Bayesian estimation of correlation matrices: Application to the Multivariate GARCH model

Transcription

1 Bayesian esimaion of correlaion marices: Applicaion o he Mulivariae GARCH model By Pierre Evarise NGUIMKEU NGUEDIA To fulfill he degree of Maser in Economics Advisor: Prof. William McCAUSLAND Deparmen of Economics Universiy of Monreal Augus, 28

2 Absrac * The correlaion marix (R) plays an imporan role in many saisical and financial models. Unforunaely, sampling he correlaion marix can be problemaic because of is posiive definie consrain and is diagonal elemens fixed a. In his work, we use he geomeric properies of he correlaion concep ogeher wih a Meropolis random-walk algorihm o esimae he consan correlaion marix in he Mulivariae GARCH models as well as is parameers. The heory of he MH algorihm used o esimae he parameers of he model and sufficien condiions o implemen i are shown in deails. We also give an overview of he mehodology used o sample he correlaion marix. Using hese algorihms, we draw all elemens of R by simulaing a common prior disribuion for all correlaions and acceping poserior draws based on a Random Walk Meropolis-Hasings accepance probabiliy. The algorihm is applied using a mulivariae GARCH model on financial ime series from hree daily sock reurns from he energy secor. We hen provide some ineresing poserior disribuions illusraion and model parameers esimaes. Key words: Correlaion marix, Bayesian inference, GARCH, Meropolis- Hasings. * The auhor would like o grealy hank Professor William McCAUSLAND for his availabiliy, his invaluable suppor and for having iniiaed him o Bayesian economerics

3 Conens I. INTRODUCTION... II. THE MULTIVARIATE GARCH MODEL... 3 II.. Model specificaion and noaions... 3 II. 2. Prior specificaion and poserior analysis... 4 III. BAYESIAN COMPUTATION OF THE MODEL PARAMETERS: THE METROPOLIS-HASTINGS ALGORITHM... 7 III.. An overview of he Meropolis-Hasings algorihm... 7 III. 2. An overview of he Correlaion marix Sampling using he geomerical properies of he linear correlaion... 8 III. 3. Sampling he full condiionals of he model parameers and correlaion marix elemens... 8 IV. APPLICATION TO FINANCIAL TIME SERIES... 9 IV.. Daa and preliminary analysis... 9 IV. 2. Parameers esimaes... IV. 3. Correlaions esimaes... 6 V. CONCLUSION... 9 References 2

4 Lis of figures Figure : Reurns of Exxon Mobil (XOM) and Toal (TOT) from 8//26 o 7/3/28 Figure 2: Reurns of Chevron (CVX) and Toal (TOT) from 8//26 o 7/3/28... Figure 3: Reurns of Chevron (CVX) and Exxon Mobil (XOM) from 8//26 o 7/3/28... Figure 4: Hisograms and draws of he poserior sample of he GARCH parameers for XOM (N=)... 2 Figure 5: Hisograms and draws of he poserior sample of he GARCH parameers for TOT (M=)... 3 Figure 6: Hisograms and draws of he poserior sample of he GARCH parameers for CVX (N=)... 5 Figure 7: Plos of hisograms and draws of he correlaion marix elemens... 7 Lis of ables Table : Parameers esimaes for XOM (M=)... 2 Table 2: Parameers esimaes for TOT (M=)... 4 Table 3: Parameers esimaes for CVX (M=)... 5 Table 5: Poserior Correlaion Marix (means and sandard deviaions) of he M-GARCH model obained wih M= acceped draws... 8

5 I. INTRODUCTION The correlaion marix (R) is direcly involved in a variey of financial models. Indeed, he imporance of risk and uncerainy in modern economic heory and he saisical properies of asse reurns have necessiaed he developmen of new economeric ime series echniques ha allow for modeling variances and correlaions. The Generalize auoregressive condiional heeroscedasic (GARCH) model (Bollerslev e al. (992)) has been very popular in modeling he volailiy of financial ime series. The exension of univariae GARCH-ype models o a mulivariae framework and he esimaion of correlaions beween asse reurns are imporan for asse pricing, risk managemen and porfolio analysis; see, for example, Gourieroux (997). Correlaions may also be useful o give subsanive informaion abou firms behavior. For example, for companies engaged in sraegies where hey expand and/or change he producs and services ha hey offer or for hybrid companies ha may no fi o a specific indusrial classificaion (for example, energy or finance), i may be of ineres o deermine wheher heir sock behavior is correlaed wih ha of a paricular class of companies (see Liechy, Liechy and Muller (24)). Sampling he correlaion marix in such models is hus necessary bu can be problemaic due o hree major problems: The larger number of parameers o be esimaed, he difficuly of he esimaion due o he posiive definieness resricions and he fixed diagonal elemens of he correlaion marix. Due o hese difficulies, several mehods of simulaion have been proposed: Barnard, McCulloch and Meng (2) suggesed using he Griddy Gibbs sampler o draw he componens of a correlaion marix one a a ime in he conex of a hierarchical shrinkage model for he marginal variances of a covariance marix.

6 Alhough he Griddy Gibbs sampler is simple o implemen, i is no compuaionally efficien. Liechy, Liechy and Muller (24) provide anoher approach o simulae all he componens of he correlaion marix one by one hrough inroducion of a laen variable. Oher similar approaches have been discussed in Bowen and Lombrano (998), Daniels and Kass (999). Belin (24) proposed a parameer-exended Meropolis-Hasings algorihm (PX-MH) for sampling R in Bayesian models wih correlaed laen variables. The seminal idea in heir mehod is ha insead of a marginal prior for R, hey specified a join prior for R and D (unidenified marginal variances) derived from some inverse Wishar disribuion. Then sampling (R;D) joinly was accomplished hrough a Meropolis Hasings algorihm. Liu and Daniels (26) propose a wo-sage parameer expanded reparameerizaion and Meropolis-Hasings (PX-RPMH) algorihm for simulaing a correlaion marix from is condiional disribuion. In sage, R can be ransformed ino a less consrained covariance marix, say Σ = DRD, such ha he poserior disribuion of Σ is an inverse Wishar disribuion. In he second sage, simulaing R in he original model is equivalen o firs simulaing Σ from he inverse Wishar disribuion in he new model, and hen ranslaing i back o R hrough he reducion funcion (R=D - ΣD - ) and acceping i based on a Meropolis-Hasings (M-H) accepance probabiliy. In his sudy, we use he geomeric properies of he linear correlaion concep ogeher wih a Meropolis random-walk algorihm o esimae he correlaion marix in he Mulivariae GARCH models as well as is parameers. The common correlaion approach (Liechy and Liechy 24) we use here assumes a common prior for all correlaions, wih he addiional resricion ha he correlaion marix is posiive definie. The former condiion is obained by simulaing a common prior disribuion for all correlaions. The laer condiion is sraighforward as i follows 2

7 from he Cholesky produc of uni vecor of he marice rows compued during he algorihm. The res of he work is organized as follows. In Secion 2, we review Mulivariae GARCH (M-GARCH) models and echniques o sample from heir respecive poserior disribuions. In Secion 3, we presen he geomeric approach for sampling a correlaion marix. This algorihm is derived for M-GARCH models wih a common correlaion prior. We also presen a Meropolis-Hasing algorihm used o sample he model parameers. Their implemenaion and properies are explained in deail. A simulaion sudy and applicaion o real daa from hree daily sock reurns from hree companies of he energy secor is repored on in Secion 4. Finally, we summarize our conclusions. II. THE MULTIVARIATE GARCH MODEL We wish o conduc Bayesian inference on a regression model wih GARCH errors. For simpliciy we use he GARCH (, ) model which is quie represenaive of GARCH models used in finance. II.. Model specificaion and noaions We consider a mulivariae financial ime series observaions y,.., T wih K elemens each, so ha ' y ( y,..., y K ) and we assume ha he observaions are of zero mean. Le y, ie: H denoe he ime-varying condiional covariance marix of Var( y I ) H Where I is he -field generaed by all he available informaion up o ime, and H is almos surely posiive definie for all. The variance elemens of H are h, for i,..., K, and he covariance elemens are h ij ij, where i 2 i i j K. 3

8 Following Bollerslev (99), he full condiional covariance marix be pariioned as: H may H SRS,..., where S denoes he K x K sochasic diagonal marix wih elemens, and R is a K x K ime-invarian marix wih elemens K r ij. Define S y. Thus, is he sandardize residual, and is assumed o be serially independenly disribued wih mean zero and variance marix R. To specify he condiional variance of y, a univariae GARCH (,) model for he marginal variances hi ( H ) ii can be defined as : y h i i i i h N(,) h 2 i i i i, i i, Researcher adoping he vech-diagonal form ypically assume ha he above equaion also apply o he condiional covariance erms hij ( H ) ij, in which h i is replaced by h ij and 2 i by i j. In our framework, we model mulivariae ime series daa by assuming a univariae GARCH (,) model for marginal variances as defined above, and compleing he model wih a srucured prior for correlaion marix, using he prior models proposed in he following paragraphs. II. 2. Prior specificaion and poserior analysis Prior specificaion In order o esimae he poserior disribuion, a prior disribuion (, R) of he unknown parameers and R needs o be specified. For Bayesian inference, i 4

9 is common o specify independen priors for and R. In absence of prior knowledge, i is also desirable o have uninformaive priors on he parameers we are esimaing. Prior specificaion on i, i and i Le denoes he parameer vecor (,, i i i i ), i,.., K, and ( ) K i i The prior densiy ( ) should respec a leas he posiiviy resricions on he parameers, ha is : i, i and i. This ensures he posiiviy of he variance h i. Also, for he y i process o be covariance-saionary, we mus impose ha: i i e al. (2). Prior specificaion on R Many possible choices of diffuse priors on R have been discussed by Bernard The commonly used is he proper joinly uniform prior: ( R ), R T Where he correlaion marix space T is a compac subspace of ( )/ 2, T T defined as follows: T r : r and R is posiive definie, i, j,..., T ij ij However, as shown by Bernard and al. (2), he poserior disribuion resuling in his prior is no easy o sample. Furhermore, his prior will end o favour marginal correlaions close o zero. Anoher commonly used uninformaive prior is he Jeffrey s prior defined by: T 2 ( R) R, R T 5

10 While his prior helps faciliae compuaions, i has been shown ha i suffers from he disadvanage of being improper. Our approach in his work uses he mehod described by Nzabandora (27) 2 for simulaing prior disribuion by an iid draw. This model assumes a common orienaion angle drawn from an arbirary disribuion from which a marix of all correlaions can be obained hrough appropriae geomerical ransformaions of he corresponding hypersphere vecors. Poserior analysis We analyze he marginal condiional poserior densiies of he model parameers and he correlaion elemens. For a sample of T observaions, he likelihood funcion derived from he M- GARGH model is given by: wih: ha is: T l( y, R) l ( y, R ) l y R H y H y 2 2 ' (, ) 2 exp l y R h R y S R S y n 2 ' (, ) 2 i 2 exp i 2 is : Poserior densiy of i, i and i The poserior densiy of ( ) K i i given he observaions y and he marix ( y, R) ( ) l( y, R ) 2 Nzabandora, W. (27). Esimaion bayesienne des marices de correlaions: sraegies de formulaion de lois a priori. Deparmen de Sciences Economiques, Universie de Monreal 6

11 Poserior densiy of R The poserior densiy of given he observaions and he parameers is: III. BAYESIAN COMPUTATION OF THE MODEL PARAMETERS: THE METROPOLIS-HASTINGS ALGORITHM In his secion, we give an overview of he Meropolis-Hasings algorihm and we describe he way of using his echnique o compue he parameers esimaes of he GARCH model. We also we also give an overview of he algorihm used o sample he correlaion Marix using he geomerical properies of he concep of linear correlaion. III.. An overview of he Meropolis-Hasings algorihm The Meropolis-Hasings algorihm is used in cases he poserior iself is hard o ake random draws from, bu a candidae generaing densiy exiss. If we adop he general noaion wih following form: as a vecor of parameers, he algorihm always akes he Sep : Choose a saring value, () Sep : Take a candidae draw, ( s ) q ( ; *). * from he candidae generaing densiy, Sep 2: Calculae an accepance probabiliy, ( s ) (, *) Sep 3: Se ( s) * wih probabiliy ( s ) (, *) and se ( s) ( s ) wih probabiliy - ( s ) (, *). Sep 4: Repea seps, 2, and 3 S imes. Sep 5: Take he average of he S draws funcion of ineres. () ( S) g( ),..., g( ), where (.) g is he These seps will lead o an esimae of E[ g( ) y ]. 7

12 In our framework, we do no have sufficien informaion o find a good approximaing densiy for he poserior. We hen use he Random Walk Chain Meropolis-Hasings algorihm where he candidae generaing densiy is chosen o wander widely, aking draws proporionaely in various regions of he poserior. III. 2. An overview of he Correlaion marix Sampling using he geomerical properies of he linear correlaion This mehod is widely discussed in Nzabandora (27). The idea is o compue uni vecors from a hypersphere and use hose vecors as rows of a marix which is used o compue a posiive definie marix, he correlaion marix. Updaing he correlaion marix requires o updae each line of he Cholesky marix obained from is Cholesky decomposiion. To achieve his, an adjusmen angle is drawn from an arbirary disribuion and new candidaes vecors are derived geomerically from hese angles. The new vecors draws are hen chosen wih respec o he accepance probabiliies compued from he associaed probabiliy disribuions. III. 3. Sampling he full condiionals of he model parameers and correlaion marix elemens Recall ha we wan o make a joinly esimaion of boh he parameers and he correlaion coefficiens of he model. Sampling he full condiionals of i, i and i Going from some saring values as described above, we sample he full condiionals of he model parameers wih an algorihm ha updaes he parameers candidaes wih a random-walk Meropolis-Hasings algorihm, aking he correlaion marix in each sep as given. 8

13 Sampling he full condiional of The correlaion marix (or, precisely, he line vecors of is Cholesky marix) is sampled wih he procedure described above, aking he model parameers in each sep of he algorihm as given. We consider he full condiionals of model above. s given by he common correlaion Recall ha for a given arge densiy, he Meropolis-Hasings algorihm consiss of a proposal densiy, which supplies candidae vecor * given he curren vecor and a probabiliy move: which deermines wheher he proposal value is acceped. The proposal densiy need no enforce he posiive definieness consrain, because ha consrain is sraighforward, since from. Where is he marix whose rows are formed by * s If his procedure is repeaed a number of ime big enough, he marices obained above follow he same prior disribuions as * s. Then, according o he Law of large Numbers, he mean of he sample correlaion marix converges almos surely hrough he expecaion of he rue correlaion marix of he model. IV. APPLICATION TO FINANCIAL TIME SERIES In his secion we presen he daa used for empirical purposes and we apply he algorihm o derive parameers esimaes and correlaion marix from he M- GARCH model. IV.. Daa and preliminary analysis For our empirical evidence, we examine he sock reurns from hree equiy securiies of he energy secor: Exxon Mobil (XOM), Toal (TOT) and Chevron (CVX). Our analysis requires hisorical prices daa. These hisorical prices are pulled from Yahoo finance websie. Our daily hisorical prices range from Augus s, 26 o July 3 s, 28. The figure below displays he evoluion of he sock reurns during he above period. 9

14 Figure : Reurns of Exxon Mobil (XOM) and Toal (TOT) from 8//26 o 7/3/ XOM TOT Figure 2: Reurns of Chevron (CVX) and Toal (TOT) from 8//26 o 7/3/ CVX TOT Figure 3: Reurns of Chevron (CVX) and Exxon Mobil (XOM) from 8//26 o 7/3/ XOM CVX The above figures presen he evoluion of reurns for our hree sock securiies in a wo-by-wo comparison: XOM/TOT, CVX/TOT and XOM/CVX. The sock prices are ransformed ino reurns hrough firs differences of he logarihms.

15 The resuls clearly display a posiive correlaion beween he asses during he period of sudy. The reurns from Toal appear o have lower variaions (or flucuaions) compared o he reurns from Mobil and Chevron which vary (or flucuae) wih fairly he same inensiy. IV. 2. Parameers esimaes Before compuing he parameers esimaes using our mehod, we esimae each equaion of our mulivariae model using he classical maximum likelihood esimaion. The resuls of his esimaion are use for comparison purposes. We launched he Meropolis chain of M= acceped draws for he parameer esimaes as well as for he correlaion coefficiens. To respec he consrain on he parameers, some of he draws had o be discarded. In sum he procedure was ouline as follows: For m= o M, - Draw he Garch parameers of each asse, given daa and correlaion marix - Draw he correlaion marix, given daa and Garch parameers of all asses End The following figures depic he hisograms of he poserior disribuions of he parameers as well as heir behaviour obained from he sampling.

16 Figure 4: Hisograms and draws of he poserior sample of he GARCH parameers for XOM (M=).2 omega 2. 5 alpha bea Table : Parameers esimaes for XOM (M=) Parameers Omega Alpha Bea Esimaes.43 (.42).8784 (.2387).766 (.67) Quaniles ML esimaes The able shows Bayesian esimaes of he GARCH parameers of XOM obained from M= draws using Meropolis random-walk. The numbers in brackes are sandard deviaions. The hree following columns display he values of he quaniles of order.5,.5 and.95 obain from he parameers disribuions. The las column shows he Maximum Likelihood esimaors of he parameers. 2

17 Poserior resuls obained from he mehod for Exxon Mobil (XOM) are displayed in Table. They are fairly closed o he resuls obained from he Maximum Likelihood esimaion, bu poserior means of, and seem a bi higher. The hisory of sampled parameers shows ha he values of he parameers are saionary around he zone of higher poserior probabiliies. Looking a he hisograms of he disribuions, we find ha for he parameers and, he algorihm did no explore he ails of he disribuion. Neverheless, he peaks of all he disribuions are locaed a he rue values of parameers. This shows ha he algorihm is converging. Figure 5: Hisograms and draws of he poserior sample of he GARCH parameers for TOT (M=).2 omega alpha bea

18 Table 2: Parameers esimaes for TOT (M=) Parameers Omega Alpha Bea Esimaes.53 (.47).944 (.246).65 (.539) Quaniles ML esimaes The able shows Bayesian esimaes of he GARCH parameers of XOM obained from M= draws using Meropolis random-walk. The numbers in bracke are sandard deviaions. The hree following columns display he values of he quaniles of order.5,.5 and.95 obained from he parameers disribuions. The las column shows he Maximum Likelihood esimaors of he parameers. Table 2 shows poserior resuls for Toal (TOT) as well as he Maximum likelihood esimaes of is GARCH parameers. The able presens poserior means (he average of he sampled values) of he model parameers and he poserior sandard deviaions of he poserior means (he sandard deviaions of he sampled values). The parameers esimaes obained by he mehod are fairly closed o he MLE s. Bu unlike he parameers of Mobil (XOM) above, he poserior mean of in his case is smaller han is MLE counerpar. The quanile of order.5 is very closed o he esimaors. This means ha he median and he mean are fairly closed, a resul ha always characerize random variables wih unimodal disribuions. The hisograms illusraed in figure 5 display he same behavior as above wih in ails and peaks closed o he rue values of he parameers. 4

19 Figure 6: Hisograms and draws of he poserior sample of he GARCH parameers for CVX (M=).2 omega 2. 5 alpha bea Table 3: Parameers esimaes for CVX (M=) Parameers Omega Alpha Bea Esimaes.388 (.47).8935 (.2373).699 (.57) Quaniles ML esimaes The able shows Bayesian esimaes of he GARCH parameers of XOM obained from M= draws using Meropolis random-walk. The numbers in bracke are sandard deviaions. The hree following columns display he values of he quaniles of order.5,.5 and.95 obain from he parameers disribuions. The las column shows he Maximum Likelihood esimaors of he parameers. 5

20 Poserior resuls obained by he sampling of he GARCH parameers of Chevron (CVX) are shown in Table 3. In addiion o he commens made in he preceding cases, he resuls show ha he esimaed values respec he posiiviy resricions on he parameers while ensuring he posiiviy of he variance of he model variables. This resul is also rue for Mobil and Toal. The hisograms of he poserior sample of he model parameers illusraed in figures 6 have he same behaviour as he previous ones described above. For all hose sampled parameers quaniles of order.5 appeared o be zero. IV. 3. Correlaions esimaes We sampled daa under Mulivariae GARCH model wih hree elemens subjec a T=54 ime poins. The empirical correlaion marix obained from he daa is: We launched he Meropolis chain of M= acceped draws wih he prior disribuion on he uni vecors s forming he rows of he Cholesky marices. The Meropolis-Hasings sep was applied one by one o all he componens of he correlaion marix. Figures 7, which plo he poserior draws of he correlaion elemens of he correlaion marix, depic he behaviour of he parameers values during he sampling. 6

21 Figure 7: Plos of hisograms and draws of he correlaion marix elemens R 2 R R 2 R R 3 R Poserior resuls obained by his sampling are shown in Table 5. The able presens poserior means (he average of he sampled values) of he correlaion marix elemens and he poserior sandard deviaions of he poserior means (he sandard deviaions of he sampled values). 7

22 Table 4: Poserior Correlaion Marix (means and sandard deviaions) of he M- GARCH model obained wih M= acceped draws XOM XOM TOT CVX.7224 (.28) TOT.724**.8829 (.7).767 (.37) CVX.882**.766** The able shows Bayesian esimaes of he GARCH Correlaion coefficiens obained from draws using Meropolis random-walk. The numbers in bracke are sandard deviaions. **The lower riangular marix shows he Empirical correlaion coefficiens. The poserior means of he elemens of he correlaion marix respec he classical resricions and ensure he posiive definieness of he correlaion marix. Furhermore, he coefficiens of his marix have values closed from heir empirical correlaion marix counerpar, and his can be explained by looking a he hisograms of he poserior sample of correlaion elemens (see Figures 7 above). because he poserior disribuions for all he coefficiens are closed o normaliy, he esimaes of he parameers and he corresponding sandard deviaions have a sraighforward inerpreaion. In summary, he resuls of our algorihm have he following feaures: - he esimaed parameers of ineres converge o a fla region of higher poserior probabiliy and hen flucuae around ha region. - Hisograms ends o have a normal shape wih peaks closed o he value of he esimaes - The Quaniles of order.5,.5 and.95 are closed o zero, o he mean and o he maximum value of he associaed parameer respecively 8

23 - The esimaed parameers are closed o heir MLE counerpars - All he consrains on parameers and correlaion coefficiens and marix are respeced a poseriori V. CONCLUSION In his sudy, we provide an empirical analysis of he Mulivariae GARCH model wih consan condiional correlaion. We use financial daa from hree daily sock reurns from he energy secor of he New York sock marke o apply he mehod. The esimaion of he parameers of he models ogeher wih he correlaion marix under consideraion is obained by using he Random Walk chain Meropolis-Hasings algorihm ogeher wih a geomerical approach of compuing he linear correlaions. We provide deailed guidelines on how o consruc he required Markov chains using Meropolis Hasings seps. Under a Bayesian framework, we hen consruc samples of he GARCH parameers and he Correlaion Marix elemens which have as saionary disribuions he poserior disribuions of he model. Finally, we find ha his algorihm is compuaionally efficien and provide good poserior samples as he esimaed parameers converge o a fla region near he rue parameers, respec he required consrains of posiiviy, posiive definieness and boundedness, and yield poserior probabiliy disribuions ha are easy o inerpre. 9

24 References [ ] Barnard, J., McCulloch, R. and Meng, X.L. (2). Modeling covariance marices in erms of sandard deviaions and correlaions wih applicaion o shrinkage. Saisica Sinica, 28-3 [ 2 ] Bollerslev T, Chou RY, Kroner KF ARCH modeling in finance A review of he heory and empirical evidence. Journal of Economerics 52: [ 3 ] Bowen, L. and Lombrano, M. (998). Bayesian inference on GARCH models using he Gibbs sampler. Economerics Journal Vol, c23-c46 [ 4 ] Chib, S. and Greenberg, E. (995). Undersanding he Meropolis-Hasings Algorihm. The American Saisician, Vol. 49, No. 4. pp [ 5 ] Chib, S. and Greenberg, E. (998). Bayesian analysis of mulivariae probi models. Biomerika 85, [ 6 ] Daniels, M.J. and Kass, R.E. (999). Nonconjugae Bayesian esimaion of covariance marices and is use in hierarchical models. Journal of he American Saisical Associaion 94, [ 7 ] Gourieroux, C. (997). ARCH Models and Financial Applicaions. New York, Springer Verlag [ 8 ] Koop, G. Bayesian Ecoomerics; Wiley [ 9 ] Liechy, J., Liechy, M., and Muller, P. (24). Bayesian Correlaion Esimaion. Biomerika 9,-4 [ ] Liu, X. & Daniels, M. (27). A new algorihm for simulaing a correlaion marix based on Parameer Expansion and Reparamerizaion. Deparmen of Saisics, Universiy of Florida [ ] Nzabandora, W. (27). Esimaion bayesienne des marices de correlaions: sraegies de formulaion de lois a priori. Deparmen de Sciences Economiques, Universie de Monreal [ 2 ] Tim Bollerslev (99). Modelling he Coherence in Shor-Run Nominal Exchange Raes: A Mulivariae Generalized Arch Model. The Review of Economics and Saisics, Vol. 72, No. 3. pp [ 3 ] Tse, Y. and Tsui, K. (2). A Mulivariae GARCH Model wih Timevarying correlaions. Deparmen of Economics, Naional Universiy of

25 Singapoore [ 4 ] Vronos, I., Dellaporas, P. and Poliis, D. (23).Inference for some Mulivariae ARCH and GARCH models.journal of Forcasing 22,

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