These authors contributed equally to this work.

This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

Financial asset returns are known to be conditionally heteroskedastic and generally non-normally distributed, fat-tailed and often skewed. These features must be taken into account to produce accurate forecasts of Value-at-Risk (VaR). We provide a comprehensive look at the problem by considering the impact that different distributional assumptions have on the accuracy of both univariate and multivariate GARCH models in out-of-sample VaR prediction. The set of analyzed distributions comprises the normal, Student, Multivariate Exponential Power and their corresponding skewed counterparts. The accuracy of the VaR forecasts is assessed by implementing standard statistical backtesting procedures used to rank the different specifications. The results show the importance of allowing for heavy-tails

Value-at-Risk (VaR) is a quantitative tool used to measure the maximum potential loss in value of a portfolio of assets over a defined period for a given probability. Specifically, VaR construction requires a quantile estimate of the far-left tail of the unconditional returns distribution. Though widely-used as a risk measure in the past, standard methods of VaR construction assuming iid-ness and normality have come under criticism due to their failure to incorporate three stylized facts of financial returns, namely

The ability to account for volatility clustering is one of the key strengths of the ARCH modelling approach developed in Engle [

Another salient feature of financial returns series is the fact that comovements between markets increase during periods of high volatility, as shown for example by Longin and Solnik [

In order to shed some light on this issue, several papers focus on direct comparison of the predictive performance of univariate and multivariate GARCH (MGARCH) models under various distributional assumptions with the aim of providing evidence in favour of one of the two approaches. Key studies in this literature include Giot and Laurent [

This paper builds on their approach by widening the set of distributions used to model the error term in both the univariate and multivariate frameworks while maintaining a generic specification for the conditional volatility. Specifically, we consider three symmetric distributions,

As for the choice of the volatility models, within the MGARCH literature we employ the Rotated BEKK (RBEKK) model of Noureldin

Both univariate and multivariate models are estimated employing the aforementioned set of distributional assumptions and their accuracy in producing out-of-sample VaR forecasts is assessed by means of statistical backtesting procedures. The selected tests include the Unconditional Coverage (UC), Independence (IND) and Conditional Coverage (CC) tests of Christoffersen [

Results from VaR backtesting show that in the multivariate setup the skew-Student clearly outperforms all other distributions. Moreover, its univariate version produces more accurate VaR forecasts than the NCT-GARCH and is able to compete with the NCT-APARCH which incorporates asymmetry into the conditional volatility specification. Overall, our results show that allowing for heavy-tails

The paper is organized as follows:

This section illustrates the key points of our theoretical framework. Namely, we outline the alternative approaches to obtain portfolio VaR forecasts using univariate and multivariate models, we describe the procedure used to construct skewed distributions from the corresponding symmetric counterparts and finally we provide an overview of the set of employed distributions, comprising their likelihood derivation.

Two things are worth mentioning. First, we focus on the portfolio VaR for a long position, implying that the predictive power of the models is linked to their ability in modelling large negative returns. Second, we define the asset allocation scheme via an

Let

The specification of the portfolio standard deviation

In the univariate case, the portfolio standard deviation is obtained as the standard deviation of the portfolio returns conditional on past portfolio returns,

In the multivariate setup, the conditioning set is made up of the entire vector of past returns:

As far as financial applications are concerned, modelling and inference based on the normal distribution have often been proven to be of limited usefulness, as it is possible to gain statistical efficiency by allowing for more involved distributions featuring heavy tails and skewness. As a way to capture higher moments, the literature offers several alternatives. For example, the multivariate noncentral

In this respect, Fernández and Steel [

We begin by defining the notion of symmetry of a standardized density used hereafter. In the univariate case, symmetry corresponds to

The idea of introducing skewness into an

The marginal

As shown in

As already mentioned, three symmetric and three asymmetric distributions are considered in the univariate and multivariate framework. Again, for sake of brevity, we only report the multivariate log-likelihood functions as the univariate can be obtained as special cases

Our dataset (used in the paper of Noureldin

Univariate descriptive statistics.

Stock | Mean | Std.dev. | Skewness | Kurtosis | KS Test | JB Test |
---|---|---|---|---|---|---|

Estimation sample: 1 February 2001 to 23 January 2007 (1500 observations) | ||||||

BAC | 0.09 | 1.09 | −0.18 | 7.45 | 0.00 | 0.00 |

JPM | 0.00 | 1.68 | 0.90 | 31.02 | 0.00 | 0.00 |

IBM | −0.04 | 1.24 | 0.01 | 5.96 | 0.01 | 0.00 |

MSFT | −0.01 | 1.37 | 0.37 | 6.01 | 0.00 | 0.00 |

XOM | −0.01 | 1.13 | 0.05 | 8.27 | 0.82 | 0.00 |

AA | 0.01 | 1.59 | 0.14 | 4.74 | 0.00 | 0.00 |

AXP | −0.02 | 1.44 | 0.33 | 7.73 | 0.00 | 0.00 |

DD | 0.02 | 1.21 | 0.37 | 6.76 | 0.21 | 0.00 |

GE | −0.01 | 1.34 | 0.13 | 7.90 | 0.02 | 0.00 |

KO | 0.01 | 0.99 | 0.16 | 5.53 | 0.00 | 0.00 |

Forecasting sample: 24 January 2007 to 30 October 2009 (700 observations) | ||||||

BAC | −0.18 | 3.95 | 0.37 | 9.36 | 0.00 | 0.00 |

JPM | 0.01 | 3.06 | 0.36 | 8.53 | 0.00 | 0.00 |

IBM | 0.08 | 1.45 | −0.02 | 6.31 | 0.00 | 0.00 |

MSFT | 0.02 | 1.60 | 0.08 | 5.90 | 0.00 | 0.00 |

XOM | 0.03 | 1.61 | −0.39 | 11.31 | 0.00 | 0.00 |

AA | −0.04 | 2.93 | −0.83 | 7.50 | 0.00 | 0.00 |

AXP | 0.04 | 3.06 | 0.22 | 6.96 | 0.00 | 0.00 |

DD | −0.04 | 1.89 | −0.12 | 5.70 | 0.00 | 0.00 |

GE | 0.02 | 2.17 | 0.21 | 8.96 | 0.00 | 0.00 |

KO | −0.03 | 1.22 | 0.07 | 7.68 | 0.06 | 0.00 |

Full sample: 1 February 2001 to 30 October 2009 (2200 observations) | ||||||

BAC | 0.01 | 2.40 | 0.33 | 21.72 | 0.00 | 0.00 |

JPM | 0.00 | 2.21 | 0.57 | 16.90 | 0.00 | 0.00 |

IBM | 0.00 | 1.31 | 0.02 | 6.24 | 0.02 | 0.00 |

MSFT | 0.00 | 1.45 | 0.25 | 6.08 | 0.00 | 0.00 |

XOM | 0.00 | 1.30 | −0.20 | 11.56 | 0.04 | 0.00 |

AA | 0.00 | 2.11 | −0.69 | 9.95 | 0.00 | 0.00 |

AXP | 0.00 | 2.09 | 0.32 | 11.23 | 0.00 | 0.00 |

DD | 0.00 | 1.46 | 0.03 | 7.25 | 0.00 | 0.00 |

GE | 0.00 | 1.65 | 0.22 | 10.85 | 0.00 | 0.00 |

KO | 0.00 | 1.07 | 0.11 | 6.89 | 0.00 | 0.00 |

Descriptive statistics of the stock return time series used in the empirical application. The three panels report the statistics for the in-sample period, the out-of-sample period and the full sample period, respectively. “KS test” and “JB test” denotes the Kolmogorov-Smirnov test and Jarque Bera test, with corresponding

Across the three panels, the values of skewness and kurtosis show that the assets are far from being unconditionally normally distributed, thus supporting the conjecture that more flexible distributional assumptions can be conducive to enhanced model performance.

To this extent, one-step ahead forecasts of the conditional portfolio variance (in the univariate case) and of the conditional covariance matrix of returns (in the multivariate one) are recursively obtained as:

For each model, the portfolio VaR forecast at

For the symmetric distributions in our analysis (normal, Student and MEP), one can easily compute the long VaR of the portfolio by applying Equation (

The models accuracy in predicting VaR is assessed using multiple statistical backtesting methods. A common starting point for this procedure is the so-called hit function, or indicator function, which is equal to

This represents the key foundation to many of the backtesting procedures developed in recent years and particularly to the accuracy tests being used in this paper. We focus on tests included in the following three categories:

Evaluation of the Frequency of Violations

Evaluation of the Independence of Violations

Evaluation of the Duration between Violations.

Denoting by

A similar useful test is the TUFF (Time Until First Failure) test (Kupiec [

The Independence test (IND) of Christoffersen [

The relevant IND test statistic can be derived as

Although the aforementioned test has received support in the literature, Christoffersen [

Formally, the CC ratio statistic can be proven to be the sum of the UC and the IND statistics:

A second test belonging to this class is the Regression-based test of Engle and Manganelli [

The idea of this approach is to regress current violations on past violations in order to test for different restrictions on the parameters of the model. That is, we estimate the linear regression model

A correctly specified model should have an expected conditional duration of

Under the null hypothesis of independent violations,

Before turning to the out of sample analysis, it is worth first looking at some parameter estimates obtained by fitting the models on the data.

Full sample parameter estimates.

Normal | Student | MEP | Skew-Normal | Skew-Student | Skew-MEP | NCT-APARCH | NCT-GARCH | |
---|---|---|---|---|---|---|---|---|

0.005 | 0.004 | 0.002 | 0.005 | 0.004 | 0.010 | 0.010 | 0.010 | |

0.05 | 0.05 | |||||||

0.90 | 0.90 | |||||||

7.20 | 5.20 | |||||||

−0.360 | −0.120 | |||||||

LogLik | −2971 | −2946 | −2965 | −2968 | −2943 | −2963 | −2944 | -2963 |

AIC | 2.703 | 2.681 | 2.698 | 2.701 | 2.679 | 2.697 | 2.680 | 2.696 |

LogLik | −32,154 | −31,352 | −32,193 | −32,121 | −31,330 | −32,067 | ||

AIC | 29.232 | 28.504 | 29.278 | 29.211 | 28.449 | 29.154 |

Note: The table reports test statistics and robust standard errors obtained from full sample parameter estimation, for

A common feature emerging from both univariate and multivariate panels is that the use of skewed distribution assumptions seems to be justified, as all asymmetric coefficients are significant at standard levels. Moreover, the direct comparison of models fit via the Akaike Information Criteria (AIC) highlights that the models incorporating skewed distributions consistently outperform their symmetric counterparts, with the skew-Student achieving the best fit. In both cases the Student has the lowest AIC among the symmetric densities, while the normal possesses the highest.

In the univariate setting, the NCT-APARCH model substantially improves over the NCT-GARCH due to the introduction of skewness into the volatility model, but it performs slightly worse than its closest competitor, the skew-Student GARCH model. This performance is even more impressive considering that the AIC penalizes the NCT-based models for only three parameters, given that remaining ones are fixed

The evolution of the parameters across re-estimations

Tail parameter evolution across re-estimations.

Panel (b) exhibits the same dynamics as before albeit at a reduced magnitude. This occurs due to the introduction of skewness which is able to capture some of the negative returns associated to the crisis, thereby reducing the need for such a large increase in tail thickness. Evidence of such

Skew parameters evolution across re-estimations.

Turning to the Student and skew-Student distributions, the dynamics are somewhat different along a number of dimensions. First, there is now a marked difference between the univariate and multivariate cases. In the former, the dynamics are similar to the MEP and skew-MEP results wherein the tails thicken at around

Unlike the MEP and skew-MEP cases, where the multivariate specification exhibits thinner tails than the univariate but both exhibit congruent dynamics, the Student and skew-Student exhibit fewer similarities between specifications. In contrast, moving from the Student to skew-Student setting reveals almost identical tail parameter dynamics. Combined with the positive skew exhibited in Panel (b) of

In the multivariate setting, the out-of-sample covariance matrix predictions are used to construct equally-weighted portfolios for the computation of the one-step-ahead VaR.

Portfolios descriptive statistics.

Univariate Models | ||||||||
---|---|---|---|---|---|---|---|---|

0.8716 | 0.8722 | 0.8662 | 0.8709 | 0.8718 | 0.8780 | 0.8222 | 0.8720 | |

0.4159 | 0.4155 | 0.3899 | 0.4134 | 0.4141 | 0.4457 | 0.3727 | 0.3852 | |

2.7805 | 2.6827 | 2.8234 | 2.8147 | 2.6965 | 2.7095 | 2.5242 | 2.5826 | |

1.4659 | 1.4601 | 1.4800 | 1.4656 | 1.4550 | 1.4573 | 1.4569 | 1.4644 | |

0.4398 | 0.4448 | 0.4139 | 0.4334 | 0.4427 | 0.4718 | 0.3745 | 0.3899 | |

3.9589 | 3.8967 | 3.9743 | 3.9656 | 3.8033 | 3.9281 | 4.2310 | 3.1567 | |

0.9021 | 0.9115 | 0.8983 | 0.9018 | 0.9115 | 0.8924 | |||

0.5226 | 0.5474 | 0.5077 | 0.5212 | 0.5478 | 0.4840 | |||

1.9067 | 1.7923 | 1.9391 | 1.9097 | 1.7935 | 1.9972 | |||

1.4721 | 1.4646 | 1.4875 | 1.4740 | 1.4549 | 1.4691 | |||

0.5226 | 0.5474 | 0.5105 | 0.5233 | 0.5479 | 0.4908 | |||

3.1733 | 3.0657 | 3.2108 | 3.1728 | 3.0674 | 3.2683 |

Note: The table reports average, minimum and maximum value of portfolio standard deviation over the in- and out-of-sample periods.

As already noted, the financial crisis features heavily in the summary statistics. Since this period is included in the forecasting sample (starting from observation 1921 according to

Overall, according to this table, the univariate and multivariate approaches deliver quite similar portfolio summary statistics. If the focus was on the predicted portfolio variance alone, then the ideal choice would be to use a univariate volatility model coupled with either the skew-Student or the NCT distribution, as they are easy to estimate and computationally faster than the multivariate specifications. Ultimately, we are interested in the models accuracy in forecasting the one-step-ahead portfolio VaR, so we move to analyse the outcomes of the statistical backtesting procedures.

All statistical tests are computed for the 5% and 1% VaR confidence level. For each portfolio we report test statistics along with their corresponding

VaR backtesting results—Univariate.

Norm | Skew-Norm | Student | Skew-Student | MEP | Skew-MEP | NCT-APARCH | NCT-GARCH | |
---|---|---|---|---|---|---|---|---|

VaR backtesting results—Multivariate.

Norm | Skew-Norm | Student | Skew-Student | MEP | Skew-MEP | |
---|---|---|---|---|---|---|

Grade comparison—VaR backtesting.

Norm | Skew-Norm | Student | Skew-Student | MEP | Skew-MEP | NCT-APARCH | NCT-GARCH | |
---|---|---|---|---|---|---|---|---|

20% | 80% | 20% | 80% | 20% | 80% | 60% | 20% | |

20% | 20% | 20% | 100% | 20% | 80% | |||

40% | 80% | 80% | 100% | 80% | 100% | 100% | 100% | |

40% | 40% | 40% | 100% | 40% | 80% |

The first distinguishing feature from the VaR backtesting results is the clear predominance of the skew-Student distribution. This holds for both the univariate and multivariate frameworks with the former producing VaR forecasts that outperform the NCT-APARCH at 5% VaR. Similarly, the skew-MEP distribution produces highly accurate VaR forecasts across the board. These findings exemplify a second feature of the results namely, the impact on VaR forecast accuracy of introducing skewness into a heavy-tailed distribution. Clearly, the performance of both the skew-Student and skew-MEP distributions improves compared to their symmetric counterparts, but this effect is less pronounced at 1% VaR. Indeed, the performance of the Student and MEP distributions improves when moving from the 5 % to the 1% VaR scenario, suggesting that despite the improvement arising from the introduction of skewness, heavy-tails remain useful in capturing larger swings in returns,

Analysis of the differences between the univariate and multivariate models reveals two key points. First, while the skew-Student and skew-MEP retain their dominance under both frameworks, the performance of their symmetric counterparts at 1% VaR is worse under the multivariate specification. Second, within the univariate framework, the skew-normal is capable of producing VaR forecasts comparable to the high performance skewed and heavy-tailed distributions at both VaR confidence levels. This does not hold in the multivariate setup where the skew-normal offers no improvements in VaR accuracy over the normal. Besides, we observe that in both frameworks the empirical failure rate (

With respect to the previous backtesting methods, the DQ test takes into account a more general temporal dependence between the series of violations and is considered the most reliable in assessing VaR accuracy

The DQ test results tell a similar story to the VaR backtesting procedures. As before, the skew-Student outperforms its competitors at both 5% and 1% VaR under both univariate and multivariate setups. The univariate skew-normal continues to produce VaR forecasts with an accuracy comparable to the top-performers. Again, this does not extend to the multivariate case.

Finally, forecast accuracy at 5% VaR again reveals a more pronounced improvement when moving from symmetric to skewed distributions than in the 1% case. Overall, introducing skewness into heavy-tailed distributions continues to offer the highest VaR forecast accuracy.

Dynamic Quantile test results—Univariate models.

Norm | Skew-Norm | Student | Skew-Student | MEP | Skew-MEP | NCT-APARCH | NCT-GARCH | |
---|---|---|---|---|---|---|---|---|

Dynamic Quantile test results—Multivariate.

Norm | Skew-norm | Student | Skew-Student | MEP | Skew-MEP | |
---|---|---|---|---|---|---|

Grade comparison—DQ tests.

Norm | Skew-Norm | Student | Skew-Student | MEP | Skew-MEP | NCT-APARCH | NCT-GARCH | |
---|---|---|---|---|---|---|---|---|

33% | 100% | 17% | 100% | 33% | 100% | 100% | 33% | |

33% | 33% | 33% | 100% | 33% | 100% | |||

33% | 83% | 83% | 83% | 33% | 100% | 83% | 100% | |

17% | 33% | 17% | 100% | 17% | 50% |

Given its importance in risk management, practitioners must be capable of forecasting the Value-at-Risk of their asset portfolios to a high degree of accuracy. This requires taking into account a number of properties of financial returns namely, non-normality, heavy-tails, skewness and the possibility of comovements between assets. In this article, we focus primarily on the effect of varying the distributional assumption used to forecast VaR. Moreover, we addressed the still open question of whether univariate or multivariate models are most appropriate for the problem of portfolio VaR forecasting.

The distributions treated in the paper comprised three symmetric and three skewed distributional assumptions (

Employing a series of standard backtesting methods to compare the distribution-based model performance, the results reveal that the skew-Student specification produces the most accurate one-step ahead VaR forecast across all multivariate specifications and is able to compete with the high-performance NCT-APARCH in the univariate setup. This finding is echoed in the univariate MEP results wherein the skewed version outperforms the symmetric across all tests and confidence levels. By contrast, the multivariate skew-MEP exhibits thinner tails than the normal and thus, performs poorly at the 1% confidence level. More generally, The test results reveal a clear hierarchy of distributional assumptions

There are several possible avenues of research extending from this work. Given the focus on distributions, we limited our attention to relatively simple parametric models namely, the RBEKK and GARCH models. As the NCT-GARCH/APARCH results showed, capturing asymmetries in returns volatility improves forecast accuracy. Considering their multivariate versions would provide a useful contribution and allow for a detailed study of the optimal model-distribution-dimension combination. Another possibility would be to consider higher forecast horizons for the VaR in order to check if the inclusion of skewness and asymmetric forms of dependence can lead to significant improvements in the long run.

Manuela Braione and Nicolas K. Scholtes acknowledge support of the “Communauté française de Belgique” through contracts “Projet d’Actions de Recherche Concertées” “12/17-045” and “13/17-055”, respectively granted by the Académie universitaire Louvain. We thank seminar participants at the 13th Journée d’économétrie at the Université Paris Ouest Nanterre La Défense along with Luc Bauwens and Florian Ielpo for useful comments. We also thank two anonymous referees and the academic editors for their comments and advice, all of which led to considerable improvements in the paper.

Manuela Braione and Nicolas K. Scholtes contributed equally to the writing of the MATLAB code as well as the preparation of the current manuscript.

The authors declare no conflict of interest.

The transformation

We report the first two moments of the univariate symmetric normal, Student and MEP distributions used to compute the log-likelihood function as given in

Note that the following relation holds:

Windows length and corresponding calendar time.

Rolling Fixed-Window | Forecast Horizon | |||
---|---|---|---|---|

1 | 1–1500 | 1 Febraury 2001–23 January 2007 | 1501–1520 | 24 January 2007–21 Febraury 2007 |

2 | 21–1520 | 2 March 2001–21 Febraury 2007 | 1521–1540 | 22 Febraury 2007–21 March 2007 |

3 | 41–1540 | 30 March 2001–21 March 2007 | 1541–1560 | 22 March 2007–19 April 2007 |

4 | 61–1560 | 30 April 2001–19 April 2007 | 1561–1580 | 20 April 2007–17 May 2007 |

5 | 81–1580 | 29 May2001–17 May 2007 | 1581–1600 | 18 May 2007–15 June 2007 |

6 | 101–1600 | 26 June 2001– 15 June 2007 | 1601–1620 | 18 June 2007–16 July 2007 |

7 | 121–1620 | 25 July 2001–16 July 2007 | 1621–1640 | 17 July 2007–13 August 2007 |

8 | 141–1640 | 22 August 2001–13 August 2007 | 1641–1660 | 14 August 2007–11 September 2007 |

9 | 161–1660 | 26 September 2001–11 September 2007 | 1661–1680 | 12 September 2007–9 October 2007 |

10 | 181–1680 | 24 October 2001–9 October 2007 | 1681–1700 | 10 October 2007–6 November 2007 |

11 | 201–1700 | 21 November 2001–6 Novembe 2007 | 1701–1720 | 7 November 2007–5 December 2007 |

12 | 221–1720 | 20 December 2001–5 December 2007 | 1721–1740 | 6 December 2007–4 January 2008 |

13 | 241–1740 | 22 January 2002–4 January 2008 | 1741–1760 | 7 January 2008–4 Febraury 2008 |

14 | 261–1760 | 20 Febraury 2002–4 Febraury 2008 | 1761–1780 | 5 Febraury 2008– 4 March 2008 |

15 | 281–1780 | 20 March 2002–4 March 2008 | 1781–1800 | 5 March 2008–2 April 2008 |

16 | 301–1800 | 18 April 2002–2 April 2008 | 1801–1820 | 3 April 2008–30 April 2008 |

17 | 321–1820 | 16 May 2002–30 May 2008 | 1821–1840 | 1 May 2008–29 May 2008 |

18 | 341–1840 | 14 June 2002–29 May 2008 | 1841–1860 | 30 May 2008–26 June 2008 |

19 | 361–1860 | 15 July 2002–26 June 2008 | 1861–1880 | 27 June 2008–25 July 2008 |

20 | 381–1880 | 12 August 2002–25 July 2008 | 1881–1900 | 28 July 2008– 22 August 2008 |

21 | 401–1900 | 10 September2002–22 August 2008 | 1901–1920 | 25 August 2008– 22 September 2008 |

22 | 421–1920 | 8 October 2002–22 September 2008 | 1921–1940 | 23 September 2008–20 October 2008 |

23 | 441–1940 | 5 November 2002–20 October 2008 | 1941–1960 | 21 October 2008–17 November 2008 |

24 | 461-1960 | 4 December 2002–17 November 2008 | 1961–1980 | 18 November 2008–16 December 2008 |

25 | 481–1980 | 3 January 2003–16 December 2008 | 1981–2000 | 17 December 2008–15 January 2009 |

26 | 501–2000 | 3 Febraury 2003–15 January 2009 | 2001–2020 | 16 January 2009–13 Febraury 2009 |

27 | 521–2020 | 4 March 2003–13 Febraury 2009 | 2021–2040 | 17 Febraury 2009–16 March 2009 |

28 | 541–2040 | 1 April 2003–16 March 2009 | 2041–2060 | 17 March 2009–14 April 2009 |

29 | 561–2060 | 30 April 2003–14 April 2009 | 2061–2080 | 15 April 2009–12 May 2009 |

30 | 581–2080 | 29 May 2003–12 May 2009 | 2081–2100 | 13 May 2009–10 June 2009 |

31 | 601–2100 | 26 June 2003–10 June 2009 | 2101–2120 | 11 June 2009–9 July 2009 |

32 | 621–2120 | 25 July 2003–9 July 2009 | 2121–2140 | 10 July 2009–6 August 2009 |

33 | 641–2140 | 22 August 2003–6 August 2009 | 2141–2160 | 7 August 2009–3 September 2009 |

34 | 661–2160 | 22 September 2003–3 September 2009 | 2161–2180 | 4 September 2009–2 October 2009 |

35 | 681-2180 | 20 October 2003–2 October 2009 | 2181–2200 | 5 October 2009–30 October 2009 |

Evolution across re-estimations of the GARCH/RBEKK parameter,

Evolution across re-estimations of the GARCH/RBEKK parameter,

Portfolio standard deviations.

VaR: Univariate models.

VaR: Multivariate models.

The main ingredients of the NCT-APARCH model discussed in the paper by Krause and Paolella [

The NCT density function is given as follows:

The evolution of the conditional variance is modeled according to the APARCH model, which allows for both heavy-tails and asymmetry. It is defined as follows:

The tests are built in an extended Markov framework that allows for higher, or

In order to make these results directly comparable with those given in

Results for both univariate and multivariate specifications are reported below.

Augmented IND and CC test results. Univariate models.

Norm | Skew-Norm | Student | Skew-Student | MEP | Skew-MEP | NCT-APARCH | NCT-GARCH | |
---|---|---|---|---|---|---|---|---|

Augmented IND and CC test results. Multivariate models.

Norm | Skew-norm | Student | Skew-Student | MEP | Skew-MEP | |
---|---|---|---|---|---|---|

Overall, adding more lags in the regression equation does not seem to crucially affect the outcome of the tests, as the new results are still in line with those obtained under

As for the CC test, results concerning the multivariate models are basically the same obtained by setting

Ultimately, univariate distributions featuring skewness and heavy tails confirm their predominance over the remaining alternatives in terms of better predictive ability. In the multivariate case this is particularly true for the skew-Student and the skew-MEP, as the skew-normal is not leading to remarkable improvements over the corresponding normal assumption.

The interested reader is referred to Ley and Paindaveine [

We thank the authors for kindly providing us their MATLAB codes.

In

In the univariate case it corresponds to

Downloaded from

We thank an anonymous referee who pointed out the possibility of better comparing the outcomes of the DQ and Christoffersen’s CC and IND tests by allowing the latter to be computed in an extended framework than the standard one described in