forecasting var models under different volatility processes and

Forecasting VaR models under di¤erent volatilityprocesses and distributions of return innovations

Yiannis Dendramis�, Giles E Spunginyand Elias Tzavalisz

November 2012

Abstract

This paper provides clear cut evidence that the out-of-sample VaR forecasting perfor-mance of alternative parametric volatility models, like EGARCH or GARCH models, canbe considerably improved if they are combined with skewed distributions of return innova-tions. The performance of these models is found to be similar to that of the EVT (extremevalue theory) approach and it is better than that of their extensions allowing for Markovregime-switching e¤ects with, or without , EGARCH e¤ects. The paper �nds that theperformance of the last approach can be also signi�cantly improved if it relies on �lteredresiduals obtained through volatility models which allow for skewed distributions of returninnovations.

.

JEL classi�cation: C10, C22, G10

Keywords: Risk measures; Value at Risk; GARCH, EGARCH and Regime-switchingmodels; Extreme value theory; Skewed distributions

The authors would like to thank Richard Baillie, George Kapetanios and Martin Solafor useful comments on early version of the paper.

�Department of Economics, Athens University of Economics & Business, Patission Str 76, Athens 104 34,E-mail: [email protected]

ySchool of Economics and Finance, Queen Mary, Univeristy of London, Mile End Road, London, E1 4NS,United Kingdom. E-Mail: [email protected]

zDepartment of Economics, Athens University of Economics and Business, Patission Str 76, Athens 104 34,Greece. E-mail: [email protected]

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1 Introduction

Value-at-Risk (or VaR for short) is a risk management concept developed in the area of portfoliorisk management which summarizes the downside risk in a single statistic (see, e.g., Christof-fersen (2003)). It is de�ned as the minimum amount of money that one could expect to lose froman investment position with a given probability over a speci�c period of time. More speci�cally,conditional on the market information set at time t, denoted as It, the VaR loss predicting valueof one-unit investment with one-period horizon t+1 is de�ned as the negative a-quantile of theconditional return distribution, i.e.

V aRt+1 (a) � �Qa (rt+1jIt) = � infxfx 2 R : Pr (rt+1 � xjIt) � ag ; 0 < a < 1;

where Qa (�) denotes the quantile function at a value (level) a, rt is the continuously compoundedrate of return for one period, de�ned as rt+1 = ln (st+1=st), and st is the asset price at time t.In other words, VaR is simply a speci�c quantile of a portfolio�s potential loss distribution overa given holding period.To obtain accurate VaR forecasts, one needs to capture well-known characteristics of asset

return distributions, such as leptokurtosis (thick tails), negative skewness, volatility cluster-ing or leverage e¤ects (see, e.g., Brooks and Persand (2003), Poon and Granger (2003), orElliott and Timmermann (2008), for a survey). In order to capture some of these character-istics, one stream of research employs parametric volatility models like the EGARCH modelof Nelson (1991), which allows for leverage e¤ects and can generate negatively skewed returndistributions,1 or the Markov regime-switching volatility model (MRS), suggested by Hamilton(1989).2 The latter model assumes that �nancial markets switch between two volatility regimes:a low and high, re�ecting bull and bear market conditions, respectively. By mixing the nor-mal distributions of asset returns corresponding to the above volatility regimes, these modelscan approximate di¤erent return distributions across a variety of asset classes, such as stocks,bonds, foreign currencies, etc. To account for negatively skewed and leptokurtic asset return dis-tributions, another stream of research combines well-known parametric volatility models, suchas the GARCH or EGARCH model with asymmetric or/and leptokurtic distributions of theasset return innovation process, like the skewed t-student distribution or the generalized errordistribution (GED).3

This paper contributes to the literature on VaR forecasting by comparing the relative per-formance of the above two di¤erent streams of research, by employing data across di¤erentasset markets. To this end, the paper introduces the Skewed-GED as a distribution of returninnovations. This is an asymmetric extension of the GED, with well-de�ned moments. To allowfor skewness, the GED is incorporated in the general framework developed by Fernandez andSteel (1998). Second, the paper extends the MRS model to allow for EGARCH e¤ects, whichcan capture leverage in stock markets. Third, the VaR forecasting performance of the abovetwo di¤erent streams of research is compared to that of the extreme value theory (EVT), whichis often used in the literature as benchmark methodology (see, e.g., Diebold et al (1997), Mc-Neil and Frey (2000), Bekiros and Georgoutsos (2005). The latter method employs parametricdensity functions, such as the generalized Pareto distribution (GPD), to model the tails of asset

1See, e.g., Karanasos and Kim (2003), Chan and Gray (2006), and Anyfantaki and Demos (2012). TheEGARCH model is found to provide better predictions of volatility like the so-called GJR asymmetric GARCHmodel of Glosten et al (1993) and the asymmetric power ARCH model of Ding et al. (1993). See, e.g., Alberget al (2008).

2See, e.g., Cai (1994), Hamilton and Susmel (1994), McLachlan and Peel (2000), Billio and Pelizzon (2000),Marcucci (2005), Samuel (2008), and Wil�ing (2009).

3See, e.g., Hansen (1994), Lambert and Laurent (2002), Angelidis et al. (2004), Kuester et al (2006), Bali etal (2008).

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returns distributions. In our analysis, the EVT method is applied to �ltered residuals based ondi¤erent distribution assumptions of return innovations and volatility models.The results of the paper lead to a number of interesting conclusions, which provide impor-

tant implications for out-of-sample VaR forecasting. First, they clearly indicate that combin-ing volatility models with asymmetric distributions of return innovations like the skewed-t, orskewed-GED, considerably improves their VaR forecast performance. Second, the EGRARCH orGARCH models combined with either of the above skewed distributions are found to outperformthe VaR forecasting performance of more complex volatility models like the MRS model with, orwithout, EGARCH e¤ects. This can be attributed to their more parsimonious parameter struc-ture. Third, we �nd that the EVT method for VaR forecasting can be signi�cantly improvedif it relies on �ltered residuals obtained through volatility models and skewed distributions ofreturn innovations. The above results are robust across di¤erent asset markets examined by thepaper, namely the US and UK stock markets, the US bond market and the FOREX marketbetween the British Sterling and the US dollar.The paper is organized as follows. Section 2 presents the alternative parametric volatility

models for VaR forecasting employed by the paper and it presents the skewed versions of theGED and t-student distributions of return innovations. It also presents the EVT method forVaR forecasting based on standardized (�ltered) residuals of asset returns. In Section 3, weconduct our empirical analysis, evaluating the out-of-sample VaR forecasting performance ofthe alternative volatility models and returns�distributions considered. The paper is concludedwith Section 4.

2 Alternative parametric volatility models and distribu-tions of return innovations

2.1 Volatility models

Consider the following model of one-period asset log-return rt+1:

rt+1 = Et(rt+1) + "t+1; with "t+1 =q�2t+1zt+1, zt+1 � IID(0; 1) (1)

where Et(rt+1) is the mean of rt+1 conditional on the current information set of the marketIt = frt; rt�1; :::g, often modelled as an AR(1) model (see, e.g., Giot and Laurent (2004)), �2t+1is the conditional variance of rt+1 and zt+1 is an IID(0; 1) innovation process. The function of�2t+1 is often modelled as a GARCH(q; p) process, i.e.

�2t+1 = �0 +

qXj=1

�j"2t+1�j +

pXj=1

j�2t+1�j . (2)

This process can allow for excess kurtosis in the unconditional distribution of asset return rt+1.On the other hand, the EGARCH(q; p) process of �2t+1, given as

�2t+1 = exp

24�0 + qXj=1

��;jzt+1�j + �j (jzt�j+1j � E jzt�j+1j)

�+

pXj=1

j ln��2t+1�j

�35 ; (3)

can also allow for negative skewness of the distribution of rt+1, by incorporating leverage ef-fects of innovations zt+1�j on variance function �2t+1. Skewness and excess kurtosis of returndistributions can be also captured by the Markov regime switching (MRS) process

3

�2t+1 =KXi=1

pi;t+1�2i , (4)

suggested by Hamilton (1989), where �2s = V art(rt+1jSt+1 = i) is the conditional variance whichcorresponds to the i-th regime of the market, and pi;t+1 = Pr (St+1 = ijIt), with

PKi=1 pi;t+1 = 1,

denotes the probability of being in regime St+1 = i, at time t + 1, conditional on informationset It. The last probability is also known as forecasted probability, as it can be �ltered fromthe data based on It. The MRS model, described by processes (1)-(4), allows for both skewnessand kurtosis of return distributions by mixing di¤erent distributions of the error terms "t�j+1at each market regime St+1 = i , for i = f1; 2; :::;Kg. In practice, we often assume two regimesof the market re�ecting the bull and bear conditions. The transition probabilities across theK�di¤erent market regimes considered by volatility function (4) are described by the followingMarkov Chain process, with (K �K) transition probability matrix:

P =

26664p11 p12 : : : p1Kp21 p22 : : :

� � �... : : :

pK1 pK2 : : : pKK

37775 , with pis = 1� psi,

where pis = Pr(St+1 = ijSt = s), for i; s 2 f1; 2; :::;Kg, denotes the transition probability fromregime i to regime s.As it stands, the MRS process of volatility function (4) does not allow for leverage e¤ects,

as it assumes that processes St+1 and zt+1 are uncorrelated. To accommodate these e¤ects infunction (4), we can assume that this function is time-varying in each regime i, �2i;t+1, i.e.

�2t+1 =kXi=1

pi;t+1�2i;t+1

and it follows EGARCH(q; p) process:

�2i;t+1 = exp

24�i;0 + qXj=1

��i;jzt�j+1 + �i;j (jzt�j+1j � E jzt�j+1j)

�+

pXj=1

i;j ln��2t�j+1

�35 , (5)

where �2t�j+1 =PK=2

i=1 pi;t�j+1�2i;t�j+1 is the value of the variance at time t � j + 1, for j =

1; 2; :::; p, conditional on the information set It�j+1 and zt�j+1 = "t�j+1=�t+1 is a NIID(0; 1)innovation process. In (5), �2t�j+1 constitutes the expected variance at time t � j + 1 which isnot a function of unobserved regime process St, and thus it is not path dependent. It can beeasily estimated based on the maximum likelihood method (see Gray (1996)).The implementation of the MRS model with, or without, EGARCH e¤ects to calculate the

VaR forecast at quantile level � of one unit of investment with return rt+1 requires a criticalvalue of the distribution of rt+1 which corresponds to this quantile level. This value can benumerically obtained from the following formula:

V aRt+1(a)Z�1

f (rt+1jIt) dr = a, (6)

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where f (rt+1jIt) denotes the conditional density function of rt+1. The analytic form of thisdensity is given as

f (rt+1jIt) =KXi=1

f (rt+1jHi;t) Pr (St+1 = ijIt) ; (7)

where Hi;t is the union of set It with regime i at time t + 1, i.e. Hi;t = It [ fSt+1 = ig, andf (rt+1jHi;t) is regime-i dependent density function. The i-th regime forecasted probabilitiesPr (St+1 = ijIt) entering density function (7) can be obtained using Hamilton�s (1989, 1994)�ltering algorithm.

2.2 Alternative distributions of return innovations zt+1In order to accommodate a wide range of skewness and kurtosis in return distributions, theparametric volatility models presented in the previous section can be combined with skewedand leptokurtic distributions of return innovations zt+1.4 The SK-t distribution employed byHansen (1994) for zt+1 is de�ned as

f(zt+1; �; �) =

8>>><>>>:bc

�1 + 1

v�2

�bzt+1+a1��

�2��(v+1=2)if zt+1 < �a=b

bc

�1 + 1

v�2

�bzt+1+a1+�

�2��(v+1=2)if zt+1 > �a=b

(8)

where 2 < � <1 , �1 < � < 1 and a; b and c are de�ned as

a = 4�c�v � 2�� 1

�; b2 = 1 + 3�2 � a2 and c =

��v+12

��v2

�p�(v � 2)

.

The parameters v and � re�ect the degrees of freedom (determining the tailness/thickness) andthe asymmetry of the SK-t distribution, respectively. This distribution reduces to the STU(standard t-student) distribution for � = 0. The distribution of zt+1 is skewed to the right, if� > 0, and to the left, if � < 0. The cumulative distribution function of the above pdf and itsquantile function required in VaR calculations is given analytically by Jondeau and Rockinger(2003).5

An alternative distribution of return innovations which can capture skewness and kurtosiscan be based on the GED. The latter has all its moments well de�ned. This distribution can beincorporated in the general framework developed by Fernandez and Steel (1998) to allow also forskewness. According to this framework, any continuous unimodal and symmetric distributioncan become asymmetric by changing its scale at each side of its mode. Let g (zt+1; v0) denotethe GED, where v0 is its tail parameter. For v0 = 2; g (zt; v0) reduces to the standardized normaldistribution N (0; 1). Introducing g (zt+1; v0) in the framework of Fernandez and Steel will givethe following skewed GED:

f�zt+1; v

0; �0�=

2

�0 + 1�0

ng�zt+1�0; v0�I(�1;0) (zt+1) + g

��0zt+1; v

0� I[0;1) (zt+1)o, (9)

4See, e.g., Hansen (1994), Harvey and Siddique (1999), Brannas and Nordman (2003), Harris et al. (2004),and Chiang and Li (2007).

5See Jondeau and Rockinger (2003), or Jondeau et al (2007).

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denoted as SK-GED, where I(�1;0) (zt+1) is an indicator function which equals 1 if zt+1 2(�1; 0), and zero otherwise. �0 is a parameter which determines the skewness of densityf�zt+1; v

0; �0�. The latter parameter controls the allocation of mass to each side of the mode of

f�zt+1; v

0; �0�. If �0 < 1, then f

�zt+1; v

0; �0�is negatively skewed while, if �0 > 1, f

�zt+1; v

0; �0�

becomes positively skewed. For �0 = 1, f�zt+1; v

0; �0�reduces to the GED. The cumulative

distribution of SK-GED and its quantile function required for VaR calculations are given asfollows:

F�zt+1; v

0; �0�=

� 21+�02

G��0zt+1; v

0� if zt+1 < 01� 2

1+�0�2G�� zt+1

�0 ; v0� if zt+1 � 0

F�1�p; v0; �0

�=

(1�0G

�1 �p2

�1 + �02

�; v0�

if p < 11+�02

��0G�1�1�p2

�1 + �0�1

�; v0�

if p � 11+�02

where, G (:; v0) and G�` (:; v0) are the cumulative distribution and quantile function of GED,respectively.

2.3 VaR forecasts based on the EVT method

For VaR forecasting, the EVT method does not consider the entire distribution of return in-novations zt+1, but instead is focused only on the left tail of this distribution. According tothis method, we rely on an unknown distribution of excess values of stock return rt+1 abovesome su¢ ciently large threshold value u. This distribution will be assumed to follow the twoparameter general Pareto distribution (GPD), given as

G(rt+1) =

8<: 1��1 + �rt+1

�

�� 1�

� 6= 0

1� exp�� rt+1

�

�� = 0

,

where � and � represent the shape (tail index) and scaling parameters of this distribution,respectively. A tail estimator of G(rt+1) for a threshold value u, denoted as Gu(rt+1), can beobtained from maximum likelihood estimates of its parameters � and �, denoted as b� and b�, i.e.

Gu(rt+1) = 1�TuT

1 +

b� (rt+1 � u)b�!� 1b�

; (10)

where Tu denotes the number of data points out of a total number of sample observations Tthat exceed threshold value u. The a quantile level of the above distribution function Gu(rt+1)is given as

G�1u (a) = u+b�b�"�

T

Tu(1� a)

��b�� 1#: (11)

The choice of the threshold u is the most important implementation issue of the EVT method-ology and still an area of active research. Selection of a threshold that is too low will givebiased parameter estimates, but a threshold that is too high will result in a large variance ofthe parameter estimates (see McNeil and Frey (2000), for discussion). In practice, to choosethreshold parameter u most of the studies rely on Akaike information criterion.The EVT method can be applied to the standardized residuals of return process, referred to

as �ltered residuals. As aptly noted by Diebold et al. (1998), this can considerably improve the

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performance of the method resulting in accurate and e¢ cient estimates of VaR. To this end,we have a choice to employ any of the parametric volatility models presented Subsection 2.1combined with a skewed and leptokurtic distribution of innovation process zt+1, presented inSubsection 2.2. In the �rst stage, this �ltering procedure requires that we �t a volatility modeland a distribution of zt+1 into asset return series rt+1 in order to obtain their �tted values,denoted as brt+1, and their volatility estimates, denoted as b�2t+1. These will be then used toobtain the standardized residuals of rt+1, de�ned as zt+1 = (rt+1 � brt+1) =qb�2t+1. Assumingthat the parametric volatility model and the distribution of zt+1 chosen are capable of explainingthe historical data well, residuals zt+1 can be seen as representing unobserved independent noisevariables, for all t. In the second stage, zt+1 will be employed to �t the GPD distribution, givenby (10). This can be done after choosing a threshold value u and having selected an appropriatequantile level of residuals zt+1 distribution, a. Given u and a, we can then identify the values ofzt+1 exceeding u and, hence, determine the number of violations zt+1 < u. This set of residualswill be employed to �t the GPD distribution and to estimate its scale and shape parameters� and �, respectively. The �tted distribution will be employed together with the a quantilelevel of Gu(rt+1), given by equation (11), to obtain a VaR forecast based on the above EVTmethodology as follows:

[V aRt+1 = ��Et(rt+1jt) + G

�1u (a)

q�2t+1

�: (12)

3 Empirical Analysis

In this section, we assess the out-of-sample VaR forecasting performance of the alternativeparametric volatility models presented in the previous section, considering a skewed t-student(SK-t) distribution and a skewed GED (SK-GED) of return innovations zt+1. We also assessthe performance of the EVT approach relying on any of these volatility models and/or returninnovation distributions, which are used to obtain the �ltered residuals. In addition to the abovevolatility models, we examine the out-of-sample performance of the exponentially weighted mov-ing average (EWMA) volatility model often used in practice to calculate VaR by RiskMetrics.This model can be thought of as a special case of the GARCH model (see, e.g., Ding and Meade(2010)).

3.1 Testing methodology

Our forecasting exercise entails comparison of forecasted VaR values with the realized daily re-turn series at di¤erent quantile levels a of the conditional distribution of asset return f(rt+1jIt).Using a set-up that is now common in the VaR literature, we evaluate the VaR forecastingperformance of the alternative volatility models considered by computing their respective vio-lation rate for the left (portfolio loss) tail of the distribution of asset returns. This rate is thenumber of instances (denoted as N) at which realized returns exceed the forecasted VaR valuein absolute terms at a predetermined level a, for a given sample of T observations. If the VaRmodel is correctly speci�ed, then this failure rate must be equal to the required coverage levela. This hypothesis can be tested based on the unconditional coverage (UC) test suggested byChristo¤ersen (1998) (see also Kupiec (1995)). To this end, the following likelihood ratio (LR)

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test statistic can be employed

LRUC = �2 lnh(1� a)T�N aN

i+ 2 ln

"�1� N

T

�T�N �N

T

�N#,

where NT is the sample violation rate. Under the null hypothesis thatNT equals the true violation

rate, this LR test statistic is asymptotically distributed as a �2(1) random variable, with onedegrees of freedom denoted in parenthesis.The UC test presented above can reject the above hypothesis if a VaR model produces

too many or too few violation events. Because this test ignores the conditional coverage (CC)at any point in time, it is often complemented with a test of conditional coverage testing ifviolation rates are serially uncorrelated (or independent). To this end, we employ the dynamicquantile (DQ) test suggested by Engle and Manganelli (2004). This is a more powerful testthan the conditional coverage (CC) test suggested by Christo¤ersen (1998), designed to testfor �rst-order dependence of the VaR violation rates and ignoring possible serial correlatione¤ects in VaR forecasts. The latter can capture a systematic behavior in violation rates andits implementation does not require a large number of violations. In particular, whereas CCcan test for serial correlation of VaR violations of lag order one, DQ statistic can test for serialcorrelation of higher order. In addition to serial correlation, DQ statistic can test if the sampleviolation rate equals the true one. To specify DQ test, de�ne the following hit variable:

Hitt+1 (a) = I (rt+1 < V aRt+1 (a))� a;

where a is a target probability level, I (rt+1 < V aRt+1 (a)) is an indicator function taking thevalue of 1 when rt+1 < V aRt+1 (a) and 0 otherwise. Similarly to UC and CC testing procedures,this variable captures the occurrence of threshold violation, or alternatively, an instance where aVaR estimate exceeds realized return. If the VaR model is correctly speci�ed, then the followingtwo hypotheses should simultaneously hold:

H1 : E [Hitt+1 (a)] = 0 and H2 : Hitt+1 (a) uncorrelated with It:

DQ test can examine jointly the validity of the above two assumptions based on the followingarti�cial regression

Hit = X�+ e;

where Hit is a (T � 1) vector of the values of variable Hitt+1 (p), X is a (T � K) matrix ofthe values of the explanatory variables which, in addition to lagged values of Hitt+1 (a) (i.e.Hitt (a) ;Hitt�1 (a) :::; Hitt�q (a)), includes the contemporaneous values of VaR, V aRt+1(a),and � is the coe¢ cients vector of these explanatory variables, including the intercept. In orderfor the hypotheses H1 and H2 to hold, the regressors in the arti�cial regression should have noexplanatory power. That is, the following null hypothesis H0: � = 0 must hold. The validityof this hypothesis can be tested by employing the following dynamic quantile test statistic

DQ =b�0X0Xb�a (1� a) ;

where b� is the least square (LS) estimator of �, which is distributed as �2K , with K degrees offreedom.

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3.2 Data

The data set used in our empirical analysis comprises daily log-returns of di¤erent asset mar-kets, namely the US and UK stock markets, the US bond market and the foreign exchange rate(FOREX) market between British pound (£ ) and US dollar ($). The log-return of these marketsrt+1 are calculated as the one-period log-di¤erences of the following indices: SP500 for the US,FTSE100 for the UK and the US corporate bond index calculated by Merrill Lynch. For theFOREX market, it is calculated as the log-di¤erence of the exchange rate de�ned as currencyunits of British pounds (£ ) per unit of US dollar ($). The above return series cover the fol-lowing date (dd/mm/yy) periods: 03/01/1995-03/02/2011 for SP500, 03/01/92-03/02/2011 forFTSE100, 02/01/1995-03/02/2011 for the US bond index and 02/01/1995-03/02/2011 for theforeign exchange. During these periods, �nancial markets witnessed several crises including thecollapse of the ".com" bubble that began in spring 2000, the collapse of Enron and WorldComcompanies in 2001, 25% annual loss of index value amid fears of war on Iraq, tensions in NorthKorea, the economic stagnation in 2003, as well as the recent �nancial crisis, triggered by thecollapse of Lehman Brothers in September 2008. These events can explain the high degree ofnegative skewness or excess kurtosis of return distributions of the above asset markets, as shownin Table 1. Note that the only market that does not exhibit a substantial degree of negativeskewness is the FOREX market between British pound and US dollar.

[Insert Table 1]

3.3 Parameter estimates

Our empirical analysis starts by presenting in Tables 2A, 2B, 2C and 2D maximum likelihood(ML) estimates of the parameters of the GARCH and EGARCH models with p = 1 and q = 1lags, the MRS and MRS-EGARCH(1,1) models with innovation process distributed as zt+1 �SK-t. In tables 3A-3C, we present parameter estimates of these models under the assumptionthat zt+1 � SK-GED.6 ;7 Finally, Table 4 presents estimates of the parameters of the generalizedPareto distribution (GPD) employed by the EVT approach. These are based on estimates ofzt+1 (�ltered residuals of) obtained through the above alternative volatility models and returninnovation distributions. Values of the log-likelihood function, denoted as LL, are given at thebottom of the tables, while standard errors are in parentheses.

[Insert Tables 2A-2D]

A number of interesting conclusions can be drawn from the results of the above tables.First, all the alternative volatility models �tted into the data provide signi�cant estimates ofthe parameters capturing the skewness and tail thickness (kurtosis) of innovation process zt+1,namely � and � for the SK-t distribution and �0 and v0 for the SK-GED. The only exception isthe FOREX market between British pound and US dollar. For this market, both the estimates

6Note that, for space considerations, we do not present parameter estimates of the GARCH, EGARCH andMRS models in case that zt+1 � NIID(0; 1):These models are not found to �t better into the data than theirspeci�cations assuming zt+1 � SK-t or zt+1 � SK-GED. See, e.g., Harris et al (2004). Also, note that we donot present results for the MRS model with GARCH e¤ects, as we found that this model performs similarly tothe MRS-EGARCH model.

7 In the estimation of the MRS-EGARCH model, coe¢ cients �1 and �1, capturing the leverage and magnitudee¤ects of zt on volatility function �2t+1, respectively, are set to be the same across the two market regimesi = f1; 2g.

9

of skewness coe¢ cients � and �0 cannot reject the hypothesis of a symmetric distribution, i.e.� = 0 and �0 = 1.8

Second, the standard EGARCH model, with zt+1 � NIID(0; 1), which implies a skeweddistribution of returns rt+1 due to leverage e¤ects, cannot capture the degree of skewness ap-peared in the data for all di¤erent markets considered. This is also true for the MRS modelor its extension allowing EGARCH e¤ects. Our results clearly indicate that the estimates ofthe skewness parameters of the SK-t and SK-GED distributions are signi�cant and large forboth of the above volatility models. These results support the view that leverage e¤ects orregime-switching in volatility models alone are not su¢ cient to capture the degrees of skewnessin asset return distributions.

[Insert Tables 3A-3C]

Third, regarding the parameter estimates of the volatility models, the results of the tablesindicate that both the GARCH and EGARCH models �nd a close to unity estimate of thevolatility persistency coe¢ cient 1, which indicates a very high persistency of shocks in volatil-ity �2t+1. This is true for all the markets examined and across the two skewed distributionsconsidered. For the EGARCH parameters, the results of Tables 2B and 3B indicate strong"sign" or "magnitude" e¤ects of asset return innovations zt on volatility function �2t+1, as wasexpected. Speci�cally, the negative value of the slope coe¢ cient �1 indicates the presence ofstrong leverage e¤ects of zt on �2t+1. However, these e¤ects are not found to be signi�cant for thebond and FOREX markets. On the other hand, the "magnitude" e¤ects of return innovationszt on �2t+1, captured through coe¢ cient �1, are found to be strong and signi�cant for all assetmarkets examined.Fourth, the estimates of the two speci�cations of the MRS model considered, i.e. without and

with EGARCH e¤ects (see results of Tables 2C-2D and Table 3C), con�rm evidence providedin the literature (see introduction) that �nancial markets can be characterized by two di¤erentvolatility regimes: the bull and bear. The �rst, denoted by "1", is the low volatility regime, whilethe second, denoted by "2", is the high volatility counterpart. These regimes are very persistent,given that the estimates of the transition probabilities between the two regimes pij are very low.The estimates of pij and those of the other parameters of the MRS models are found to hardlychange across the two skewed distributions of the innovation process zt+1 considered. As wasexpected, the estimates of slope coe¢ cients �1 and �1 of the MRS-EGARCH model, capturingthe "size" and "magnitude" e¤ects of innovation process zt on volatility functions �2(i);t+1 ofeach regime i; are analogous to those of the standard EGARCH model, which does not allowfor regime switching (see Tables 2A and 3A). This is also true for the persistency coe¢ cientof variance �2i;t, at each regime i, (i);1. This coe¢ cient is found to be close to unity, for all�nancial markets examined and across the two regimes "1" and "2". The bigger than unityestimate of (i);2 for the FOREX market in regime "2" does not imply an explosive volatilityfunction.9

Finally, the results of the parameter estimates of the generalized Pareto distribution (GPD)assumed for innovation process zt+1 under the the EVT method, reported in Table 4, indicatethat the estimates of �, which is the tail index of the GPD, are not di¤erent than zero. Azero value of � corresponds to distributions of zt+1 whose tails decay exponentially, which is

8Recall that the estimates of the skewness parameters of innovation process zt+1 for SK-t distribution and theSK-GED, namely � and �0, can not be compared to each other. One can derive � from �0 through relationship?????

9Note that evidence of non-stationary (or explosive) volatility in a regime i of the MRS-EGARCH model, fori = f1; 2g, should not be interpreted as evidence that this model is globally non-stationary. The global MRSmodel stationarity is ensured by a non-zero transition probability between the two regimes (see, e.g., Holst et al(1994) or Bauwens et al (2006)).

10

a feature of the normal and lognormal distributions. The only exception is the estimate of �for the UK, which is found to be positive and signi�cantly di¤erent than zero. Such a value of� means that the distribution of zt+1 is heavily tailed, which is a feature of the t-student andPareto distributions. Note that the above results are consistent across all the volatility modelsemployed, which standardize the residuals of return series rt+1, denoted zt+1. The choice of thethreshold u, used to estimate the parameters of the GDP, namely � and �, is taken to be the10-th percentile of the distribution of zt+1, as often suggested in the literature.10 In our analysis,we have also considered several other values for u, but these lead to a similar conclusion that10-th percentile value is the most suitable choice. This has been also con�rmed by the Akaikeinformation criterion.

[Insert Table 4]

3.4 Evaluation of VaR forecasts

To evaluate the relative VaR forecasting performance of the alternative volatility models andreturn innovation distributions estimated in the previous section, in this section we presentthe results of our out-of-sample forecasting exercise.11 This exercise also considers the case ofnormally distributed return innovations. This is done for the EGARCH model, which implies askewed unconditional distribution of asset returns, and for the EVT method which often relieson this distribution to obtain �ltered residuals. The comparison of VaR forecasts between thealternative distributions of innovations zt+1 presented above will enable us to draw interestingconclusions about the need to employ the SK-t and/or SK-GED distributions in modellingasset return distributions and for VaR forecasting.Our out-of-sample exercise is conducted by, �rst, obtaining parameter estimates of the al-

ternative volatility models considered based on a rolling over estimation procedure using asu¢ ciently large sample window of approximately 1800 observations. The �rst window startsat the beginning of the sample and ends at the last trading day of year 2001. We then use theseestimates to obtain one-step ahead, out-of-sample forecast values of the mean and volatility ofreturn rt+1 conditional on information set It. This window is then moved one step (observa-tion) forward, and the above procedure is repeated for each of the remaining time-points of thesample.The obtained out-of-sample VaR forecasts, described above, imply time-varying values of the

quantile levels a in the cases of the SK-t and SK-GED distributions of innovation process zt+1,since new estimates of the shape parameters of these distributions are obtained from each samplewindow of our rolling over estimation and forecasting procedures. For the MRS model, this isalso done for the case that zt+1 � NIID(0; 1), based on the procedure described in Section 2.Lastly, in order to fully assess the forecasting VaR ability of the volatility models under theinvestigation, in addition to generating one-day VaR forecasts for both 5% and 1% quantilelevels of the conditional distribution of return rt+1, recommended by the Basel Committee,we also introduce the 2:5% quantile level. This can provide an enriched assessment of therelative forecasting performance of the alternative volatility models at the entire lower tailof the conditional distribution of returns rt+1, under the di¤erent distributions of innovationprocess zt+1 considered.

10See McNeil and Frey (2000) for extensive discussion of the threshold choice. Note that, in our analysis, wehave also considered several other values for u. But, these lead to a similar conclusion which is consistent withthe literature, that 10-th percentile value is the most suitable choice. This has been also con�rmed by standardinformation criteria like the Akaike.11 In-sample forecasts of the models can be found in a previous version of the paper.

11

3.4.1 Forecasting performance

The results of our forecasting VaR exercise are reported in Tables 5A-5C. These correspondto a = f1%, 2.5%,5% } quantile levels of asset return distributions, respectively. The tablesreport both the target and the model generated number of violations, as well as their percentagerates for all the alternative volatility models and return innovation distributions considered. Inaddition to these metrics, the tables also report probability values (p-values) for test statisticsLRUC and DQ, presented in the previous section. The last test statistic considers the intercept,seven lagged violations and a contemporaneous VaR estimate, V aRt+1, as explanatory variables.It is chi-squared distributed with nine degrees of freedom.

[Insert Tables 3A-3C]

The results of Tables 5A-5C lead to a number of interesting conclusions, which have impor-tant implications on choosing the appropriate method of calculating VaR in practice. First, theVaR forecasting performance of all alternative parametric volatility models examined, whichassume a NIID(0; 1) innovation process zt+1, is much worse than that of their speci�cationsconsidering that zt+1 � SK-t and SK-GED. This is true for all levels of quantiles and the�nancial markets examined. This result can be supported by all the forecasting performancemetrics and test statistics reported in the tables, namely the violation numbers and their per-centage rates, as well as statistics LRUC and DQ. It is also true even for the two MRS modelsconsidered, which assume a mixture of distributions of asset returns rt+1, or the EVT approach,which focuses on the left tail of the distribution of rt+1. A skewed distribution of return inno-vations zt+1 of the volatility process used to obtain the �ltered residuals of the EVT approachis found to considerably improve the VaR forecasting performance of this method, for most ofthe markets considered.Second, the violation numbers and their percentage rates, as well as the values of statistics

LRUC and DQ reported in tables clearly indicate that, between the alternative volatility modelsconsidered under zt+1 � SK-t and/or zt+1 � SK-GED, the EGARCH and/or GARCH modelsperform better than the MRS model and its extension with EGARCH e¤ects, especially at the1% quantile of the distribution of return rt+1. For the 1% quantile level, the EGARCH modelseems to slightly outperform the GARCH in cases of the S&P500 and US bond market, whilethe opposite happens in case of the FTSE100 and FOREX indices. At 1% quantile level, bothmodels are found to provide better VaR forecasts, when zt+1 � SK-t. This is not true forthe quantile levels of 2.5% and 5.0%. In this case, the GARCH model with zt+1 � SK-GEDis found to outperform the EGARCH model, under zt+1 � SK-t and/or zt+1 � SK-GED.Note that the performance of both of the above two parametric volatility models with skeweddistributions of return innovations zt+1 is very close to that of the EVT approach, across alldi¤erent levels of � considered.The outperformance of the above volatility models with zt+1 � SK-GED. at 2.5% and

5.0%, compared to zt+1 � SK-t, may be attributed to the fact that the symmetric GED, whichis transformed to SK-GED, has all its moments well de�ned. This does not happen for thet-student distribution, which is the symmetric version of the Sk-t.The better performance of the GARCH and/or EGARCH models under zt+1 � SK-t and/or

zt+1 � SK-GED compared to the MRS models with, or without, EGARCH e¤ects can be at-tributed to the more parsimonious parameter structure of the former. This can reduce thee¤ects of parameter estimates�uncertainty on the VaR out-of-sample forecasts. Note that theMRS model-based VaR forecasts rely also on estimates of the probabilities of the current mar-ket regime, denoted as Pr (St+1 = ijIt), for i = f1; 2g, which can generate another source of

12

uncertainty in forecasting VaR.12

Third, between the two speci�cations of the MRS model considered in our analysis, i.e. itsstandard speci�cation and that extended for EGARCH e¤ects, the second is found to clearlyoutperform the �rst, for all levels of � and markets considered. The MRS model with EGARCHe¤ects and a skewed distribution of innovations zt+1 constitutes a considerable improvementupon the standard MRS model. For the 2.5% and 5% quantiles of the return distributionsexamined, its VaR performance becomes close to that of the GARCH and EGARCH models withzt+1 � SK-t and zt+1 � SK-GED. This can be obviously attributed to the richer dynamicsallowed by the MRS model with EGARCH e¤ects. Similar conclusions can be supported byour data for extensions of the MRS model allowing for GARCH e¤ects. These results are notreported for reasons of space.Finally, note that the EGARCH and GARCH volatility models with skewed distributed

innovations zt+1 are also signi�cantly superior than that of the EWMA volatility model, usedby RiskMetrics. This is true for both values of the persistency coe¢ cients �=0:99 and � = 0:94of the EWMA model reported in the table, which are frequently assumed in practice.

4 Concluding remarks

This paper evaluates the out-of-sample VaR forecasting performance of di¤erent parametricvolatility models, namely the GARCH, EGARCH and MRS (Markov regime-switching) models,frequently used in practice. This is done under the assumption that the distributions of assetreturn innovations are skewed and leptokurtic. To capture degrees of skewness and kurtosisin asset markets, we consider skewed versions of t-student and GED distributions for returninnovations. These distributions have the advantage of simplicity. Their shape parameters havea clear cut interpretation.The EGARCH model is considered as a useful vehicle contributing to the accuracy of the

VaR forecasting since it allows for leverage e¤ects, which can generate negative skewness inreturn distributions by its own. The MRS model combined with EGARCH e¤ects is consideredas a powerful framework used to accurately approximate asset return distributions by meansof a mixture of normal distributions, while allowing for leverage e¤ects in each market regime.All the above volatility models are compared to the EVT method of forecasting VaR, which isconsidered as a benchmark approach in the literature. This method models only the tails ofasset return distributions.The results of the paper provide a number of interesting conclusions, with useful practical

implications. They show that the GARCH or EGARCH volatility models combined with askewed distribution of return innovations, like the skewed t-student or the Skewed-GED, canprovide satisfactory VaR forecasts. These are very closed to those produced by the EVT ap-proach which is based on �ltered residuals. To improve its VaR forecasting ability, the MRSmodel needs to be extended by EGARCH (or GARCH) e¤ects in each regime, with a skeweddistribution of return innovations. Such a distribution can also improve the performance of theEVT method relying on �ltered residuals.The above results indicate the following. Firstly, leverage e¤ects, as captured by the EGARCH

model or the mixture of normal distributions of asset returns with the MRS model, alone are

12An analogous conclusion is reached by Yu (2006), who found that extensions of the GARCH model by Poisson-distributed jumps requiring more parameters to be estimated do not improve the VaR forecasting performance ofthis model, compared to that of simpler models like the GARCH model with SK-t distributed return innovationszt+1.

13

not adequate enough to capture the distribution tail features of asset returns. This is impor-tant, as the latter features have a direct impact on the VaR forecasts. Secondly, the skewedt-student and GED distributions constitute excellent tools in modelling distribution features ofasset returns. These results are robust across di¤erent �nancial markets considered.

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Table 1: Descriptive statistics of daily log-returns (in %)FTSE100 SP500 FOREX Bond Market

Mean 0.0165 0.0258 0.0007 0.0264Median 0.0521 0.0699 0.0025 0.0315St. Dev 1.2217 1.2711 0.5675 0.3698Skewness -0.1284 -0.2037 -0.0351 -0.5329Kurtosis 9.0274 10.9704 7.6601 9.5906no. obs 4064 4051 4199 4199

16

Table 2A: Estimates of GARCH (1,1) with zt+1 � SK-t(v; �)Model: rt+1 = a0 + a1rt + "t+1; with "t+1 =

q�2t+1zt+1 and

�2t+1 = �0 + �1"2t + 1�

2t

FTSE100 SP500 Bond Market FOREX�0 0:0493 0:0638 0:0282 0:0078

(0:0122) (0:0135) (0:0049) (0:0074)�1 �0:0314 �0:0426 0:0183 0:0290

(0:0182) (0:0162) (0:0152) (0:0163)

�0 0:0093 0:0080 0:0009 0:0013(0:0027) (0:0025) (0:0003) (0:0006)

�1 0:0859 0:0800 0:0382 0:0365(0:0093) (0:0107) (0:0059) (0:0052)

1 0:9082 0:9171 0:9551 0:9598(0:0096) (0:0104) (0:0069) (0:0058)

v 15:9995 7:7142 6:8856 7:6550(2:9392) (0:6928) (0:7278) (0:8748)

� �0:1147 �0:0918 �0:0509 �0:0020(0:0231) (0:0187) (0:0212) (0:0206)

LL �5701:30 �5769:73 �1345:51 �3125:32Notes: Standard errors are in parentheses. LL is the log-likelihood value.

Table 2B: Estimates of EGARCH(1,1) with zt+1 � SK-t(v; �)Model: rt+1 = a0 + a1rt + "t+1; with "t+1 =

q�2t+1zt+1 and

�2t+1 = exp��0 + �1zt + �1 (jztj � E jztj) + 1 ln

��2t��

FTSE100 SP500 Bond Market FOREXSK-t SK-t SK-t SK-t

�0 0:0155 0:0347 0:0285 0:0051(0:0648) (0:0140) (0:0049) (0:0047)

�1 �0:0198 �0:0344 0:0201 0:0291(0:0267) (0:0082) (0:0160) (0:0114)

�0 �0:0006 �0:0008 �0:0198 �0:0087(0:0050) (0:0014) (0:0063) (0:0033)

�1 �0:1068 �0:1205 �0:0031 �0:0074(0:0106) (0:0102) (0:0061) (0:0073)

�1 0:1134 0:1196 0:0932 0:0877(0:0210) (0:0128) (0:0127) (0:0115)

1 0:9869 0:9833 0:9916 0:9938(0:0026) (0:0025) (0:0028) (0:0023)

v 18:800 9:071 6:7130 7:5355(4:9871) (1:262) (0:6749) (1:0317)

� �0:1300 �0:1138 �0:0491 �0:0062(0:0237) (�0:0205) (0:021) (0:0203)


17

Table 2C: Estimates of the MRS model with zt+1 � SK-t(v; �)Model: rt+1 = a0 + a1rt + "t+1; where "t+1 =

q�2t+1zt+1


(0:0135) (0:0142) (0:0901) (0:0081)�1 �0:0165 �0:0552 0:0206 0:0171

(0:0517) (0:0139) (0:0362) (0:0115)

�21 0:5071 0:5332 0:0826 0:2350(0:0301) (0:0316) (0:0087) (0:0085)

�22 2:7240 2:7011 0:2450 0:9442(0:1605) (0:1874) (0:0482) (0:0946)

p12 0:0076 0:0069 0:0041 0:0012(0:0026) (0:0022) (0:0047) (0:0007)

p21 0:0110 0:0081 0:0088 0:0088(0:0035) (0:0031) (0:0021) (0:0050)

v 8:0748 5:2060 6:7431 7:0652(1:0595) (0:4984) (0:6940) (0:8261)

� �0:0463 �0:0436 �0:0479 �0:0108(0:0222) (0:0210) (0:0150) (0:0199)


18

Table 2D: Estimates of the MRS-EGARCH(1,1) model with zt+1 � SK-t(v; �)Model: rt+1 = a0 + a1rt + "t+1; where "t+1 =

q�2t+1zt+1

and �2(i);t+1 = exph�(i);0 + �1zt + �1 (jztj � E jztj) + (i);1 ln

��2t�i, for i = f1; 2g.


(0:0130) (0:0131) (0:0050) (0:0076)�1 �0:0190 �0:0313 0:0147 0:0272

(0:0159) (0:0162) (0:0148) (0:0153)

p12 0:0040 0:0027 0:0084 0:0002(0:0017) (0:0019) (0:0077) (0:0004)

p21 0:0023 0:0025 0:0173 0:0713(0:0009) (0:0013) (0:0613) (0:0622)

�(1);0 �0:2204 �0:1691 �0:0528 �0:0593(0:0506) (0:0382) (0:0193) (0:0115)

�(2);0 �0:0065 0:0015 0:1852 0:9107

(0:0194) (0:0243) (0:1820) (0:3797)�1 �0:1513 �0:1622 �0:0120 �0:0006

(0:0114) (0:0132) (0:0086) (0:0065)�1 0:0655 0:0690 0:0250 0:0600

(0:0163) (0:0151) (0:0220) (0:0156) (1);1 0:8679 0:9063 1:0002 0:9921

(0:0394) (0:0279) (0:0068) (0:0025) (2);1 0:9671 0:9652 0:8475 1:2383

(0:0061) (0:0074) (0:0992) (0:2070)v 22:3580 10:3492 7:1984 7:8389

(5:9892) (1:6094) (0:7661) (0:9460)� �0:1292 �0:1072 �0:0594 �0:0057

(0:0226) (0:0213) (0:0219) (0:0208)LL �5615:25 �5673:70 �1339:62 �3125:11

Notes: Standard errors are in parentheses. LL is the log-likelihood value.

19

Table 3A: Estimates of GARCH (1,1) with zt+1 � SK-GED(v0; �0)Model: rt+1 = a0 + a1rt + "t+1; where "t+1 =

q�2t+1zt+1 and

�2t+1 = �0 + �1"2t + 1�

2t

FTSE100 SP500 Bond Market FOREX�0 0.18604 0.1917 0.041357 0.013415

(0.021847) 0.0169 0.0062511 0.010875�1 -0.035876 -0.0556 0.021648 0.0245

0.016401 0.0139 0.014834 0.014603

�0 0.0087569 0.0087 0.00094858 0.0016860.002595 0.0027 0.00033078 0.00063743

�1 0.086343 0.0839 0.040415 0.0368290.0090149 0.0087 0.0063113 0.0053239

1 0.90485 0.9095 0.95256 0.957960.0095556 0.0093 0.0073981 0.0061435

v0 1.7071 1.4186 1.3667 1.40930.055325 0.0283 0.041112 0.039921

�0 0.88725 0.8905 0.96802 0.99020.014903 0.0131 0.013472 0.015294

LL -5689.5293 -5.7489e+003 -1349.7994 -3120.403679

Table 3B: Estimates of EGARCH(1,1) with zt+1 � SK-GED(v0; �0)Model: rt+1 = a0 + a1rt + "t+1; where "t+1 =

q�2t+1zt+1 and

�2t+1 = exp��0 + �1zt + �1 (jztj � E jztj) + 1 ln

��2t��


(0.03208) (0.019494) (0.0072633) (0.010322)�1 -0.020277 -0.040909 0.022405 0.024949

0.0072274 0.015551 0.010108 0.015099

�0 -0.10201 -0.10475 -0.094431 -0.0774990.011068 0.010288 0.014204 0.0091958

�1 -0.093576 -0.10917 -0.0047155 -0.00712740.0073681 0.010439 0.0070215 0.0027407

�1 0.11482 0.11825 0.099849 0.089720.013974 0.012509 0.013392 0.010372

1 0.99172 0.98901 0.99124 0.99280.00294 0.0026237 0.0030211 0.0023094

v0 1.7399 1.4679 1.3576 1.40110.058419 0.045624 0.041783 0.043358

�0 0.92245 0.91796 0.96886 0.99640.015838 0.014772 0.015363 0.015616

LL -5651.1909 -5701.198 -1353.3004 -3127.2741

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Table 3C: Estimates of the MRS volatility model with zt+1 � SK-GED(v0; �0)Model: rt+1 = a0 + a1rt + "t+1; where "t+1 =

q�2t+1zt+1


(0.0226160 (0.020209) (0.00802930 (0.013921)�1 -0.015528 -0.067273 0.023203 0.014442

0.017324 0.031752 0.013974 0.014175

�21 0.52189 0.52848 0.084202 0.231730.029486 0.0882 0.0041899 0.00758

�22 2.9603 2.8647 0.26608 0.985430.1947 0.66485 0.021237 0.082484

p12 0.0075828 0.0070598 0.0039012 0.00129190.0021019 0.0023435 0.0015664 0.00072149

p21 0.01262 0.0095762 0.010105 0.00990280.0035782 0.0032772 0.0038034 0.00518

v0 1.5068 1.2578 1.3648 1.3730.04885 0.043524 0.043229 0.04273

�0 0.94232 0.93248 0.97115 0.989690.016484 0.044539 0.016138 0.018229

LL 5863.0295 -5925.1729 -1399.6181 -3199.57879

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Table 4: Estimates of the GPD parameters � and � of the EVT method

GPD : G(r) =

8<: 1��1 + �r

�

�� 1�

� 6= 0

1� exp�� r�

�� = 0

EGARCH(SK-t) GARCH(SK-t) MRS-EGARCH(SK-t) EGARCH(SK-GED) GARCH(SK-GED)FTSE100

� �0:1006 �0:0783 �0:0667 �0:0489 �0:0629(0:0353) (0:0446) (0:0355) (0:0364) (0:0421)

� 0:6671 0:6428 0:6200 0:6248 0:6282(0:0405) (0:0432) (0:0370) (0:0388) (0:0394)

LL �197:35 �194:80 �184:86 �195:20 �191:74SP500

� 0:0238 0:0763 0:0176 0:0216 0:0423(0:0431) (0:0492) (0:0462) (0:0448) (0:0466)

� 0:6242 0:5707 0:6014 0:6347 0:6142(0:0397) (0:0394) (0:0406) (0:0425) (0:0417)

LL �223:76 �208:73 �206:19 �229:61 �224:7254Bond Market

� 0:0337 0:0249 0:0497 0:0259 0:0317(0:0439) (0:0455) (0:0451) (0:0446) (0:0447)

� 0:6462 0:6568 0:6288 0:6561 0:6465(0:0424) (0:0433) (0:0423) (0:0437) (0:0427)

LL �250:71 �253:92 �246:01 �253:89 �250:12FOREX

� �0:0256 �0:0501 �0:0285 �0:0291 �0:0569(0:0451) (0:0468) (0:0444) (0:0452) (0:0438)

� 0:5627 0:5767 0:5622 0:5679 0:5861(0:0377) (0:0363) (0:0370) (0:0375) (0:0386)

LL �167:73 �167:75 �166:18 �170:13 �171:70Notes: Standard errors are in parentheses. LL is the log-likelihood value.

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Table 5A: Out-of-sample VaR forecasts (Viol. Target = 1.0% )

Model/Filter Dist. Act:V iol:(%) Act:V iol:(no) LRUC DQ____V aR t+1

FTSE100 (Viol. Target no. = 22.97)EGARCH Normal 1:83 42 0:00 0:00 �2:56

SK-t 1:04 24 0:83 0:71 �2:81SK-GED 1:22 28 0:31 0:43 �2:80

GARCH SK-t 1:31 30 0:16 0:00 �2:90SK-GED 1:13 26 0:53 0:68 �2:98

MRS Normal 1:92 44 0:00 0:00 �2:87SK-t 1:61 37 0:01 0:00 �3:07

SK-GED 1:39 32 0:07 0:00 �3:15MRS-EGARCH Normal 1:78 41 0:00 0:02 �2:64

SK-t 1:18 27 0:41 0:47 �2:87EVT/EGARCH Normal 1:00 23 0:99 0:68 �2:87EVT/EGARCH SK-t 0:96 22 0:84 0:89 �2:87EVT/EGARCH SK-GED 0:96 22 0:84 0:85 �2:91EVT/GARCH SK-t 1:13 26 0:53 0:73 �3:04EVT/GARCH SK-GED 1:05 24 0:83 0:69 �3:06

EVT/MRS-EGARCH Normal 0:96 22 0:84 0:89 �2:94SK-t 1:18 27 0:41 0:05 �2:92

EWMA � = :99 1:95 45 0:00 0:00 �2:85� = :94 2:22 51 0:00 0:00 �2:66


SP500 (Viol. Target no. = 22.88)EGARCH Normal 2:27 52 0:00 0:00 �2:61

SK-t 1:35 31 0:11 0:00 �2:92SK-GED 1:27 29 0:22 0:01 �2:96

GARCH SK-t 0:96 22 0:85 0:01 �3:03SK-GED 0:92 21 0:69 0:02 �3:17

MRS Normal 1:84 42: 0:00 0:00 �2:85SK-t 1:40 32 0:07 0:00 �3:04




EWMA � = :99 1:79 41 0:00 0:00 �2:88� = :94 2:05 47 0:00 0:00 �2:69

23


Bond Market (Viol. Target no. = 23.73)EGARCH Normal 1:94 46 0:00 0:00 �0:84

SK-t 1:18 28 0:39 0:05 �0:95SK-GED 1:26 30 0:21 0:07 �0:95

GARCH SK-t 1:22 29 0:29 0:07 �0:97SK-GED 1:31 31 0:15 0:07 �0:97

MRS Normal 2:02 48 0:00 0:00 �0:85SK-t 1:64 39 0:00 0:00 �0:93




EWT � = :99 1:98 47 0:00 0:00 �0:89� = :94 1:72 41 0:00 0:00 �0:87


FOREX (Viol. Target no. = 23.73)EGARCH Normal 1:77 42 0:00 0:00 �1:32

SK-t 1:47 35 0:03 0:06 �1:42SK-GED 1:31 31 0:15 0:06 �1:45

GARCH SK-t 1:22 29 0:29 0:51 �1:43SK-GED 1:22 29 0:29 0:50 �1:47

MRS Normal 1:94 46 0:00 0:00 �1:35SK-t 1:60 38 0:01 0:00 �1:42




EWMA � = :99 1:64 39 0:00 0:00 �1:38� = :94 1:81 43 0:00 0:00 �1:36

Notes: The table summarizes the results of a rolling over out-of-sample VaR forecasting exercisebased on a sample window of approximately 1800 observations."Dist." stands for distribution, "Act.Viol." for actual violations, "viol. target no." for the number of target violations. SK-t is the skewed

24

t-student distribution, while SK �GED is the skewed GED. The table reports p-values of the LRUCand DQ test statistics, which are distributed as Chi-squared with one and none degrees of freedom,

respectively.____V aR t+1is the average over the whole sample one-period ahead VaR forecast, V aRt+1.

Table 5B: Out-of-sample VaR forecasts (Viol. Target = 2.5% )

Model/Filter Dist. Act:V iol:(%) Act:V iol:(%) LRUC DQ____V aR t+1

FTSE100 (Viol. target no = 57.42)EGARCH Normal 3:74 86 0:00 0:00 �2:16

SK-t 2:87 66 0:26 0:46 �2:30SK-GED 3:00 69 0:13 0:38 �2:33

GARCH Sk-t 2:87 66 0:26 0:14 �2:37SK-GED 2:70 62 0:55 0:19 �2:48

MRS Normal 3:53 81 0:00 0:00 �2:35SK-t 3:05 70 0:10 0:00 �2:43




EWMA � = :99 3:30 76 0:02 0:00 �2:40� = :94 3:52 81 0:00 0:00 �2:24

25


SP500 (Viol. target no = 57.20)EGARCH Normal 3:5 80 0:00 0:21 �2:20

SK-t 2:93 67 0:20 0:07 �2:35SK-GED 2:71 62 0:53 0:31 �2:42

GARCH SK-t 3:02 69 0:13 0:00 �2:42SK-GED 2:45 56 0:87 0:51 �2:57

MRS Normal 3:32 76 0:02 0:00 �2:34SK-t 3:23 74 0:03 0:00 �2:36




EWMA � = :99 3:19 73 0:04 0:00 �2:42� = :94 3:84 88 0:00 0:00 �2:26


Bond Market (Viol. Target no. = 59.32)EGARCH Normal 3:33 79 0:01 0:21 �0:71

SK-t 2:65 63 0:63 0:58 �0:75SK-GED 2:53 60 0:93 0:81 �0:76

GARCH SK-t 2:61 62 0:73 0:59 �0:76SK-GED 2:53 60 0:93 0:46 �0:77

MRS Normal 3:62 86 0:00 0:01 �0:69SK-t 3:29 78 0:02 0:26 �0:72




EWMA � = :99 3:07 73 0:08 0:14 �0:75� = :94 3:03 72 0:11 0:10 �0:73

26


FOREX (Viol. target no = 59.32)EGARCH Normal 3:46 82 0:01 0:02 �1:12EGARCH SK-t 3:03 72 0:11 0:47 �1:14

SK-GED 2:87 68 0:26 0:55 �1:17GARCH SK-t 3:03 72 0:10 0:29 �1:15

SK-GED 2:70 64 0:54 0:78 �1:19MRS Normal 3:75 89 0:00 0:00 �1:11

SK-t 3:54 84 0:00 0:00 �1:13SK-GED 3:46 82 0:01 0:00 �1:17

MRS-EGARCH Normal 2:95 70 0:17 0:15 �1:16SK-t 3:67 87 0:00 0:00 �1:13

EVT/EGARCH Normal 3:03 72 0:11 0:17 �1:16EVT/EGARCH SK-t 2:95 70 0:17 0:67 �1:16EVT/EGARCH SK-GED 3:03 72 0:11 0:36 �1:16EVT/GARCH SK-t 2:82 67 0:32 0:64 �1:17EVT/GARCH SK-GED 2:99 71 0:14 0:31 �1:17


EWMA � = :99 3:49 83 0:00 0:00 �1:17� = :94 3:37 80 0:01 0:02 �1:14

Notes: See Table 5A

Table 5C: Out-of-sample VaR forecasts (Viol. Target =5.0% )


FTSE100 (Viol. Target no. = 114.85)EGARCH Normal 6:14 141 0:02 0:06 �1:81

SK-t 5:53 127 0:25 0:30 �1:89SK-GED 5:48 126 0:29 0:51 �1:93

GARCH SK-t 5:49 126 0:29 0:25 �1:93SK-GED 5:22 120 0:62 0:40 �2:04

MRS Normal 5:44 125 0:34 0:00 �1:93SK-t 5:27 121 0:56 0:00 �1:94




EWMA � = :99 4:91 113 0:86 0:00 �2:02� = :94 5:96 137 0:04 0:01 �1:88

27


SP500 (Viol. Target no. = 114.40)EGARCH Normal 5:86 134 0:07 0:17 �1:84

SK-t 5:42 124 0:36 0:62 �1:91SK-GED 5:29 121 0:53 0:44 �1:98

GARCH SK-t 5:38 123 0:41 0:71 �1:94SK-GED 4:89 112 0:82 0:64 �2:08

MRS Normal 5:24 120 0:59 0:00 �1:92SK-t 5:55 127 0:23 0:00 �1:87




EWMA � = :99 5:20 119 0:66 0:00 �2:03� = :94 5:72 131 0:12 0:20 �1:90


Bond Market (Viol. Target no. = 118.65)EGARCH Normal 5:94 141 0:04 0:16 �0:59EGARCH SK-t 5:77 137 0:09 0:34 �0:59

SK-GED 5:35 127 0:44 0:82 �0:61GARCH SK-t 5:56 132 0:00 0:59 �0:60

SK-GED 5:27 125 0:55 0:74 �0:62MRS Normal 6:41 152 0:00 0:06 �0:56

SK-t 6:19 147 0:01 0:00 �0:57SK-GED 6:07 144 0:02 0:02 �0:58

MRS-EGARCH Normal 6:28 149 0:01 0:08 �0:57SK-t 5:86 139 0:06 0:15 �0:59

EVT/EGARCH Normal 5:48 130 0:29 0:53 �0:60EVT/EGARCH SK-t 5:44 129 0:34 0:48 �0:60EVT/EGARCH SK-GED 5:35 127 0:44 0:72 �0:60EVT/GARCH SK-t 5:27 125 0:55 0:81 �0:61EVT/GARCH SK-GED 5:23 124 0:62 0:52 �0:61


EWMA � = :99 4:76 113 0:59 0:23 �0:63� = :94 5:30 126 0:49 0:46 �0:62

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FOREX (Viol. target no. = 118.65)EGARCH Normal 5:90 140 0:05 0:68 �0:93

SK-t 6:07 144 0:02 0:49 �0:93SK-GED 5:90 140 0:05 0:77 �0:95

GARCH SK-t 5:86 139 0:06 0:62 �0:93SK-GED 5:31 126 0:49 0:96 �0:96

MRS Normal 6:45 153 0:00 0:00 �0:91SK-t 6:36 151 0:00 0:00 �0:91




EWMA � = :99 5:60 133 0:18 0:00 �0:98� = :94 5:48 130 0:29 0:69 �0:96

Notes: See Table 5A

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forecasting var models under different volatility processes and

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