan, J.-M. (2012). QML estimation of a class of multivariate asym- metric GARCH models.Econometric Theory, 28(1):179–206

    2012

  9. [17]

    (2016).Deep Learning

    Goodfellow, I., Bengio, Y., and Courville, A. (2016).Deep Learning. MIT Press

    2016

  10. [18]

    M., Laurent, S., and Violante, F

    Hafner, C. M., Laurent, S., and Violante, F. (2017). Weak diffusion limits of dynamic conditional correlation models.Econometric Theory, 33(3):691–716

    2017

  11. [19]

    Hafner, C. M. and Preminger, A. (2009). Asymptotic theory for a factor GARCH model. Econometric Theory, 25(2):336–363

    2009

  12. [20]

    Hafner, C. M. and Rombouts, J. V. (2007). Semiparametric multivariate volatility models. Econometric Theory, 23(2):251–280

    2007

  13. [21]

    R., Lunde, A., and Nason, J

    Hansen, P. R., Lunde, A., and Nason, J. M. (2011). The Model Confidence Set.Econo- metrica, 79(2):453–497

    2011

  14. [22]

    and Schmidhuber, J

    Hochreiter, S. and Schmidhuber, J. (1997). Long short-term memory.Neural Computa- tion, 9(8):1735–1780

    1997

  15. [23]

    and Bassett, G

    Koenker, R. and Bassett, G. (1978). Regression quantiles.Econometrica: journal of the Econometric Society, pages 33–50

    1978

  16. [24]

    Ku, Y.-H. H. (2008). Student-t distribution based VAR-MGARCH: an application of the DCC model on international portfolio risk management.Applied Economics, 40(13):1685–1697

    2008

  17. [25]

    and Sheu, H.-J

    Lai, Y.-S. and Sheu, H.-J. (2011). On the importance of asymmetries for dynamic hedging during the subprime crisis.Applied Financial Economics, 21(11):801–813

    2011

  18. [26]

    and Wolf, M

    Ledoit, O. and Wolf, M. (2012). Nonlinear shrinkage estimation of large-dimensional covariance matrices.Annals of Statistics, 40(2):1024–1060

    2012

  19. [27]

    and Wolf, M

    Ledoit, O. and Wolf, M. (2015). Spectrum estimation: A unified framework for covariance matrix estimation and PCA in large dimensions.Journal of Multivariate Analysis, 139:360–384. 44

    2015

  20. [28]

    Liu, Y. (2019). Novel volatility forecasting using deep learning–long short term memory recurrent neural networks.Expert Systems with Applications, 132:99–109

    2019

  21. [29]

    and Pedersen, R

    Matsui, M. and Pedersen, R. S. (2022). Characterization of the tail behavior of a class of BEKK processes: A stochastic recurrence equation approach.Econometric Theory, 38(1):1–34

    2022

  22. [30]

    McAleer, M., Chan, F., Hoti, S., and Lieberman, O. (2008). Generalized autoregressive conditional correlation.Econometric Theory, 24(6):1554–1583

    2008

  23. [31]

    Nguyen, T.-N., Tran, M.-N., and Kohn, R. (2022). Recurrent Conditional Heteroskedas- ticity.Journal of Applied Econometrics, 37(5):1031–1054

    2022

  24. [32]

    and Ribarits, E

    Scherrer, W. and Ribarits, E. (2007). On the parametrization of multivariate GARCH models.Econometric Theory, 23(3):464–484

    2007

  25. [33]

    and Teräsvirta, T

    Silvennoinen, A. and Teräsvirta, T. (2009). Multivariate GARCH models.Handbook of Financial Time Series, pages 201–229

    2009

  26. [34]

    Taylor, J. W. (2019). Forecasting value at risk and expected shortfall using a semipara- metric approach based on the asymmetric Laplace distribution.Journal of Business & Economic Statistics, 37(1):121–133

    2019

  27. [35]

    Taylor, S. J. (1994). Modeling stochastic volatility: A review and comparative study. Mathematical Finance, 4(2):183–204. 45

    1994

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