Modelling Higher Moments using a simplified Multivariate ARCD: An Application to International Equity and Currency Portfolio VaR

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1 Modelling Higher Momens using a simplified Mulivariae ARCD: An Applicaion o Inernaional Equiy and Currency Porfolio VaR Elefheria Kosika Raphael N. Markellos Ahens Universiy of Economics and Business Absrac. This paper proposes a new approach o esimae ime varying condiional variance and covariance marices allowing for he impac of higher momens in he framework of he Auoregressive Condiional Densiy (ARCD) model. The proposed mehod is based on he esimaion of only univariae ARCD models and is numerically feasible and easier o esimae han exising complicaed mulivariae volailiy processes ha ofen suffer from unrealisic a priori resricions and convergence problems. An empirical applicaion of he new model is provided o forecas he VaR of aggregae equiy porfolios for he US and UK and foreign exchange porfolio for EUR and GBP agains USD and is compared o GARCH and BEKK models. Our resuls, using boh saisical and economic crieria, sugges ha he simplified mulivariae version of ARCD performs a leas well as he oher wo models indicaing he higher momens imporance in volailiy forecasing and VaR calculaion. Keywords : VaR, GARCH, BEKK, Simplified Mulivariae ARCD. EFMA Classificaion Codes: 370, 380, 450 1

2 Elefheria Kosika is PhD Candidae, Deparmen of Managemen Science and Technology, Ahens Universiy of Economics and Business, Greece, Tel ; Fax Corresponding auhor is Raphael Markellos, Senior Lecurer, Deparmen of Managemen Science and Technology, Ahens Universiy of Economics and Business, Office 915, 47A Evelpidon Sr , Ahens, Greece; Visiing Research Fellow, Cenre for Research in Inernaional Economics and Finance (CIFER), Loughborough Universiy, UK, Tel ; Fax Inroducion Undersanding and esimaing ime varying condiional variances and covariances is imporan for many issues in finance since here are many applicaions ha rely on mulivariae covariance models. I is essenial, for opimal hedging, asse allocaion, derivaives pricing and risk managemen, he accurae modelling and forecas of he asses reurns co-movemen. Bollerslev, Engle and Wooldridge (1988), Cecchei (1988), Myers and Thompson (1989), Baillie and Myers (1991), Kroner and Sulan (1993), Ng and Kroner (1998), argue ha financial prices are characerized by ime varying variances and covariances, presening a variey of mulivariae GARCH models. Bollerslev, Engle and Wooldridge (1988) suggesed he VEC model and he diagonal VEC (DVEC) model in which he variances depend only on heir pas squared errors and he covariances on heir pas cross-producs of errors. Given he excessively large parameers needed o esimae he VEC model and he necessiy o impose srong resricions on he parameers Engle and Kroner (1995) proposed he BEKK paramerizaion avoiding unrealisic assumpions such as ha he correlaion beween he condiional variances is consan (Consan correlaion model by Bollerslev 1990), and guaraneeing ha he ime varying covariance marix is posiive definie. Addiional models can be found in Engle, Ng and Rohschild (1990b) who proposed facor models (FGARCH), in Alexander and Chibumba (1997) who proposed he orhogonal GARCH models (O-GARCH), in Tse and Tsui (00) and of Engle (00) who suggesed he Dynamic Condiional Correlaion models (DCC). All he above models assume ha asse reurns are joinly normally disribued ignoring he fac of asymmery in volailiy and covariance, fa ailness and skewness. However, asymmery and skewness in disribuion, is found in many financial asses since heir reurn disribuions depar far away from normaliy. For insance, French, Schwer and Sambaugh (1987) rejeced normaliy claiming significan condiional skewness in daily residuals of he SP500 reurns, Hong (1988) found abnormally high kurosis in daily NYSE sock reurns, Harvey (1995) observed deviaions form normaliy in

3 emerging sock markes indices, Harvey and Siddique (1999) showed ha condiional skewness is imporan and consisen wih asymmeric variance in daily, weekly and monhly reurns of seleced markes. Since here is well esablished sylized evidence ha financial reurns exhibi fa ails and skewness, a lo of sudies focused on using of non normal disribuions o beer model his excess kurosis and skewness. More specifically, in he univariae framework, a large variey of condiional densiies has been employed o accommodae he asymmery and fa ailness. Hansen (1994) was he firs o propose a Skew-Suden disribuion which allows for condiional higher momens. Recenly, Harvey and Siddique (00a, 00b), Jondeau and Rockinger (006) and Yan (005), Brooks, Burke and Persand (005) among ohers, have discussed ways o joinly esimae ime varying condiional variance and skewness, bu heir resuling formulaion is difficul o be implemened, moreover in a mulivariae exension. More precisely, none of he popular mulivariae models are compaible wih he skewness and kurosis of asse reurns since hey assume mulivariae normaliy. A few sudies exis on he higher momens modelling in mulivariae approaches. Harvey, Ruiz and Shepard (1994) and Fiorenini, Senana and Galzolari (003) replace mulivariae Gaussian densiy wih suden densiy by leing condiional innovaions o follow a Suden- disribuion. Sahu, Dey, and Branco (001), and Bawens and Lauren (005) propose a mulivariae skew Suden densiy wih suppor on he full Euclidian space. Their main finding is ha his densiy improves he qualiy of ou of sample VaR forecass. More recenly, Hafner and Rombous (004) and Rombous and Verbeek (005) apply a mulivariae semi parameric GARCH esimaion echnique o capure higher momens showing ha in wihin sample porfolios VaR he model s superioriy and robusness is confirmed. Azzallini (1996) and De Luca, Genon and Loperfido (006) propose he mulivariae Skew- GARCH model including a parameer o conrol skewness. Lee and Long (005) inroduce copula-based mulivariae GARCH, he C-MGARCH wih uncorrelaed dependen errors, arguing ha in erms of in sample model selecion and ou of sample mulivariae densiy forecas, he choice of copula funcions is more imporan han he volailiy models. The main drawback of he above models is ha are raher complex, and suffer from a large parameers esimaion and convergence problems. In his paper, we propose an alernaive, simplified mulivariae model, he simplified Mulivariae Auoregressive Condiional Densiy Model (S-ARCD) which is compaible 3

4 wih he skewness and kurosis of he financial reurns and is easy o be implemened increasing he compuaional efficiency. I is based on he Auoregressive Condiional Densiy Model (ARCD) proposed by Hansen (1994) and involves he esimaion only of he univariae specificaion of he above model. The condiional variances are calculaed by he simple univariae models, and he condiional covariance is hen impued from hese variance esimaes. We illusrae he S-ARCD o forecas he VaR of aggregae equiy porfolios for he US and UK and foreign exchange porfolio for EUR and GBP agains USD and is compared o he ad hoc mulivariae version of GARCH (Wang, Yao, 005) and BEKK models. Our resuls, using boh saisical and economic crieria, sugges ha he simplified mulivariae version of ARCD performs a leas well as he oher wo models indicaing he higher momens imporance in volailiy forecasing and VaR calculaion. The remainder of he paper is organized as follows: he nex secion inroduces S- ARCD and briefly describes BEKK and he ad hoc mulivariae version of GARCH (Wang, Yao, 005) models. The hird secion describes he daa and he empirical resuls on he VaR esimaion. The nex secion compares he VaR performance of he alernaive models. The final secion concludes he paper. 1. Mehodology This secion describes he hree models under consideraion: he ad hoc mulivariae version of GARCH, BEKK and S-ARCD. The firs wo models are presened briefly since here is exensive descripion in he academic lieraure. 1.1 The ad hoc mulivariae of GARCH Wang and Yao (005) firs proposed an ad hoc mulivariae mehod using univariae GARCH models in order o allow he reurn covariance marix o vary over ime. More precisely, using he popular GARCH(1,1) specificaion, he condiional variances of wo reurn series y 1,, y,,can be modelled respecively as: h = c + a ε + β h 1, , , -1 (1a) h = c + a ε + β h (1b), 1, -1, -1 4

5 where ε and are he lagged squared residuals from he condiional mean equaions for he spo and fuures reurns respecively. Then, o model he condiional covariance, he following seps are implemened. Firsly, a new series is consruced as: x 1, 1 ε, 1 h 1, 1, ( y1, + y, ) =. Secondly, he condiional variance of he new series is esimaed from anoher univariae GARCH(1,1) as: h = c + a ε + β h 1, 1 1 1, , -1 (1c) Finally, he ime varying condiional covariance of reurns is given by he equaion: σ ( h1, + h, ) = h () 1, 1, 1. Mulivariae GARCH Engle and Kroner (1995), among ohers, have relaxed his assumpion by proposing a mulivariae GARCH process. Using a BEKK represenaion, he condiional variance marix is he following: H = CC ' + Aε ε ' A + BH B (3) where C, Α, Β are n n marices, wih C being upper riangular, symmeric and posiive definie. The condiional variance marix is posiive definie since he second and hird erms in he above equaion (5) are expressed in quadraic forms. This means ha no oher consrains for he marices Α and Β are necessary. For he case of he bivariae GARCH(1,1), he BEKK model is esimaed in a resriced form wih C as a lower riangular marix, and, Α, Β being diagonal marices. This can be expressed by he following equaions: H -1 h11, h1, c ε ε ε 11 0 c11 c a1 0 1, 1 1, 1, 1 a1 0 = + + ε ε ε h1,, , 1, 1, 1 0 h c c c a a or, β 1 0 h11, 1 h1, 1 β β h β 1, 1, 1 0 h (4a) 5

6 h = c + a ε + β h 11, , , 1 h = c + c + a ε + β h, 1, 1, 1 h = c c + a a ε ε + β β h 1, , 1, 1 1 1, 1 (4b) where c11, c, c1, are consans and ε 1, 1 ε, 1 are he lagged residuals from he condiional mean equaion for he spo and fuures reurns respecively. 1.3 The Simplified Mulivariae Auoregressive Condiional Densiy Model Le y 1, and y, wo univariae discree ime real-valued sochasic processes, (i.e. he rae of reurn of an asse or marke porfolio) and I is he informaion se a ime, which encompasses y i, and all he pas realizaions of he process y i, where i=1,. Then, he condiional mean reurns are denoed as: µ E y I ), 1. = ( 1, 1 µ E y I ) and he condiional covariance marix of y 1, and y, is given by:. = (, 1 h 11, h1, Var ( y 1,, y, I 1) = = h1, h, (5) The goal is o model he elemens of he condiional covariance marix aking ino consideraion he ime varying skewness and kurosis. Our approach is based on Hansen (1994) Auoregressive Condiional Densiy Model (ARCD), who proposed a generalizaion of he Suden- disribuion o capure asymmery and fa ailness, involving only univariae modeling. Alhough alernaive skewed Suden- disribuions have been considered in he lieraure (eg., Jondeau and Rockinger, 003, 006; Harvey and Siddique, 1999, 00a, 00b), we seleced his specificaion because i has a clear compuaional advanage over compeing models (e.g., see Harvey and Siddique, 1999; Brooks and Persand, 005) and he variaion in he shape parameers may be smaller and easier o manage numerically. Also, only few parameers are esimaed in each model and, generally, i is easier o implemen han oher mulivariae models such as BEKK, Vech or sochasic variance models. An alernaive approach, he simplified mulivariae GARCH, has been presened by Harris, Soja and Tucker (007) in order o esimae he minimumvariance hedge raio for he FTSE 100 index porfolio. In his paper, he proposed mehod involves wo seps and is based on Wang and Yao (005) mehod. Firsly, 6

7 he condiional variances are esimaed using he following univariae form of he ARCD model s disribuion densiy funcion: 1 bz + k η - 1-λ -( η + 1) f ( z η, λ) = bd (1 + ( ( ) ) for < -k z b and (6) -( η + 1) 1 bz + k f ( z η, λ) = bd (1 + ( ( ) ) for z f -k η - 1+ λ b where < η <, -1 < λ < 1, η are he degrees of freedom, λ is he skewness, while k, b, d are consan parameers defined by he following equaions: η - k = 4 λd ( ), b = 1+ 3λ - k, η -1 η + 1) Γ( ) d = (7) η π( η - ) Γ( ) More specifically, he degrees of freedom η and skewness λ are specified as following: η = γ + γ ε + γ ε (8a) λ = λ0 + λε λε -1 (8b) Jondeau and Rockinger (003) have presened he exac formulas for he calculaion of he kurosis and skewness. The condiional log-likelihood of he full ARCD model is calculaed as: 1 LLK f z h (9) T = log ( ; η) - log = max( p, q) + 1 The ARCD shape parameers η can be esimaed by sandard ieraive mehods by assuming arbirary reasonable iniial valueη 1. I is advisable o compue robus 7

8 sandard errors since hey generae asympoically valid confidence inervals for he pseudo rue parameer values which minimize he informaion disance beween he rue probabiliy and he quasi-likelihood measure. In his manner, we can achieve he maximum possible accuracy in our resuls. Finally, he Nyblom L-saisic, for esing he consancy of he esimaed parameers, akes he form: 1 = n G i Lk n V = 1 ii (10) where G i are he cumulaive scores and V ii is he ih diagonal elemen of he esimaed variance. The L-saisic is used o es he saionariy of he parameers of he disribuion funcion and can be considered as he LM es of he null hypohesis ha he parameers are sable. Asympoic criical values for he Nyblom es and an exensive analysis have been presened in Hansen (1990). The condiional variances are given by: h = c + a (ε + β 1, , -1 h1, 1) 1h1, 1,, -1 h, 1) h, 1 (11a) h = c + a (ε + β (11b) Secondly, he condiional covariance σ ij, is esimaed following he mehod proposed by Wang and Yao (005) consrucing a new series wih he general form of: ( y1, + y, ) ω 1, = for he i<j elemens of he equaion (5) where i,j =1,. The condiional variance of he above new series is esimaed as h = Var( ω I ) 1, 1, 1 using he univariae version of he ARCD model described above using he following equaion: h = c + a (ε + β 1, 1 1 1, -1 h1, 1) 1 1, 1 h (11c) Then, for all 1 i < j he condiional covariance is calculaed using he following equaion: ( h1, + h, ) σ 1, = h1,. (1) The above ideniy has been proposed by Wang and Yao (005) in order o derive esimaors of he covariance marix when here is no mulivariae exension of he underlying univariae model. Overall, he simplified ARCD model involves he 8

9 esimaion of only univariae ARCD models. Therefore i is easier and compuaionally simpler o be implemened han Vech and BEKK models avoiding overparamerizaion since only a few parameers are esimaed in each model, and he maximum likelihood converges more efficienly. Also, here are no resricions for he coefficiens of he condiional covariance unlike diagonal Vech, BEKK and consan correlaion models covariance marix elemens. However, he above flexible srucure of he S-ARCD model comes wih some disadvanage. The resuling esimae of he condiional correlaion marix is no necessarily posiive semidefinie since here is no possibiliy o se any resricions in he condiional covariance coefficiens. There several procedures o encouner his problem. Harris, Soja and Tucker (007) propose hree simple echniques o ensure he esimaed correlaion marix posiive semi definieness Daa Descripion and Empirical Analysis The daa used in he presen sudy includes daily closing prices for he UK Financial Times Sock Exchange Index (FTSE), and he US Dow Jοnes Index (DJ) and daily spo prices of wo exchange series, he EUR and he GBP agains USD. The sample covers he period from 3 January 000 unil 30 June 007 for all four daa series, a oal of 195 observaions for boh he exchange raes and 1861 for he FTSE and Dow Jones indices respecively. The las 100 observaions for each series are lef for ex ane (ou of sample) porfolio VaR esimaion. Descripive saisics of logarihmic reurns of all series daa under sudy are provided in Table 1. There is srong evidence ha all series are non-normally disribued wih high peaks and fa ails. For all series, here is negaive asymmery in he disribuion. The Ljung-Box pormaneau es for all series excep FTSE shows no significan auocorrelaion while he ARCH-LM es for serial correlaion in squared reurns reveals volailiy clusering in all series and more significanly in equiy indices. Table 1. Descripive Saisics of spo and fuures index reurns Obs. Mean S. Dev. Skewness Κurosis JB Q 1 (10) ARCH(4) Equiy Indices 1 A deailed analysis can be found in Harris, Soya and Tucker paper: A simplified approach o modelling he co-movemen of asse reurns published in The Journal of Fuures Markes (007). 9

10 FTSE 1, DJ 1, FX Spo agains USD EUR 1, GBP 1, Jarque-Bera (JB) is asympoically disribued as a Chi-squared wih degrees of freedom under he null hypohesis of normaliy. The Q 1 (k) represens he Ljung-Box pormaneau es of he reurn series. ARCH(4) saisic ess he null hypohesis ha he firs four parial auocorrelaion of squared reurns are zero. For he empirical implemenaion of he simplified ARCD he condiional variances are esimaed using he simple version of ARCD model applying equaions 11a and 11b for each daa series separaely. We hen consruc he new series r F + D = ( r + r ) for he equiy indicess FTSE and DJ so as o esimae he condiional covariance by equaion 1. The condiional variance of he new series r F + D is esimaed applying anoher univariae version of ARCD. The above procedure is implemened, also, for he foreign exchange series currencies EUR and GBP. The new series, from he currencies EUR and GBP agains USD, r ( r + r ) EUR GBP E + G= is consruced and used for he condiional covariance calculaion. The esimaed parameers of he simplified mulivariae ARCD S-ARCD model for he DJ and FTSE indices and he EUR and GBP currencies are presened in Table. The simplified ARCD model esimaion resuls are presened in Table. We repor he condiional variance, degrees of freedom, skewness and Nyblom es values. For he condiional variances, he condiional degrees of freedom and he condiional skewness, he Nyblom L es indicaes ha he parameers are all sable since he es saisic is less ha he 1% level criical value of This is also confirmed by he join Nyblom es which is smaller han he 1% criical value of.8. Overall, he coefficiens for he condiional degrees of freedom and he condiional skewness seem o be highly significan implying ha he simplified ARCD model is well specified and fis he daa capuring he higher momens ime variaion. A log likelihood raio raio (LR) saisic is applied o es he null hypohesis ha he series follow he normal disribuion agains he alernaive of ime varying higher momens. Since he normal disribuion (of GARCH model) is nesed o he skewed 10 FTSE DJ

11 Suden- disribuion of he simplified ARCD model, he LR saisic is calculaed by he following formula: LR= -[ LGARCH ( LARCD where LGARCH and LARCD are he absolue values of he maximum values of he log likelihood funcions under he normal disribuion and he skewed Suden- disribuion respecively. As shown in Table, all LR values are greaer han heir criical value of 9.1 a 1% significance level, srongly rejecing he null hypohesis of ime invarian shape parameers which GARCH assumes, implying ha he empirical disribuion of daa reurns do no follow a normal disribuion. Overall, he simplified ARCD model seems o fi he daa beer compared o he ad hoc GARCH(1,1) model since he Nyblom Join es saisic for he sabiliy of he parameers of GARCH(1,1) model is rejeced a 1% significance level, evidence ha a furher dynamic specificaion is needed. Table. S-ARCD esimaes for FTSE, DJ, (F+D), EUR, GBP, (E+G) S-ARCD FTSE DJ (F+D) EUR GBP (E+G) Condiional Variance c 11, c, c (0.0033) (0.0033) (0.003) (0.014) (0.0014) (0.0008) 1 ε, - 1-h 1, - 1, ε, - 1-h, - 1, 1 ε, - 1-1, -1 h (0.0137) (0.0137) (0.0176) (0.0061) (0.0061) (0.0073) h 1, -1, h, -1, h , -1 (0.0056) Nyblom L σ es (0.0055) (0.0073) (0.0416) (0.0416) (0.003) c,, c c 1 ε 1, - 1-h1, - 1, ε, - 1 -h,, ε, - 1-1, - 1 1, -1 h h, h 1, h , -, - Condiional Degrees of Freedom γ (0.396) ε -1 (γ 1 ) (0.5764) ε -1 (γ ) Nyblom L σ es (0.1538) (0.396) (1.0053) (0.0358) (0.6614) (0.816) 0.97 (0.006) (1.3801) (.75) (0.9169) (0.3801) (0.4100) (0.0537) (0.8161) (0.5764) (0.1538) γ ε -1 (γ 1 ) ε -1 (γ ) Condiional Skewness 11

12 λ (0.0797) ε -1 (λ 1 ) 0.01 (0.0067) ε -1 (λ ) Nyblom L σ es (0.0041) (0.0754) (0.068) (0.0409) (0.0830) (0.051) (0.0031) (0.044) (0.105) (0.0809) (0.044) (0.015) (0.0809) (0.0033) 0.14 (0.0154) (0.0661) λ ε -1 (λ 1 ) ε -1 (λ ) Log Likelihood LR es Nyblom Join Tes Numbers in brackes under he parameer esimaes give he sandard errors values. Nyblom L saisic has been inroduced by Nyblom (1989) and modified by Hansen (1990) for esing he consancy of he esimaed parameers. I akes he form: L i =1/n*(Σ(G i / Ṽ ii ) where G i are he cumulaive scores, Ṽ ii is he i h diagonal elemen of he esimae variance Ṽ and can be considered as he LM es of he null hypohesis ha all parameers are sable. The asympoic criical values for he Nyblom es have been presened in Hansen (1990). For he Nyblom es, he 1% cri ical value is equal o 0. 75, and for he Nyblom Join es, he 1% criical value is equal o. 8. LR saisic is he likelihood raio es of he null hypohesis of normal disribuion agains he alernaive ha he daa reurns follow a skewed suden disribuion. The LR saisic is asympoically disribued as a Chi-squared wih degrees of freedom and is criical value a 1% significance level is 9.1. Also, he ad hoc mulivariae version of GARCH (Wang, Yao, 005) and BEKK models are esimaed for all daa. Tables 3 and 4 presen he parameers for he ad hoc mulivariae version of GARCH and BEKK models respecively. For boh GARCH(1,1) and Bivariae GARCH(1,1) models, all coefficiens are posiive and saisically significan. Especially, for he simple univariae GARCH(1,1), he near-uniy sum of he coefficiens suggess very high persisence in he condiional variances. Table 3. Ad-Hoc GARCH(1,1) esimaes for DJ, FTSE, (F+D), EUR, GBP, (E+G) Ad Hoc GARCH(1,1) FTSE DJ (F+D) EUR GBP (E+G) c,, c , c 1 (0.0038) (0.001) (0.001) (0.0005) (0.0187) (0.0008) ε, ε, 1, -1, -1 ε 1, (0.013) (0.0087) (0.0096) (0.0043) (0.004) (0.0058) h1, -1, -1 h, h1, (0.0138) (0.0103) (0.0110) (0.0045) (0.0440) (0.0073) LL Nyblom Join Tes

13 Sandard errors appear in brackes. For he Nyblom Join es, he 1% criical value is equal o.8. Table 4. BEKK(1,1) esimaes for (FTSE, DJ), (EUR, GBP) BEKK(1,1) (FTSE, DJ) (EUR,GBP) c (0.0194) (0.010) β (0.0030) (0.0017) a (0.0117) (0.0103) c (0.0188) c (0.0151) β (0.006) a (0.0105) (0.010) (0.009) (0.0117) (0.0177) LL Sandard errors appear in brackes. In he following figures 1a, 1b, 1c he fied condiional covariance is ploed for he hree mulivariae models. From a firs view, for boh porfolios, he EUR-GBP currencies porfolio and he Dow-FTSE indices porfolio, he magniudes and paerns of he ime varying condiional covariance obained from S-ARCD are similar o hese capured from he oher wo mulivariae models ad hoc GARCH(1,1) and BEKK(1,1). As a resul, a more sophisicaed evaluaion approach mus be developed in order o examine he performance of he hree mulivariae models. Figure 1a: Time varying Condiional Covariances for S-ARCD model. 13

14 Figure 1b: Time varying Condiional Covariances for ad hoc GARCH(1,1) model. Figure 1c: Time varying Condiional Covariances for BEKK(1,1) model. 4. Evaluaion Two approaches are employed in order o evaluae he performance of he S-ARCD model agains he oher wo mulivariae models ad-hoc GARCH(1,1) and BEKK(1,1) models. Firsly, a saisical mehod is implemened using a regression model and secondly an economic approach is used o esimae he volailiy for he VaR calculaion of he aggregae equiy porfolios and he foreign exchange porfolios. For he saisical evaluaion, he Harris, Soja and Tucker (007) proposed a regression so as o es he condiional unbiasedness of he esimaed condiional covariance marix. We compare he esimaed condiional variances and covariance h h σ versus o he realized condiional variances and covariance which are 1,,,, 1,, 14

15 he squares and he cross producs of he residuals esimaed by he relaive mulivariae model for each series ˆ ε ˆ 1,, ε,, respecively, using he following Ordinary Leas Squares (OLS) regressions: ˆε = θ + θ σ + v 1, 1,0 1,1 1, 1, ˆ ε = θ + θ σ + v,,0,1,, ˆ ε ˆ ε = θ + θ σ + v 1,, 1,0 1,1 1, 1, (13) The above regressions are esed using an F-saisic. When he null hypohesis of a zero inercep and a slope coefficien equal o one is no rejeced hen he mulivariae model under esing is well specified and correcly defined (Andersen and Bollerslev 1998, Harris, Soja and Tucker, 007). For he economic approach, we compue ou of sample one day period VaR forecass using he variance covariance approach (VCV) for boh equiies and foreign exchange porfolios, since he VCV approach considers and reveals direcly he volailiy and correlaion effec in he Value a risk (VaR) esimaion. The volailiy is updaed as in Hull and Whie (1988) procedure in order o capure he volailiy clusering. The more accurae and efficien variance covariance esimaion (VCV) is he one which gives he lower level of capial o cover agains unexpeced porfolio s losses and also he smaller average deviaion beween he esimaed VaR and he acual reurn. Brooks and Persand (003) showed ha he forecased porfolio s VaR based on he VCV approach is calculaed as: a VaR (T,a ) = F h i i i p, ,T p,+ 1,T where i=s-arcd, BEKK, ad-hoc GARCH(1,1), T is he forecas horizon, here equal o (14) 1 day period, α is he desired confidence level, F i,t 1 is he inverse of he cumulaive disribuion funcion and h i p,,t is he porfolio s forecased condiional variance which is given by he following ype: h = a h + b h + ab σ p, + 1 1, + 1, + 1 1, + 1 (14a) We resric our aenion o he variance covariance approach bu also and oher mehodologies such as hisorical simulaion or parameric Riskmerics approach could be applied as well. 15

16 where, h h 1,+ 1,+ 1 are he forecased condiional variances esimaed form he hree models for he indices and foreign currencies respecively and σ 1, + 1 is he forecased condiional covariance of he wo indices or he currencies esimaed form he respecive model, wih a and b being he proporion invesed in each asse. In his sudy he weighs a and b are each equal o 0.5, while he cumulaive disribuion funcion is he normal disribuion and he significance confidence level is chosen as 5% and 1% which corresponds o a value of δ equal o 1.65 and.33 for he normal disribuion respecively. In order o compare he VaR forecass accuracy esimaed by he hree models he following measures for VaR evaluaion are performed: Uncondiional Coverage Kupiec (1995) proposed an uncondiional es (LR un ) so as o es he proporion of imes VaR is exceeded in a given sample and under he correc VaR model wih he null hypohesis ha expeced violaion frequency is equal o he desired significance level. The LR un follows an asympoic chi-square disribuion wih one degree of freedom χ (1) and compues he appropriae likelihood raio saisic as: LR = ln (1 p) p + ln (1 N / T ) ( N / T) (15) un T N N T N N where T is he sample size, N is he number of failures or violaions, and p is he desired significance. Condiional Coverage Chrisofferson (1998) developed a es saisic (LR ind ) o accoun for uncondiional coverage and also for serial independence of VaR esimaes. This is very useful since we can conclude if a model rejecion is due he uncondiional coverage failure or clusering of he excepions or boh. For esing he independence of he VaR violaions, he saisic is asympoically χ disribued wih one degree of freedom and is derived as: [ π π π π π π π ] ind LR = v ln( /(1 )) + v ln((1 )/ ) + v ln( /(1 )) + v ln(1 )/ π ) (16) where v ij is he number of observaions of I wih value i followed by j, π 00 = v00 /( v00 + v01), 10 v10 v10 v11 π = /( + ), π = ( v01 + v 11)/( ν00 + ν01 + ν10 + ν11). Τhe indicaor I is consruced as: = 1, if exceedence occurs I. 0, if no exceedence occurs 16

17 The join es for condiional coverage capuring boh uncondiional coverage and he independence is simply given by he sum of he above individual ess and follows a χ disribuion wih wo degrees of freedom: cc un ind LR = LR + LR (16a). Roo Mean Square Error (RMSE) In VaR models evaluaion, he roo mean square error is a frequenly used measure of he difference beween he VaR esimaed values and he acually observed porfolios reurns. The model wih he smaller RMSE is considered as he mos accurae VaR forecasing model. I is defined as: 1 i RMSE = ( r + 1 VaR p, +1 T ) T = 0 (17) Sandard Deviaion of Capial Employed The Economic Capial or Capial Employed is considered as he amoun o be se aside in order o cover mos of he poenial losses a a predeermined level. Is sandard deviaion is calculaed as: 1 i i SD ( VaR ) = ( VaR p, + 1 VaRp) T T = 0 (18) i wherevar is he average esimaed porfolio VaR given by he following ype: i 1 = T VaRp VaR p T p = 0 i, + 1. The lower he sandard deviaion of he capial employed, he mos accurae is he model used for he VaR calculaion since he uncerainy of he compulsory capial used o cover he unexpeced porfolio s losses is reduced. 5. Resuls Summary saisics for he esimaed covariances σ ˆF, D,, σ,, ˆE G from he hree models are repored in Tables 5a and 5b, where σˆf, D, is he esimaed covariance for he FTSE and Dow equiy indices porfolio and σˆe, G, is he esimaed covariance for he EUR and GBP currencies porfolio. For he equiy indices, he S-ARCD model has he highes sandard deviaion, while for he exchange rae series he BEKK(1,1) model is he one which gives he lowes level of volailiy. Table 5a. Descripive Saisics of he esimaed covariance σ ˆF, D, 17

18 σ ˆF, D, Mean S. Dev. Skewness Κurosis Min Max S-ARCD Ad-Hoc GARCH(1,1) BEKK(1,1) Table 5b. Descripive Saisics of he esimaed covariance σ ˆE, G, σ ˆE, G, Mean S. Dev. Skewness Κurosis Min Max S-ARCD Ad-Hoc GARCH(1,1) BEKK(1,1) In Tables 6a, 6b he descripive saisics of he condiional correlaions, esimaed by he hree mulivariae models, are presened. The BEKK(1,1) model esimaes correlaion wih he lowes variabiliy, while he S-ARCD follows. The Ad-Hoc GARCH(1,1) model gives he mos variable mulivariae correlaion for boh daa series since he esimaed sandard deviaions are he highes. Obviously, he fied correlaion process for all he hree models remains beween -1< ρ <1 and 1< ρ <1, meaning ha he resuling esimaed correlaion marix saisfies he condiion for posiive semi-definieness for boh equiy indices and foreign currency daa. ˆE, G, Table 6a. Descripive Saisics of he esimaed correlaion ρ ˆF, D, ˆF, D, ρ ˆF, D, Mean S. Dev. Skewness Κurosis Min Max S-ARCD Ad-Hoc GARCH(1,1) BEKK(1,1) Table 6b. Descripive Saisics of he esimaed correlaion ρ ˆE, G, ρ ˆE, G, Mean S. Dev. Skewness Κurosis Min Max 18

19 S-ARCD Ad-Hoc GARCH(1,1) BEKK(1,1) In Tables 7a, 7b he correlaion marix beween he hree models is presened. More precisely, for he equiy indices daa he models S-ARCD and Ad-hoc GARCH(1,1) have he highes correlaion reflecing he fac ha are based on he same heory framework, bu he S-ARCD has higher correlaion wih BEKK(1,1) han he Ad-hoc model. Indeed, for he currency series, he highes correlaion is beween S-ARCD and BEKK(1,1) imposing ha ime varying higher momens such as skewness and kurosis play imporan role in he esimaion of mulivariae variance covariance marix and mus no ignoring hem such as in he case of Ad-hoc GARCH(1,1) model which has he lowes correlaion wih BEKK(1,1) model for boh cases. Table 7a. Correlaion marix for he esimaed correlaion ρ ˆF, D, ρ ˆF, D, S-ARCD Ad-Hoc GARCH(1,1) BEKK(1,1) S-ARCD Ad-Hoc GARCH(1,1) BEKK(1,1) Table 7b. Correlaion marix for he esimaed correlaion ρ ˆE, G, ρ ˆE, G, S-ARCD Ad-Hoc GARCH(1,1) BEKK(1,1) S-ARCD Ad-Hoc GARCH(1,1) BEKK(1,1) The resuls of he regressions for he saisical evaluaion are presened in he following Tables 8a and 8b. The null hypohesis of he uncondiional unbiasedness for all he covariance marix elemens is examined by he regressions (13) using an F-saisic in order o confirm if a mulivariae model is well and correcly specified. For he equiy indices, and heir esimaed variancesσ, σ F, D,, he null hypohesis of uncondiional unbiasedness is acceped for all mulivariae models, while for heir esimaed covariance, all he mulivariae models rejec he uncondiional 19

20 unbiasedness, wih he BEKK(1,1) model o have he weakes rejecion. For he currency daa, boh S-ARCD and BEKK(1,1) models accep he uncondiional unbiasedness for all he hree elemens of he condiional variance covariance marix, while he ad-hoc GARCH(1,1) model rejec again he null hypohesis. Overall, he BEKK(1,1) and he S-ARCD models are he ones wih he bes saisical evaluaion performance since he uncondiional unbiasedness condiion is rejeced only in one from he six cases. Table 8a. Uncondiional Unbiasedness esing Regressions for he equiy indices ˆ ε = θ + θ σ + v S-ARCD Ad-Hoc GARCH(1,1) BEKK(1,1) F, F,0 F,1 F, F, ˆF θ ,0 ˆF θ ,1 F-saisic ˆ D, = D,0 + D,1 D, + v D, ε θ θ σ ˆD,0 θ ˆD,1 θ F-saisic ˆ ε ˆ ε = θ + θ σ + v F, D, FD,0 FD,1 FD, FD, ˆFD,0 θ ˆFD,1 θ F-saisic The F saisic ess he null hypohesis ha ˆθ i,0 =0 and ˆθ i,1 =1 where i=f, D, FD and has an F(,1861) disribuion wih criical value 3.00 a he 5% significance level. Table 8b. Uncondiional Unbiasedness esing Regressions for he currency series ˆ ε = θ + θ σ + v S-ARCD Ad-Hoc GARCH(1,1) BEKK(1,1) E, E,0 E,1 E, E, ˆE,0 θ

21 ˆE,1 θ F-saisic ˆ ε = θ + θ σ + v, G, G,0 G,1 G, G ˆG,0 θ ˆG,1 θ F-saisic ˆ ε ˆ ε = θ + θ σ + v E, G, EG,0 EG,1 EG, EG, ˆEG,0 θ ˆEG,1 θ F-saisic The F saisic ess he null hypohesis ha ˆθ i,0 =0 and ˆθ i,1 =1 where i=e, G, EG and has an F(,195) disribuion wih criical value 3.00 a he 5% significance level. A rolling window of 100 observaions is used for he ou of sample esimaion of each model. The model which has he greaer percenage of VaR exceedences in he ou of sample period he highes RMSE and he highes sandard deviaion of he capial employed is ranked as he wors model, while he mos accurae is he one wih he lower percenage of exceedences and also wih he lowes RMSE and sandard deviaion of he capial employed. Tables 9a and 9b and ables 10a and 10b repor he esimaed five measures for he ou of sample VaR evaluaion of boh equiy and currency porfolios a he 95% and 99% levels respecively. Table 9a. Ou of sample VaR evaluaion measures for he equiy indices porfolio a 95% confidence level. FTSE-DJ Porfolio 95% conf. level un LR ind LR cc LR RMSE SD(VaR) 1

22 S-ARCD Ad-Hoc GARCH(1,1) BEKK(1,1) un The LR saisic ess he null hypohesis ha he proporion of VaR exceedences is equal o he nominal significance level and has an chi squared disribuion wih criical value 3.84 a ind he 5% significance level. The LR ess he null hypohesis ha he VaR exceedences are serially uncorrelaed and has an chi squared disribuion wih criical value 3.84 a he 5% cc significance level. The LR ess he null hypohesis of boh uncondiional coverage and ha he VaR exceedences are serially uncorrelaed and has an chi squared disribuion wih criical value 5.99 a he 5%. Table 9b. Ou of sample VaR evaluaion measures for he equiy indices porfolio a 99% confidence level. FTSE-DJ Porfolio 99% conf. level un LR ind LR cc LR RMSE SD(VaR) S-ARCD Ad-Hoc GARCH(1,1) BEKK(1,1) un The LR saisic ess he null hypohesis ha he proporion of VaR exceedences is equal o he nominal significance level and has an chi squared disribuion wih criical value 6.63 a ind he 1% significance level. The LR ess he null hypohesis ha he VaR exceedences are serially uncorrelaed and has an chi squared disribuion wih criical value 6.63 a he 1% cc significance level. The LR ess he null hypohesis of boh uncondiional coverage and ha he VaR exceedences are serially uncorrelaed and has an chi squared disribuion wih criical value 9.1 a he 1%. Table 10a. Ou of sample VaR evaluaion measures for he foreign currencies porfolio a 95% confidence level. EUR-GBP Porfolio 95% conf. level un LR ind LR cc LR RMSE SD(VaR) S-ARCD Ad-Hoc GARCH(1,1) E BEKK(1,1) un The LR saisic ess he null hypohesis ha he proporion of VaR exceedences is equal o he nominal significance level and has an chi squared disribuion wih criical value 3.84 a ind he 5% significance level. The LR ess he null hypohesis ha he VaR exceedences are serially uncorrelaed and has an chi squared disribuion wih criical value 3.84 a he 5% cc significance level. The LR ess he null hypohesis of boh uncondiional coverage and ha he VaR exceedences are serially uncorrelaed and has an chi squared disribuion wih criical value 5.99 a he 5%.

23 Table 10b. Ou of sample VaR evaluaion measures for he foreign currencies porfolio a 99% confidence level. EUR-GBP Porfolio 99% conf. level un LR ind LR cc LR RMSE SD(VaR) S-ARCD E Ad-Hoc GARCH(1,1) E BEKK(1,1) E un The LR saisic ess he null hypohesis ha he proporion of VaR exceedences is equal o he nominal significance level and has an chi squared disribuion wih criical value 6.63 a ind he 1% significance level. The LR ess he null hypohesis ha he VaR exceedences are serially uncorrelaed and has an chi squared disribuion wih criical value 6.63 a he 1% cc significance level. The LR ess he null hypohesis of boh uncondiional coverage and ha he VaR exceedences are serially uncorrelaed and has an chi squared disribuion wih criical value 9.1 a he 1%. All models provide correc uncondiional and condiional coverage close o he un ind cc significance levels since heir LR, LR, and LR values do no violae boh he 5% and 1% olerance levels, and herefore heir VaR forecass are adequae. More precisely, he S-ARCD model in he equiy indices porfolio achieves he required coverage in he ou of sample period, meaning ha here is no wase capial, while he ad-hoc GARCH(1,1) model is he one in he foreign currency porfolio. Summarizing he resuls for he uncondiional and condiional coverage, he VaR predicions obained from he hree models are all wihin he percenage exceedences hreshold. Since, all models have a good uncondiional and condiional coverage performance, he examinaion of he roo mean square deviaion and he sandard deviaion of he capial employed is required. For boh equiy and foreign currency porfolios, he S-ARCD model provides he lower RMSE a 5% and 1% levels reflecing is enhanced efficiency in he presence of lepokurosis since i capures ime varying skewness and kurosis, while he BEKK follows. Considering he capial employed, he S-ARCD model produces again he lowes sandard deviaion of he capial employed allowing he uncerainy, over he required capial reserved o cover agains unexpeced adverse price movemens, o be highly reduced. This is very imporan for risk managers and invesors since heir capial can be allocaed in oher more profiable asses. 5. Conclusions 3

24 This paper proposes a simple and effecive mulivariae version of ARCD model he S-ARCD o model condiional covariance processes which allows for ime variaion in higher momens and is consisen wih boh asymmeries and fa-ails ha are ypically observed in financial daa. Empirical resuls using he equiy indices FTSE, DJ and he foreign currencies EUR and GBP agains USD sugges ha a significan ARCD model can be esimaed for all series. Moreover, VaR esimaes via he S-ARCD offer superior ou-of-sample performance compared o he BEKK and ad-hoc GARCH(1,1) models respecively implying ha a pracical and compuaionally easier esimaion of he condiional covariance approach can be obained considering also he ime variaion of he higher momens. To he bes of our knowledge ha for he firs ime ARCD model is used for condiional covariance esimaion. On he basis of our resuls and he flexibiliy ha he S-ARCD offers, we believe ha furher empirical research is jusified. 4

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