# Forecasting Value-at-Risk under Different Distributional Assumptions

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## Abstract

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## 1. Introduction

## 2. Theoretical Framework

#### 2.1. VaR Estimation

#### 2.2. Constructing Skew Densities

**Definition 1.**Given a random vector $z={({z}_{1},\dots ,{z}_{N})}^{\prime}$ with multivariate symmetric standardized distribution $g(z;\eta )$ following Equation (6), the standardized skewed density $f(z|\mathit{\xi};\eta )$ with vector of asymmetry parameters $\mathit{\xi}={({\xi}_{1},\dots {\xi}_{N})}^{\prime}$, is obtained as:

**ξ**(and, where is the case, of the shape parameter η). Given that the elements of ${z}^{\u2605}$ are uncorrelated (since those of x are uncorrelated by assumption), standardization of ${z}^{\u2605}$ is achieved by the following transformation:

#### 2.3. Distributions

**ψ**is the finite-dimensional vector of model parameters and $f(\xb7|\mathit{\psi},{\mathcal{F}}_{t-1})$ denotes the assumed conditional density function of the portfolio returns, in the univariate case, or of the asset return vector in the multivariate one.

**Multivariate normal distribution**This is the most commonly employed distribution in the literature as it is uniquely identified by its conditional first and second moments, which renders ML estimation much simpler from a computational point of view. In addition, given that the score of the normal log-likelihood function has the martingale difference property when the first two conditional moments are correctly specified, the Quasi Maximum Likelihood (QML) estimates are still consistent and asymptotically normal even if the true DGP is not normally-distributed Bollerslev and Wooldridge [30]. The log-likelihood function for T observations is expressed as follows

**Multivariate Student distribution**The Student distribution is a symmetric and bell-shaped distribution, with heavier tails than the normal. Under the multivariate Student assumption, the log-likelihood function is obtained as

**Multivariate Exponential Power (MEP) distribution**This distribution belongs to the Kotz family of distributions (a particular class of symmetric and elliptical distributions discussed extensively in Fang et al. [31]) and is known to have several equivalent definitions in the literature. It can also include both the normal and the Laplace as special cases, as a function of the value of the non-normality parameter β dictating the tail-behaviour of the distribution. Given its simple implementation, in this paper we consider the pdf given in Solaro [32], which gives rise to the following log-likelihood function:

**Multivariate skew-normal distribution**The multivariate skew-normal is the first non-symmetric distribution we consider herein; it accounts for the skewness of the return distribution without taking into account its kurtosis (as it does not involve a tail parameter). Applying Definition 1 we derive the skew-normal density function, with corresponding log-likelihood function equal to

**Multivariate skew-Student distributio**n Applying the same procedure as for the skew-normal, the log-likelihood function of the skew-Student distribution is given by the following expression:

**Multivariate skew-MEP distribution**The log-likelihood function to be maximized is given by

## 3. Empirical Application

#### 3.1. Data and Forecasting Scheme

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 |

#### 3.2. Testing the Accuracy of VaR Forecasts

- Evaluation of the Frequency of Violations
- Evaluation of the Independence of Violations
- Evaluation of the Duration between Violations.

**Frequency of Violations**The first way of testing the VaR accuracy is to test the number or the frequency of margin exceedances. A test designed to this aim is the Kupiec test (Kupiec [35]), also known as the Unconditional Coverage (UC) test. Its null hypothesis is simply that the percentage of violated VaR forecasts or failure rate p is consistent with the given confidence level α, i.e., ${H}_{0}:p=\alpha $.

**Independence of Violations**A limitation of the Kupiec test is that it is only concerned with the coverage of the VaR estimates without accounting for any clustering of the violations. This aspect is crucial for VaR practitioners, as large losses occurring in rapid succession are more likely to lead to disastrous events than individual exceptions.

**Duration between Violations**One of the drawbacks of Christoffersen’s CC test is that it is not capable of capturing dependence in all forms, since it only considers the dependence of observations between two successive days. To address this, Christofferson and Pelletier [38] introduced the Duration-Based test of independence (DBI), which is an improved test for both independence and coverage. Its basic intuition is that if exceptions are completely independent of each other, then the upcoming VaR violations should be independent of the time that has elapsed since the occurrence of the last exceedance (Campbell [36]). The duration (in days) between two exceptions is defined via the no-hit-duration ${D}_{i}={t}_{i}-{t}_{i-1}$, where ${t}_{i}$ is the day of ${i}^{\mathrm{th}}$ violation.

## 4. Results

#### 4.1. Parameter Estimates

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

Univariate models | ||||||||

$\overline{\sigma}$ | 0.005 | 0.004 | 0.002 | 0.005 | 0.004 | 0.010 | 0.010 | 0.010 |

${a}_{1}$ | $\underset{(0.016)}{0.07}$ | $\underset{(0.013)}{0.06}$ | $\underset{(0.016)}{0.07}$ | $\underset{(0.029)}{0.07}$ | $\underset{(0.013)}{0.06}$ | $\underset{(0.015)}{0.07}$ | 0.05 | 0.05 |

${b}_{1}$ | $\underset{(0.017)}{0.92}$ | $\underset{(0.013)}{0.93}$ | $\underset{(0.017)}{0.92}$ | $\underset{(0.031)}{0.92}$ | $\underset{(0.014)}{0.93}$ | $\underset{(0.017)}{0.93}$ | 0.90 | 0.90 |

ν | $\underset{(1.358)}{8.44}$ | $\underset{(1.366)}{8.42}$ | 7.20 | 5.20 | ||||

γ | −0.360 | −0.120 | ||||||

β | $\underset{(0.041)}{1.875}$ | $\underset{(0.038)}{1.873}$ | ||||||

$\overline{\xi}$ | $\underset{(0.026)}{0.938}$ | $\underset{(0.023)}{0.935}$ | $\underset{(0.02)}{0.946}$ | |||||

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 |

Multivariate models | ||||||||

${a}_{2}$ | $\underset{(0.001)}{0.021}$ | $\underset{(0.001)}{0.015}$ | $\underset{(0.001)}{0.022}$ | $\underset{(0.002)}{0.032}$ | $\underset{(0.001)}{0.015}$ | $\underset{(0.001)}{0.026}$ | ||

${b}_{2}$ | $\underset{(0.001)}{0.976}$ | $\underset{(0.001)}{0.983}$ | $\underset{(0.001)}{0.976}$ | $\underset{(0.002)}{0.966}$ | $\underset{(0.001)}{0.983}$ | $\underset{(0.002)}{0.972}$ | ||

ν | $\underset{(0.370)}{8.42}$ | $\underset{(0.37)}{8.30}$ | ||||||

$\overline{\xi}$ | $\underset{(0.035)}{1.016}$ | $\underset{(0.022)}{1.018}$ | $\underset{(0.034)}{1.018}$ | |||||

β | $\underset{(0.013)}{1.919}$ | $\underset{(0.031)}{2.036}$ | ||||||

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 |

#### 4.2. VaR Backtesting Results

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

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

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

${\overline{\sigma}}_{p}$ | 0.8716 | 0.8722 | 0.8662 | 0.8709 | 0.8718 | 0.8780 | 0.8222 | 0.8720 |

$min\left\{{\sigma}_{p}\right\}$ | 0.4159 | 0.4155 | 0.3899 | 0.4134 | 0.4141 | 0.4457 | 0.3727 | 0.3852 |

$max\left\{{\sigma}_{p}\right\}$ | 2.7805 | 2.6827 | 2.8234 | 2.8147 | 2.6965 | 2.7095 | 2.5242 | 2.5826 |

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

${\overline{\sigma}}_{p}$ | 1.4659 | 1.4601 | 1.4800 | 1.4656 | 1.4550 | 1.4573 | 1.4569 | 1.4644 |

$min\left\{{\sigma}_{p}\right\}$ | 0.4398 | 0.4448 | 0.4139 | 0.4334 | 0.4427 | 0.4718 | 0.3745 | 0.3899 |

$max\left\{{\sigma}_{p}\right\}$ | 3.9589 | 3.8967 | 3.9743 | 3.9656 | 3.8033 | 3.9281 | 4.2310 | 3.1567 |

Multivariate Models | ||||||||

Normal | Student | MEP | Skew-Normal | Skew-Student | Skew-MEP | |||

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

${\overline{\sigma}}_{p}$ | 0.9021 | 0.9115 | 0.8983 | 0.9018 | 0.9115 | 0.8924 | ||

$min\left\{{\sigma}_{p}\right\}$ | 0.5226 | 0.5474 | 0.5077 | 0.5212 | 0.5478 | 0.4840 | ||

$max\left\{{\sigma}_{p}\right\}$ | 1.9067 | 1.7923 | 1.9391 | 1.9097 | 1.7935 | 1.9972 | ||

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

${\overline{\sigma}}_{p}$ | 1.4721 | 1.4646 | 1.4875 | 1.4740 | 1.4549 | 1.4691 | ||

$min\left\{{\sigma}_{p}\right\}$ | 0.5226 | 0.5474 | 0.5105 | 0.5233 | 0.5479 | 0.4908 | ||

$max\left\{{\sigma}_{p}\right\}$ | 3.1733 | 3.0657 | 3.2108 | 3.1728 | 3.0674 | 3.2683 |

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

5% VaR | ||||||||

violation/frequency | $\underset{(0.074)}{52}$ | $\underset{(0.060)}{42}$ | $\underset{(0.076)}{53}$ | $\underset{(0.041)}{29}$ | $\underset{(0.074)}{52}$ | $\underset{(0.057)}{40}$ | $\underset{(0.076)}{53}$ | $\underset{(0.036)}{25}$ |

TUFF | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{5.991}$ | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{5.991}$ | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{5.991}$ | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{5.991}$ | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{5.991}$ | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{5.991}$ | $\underset{(0.883)}{0.022}$ | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{5.991}$ |

UC | $\underset{\left(\mathbf{0}.\mathbf{006}\right)}{7.611}$ | $\underset{(0.239)}{1.389}$ | $\underset{\left(\mathbf{0}.\mathbf{004}\right)}{8.476}$ | $\underset{(0.284)}{1.147}$ | $\underset{\left(\mathbf{0}.\mathbf{006}\right)}{7.611}$ | $\underset{(0.396)}{0.720}$ | $\underset{(0.068)}{3.326}$ | $\underset{\left(\mathbf{0}.\mathbf{004}\right)}{8.476}$ |

IND | $\underset{\left(\mathbf{0}.\mathbf{003}\right)}{8.829}$ | $\underset{(0.104)}{2.646}$ | $\underset{\left(\mathbf{0}.\mathbf{003}\right)}{8.815}$ | $\underset{(0.283)}{1.155}$ | $\underset{\left(\mathbf{0}.\mathbf{005}\right)}{7.870}$ | $\underset{(0.195)}{1.676}$ | $\underset{\left(\mathbf{0}.\mathbf{023}\right)}{5.151}$ | $\underset{\left(\mathbf{0}.\mathbf{002}\right)}{9.861}$ |

CC | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{16.440}$ | $\underset{(0.133)}{4.035}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{17.290}$ | $\underset{(0.316)}{2.302}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{15.481}$ | $\underset{(0.302)}{2.397}$ | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{8.477}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{18.337}$ |

DBI | $\underset{(0.880)}{0.023}$ | $\underset{(0.778)}{0.079}$ | $\underset{(0.768)}{0.087}$ | $\underset{(0.362)}{0.830}$ | $\underset{(0.831)}{0.046}$ | $\underset{(0.751)}{0.100}$ | $\underset{(0.372)}{0.797}$ | $\underset{(0.582)}{0.303}$ |

1% VaR | ||||||||

violation/frequency | $\underset{(0.021)}{15}$ | $\underset{(0.017)}{12}$ | $\underset{(0.017)}{12}$ | $\underset{(0.007)}{5}$ | $\underset{(0.020)}{14}$ | $\underset{(0.016)}{11}$ | $\underset{(0.016)}{11}$ | $\underset{(0.011)}{8}$ |

TUFF | $\underset{(0.232)}{1.426}$ | $\underset{(0.232)}{1.426}$ | $\underset{(0.232)}{1.426}$ | $\underset{(0.232)}{1.426}$ | $\underset{(0.232)}{1.426}$ | $\underset{(0.232)}{1.426}$ | $\underset{(0.232)}{1.426}$ | $\underset{(0.232)}{1.426}$ |

UC | $\underset{\left(\mathbf{0}.\mathbf{008}\right)}{6.957}$ | $\underset{(0.085)}{2.972}$ | $\underset{(0.085)}{2.972}$ | $\underset{(0.423)}{0.641}$ | $\underset{(0.019)}{5.479}$ | $\underset{(0.161)}{1.967}$ | $\underset{(0.710)}{0.138}$ | $\underset{(0.161)}{1.967}$ |

IND | $\underset{\left(\mathbf{0}.\mathbf{006}\right)}{7.638}$ | $\underset{(0.065)}{3.406}$ | $\underset{(0.065)}{3.406}$ | $\underset{(0.400)}{0.707}$ | $\underset{(0.014)}{6.072}$ | $\underset{(0.127)}{2.330}$ | $\underset{(0.568)}{0.326}$ | $\underset{(0.127)}{2.330}$ |

CC | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{14.595}$ | $\underset{\left(\mathbf{0}.\mathbf{041}\right)}{6.378}$ | $\underset{\left(\mathbf{0}.\mathbf{041}\right)}{6.378}$ | $\underset{(0.510)}{1.348}$ | $\underset{\left(\mathbf{0}.\mathbf{003}\right)}{11.551}$ | $\underset{(0.117)}{4.297}$ | $\underset{(0.793)}{0.464}$ | $\underset{(0.117)}{4.297}$ |

DBI | $\underset{(0.472)}{0.518}$ | $\underset{(0.723)}{0.126}$ | $\underset{(0.969)}{0.002}$ | $\underset{(0.985)}{0.000}$ | $\underset{(0.459)}{0.547}$ | $\underset{(0.596)}{0.281}$ | $\underset{(0.479)}{0.501}$ | $\underset{(0.596)}{0.281}$ |

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

5% VaR | ||||||

violation/frequency | $\underset{(0.076)}{53}$ | $\underset{(0.073)}{51}$ | $\underset{(0.080)}{56}$ | $\underset{(0.054)}{38}$ | $\underset{(0.074)}{52}$ | $\underset{(0.063)}{44}$ |

TUFF | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{5.991}$ | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{5.991}$ | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{5.991}$ | $\underset{(0.883)}{0.022}$ | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{5.991}$ | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{5.991}$ |

UC | $\underset{\left(\mathbf{0}.\mathbf{004}\right)}{8.476}$ | $\underset{\left(\mathbf{0}.\mathbf{009}\right)}{6.789}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{11.311}$ | $\underset{(0.608)}{0.264}$ | $\underset{\left(\mathbf{0}.\mathbf{006}\right)}{7.611}$ | $\underset{(0.133)}{2.260}$ |

IND | $\underset{\left(\mathbf{0}.\mathbf{002}\right)}{9.861}$ | $\underset{\left(\mathbf{0}.\mathbf{005}\right)}{7.849}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{13.261}$ | $\underset{(0.601)}{0.273}$ | $\underset{\left(\mathbf{0}.\mathbf{003}\right)}{8.829}$ | $\underset{(0.113)}{2.516}$ |

CC | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{18.337}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{14.637}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{24.572}$ | $\underset{(0.765)}{0.536}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{16.440}$ | $\underset{(0.092)}{4.776}$ |

DBI | $\underset{(0.354)}{0.858}$ | $\underset{(0.366)}{0.818}$ | $\underset{(0.305)}{1.052}$ | $\underset{(0.221)}{1.501}$ | $\underset{(0.372)}{0.797}$ | $\underset{(0.207)}{1.593}$ |

1% VaR | ||||||

violation/frequency | $\underset{(0.027)}{19}$ | $\underset{(0.026)}{18}$ | $\underset{(0.024)}{17}$ | $\underset{(0.011)}{8}$ | $\underset{(0.024)}{17}$ | $\underset{(0.021)}{15}$ |

TUFF | $\underset{(0.232)}{1.426}$ | $\underset{(0.232)}{1.426}$ | $\underset{(0.232)}{1.426}$ | $\underset{(0.232)}{1.426}$ | $\underset{(0.232)}{1.426}$ | $\underset{(0.232)}{1.426}$ |

UC | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{14.153}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{12.176}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{10.313}$ | $\underset{(0.710)}{0.138}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{10.313}$ | $\underset{\left(\mathbf{0}.\mathbf{044}\right)}{4.051}$ |

IND | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{14.568}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{13.106}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{11.190}$ | $\underset{(0.568)}{0.326}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{10.978}$ | $\underset{\left(0.199\right)}{1.652}$ |

CC | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{28.721}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{25.282}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{21.503}$ | $\underset{(0.793)}{0.464}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{21.292}$ | $\underset{\left(0.057\right)}{5.702}$ |

DBI | $\underset{(0.078)}{3.109}$ | $\underset{(0.342)}{0.902}$ | $\underset{(0.055)}{3.798}$ | $\underset{(0.595)}{0.282}$ | $\underset{(0.210)}{1.573}$ | $\underset{(0.379)}{0.774}$ |

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

5% VaR | ||||||||

Univariate grade | 20% | 80% | 20% | 80% | 20% | 80% | 60% | 20% |

Multivariate grade | 20% | 20% | 20% | 100% | 20% | 80% | ||

1% VaR | ||||||||

Univariate grade | 40% | 80% | 80% | 100% | 80% | 100% | 100% | 100% |

Multivariate grade | 40% | 40% | 40% | 100% | 40% | 80% |

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

5% VaR | ||||||||

K = 1 | ||||||||

$D{Q}_{UC}$ | $\underset{\left(\mathbf{0}.\mathbf{002}\right)}{9.319}$ | $\underset{(0.207)}{1.594}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{10.092}$ | $\underset{(0.299)}{1.079}$ | $\underset{\left(\mathbf{0}.\mathbf{003}\right)}{8.965}$ | $\underset{(0.368)}{0.809}$ | $\underset{(0.074)}{3.188}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{10.485}$ |

$D{Q}_{IND}$ | $\underset{(0.231)}{1.433}$ | $\underset{(0.280)}{1.168}$ | $\underset{(0.533)}{0.389}$ | $\underset{(0.886)}{0.020}$ | $\underset{(0.596)}{0.281}$ | $\underset{(0.352)}{0.867}$ | $\underset{(0.403)}{0.698}$ | $\underset{(0.199)}{1.651}$ |

$D{Q}_{CC}$ | $\underset{\left(\mathbf{0}.\mathbf{006}\right)}{10.189}$ | $\underset{(0.264)}{2.665}$ | $\underset{\left(\mathbf{0}.\mathbf{006}\right)}{10.201}$ | $\underset{(0.581)}{1.087}$ | $\underset{\left(\mathbf{0}.\mathbf{011}\right)}{9.036}$ | $\underset{(0.441)}{1.636}$ | $\underset{(0.159)}{3.680}$ | $\underset{\left(\mathbf{0}.\mathbf{003}\right)}{11.463}$ |

K = 3 | ||||||||

$D{Q}_{UC}$ | $\underset{\left(\mathbf{0}.\mathbf{008}\right)}{7.028}$ | $\underset{(0.274)}{1.196}$ | $\underset{\left(\mathbf{0}.\mathbf{009}\right)}{6.821}$ | $\underset{(0.265)}{1.240}$ | $\underset{\left(\mathbf{0}.\mathbf{013}\right)}{6.193}$ | $\underset{(0.469)}{0.525}$ | $\underset{(0.082)}{3.024}$ | $\underset{\left(\mathbf{0}.\mathbf{005}\right)}{8.068}$ |

$D{Q}_{IND}$ | $\underset{(0.426)}{0.633}$ | $\underset{(0.616)}{0.251}$ | $\underset{\left(\mathbf{0}.\mathbf{033}\right)}{4.536}$ | $\underset{(0.399)}{0.712}$ | $\underset{(0.053)}{3.738}$ | $\underset{(0.838)}{0.042}$ | $\underset{(0.749)}{0.102}$ | $\underset{(0.537)}{0.382}$ |

$D{Q}_{CC}$ | $\underset{\left(\mathbf{0}.\mathbf{015}\right)}{12.280}$ | $\underset{(0.663)}{2.398}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{18.210}$ | $\underset{(0.621)}{2.633}$ | $\underset{\left(\mathbf{0}.\mathbf{004}\right)}{15.467}$ | $\underset{(0.805)}{1.624}$ | $\underset{(0.265)}{5.220}$ | $\underset{\left(\mathbf{0}.\mathbf{011}\right)}{12.955}$ |

1% VaR | ||||||||

K = 1 | ||||||||

$D{Q}_{UC}$ | $\underset{\left(\mathbf{0}.\mathbf{002}\right)}{9.622}$ | $\underset{(0.053)}{3.744}$ | $\underset{(0.055)}{3.744}$ | $\underset{(0.446)}{0.580}$ | $\underset{\left(\mathbf{0}.\mathbf{007}\right)}{7.357}$ | $\underset{(0.122)}{2.393}$ | $\underset{(0.698)}{0.151}$ | $\underset{(0.122)}{2.393}$ |

$D{Q}_{IND}$ | $\underset{(0.398)}{0.713}$ | $\underset{(0.547)}{0.363}$ | $\underset{(0.557)}{0.363}$ | $\underset{(0.872)}{0.026}$ | $\underset{(0.447)}{0.579}$ | $\underset{(0.597)}{0.280}$ | $\underset{(0.744)}{0.107}$ | $\underset{(0.597)}{0.280}$ |

$D{Q}_{CC}$ | $\underset{\left(\mathbf{0}.\mathbf{007}\right)}{9.985}$ | $\underset{(0.136)}{3.991}$ | $\underset{(0.136)}{3.991}$ | $\underset{(0.741)}{0.598}$ | $\underset{(\mathbf{0}.\mathbf{021})}{7.680}$ | $\underset{(0.272)}{2.603}$ | $\underset{(0.881)}{0.254}$ | $\underset{(0.272)}{2.603}$ |

K = 3 | ||||||||

$D{Q}_{UC}$ | $\underset{\left(\mathbf{0}.\mathbf{003}\right)}{9.078}$ | $\underset{(0.066)}{3.376}$ | $\underset{(0.066)}{3.376}$ | $\underset{(0.541)}{0.373}$ | $\underset{\left(\mathbf{0}.\mathbf{009}\right)}{6.848}$ | $\underset{(0.146)}{2.118}$ | $\underset{(0.723)}{0.126}$ | $\underset{(0.146)}{2.118}$ |

$D{Q}_{IND}$ | $\underset{(0.910)}{0.013}$ | $\underset{(0.468)}{0.528}$ | $\underset{(0.468)}{0.528}$ | $\underset{(0.432)}{0.617}$ | $\underset{(0.751)}{0.100}$ | $\underset{(0.349)}{0.876}$ | $\underset{(0.106)}{2.605}$ | $\underset{(0.349)}{0.876}$ |

$D{Q}_{CC}$ | $\underset{\left(\mathbf{0}.\mathbf{008}\right)}{13.749}$ | $\underset{(\mathbf{0}.\mathbf{047})}{9.654}$ | $\underset{(\mathbf{0}.\mathbf{047})}{9.654}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{19.505}$ | $\underset{(\mathbf{0}.\mathbf{018})}{11.961}$ | $\underset{(0.057)}{9.171}$ | $\underset{\left(\mathbf{0}.\mathbf{028}\right)}{10.840}$ | $\underset{(0.057)}{9.171}$ |

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

5% VaR | ||||||

K = 1 | ||||||

$D{Q}_{UC}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{10.485}$ | $\underset{\left(\mathbf{0}.\mathbf{004}\right)}{8.226}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{14.416}$ | $\underset{(0.597)}{0.280}$ | $\underset{\left(\mathbf{0}.\mathbf{002}\right)}{9.319}$ | $\underset{(0.111)}{2.538}$ |

$D{Q}_{IND}$ | $\underset{(0.199)}{1.651}$ | $\underset{(0.267)}{1.232}$ | $\underset{(0.121)}{2.406}$ | $\underset{(0.993)}{0.000}$ | $\underset{(0.231)}{1.433}$ | $\underset{(0.610)}{0.261}$ |

$D{Q}_{CC}$ | $\underset{\left(\mathbf{0}.\mathbf{003}\right)}{11.463}$ | $\underset{\left(\mathbf{0}.\mathbf{011}\right)}{8.990}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{15.751}$ | $\underset{(0.869)}{0.280}$ | $\underset{\left(\mathbf{0}.\mathbf{006}\right)}{10.189}$ | $\underset{(0.256)}{2.727}$ |

K = 3 | ||||||

$D{Q}_{UC}$ | $\underset{\left(\mathbf{0}.\mathbf{005}\right)}{7.733}$ | $\underset{\left(\mathbf{0}.\mathbf{014}\right)}{6.096}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{10.767}$ | $\underset{(0.599)}{0.277}$ | $\underset{\left(\mathbf{0}.\mathbf{008}\right)}{7.028}$ | $\underset{(0.205)}{1.607}$ |

$D{Q}_{IND}$ | $\underset{(0.316)}{1.006}$ | $\underset{(0.339)}{0.914}$ | $\underset{(0.406)}{0.690}$ | $\underset{(0.374)}{0.789}$ | $\underset{(0.426)}{0.633}$ | $\underset{(0.076)}{3.148}$ |

$D{Q}_{CC}$ | $\underset{\left(\mathbf{0}.\mathbf{006}\right)}{14.578}$ | $\underset{\left(\mathbf{0}.\mathbf{019}\right)}{11.839}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{18.650}$ | $\underset{(0.616)}{2.662}$ | $\underset{\left(\mathbf{0}.\mathbf{015}\right)}{12.280}$ | $\underset{(0.056)}{9.201}$ |

1% VaR | ||||||

K = 1 | ||||||

$D{Q}_{UC}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{19.549}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{18.210}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{15.080}$ | $\underset{(0.698)}{0.151}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{13.358}$ | $\underset{\left(\mathbf{0}.\mathbf{002}\right)}{9.622}$ |

$D{Q}_{IND}$ | $\underset{(0.258)}{1.278}$ | $\underset{(0.280)}{1.167}$ | $\underset{(0.308)}{1.041}$ | $\underset{(0.744)}{0.107}$ | $\underset{(0.148)}{2.095}$ | $\underset{(0.398)}{0.713}$ |

$D{Q}_{CC}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{22.121}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{18.684}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{15.521}$ | $\underset{(0.881)}{0.254}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{16.575}$ | $\underset{\left(\mathbf{0}.\mathbf{007}\right)}{9.985}$ |

K = 3 | ||||||

$D{Q}_{UC}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{13.935}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{14.056}$ | $\underset{\left(\mathbf{0}.\mathbf{002}\right)}{9.885}$ | $\underset{(0.723)}{0.126}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{10.157}$ | $\underset{\left(\mathbf{0}.\mathbf{009}\right)}{6.809}$ |

$D{Q}_{IND}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{20.104}$ | $\underset{(0.094)}{6.375}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{15.843}$ | $\underset{(0.106)}{2.605}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{14.404}$ | $\underset{\left(0.255\right)}{4.056}$ |

$D{Q}_{CC}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{44.966}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{35.949}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{47.702}$ | $\underset{(0.065)}{8.842}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{33.192}$ | $\underset{(0.054)}{9.271}$ |

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

5% VaR | ||||||||

Univariate grade | 33% | 100% | 17% | 100% | 33% | 100% | 100% | 33% |

Multivariate grade | 33% | 33% | 33% | 100% | 33% | 100% | ||

1% VaR | ||||||||

Univariate grade | 33% | 83% | 83% | 83% | 33% | 100% | 83% | 100% |

Multivariate grade | 17% | 33% | 17% | 100% | 17% | 50% |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendixes

## A. Derivations

#### A.1. Transformation

#### A.2. Distributions Moments

**Skew-Normal**

**Skew-Student**

**Skew-MEP**

## B. Tables

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

It. | Observations | Days | Observations | Days |

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 |

## C. Figures

## D. The Univariate NCT-APARCH Model

## E. Backtesting VaR: Augmented Independence and Conditional Coverage Tests

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

5% VaR | ||||||||

IND (K = 3) | $\underset{(0.358)}{0.843}$ | $\underset{(0.699)}{0.1494}$ | $\underset{(0.130)}{2.294}$ | $\underset{(0.299)}{1.0807}$ | $\underset{(0.179)}{1.802}$ | $\underset{(0.924)}{0.0091}$ | $\underset{(0.670)}{0.182}$ | $\underset{(0.448)}{0.576}$ |

CC (K = 3) | $\underset{\left(\mathbf{0}.\mathbf{020}\right)}{7.779}$ | $\underset{(0.540)}{1.233}$ | $\underset{\left(\mathbf{0}.\mathbf{007}\right)}{10.062}$ | $\underset{(0.273)}{2.596}$ | $\underset{\left(\mathbf{0}.\mathbf{013}\right)}{8.739}$ | $\underset{(0.775)}{0.511}$ | $\underset{(0.181)}{3.419}$ | $\underset{\left(\mathbf{0}.\mathbf{015}\right)}{8.344}$ |

1% VaR | ||||||||

IND (K = 3) | $\underset{(0.955)}{0.003}$ | $\underset{(0.626)}{0.2371}$ | $\underset{(0.626)}{0.237}$ | $\underset{(0.082)}{3.0345}$ | $\underset{(0.844)}{0.038}$ | $\underset{(0.523)}{0.408}$ | $\underset{(0.256)}{1.288}$ | $\underset{(0.523)}{0.408}$ |

CC (K = 3) | $\underset{\left(\mathbf{0}.\mathbf{030}\right)}{7.030}$ | $\underset{(0.197)}{3.253}$ | $\underset{(0.197)}{3.253}$ | $\underset{(0.161)}{3.658}$ | $\underset{(0.061)}{5.579}$ | $\underset{(0.300)}{2.410}$ | $\underset{(0.488)}{1.436}$ | $\underset{(0.300)}{2.410}$ |

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

5% VaR | ||||||

IND (K = 3) | $\underset{(0.272)}{1.204}$ | $\underset{(0.293)}{1.105}$ | $\underset{(0.332)}{0.940}$ | $\underset{(0.285)}{1.142}$ | $\underset{(0.358)}{0.843}$ | $\underset{(0.063)}{3.453}$ |

CC (K = 3) | $\underset{\left(\mathbf{0}.\mathbf{011}\right)}{8.972}$ | $\underset{\left(\mathbf{0}.\mathbf{027}\right)}{7.251}$ | $\underset{\left(\mathbf{0}.\mathbf{003}\right)}{11.451}$ | $\underset{(0.488)}{1.434}$ | $\underset{\left(\mathbf{0}.\mathbf{020}\right)}{7.779}$ | $\underset{(0.070)}{5.327}$ |

1% VaR | ||||||

IND (K = 3) | $\underset{(0.012)}{6.349}$ | $\underset{(0.161)}{1.965}$ | $\underset{(0.029)}{4.765}$ | $\underset{(0.256)}{1.288}$ | $\underset{(0.027)}{4.896}$ | $\underset{(0.072)}{3.234}$ |

CC (K = 3) | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{20.608}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{14.237}$ | $\underset{\left(\mathbf{0}.\mathbf{001}\right)}{15.166}$ | $\underset{(0.488)}{1.436}$ | $\underset{\left(\mathbf{0}.\mathbf{000}\right)}{15.298}$ | $\underset{\left(0.058\right)}{5.671}$ |

## References

- R. Engle. “Autoregressive Conditional Heteroskedasticity with Estimates of United Kindgom Heteroskedasticity.” Econometrica 50 (1982): 987–1007. [Google Scholar] [CrossRef]
- T. Bollerslev. “Generalized Autoregressive Conditional Heteroskedasticity.” J. Econom. 31 (1986): 307–327. [Google Scholar] [CrossRef]
- T. Angelidis, A. Benos, and S. Degiannakis. “The Use of GARCH Models in VaR Estimation.” Stat. Methodol. 1 (2004): 105–128. [Google Scholar] [CrossRef]
- D.B. Nelson. “Conditional heteroskedasticity in asset returns: A new approach.” Econometrica 59 (1991): 347–370. [Google Scholar] [CrossRef]
- S. Mittnik, and M. Paolella. “Conditional density and value-at-risk prediction of Asian currency exchange rates.” J. Forecast. 19 (2000): 313–333. [Google Scholar] [CrossRef]
- Z. Ding, C.W. Granger, and R.F. Engle. “A long memory property of stock market returns and a new model.” J. Empir. Financ. 1 (1993): 83–106. [Google Scholar] [CrossRef]
- K. Kuester, S. Mittnik, and M. Paolella. “Value-at-risk prediction: A comparison of alternative strategies.” J. Financ. Econom. 4 (2006): 53–89. [Google Scholar] [CrossRef]
- F. Longin, and B. Solnik. “Is the correlation in international equity returns constant: 1960–1990? ” J. Int. Money Financ. 14 (1995): 3–26. [Google Scholar] [CrossRef]
- C. Brooks, and G. Persand. “Value-at-risk and market crashes.” J. Risk 2 (2000): 5–26. [Google Scholar]
- L. Bauwens, S. Laurent, and J. Rombouts. “Multivariate GARCH models: A Survey.” J. Appl. Econom. 21 (2006): 79–109. [Google Scholar] [CrossRef]
- P. Giot, and S. Laurent. “Value-at-risk for long and short trading positions.” J. Appl. Econom. 18 (2003): 641–663. [Google Scholar] [CrossRef]
- A.A. Santos, F.J. Nogales, and E. Ruiz. “Comparing univariate and multivariate models to forecast portfolio value-at-risk.” J. Financ. Econom. 11 (2013): 400–441. [Google Scholar] [CrossRef]
- L. Bauwens, and S. Laurent. “A New Class of Multivariate Skew Densities, with Application to GARCH Models.” J. Bus. Econ. Stat. 23 (2005): 346–354. [Google Scholar] [CrossRef]
- C. Ley, and D. Paindaveine. “Multivariate skewing mechanisms: A unified perspective based on the transformation approach.” Stat. Probab. Lett. 80 (2010): 1685–1694. [Google Scholar] [CrossRef]
- D. Noureldin, N. Shephard, and K. Sheppard. “Multivariate rotated ARCH models.” J. Econom. 179 (2014): 16–30. [Google Scholar] [CrossRef]
- J. Krause, and M.S. Paolella. “A fast, accurate method for value-at-risk and expected shortfall.” Econometrics 2 (2014): 98–122. [Google Scholar] [CrossRef][Green Version]
- P.F. Christoffersen. “Evaluating interval forecasts.” Int. Econ. Rev. 39 (1998): 841–862. [Google Scholar] [CrossRef]
- E. Dumitrescu, C. Hurlin, and V. Pham. “Backtesting Value-at-Risk: From Dynamic Quantile to Dynamic Binary Tests.” Finance 33 (1995): 79–111. [Google Scholar]
- P.H. Kupiec. “Techniques for verifying the accuracy of risk measurement models.” J. Deriv. 3 (1995): 73–84. [Google Scholar] [CrossRef]
- R.F. Engle, and S. Manganelli. “CAViaR: Conditional autoregressive value at risk by regression quantiles.” J. Bus. Econ. Stat. 22 (2004): 367–381. [Google Scholar] [CrossRef]
- V. DeMiguel, L. Garlappi, and R. Uppal. “Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? ” Rev. Financ. Stud. 22 (2009): 1915–1953. [Google Scholar] [CrossRef]
- J. Tu, and G. Zhou. “Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies.” J. Financ. Econ. 99 (2011): 204–215. [Google Scholar] [CrossRef][Green Version]
- S.J. Brown, I. Hwang, and F. In. “Why Optimal Diversification Cannot Outperform Naive Diversification: Evidence from Tail Risk Exposure.” , 2013. [Google Scholar] [CrossRef]
- C. Fugazza, M. Guidolin, and G. Nicodano. “Equally Weighted vs. Long-Run Optimal Portfolios.” Eur. Financ. Manag. 21 (2014): 742–789. [Google Scholar] [CrossRef]
- L. Bauwens, W. Omrane, and E. Rengifo. Intra-Daily FX Optimal Portfolio Allocation. CORE Discussion Paper 2006/10; Louvain-la-Neuve, Belgium: CORE, 2006. [Google Scholar]
- R.F. Engle, and K.F. Kroner. “Multivariate simultaneous generalized ARCH.” Econometr. Theory 11 (1995): 122–150. [Google Scholar] [CrossRef]
- O. Barndorff-Nielsen. “Exponentially decreasing distributions for the logarithm of particle size.” Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 353 (1977): 401–419. [Google Scholar] [CrossRef]
- M.S. Paolella, and P. Polak. “COMFORT: A Common Market Factor Non-Gaussian Returns Model.” J. Econom. 187 (2015): 593–605. [Google Scholar] [CrossRef]
- C. Fernández, and M.F. Steel. “On Bayesian modeling of fat tails and skewness.” J. Am. Stat. Assoc. 93 (1998): 359–371. [Google Scholar]
- T. Bollerslev, and J. Wooldridge. “Quasi-Maximum Likelihood Estimation and Inference in Dynamic Models with Time-Varying Covariances.” Econom. Rev. 1 (1992): 143–172. [Google Scholar] [CrossRef]
- K. Fang, S. Kotz, and K. Ng. Symmetric Multivariate and Related Distributions. London, UK: Chapman and Hall, 1990. [Google Scholar]
- N. Solaro. “Random variate generation from multivariate exponential power distribution.” Stat. Appl. II 2 (2004): 25–44. [Google Scholar]
- D. Noureldin, N. Shephard, and K. Sheppard. “Multivariate high-frequency-based volatility (HEAVY) models.” J. Appl. Econom. 27 (2012): 907–933. [Google Scholar] [CrossRef]
- P. Christoffersen. “Evaluating Interval Forecasts.” Int. Eco. Rev. 39 (1998): 841–862. [Google Scholar] [CrossRef]
- P. Kupiec. “Techniques for Verifying the Accuracy of Risk Management Models.” J. Deriv. 23 (1995): 73–84. [Google Scholar] [CrossRef]
- S. Campbell. “A Review of Backtesting and Backtesting Procedures.” J. Risk 9 (2006): 1–17. [Google Scholar]
- R. Engle, and S. Manganelli. “CaViaR: Conditional Autoregressive Value at Risk by Regression Quantiles.” J. Bus. Econ. Stat. 22 (2004): 367–381. [Google Scholar] [CrossRef]
- P. Christofferson, and D. Pelletier. “Backtesting Value-at-Risk: A Duration-Based Approach.” J. Financ. Econom. 2 (2004): 84–108. [Google Scholar] [CrossRef]
- T. Pajhede. Backtesting Value-at-Risk: The generalized Markov Test. U. Copenhagen Economics Discussion Paper 15-18; Copenhagen, Denmark: University of Copenhagen, 2015. [Google Scholar]

^{1.}The interested reader is referred to Ley and Paindaveine [14] who provide a detailed overview of this approach.^{2.}We thank the authors for kindly providing us their MATLAB codes.^{3.}In Appendix D we provide a brief overwiew of the noncentral t distribution and its likelihood derivation.^{4.}In the univariate case it corresponds to $\mathit{\psi}={({a}_{1},{b}_{1})}^{\prime}$.^{5.}Downloaded from`http://realized.oxford-man.ox.ac.uk/data/download`.^{6.}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 Section 3.2. As results did not lead to significant improvements, this issue is briefly covered in Appendix E.

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**MDPI and ACS Style**

Braione, M.; Scholtes, N.K.
Forecasting Value-at-Risk under Different Distributional Assumptions. *Econometrics* **2016**, *4*, 3.
https://doi.org/10.3390/econometrics4010003

**AMA Style**

Braione M, Scholtes NK.
Forecasting Value-at-Risk under Different Distributional Assumptions. *Econometrics*. 2016; 4(1):3.
https://doi.org/10.3390/econometrics4010003

**Chicago/Turabian Style**

Braione, Manuela, and Nicolas K. Scholtes.
2016. "Forecasting Value-at-Risk under Different Distributional Assumptions" *Econometrics* 4, no. 1: 3.
https://doi.org/10.3390/econometrics4010003