DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń Ryszard Doman Adam Mickiewicz University in Poznań

Transcription

1 DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus Universiy Toruń Inroducion Adam Mickiewicz Universiy in Poznań Measuring Condiional Dependence of Polish Financial Reurns Idenificaion of condiional dependence srucure beween financial insrumens undoubedly belongs o he main challenges of modern finance and insurance. Among ohers, i is a crucial ask in pricing baske derivaives or in porfolio risk managemen. The basic heoreical framework applied in his area is ha of mulivariae volailiy model describing he volailiy of several ime series joinly in order o exploi possible connecion beween heir dynamics. Very useful and popular ools for modeling he dynamic condiional covariances or correlaions among various asses are parameric mulivariae GARCH (MGARCH) models (see a survey by Bauwens e al., 23). In heir convenional form MGARCH models assume ha he sandardized innovaions follow mulivariae Gaussian process. In a more general seing, usually a mulivariae ellipical disribuion for he innovaion is allowed. The mulivariae normal disribuion is, however, no consisen wih such sylized facs abou financial reurn disribuions like asymmery and ail-faness, and he mos common non- Gaussian ellipical disribuion mulivariae Suden s imposes, ofen unrealisically, he same degrees of freedom for all marginal disribuions. Recenly, Lee and Long (25) proposed a new model named Copula-based Mulivariae GARCH model (C-MGARCH) which permis modeling condiional correlaions and possible more hidden dependence separaely and simulaneously in he case where he sandardized residuals of considered financial reurns are allowed o be non-ellipically disribued and dependen. The class of C- MGARCH models includes MGARCH models as special cases bu because of incorporaing a copula apparaus C-MGARCH models can capure and describe he dependence srucure ha may be negleced by he condiional covariance. Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House

2 6 In his paper we apply C-MGARCH mehodology o model condiional dependency beween pairs of seleced Polish financial reurns. We compare he dynamic condiional correlaions esimaed by means of C- model belonging o he C-MGARCH class wih hose obained wih Engle s model (Engle 22). We also compare he 1-day ahead condiional correlaion forecass calculaed wih and C- models. In addiion, by using Euclidean marix norm, we evaluaed how he implied condiional covariance forecass fi he marices of cross producs of acually realized daily reurns. Our main finding is ha he condiional correlaions obained wih he applied C- model (for all he considered pairs hey are almos everywhere posiive) are, in fac oally, much lower han he ones esimaed wih Engle s model. As regards he poin forecass of marices of cross producs of daily reurns, we find ha models are beer in ha ask bu, alogeher, we find he resuls no very impressive. 2. Mulivariae Parameric Volailiy Models For a mulivariae reurn series r ( r, K, ), consider he decomposiion = 1, r k, r = μ + y, (1) where μ = E( r Ω 1) and Ω 1 is he informaion se available a ime 1. A general mulivariae volailiy model for he residual process y is given by he equaion y = 1/ 2 ε, (2) H where E ( y y Ω 1) = H and hus E( ε ε Ω 1) = I. A mulivariae GARCH (MGARCH) model can be obained by describing a specific parameerizaion for he condiional covariance marix H. There exis many such parameerizaions (Bauwens e al., 23). In his paper we resric ourselves o a very simple parameerizaion (dynamic condiional correlaion) proposed by Engle (22) and is copula-based exension C- (Lee and Long, 25). The idea of model is o consider he evoluion no of he condiional covariance marix H bu raher he condiional correlaion marix R. The -GARCH model is described by he following specificaion y 1 Ω ~ N (, H ), (3) Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House = ( ρij, hii, h jj ) ( h, K h ) H D R D, =, (4) D diag 11,, kk, =, (5)

3 Measuring Condiional Dependence of Polish Financial Reurns 61 h q p = ii, ωi αij yi, j βij j= 1 j= 1 h ii, j ( ( )) diag Q Q ( diag( Q )) =, (6) R, (7) M N M N Q = + 1 α m βn Q α mu-mu-m + βnq n, (8) m= 1 n= 1 m= 1 n= 1 1 where u = D y, and Q is he uncondiional covariance marix of he series u. In he following, we apply he simples model wih M = N = p = q =1. The model can be esimaed by wo-sep maximum likelihood mehod (Engle, 22). 3. Copula-based Exensions of MGARCH Models Le η 1, η 2 be random variables wih disribuion funcions F and G. If F and G are coninuous hen he copula of ( η 1, η2 ) is he funcion C wih domain [,1] [,1] which is he (resriced o he square [,1] [,1] ) join disribuion funcion of he variables U = F( η 1 ) and V = G( η 2 ). If H is he join disribuion funcion of ( η 1, η2 ) hen, in he above siuaion, by Sklar s heorem (1959), here exiss he unique copula C such ha H ( x, y) = C( F( x), G( y)). (9) Thus copula allows o decompose he join disribuion ino wo pars: marginal disribuions and dependence srucure. We are no going o repor deails concerning general properies and applicaions of copulas, referring o (Nelsen, 1999). The main improvemen in MGARCH model srucure proposed by Lee and Long (25) consiss in rejecing he very resricive condiion (3) and posulaing insead ha 2 ε = Σ 1/ η, η η, η ), (1) = ( 1, 2, η Ω ~ C ( F ( ), G ( ); θ ), (11) 1 where C is possible ime-varying copula of ( η 1,, η2, ) dependen on parameer vecor θ and he marginal disribuion funcions F and G of ( η 1,, η2, ) are also allowed o vary in ime. The main advanage of he repored approach is ha he ε are sill uncorrelaed bu can be dependen and he dependence srucure is conrolled by hidden variables η, ) and heir copula. The off- Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House ( 1, η2, C

4 62 diagonal elemen σ 12, of he covariance marix Σ of η is deermined by he copula and he marginal disribuion funcions F and G. By Heoffding s Lemma (Lehmann, 1966) i can be compued by he formula σ E η η Ω ) = ( C ( F ( x), G ( y)) F ( x) G ( y) )dxdy. (12) 12, = ( 1, 2, 1 2 R The log-likelihood funcion for η has he form L Θ, η ) = ln( f ( η )) + ln( g ( η )) + ln( c ( F ( η ), G ( η ))), (13) ( 1, 2, 1, 2, where f, g, and c are he corresponding densiy funcions. One can easily derive he corresponding formula for r, aking ino accoun ha r = μ + y 1/ 2 1/ 2 and y = H Σ η. The C- model reduces o he convenional model when C = C is he produc copula and he marginal disribuions are sandard normal. 4. The Daa and Model Specificaion The daa we analyze consis of daily reurns on wo exchange raes EUR/PLN and USD/PLN, and hree sub-indices of he sock index WIG published by he Warsaw Sock Exchange. The sub-indices under scruiny are WIG-consrucion (WIG-con), WIG-IT and WIG-food. All observaions are from he period November 17, 2 March 23, 25. They are divided ino wo groups. The firs 99 observaions were used for in-sample esimaion. On he base of he remaining 12 reurns from November 2, 24 we have done one-day-ahead forecasing. Our analysis concerns he reurns defined as r = 1(ln P ln P 1), where P is he closing quoaion on day. Basic descripive saisics for he reurn series are presened in Table 1. Table 1. Descripive saisics for he reurn series (November 17, 2 March 23, 25) EUR/PLN USD/PLN WIG-con WIG-IT WIG-food Mean Sd. Dev Min Max Skewness Kurosis Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House

5 Measuring Condiional Dependence of Polish Financial Reurns 63 We esimaed he models given by he equaions (1) (8). For our C- models, we assumed he sandardized Suden s disribuion wih ν 1 and ν 2 degrees of freedom for he marginals η 1, = η1 and η 2, = η2. As he copula of η, ) we have chosen Frank copula ( 1 η2 1 (1 exp( θu))(1 exp( θv)) C Frank ( u, v; θ ) = ln 1. (14) θ 1 exp( θ ) 5. Empirical Resuls The esimaed parameers of he and C- models are presened in ables 2 and 3. Prior o he parameer esimaion we filered he pairs of raw reurn series by means of VAR(1) model and so we could assume μ =. We do no repor here he VAR(1) parameers, as well as hose of he univariae GARCH models. In he case of C- model, due o he join esimaion, we had o replace he equaion of form (8) by he following one Q α. (15) = CC + u-1 u-1 + β Q 1 Table 2. Parameers of he esimaed models parameers α β EUR/PLN USD/PLN.225 (.95).943 (.249) WIG-con WIG-IT.149 (.86).9751 (.167) WIG-con WIG-food.272 (.154).9425 (.371) WIG-IT WIG-food.77 (.48).992 (.28) Table 3. Parameers of he esimaed C- models C- EUR/PLN WIG-con WIG-con WIG-IT parameers USD/PLN WIG-IT WIG-food WIG-food ν 1 (.6594) (.4288) (.4498) (1.474) ν 2 (.9974) (.6722) (983) (.6931) α (.246) (.32) (.269) (.83) β (.216) (.21) (.912) (437) θ (994) (.4228) (.4242) (.4161) C σ Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House 12

6 64 I is no obvious o predic from he values of he parameers how he corresponding series of dynamic condiional correlaions can differ from each oher. As we can see in he following figures 1-4, he esimaed condiional correlaions from C- models are much lower han hose from models for almos all he ime. The same is also rue for he forecass presened in figures 5-8. We also ried o check how he condiional covariance marices implied by he obained condiional correlaion forecas fi he marices of cross producs of he acually realized daily reurns bu we found ha resul no very impressive and do no repor i here C Fig. 1. Comparison of he condiional correlaions for EUR/PLN and USD/PLN C- Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House Fig. 2. Comparison of he condiional correlaions for WIG-con and WIG-IT

7 Measuring Condiional Dependence of Polish Financial Reurns C_ Fig. 3. Comparison of he condiional correlaions for WIG-con and WIG-food.7.6 C Fig. 4. Comparison of he condiional correlaions for WIG-IT and WIG-food C Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House Fig. 5. Forecass of he condiional correlaions for EUR/PLN and USD/PLN

8 C Fig. 6. Forecass of he condiional correlaions for WIG-con and WIG-IT C Fig. 7. Forecass of he condiional correlaions for WIG-con and WIG-food C- Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House Fig. 8. Forecass of he condiional correlaions for WIG-IT and WIG-food

9 Measuring Condiional Dependence of Polish Financial Reurns Conclusions In his paper we model condiional dependence for pairs of seleced Polish financial reurns using he C- model. The model belongs o he class C- MGARCH ha includes he convenional mulivariae GARCH models and by incorporaing copula mehodology allows o model he condiional correlaions and he remaining dependence separaely and simulaneously wihou limiaion o ellipically disribued errors. We find ha he copula-based models applied by us produce much lower esimaes of he condiional correlaion han he sandard models do. In our opinion, a possible explanaion of his phenomenon, apar from he wo-sage esimaion imperfecions, is ha he copula improved model srucure managed o idenify and separae he dependence ha was misakenly qualified as linear condiional correlaion by he simpler model. Our resuls regarding he applicaion of C- and models o forecasing he marices of reurn cross producs are no very impressive hough he simpler model has gone beer. We suppose ha he resuls in his subjec could look more ineresing if we would compare he forecass wih he so-called daily realized covariance (Andersen e al. 25) marices calculaed on he basis of inraday quoaions. References Andersen, T.G., Bollerslev, T., Chrisoffersen, P.F., Diebold, F.X. (25), Pracical Volailiy and Correlaions Modeling for Financial Marke Risk Managemen, Penn Insiue for Economic Research Working Paper 5-7. Bauwens, E. Lauren, S. Rombous, J.V.K. (23), Mulivariae GARCH Models: A Survey, Core Discusion Paper 23/1. Engle, R.F. (22), Dynamic Condiional Correlaion: A Simple Class of Mulivariae GARCH Models, Journal of Business and Economic Saisics 2, Lee, T.H., Long, X. (25), Copula-based Mulivariae GARCH Model wih Uncorrelaed Dependen Sandardized Reurns, Deparmen of Economics, Universiy of California, Riverside. Lehmann, E.L. (1966), Some Conceps of Dependence, Annals of Mahemaical Saisics, 37, Nelsen, R.B. (1999), An Inroducion o Copulas, Springer Verlag, New York. Sklar, A. (1959), Foncions de repariion à n dimensions e leurs marges, Publicaons de Insiu Saisique de Universie de Paris 8, Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House