Comparison of Multivariate GARCH Models with Application to Zero-Coupon Bond Volatility

Transcription

1 Deparmen of Saisics Comparison of Mulivariae GARCH Models wih Applicaion o Zero-Coupon Bond Volailiy Wenjing Su and Yiyu Huang Maser Thesis 15 ECTS Spring Semeser 2010 Supervisor: Professor Björn Holmquis

2 Acknowledgemens Firs and foremos, we appreciae very much our supervisor Professor Björn Holmquis for his consisen dedicaion work on he guidance of our hesis. Your academic spiri and willingness of moivaing us o work had grea influence on our hesis and also made lasing impression on us. In addiion, we feel graeful o he res of our hesis commiee. You all provided valuable suggesions and poined ou somewhere in our hesis worhy of modificaion, which would surely meliorae our hesis. During he period of sudying his program, we me so many fanasic classmaes. We really enjoyed he whole process of cooperaing, discussing and coordinaing wih you. Thank you so much all! Bes luck for your fuure.

3 Table of Conens 1 Inroducion.1 2 Model Specificaion and Esimaion Mehodology ARCH models GARCH models General form of GARCH models GARCH (1,1) models Exensions of he GARCH models Mulivariae GARCH models Formulaions of mulivariae GARCH models Esimaion of MGARCH models Diagnosics of MGARCH models Forecasing.12 3 Consrucion of Mulivariae GARCH Models Daa Descripion Mulivariae-GARCH modeling Model esimaion BEKK models DCC models Model Diagnosics Diagnosics of BEKK models Diagnosics of DCC models Comparison of BEKK and DCC models Forecasing Conclusion and Fuure Work..49 References... 50

4 Absrac The purpose of his hesis is o invesigae differen formulaions of mulivariae GARCH models and o apply wo of he popular ones he BEKK- GARCH model and he DCC- GARCH model in evaluaing he volailiy of a porfolio of zero-coupon bonds. Mulivariae GARCH models are considered as one of he mos useful ools for analyzing and forecasing he volailiy of ime series when volailiy flucuaes over ime. This feaure demonsraes is availabiliy in modeling he co-movemen of mulivariae ime series wih varying condiional covariance marix. From his poin of view, firsly we focus on undersanding he model specificaions of several widely used mulivariae GARCH models so as o selec appropriae models; and hen consruc he BEKK form and he DCC form separaely by employing he financial daa obained from he websie of he European Cenral Bank. The nex work is dedicaed o diagnose he goodness of fi of he esablished models even hough here are comparaively few ess specific o mulivariae models according o previous lieraures. On op of hose, we compare he fiing performance of hese wo forms and predic he fuure dynamics of our daa on he ground of hese wo models.

5 1. Inroducion Wih he increase in he complexiy of he insrumens in he risk managemen field, huge demands for he various models which can simulae and reflec he characerisics of he financial ime series have expanded. One of he significan feaures of financial daa ha has won much aenion is he volailiy; because i is a numerical measure of he risk faced by individual invesors and financial insiuions. I is well known ha he volailiy of financial daa ofen varies over ime and ends o cluser in periods, i.e., high volailiy is usually followed by high volailiy, and low volailiy by low volailiy. This phenomenon corresponds o he flucuaing volailiy. The Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) model and is exensions have been proved o be able o capure he volailiy clusering and predic volailiies in he fuure. Specifically, when analyzing he co-movemens of financial reurns, i is always essenial o esimae, consruc, evaluae, and forecas he co-volailiy dynamics of asse reurns in a porfolio. This ask can be fulfilled by mulivariae GARCH (MGARCH) models. The developmen of MGARCH models could be hough as a grea breakhrough agains he curse of dimensionaliy in he financial modeling. Many differen formulaions have been consruced parsimoniously and sill remain necessary flexibiliy. MGARCH models can be applied o asse pricing, porfolio heory, VaR esimaion and risk managemen or diversificaion, which require he volailiies and co-volailiies of several markes [Bauwens e al., 2006]. In his hesis, MGARCH models are esimaed for volailiy and co-volailiy of hree zero coupon bond prices wih differen mauriies. The daa is provided by he websie of he European Cenral Bank (ECB) which is he insiuion of he European Union asked wih adminisraing he moneary policy of he 16 EU member saes aking par in he Eurozone. 1

6 A zero coupon bond is a non-coupon-bearing bond ha pays face value a he ime of mauriy even hough i is bough a a price lower han is face value. I has no reinvesmen risk and is more sensiive o ineres rae change han coupon-bearing bonds. Due o hese feaures, zero-coupon bonds can be easily used o creae any ype of cash flow sream and hus mach asse cash flows wih liabiliy cash flows (e.g. o provide for college expenses, house-purchase down paymen, or oher liabiliy funding.), and are used by pension funds and insurance companies o offse, or immunize he ineres rae risk of hese firms' long-erm liabiliies. Moreover, he reurn of zero coupon bond, referred o as zero rae, is a fundamenal elemen in he field of fix-income pricing and risk evaluaion. By using cash-flow-mapping mehod [Hull, 2005], any fixed cash flow can be mapped o a porfolio consising of a few zero coupon bonds, which mach he cash flow s reurn and volailiy. This viewpoin exemplifies how o generalize he specific zero coupon bond volailiies ino a general case. I also moivaes our sudy o model volailiy and co-volailiy of hree zero-coupon bonds wih differen mauriies of 6 monh, 1 year and 2 year. In laer secion, we esimaed wo MGARCH models based on he BEKK form and DCC form by he quasi - maximum likelihood mehod and also esed he goodness of fi of hese models. The reminder of his hesis is organized as follows. Secion 2 reviews MGARCH models, including is differen forms, diagnosics and he forecasing. In secion 3 we presen he BEKK and DCC MGARCH models of volailiy and co-volailiy of ECB zero coupon bond daa se. Secion 4 provides conclusions and furher work. 2

7 2. Model Specificaion and Esimaion Mehodology In order o accuraely capure he characerisic heeroskedasiciy of many financial daa, which refers o he fac ha he marke volailiy varies and ends o cluser in periods of high volailiy as well as periods of low volailiy, he ARCH model was inroduced by Engle (1982). Even hough his model capures he varying volailiy of financial ime series in conras wih he consan volailiy in previous research, here was sill need for a beer model o measure risk which is refleced as he volailiy. This secion mainly concerns a more generalized model of he ARCH model from he univariae case o mulivariae cases. 2.1 ARCH models The mean process of ARCH models can be expressed by r., = 1,,T (1) Here, μ is he mean of he ime series r and ε denoes is residual. T is he number of observaions. Regarding he residuals variance process of ARCH models, assume ε =σ z, where z ~ N(0,1) and he series σ 2 are modeled by, (2) q 2 q where α 0 > 0 and α i 0, i > 0. residuals. I specifies a sochasic process for he residuals and predics he average size of he However, i has is own drawbacks in ha he assumpion ha posiive and negaive 3

8 shocks have he same effecs on volailiy goes conrary o he realiy. I is very common ha he price of a financial asse responds differenly o posiive and negaive shocks [Paul, 2007]. In addiion, i is always he case ha ARCH models require he esimaion of a large number of parameers as a high order of ARCH erms has o be seleced for he purpose of caching he dynamic of he condiional variance. 2.2 GARCH models The following subsecions inroduce he general formulaion of a univariae GARCH model, he mos widely used GARCH form GARCH (1, 1) and some exensions General form of GARCH models In view of he ARCH model s limiaions, Bollerslev (1986) proposed he Generalized ARCH model (GARCH), in which he condiional variance saisfies he following form (3) q 2 q p 2 p where 0 and 0. i i In GARCH models, residuals lags can be replaced by a limied number of lags of condiional variances, which simplifies he lag srucure and as well he esimaion process of coefficiens GARCH (1, 1) models The mos frequenly used GARCH model is he GARCH (1, 1) model. In GARCH (1, 1), he condiional variance marix is calculaed from a long-run average variance rae, V L, and also from he lag erms n1 and n1. The equaion of he condiional variance for GARCH (1, 1) is (4) n VL n1 n1 4

9 where is he weigh assigned o V L, is he weigh assigned o 2 n1, and is he weigh assigned o 2 n1. In addiion, he weighs sum o one, ha is, 1 (5) The GARCH (1, 1) models specifies ha 2 n is based on he mos recen observaion of 2 2 n and he mos recen variance rae n1. Seing VL, he GARCH (1, 1) model can be rewrien as (6) 2 n 2 n1 2 n1 This is he form ha is usually used for he esimaion of parameers in he univariae case Exensions of he GARCH models There are many exensions of he sandard GARCH models 1. Nonlinear GARCH (NGARCH) was proposed by Engle and Ng in The condiional covariance equaion is in he form ), where,, 0. The ( inegraed GARCH (IGARCH) is a resriced version of he GARCH model, where he sum of all he parameers sum up o one. The exponenial GARCH (EGARCH) inroduced by Nelson (1991) is o model he logarihm of he variance raher han he level. The GARCH-in-mean (GARCH-M) model adds a heeroskedasiciy erm ino he mean equaion. The quadraic GARCH (QGARCH) model can handle asymmeric effecs of posiive and negaive shocks. The Glosen-Jagannahan-Runkle GARCH (GJR-GARCH) model (1993) can also model asymmery in he GARCH process. The hreshold GARCH (TGARCH) model is similar o GJR-GARCH wih he specificaion on condiional sandard deviaion insead of condiional variance. Family GARCH (FGARCH) by Henschel (1995) is an omnibus model ha is a mix of oher symmeric 1 hp://en.wikipedia.org/wiki/auoregressive_condiional_heeroskedasiciy 5

10 or asymmeric GARCH models. 2.3 Mulivariae GARCH models The basic idea o exend univariae GARCH models o mulivariae GARCH models is ha i is significan o predic he dependence in he comovemens of asse reurns in a porfolio. To recognize his feaure hrough a mulivariae model would generae a more reliable model han separae univariae models. In he firs place, one should consider wha specificaion of an MGARCH model should be imposed. On he one hand, i should be flexible enough o sae he dynamics of he condiional variances and covariances. On he oher hand, as he number of parameers in an MGARCH model increases rapidly along wih he dimension of he model, he specificaion should be parsimonious o simplify he model esimaion and also reach he purpose of easy inerpreaion of he model parameers. However, parsimony may reduce he number of parameers, in which siuaion he relevan dynamics in he covariance marix canno be capured. So i is imporan o ge a balance beween he parsimony and he flexibiliy when designing he mulivariae GARCH model specificaions. Anoher feaure ha mulivariae GARCH models mus saisfy is ha he covariance marix should be posiive definie Formulaions of Mulivariae GARCH models This secion emphasizes on giving a brief inroducion o several differen mulivariae GARCH models. VEC/DVEC-GARCH models The firs MGARCH model was inroduced by Bollerslev, Engle and Wooldridge in 1988, which is called VEC model. I is much general compared o he subsequen formulaions. In he VEC model, every condiional variance and covariance is a funcion of all lagged condiional variances and covariances, as well as lagged squared reurns 6

11 and cross-producs of reurns. The model can be expressed below: q j1 j ' j p vech( H ) c A vech( ) B vech( H ), (7) j j1 j j where vech ( ) is an operaor ha sacks he columns of he lower riangular par of is argumen square marix, H is he covariance marix of he residuals, N presens he number of variables, is he index of he h observaion, c is an N(N+1)/2 1 vecor, A j and B j are N(N+1)/2 N(N+1)/2 parameer marices and is an N 1 vecor. The condiion for H o be posiive definie for all is no resricive. In addiion, he number of parameers equals (p+q) (N(N+1)/2) 2 +N(N+1)/2, which is large. Furhermore, i demands a large quaniy of compuaion. The DVEC model, he resriced version of VEC, was also proposed by Bollerslev, e al (1988). I assumes he A j and B j in equaion (7) are diagonal marices, which makes i possible for H o be posiive definie for all. Also, he esimaion process proceeds much smoohly compared o he complee VEC model. However, he DVEC model wih (p+q+1) N (N+1)/2 parameers is oo resricive since i does no ake ino accoun he ineracion beween differen condiional variances and covariances. BEKK-GARCH models To ensure posiive definieness, a new parameerizaion of he condiional variance marix H was defined by Baba, Engle, Kraf and Kroner (1990) and became known as he BEKK model, which is viewed as anoher resriced version of he VEC model. I achieves he posiive definieness of he condiional covariance by formulaing he model in a way ha his propery is implied by he model srucure. The form of he BEKK model is as follows H CC q K A A B H B (8) kj j j kj j1 k1 j1 k1 p K kj j kj where A kj, B kj, and C are N N parameer marices, and C is a lower riangular marix. 7

12 The purpose of decomposing he consan erm ino a produc of wo riangular marices is o guaranee he posiive semi-definieness of H. Whenever K > 1 an idenificaion problem would be generaed for he reason ha here are no only a single parameerizaion ha can obain he same represenaion of he model. The firs-order BEKK model is H CC A A BH B. (9) The BEKK model also has is diagonal form by assuming A kj, B kj marices are diagonal. I is a resriced version of he DVEC model. The mos resriced version of he diagonal BEKK model is he scalar BEKK one wih A = ai and B = bi where a and b are scalars. Esimaion of a BEKK model sill bears large compuaions due o several marix ransposiions. The number of parameers of he complee BEKK model is (p+q)kn 2 +N(N+1)/2. Even in he diagonal one, he number of parameers soon reduces o (p+q) K N+N (N+1)/2, bu i is sill large. The BEKK form is no linear in parameers, which makes he convergence of he model difficul. However, he srong poin lies in ha he model srucure auomaically guaranees he posiive definieness of H.. Under he overall consideraion, i is ypically assumed ha p = q = K = 1 in BEKK form s applicaion. Consan Condiional Correlaions (CCC) models The Consan Condiional Correlaion model was inroduced by Bollerslev in 1990 o primarily model he condiional covariance marix indirecly by esimaing he condiional correlaion marix. The condiional correlaion is assumed o be consan while he condiional variances are varying. Obviously, his assumpion is impracical for real financial ime series. Then cerain modificaions were made grounded on his form [Annasiina and Timo, 2008]. Dynamic Condiional Correlaions (DCC) models 8

13 The Dynamic Condiional Correlaion model was proposed by Engle in I is a nonlinear combinaion of univariae GARCH models and i is also a generalized version of he CCC model. The form of Engle s DCC model is as follows: H D R D (10) where D diag ( h,, h 1/ 2 1/ 2 11 NN ) and each h ii is described by a univariae GARCH model. Furher, R 1/ 2 1/ 2 1/ 2 1/ 2 diag ( q,, q ) Q diag ( q,, q ), 11 NN 11 NN where Q = (q ij ) is he N N symmeric posiive definie marix which has he form: Q. (11) ( 1 ) Q u 1u 1 Q 1 Here, u / h, and are non-negaive scalars ha 1, Q is he i i ii N N uncondiional variance marix of u. The shorcoming of he model is ha all condiional correlaions follow he same dynamic srucure. The number of parameers o be esimaed is (N+1) (N+4)/2, which is relaively smaller han he complee BEKK form wih he same dimension when N is small. When N is large, he esimaion of he DCC model can be performed by a wo-sep procedure which decreases he complexiy of he esimaion process. In brief, in he firs place, he condiional variance is esimaed via univariae GARCH model for each variable. The nex sep is o esimae he parameers for he condiional correlaion. The DCC model can make he covariance marix posiive definie a any poin in ime. Oher mulivariae forms To overcome he difficuly of large number of parameers, he O-GARCH model was proposed by Alexander in I ries o express a mulivariae GARCH in erms 9

14 of univariae ones. The advanage of his model is ha he flucuaing volailiy can be explained by a few principle componens. One of he disadvanages is ha i is usually uncerain wheher he uncondiional variances have he coheren scaling. Anoher mulivariae GARCH model GO-GARCH model is proposed by Bauwens e al. in Esimaion of MGARCH models The mos usual way o esimae he condiional covariance marix in he MGARCH model is by he quasi maximum likelihood mehod. Le H (θ) be a posiive definie N N condiional covariance marix of some N 1 residual vecor ε, parameerized by he vecor θ. Denoing he available informaion a ime by Ƒ, we have E -1 [ε Ƒ -1 ] = 0; (12) E -1 [ε ε Ƒ -1 ] = H (θ). (13) Generally he condiional covariance marix H (θ) is well specified based on a cerain MGARCH model. Suppose here is an underlying parameer vecor θ 0 which one wans o esimae using a given sample of T observaions. The quasi maximum likelihood (QML) approach esimaes θ 0 by maximizing he Gaussian log likelihood funcion T T N T log LT ( ) log(2 ) log H ' H. (14) One needs o noice is assumpion ha he ime series reaed should be saionary and he disribuion of is residual is pre-defined as a condiional Gaussian disribuion. The laer assumpion can meanwhile give us hins on how o check he adequacy of he esablished MGARCH model Diagnosics of MGARCH models The check of he adequacy of MGARCH models is essenial in idenifying wheher a well specified MGARCH model can aain reliable esimaes and inference. 10

15 Graphical diagnosics for MGARCH models can be fulfilled by examining plos of he sample auocorrelaion (ACF) and he sample cross correlaion funcions (XCF). To ensure he inference from he esimaed parameers in he MGARCH model is enough valid, he residuals should be exhibied as a se of whie noise wih feaures like expeced zero mean vecor, no auocorrelaions, consan variance, and normal disribuion of he residuals. The auocorrelaion and cross correlaion funcions for he squared process are shown o be useful in idenifying and checking ime series behavior in he condiional variance equaion of he GARCH form. In he lieraure, several ess have been developed o es he auocorrelaion no maer in univariae or mulivariae form. Box and Pierce derived a goodness-of-fi es, called he pormaneau es. I may be he mos popular one among all he diagnosics for condiional heeroscedasiciy models. The es saisic may be expressed as a funcion of he covariances beween he residuals of he fied model [Hosking, 1980]. A mulivariae version is given by HM M ( M ) T ( T j) r CY (0) CY ( j) CY (0) CY ( j), (15) j1 where T is he number of observaions, C Y (j) is he sample auocovariance marix of order j and Y vech y y). ( 2 2 The disribuion of HM(M) is he asympoical ( K M ) under he null hypohesis ha here is no MGARCH effecs. Bu sill, he fac is ha very few ess are adapable o mulivariae models even hough here are many diagnosic ess dealing wih univariae models. To summarize, once he model is assumed o cach he dynamics of he ime series, he sandardized residual zˆ ˆ ˆ should saisfy he following condiions 1/ 2 H [Bauwens e al., 2006]: 11

16 1) E( zˆ zˆ) I N ; (16) 2 2 2) Cov ( zˆ i, zˆ j ) 0, for all pairs of he variable index i j; (17) 2 2 3) Cov ( zˆ i, zˆ j, k ) 0, for k > 0. (18) Tesing 1) would find he misspecificaion in he condiional mean; esing 2) is o verify wheher he condiional disribuion is Gaussian; he purpose of esing 3) is o check he adequacy of he dynamic specificaion of H even wihou knowing he validiy of he assumpion on he disribuion of z. Concerning he comparison of he BEKK-GARCH model and he DCC-GARCH model, he mean absolue error (MAE) is used o evaluae he fiing performance of boh models Forecasing In he class of mulivariae ARCH/GARCH models and heir exensions, he covariance marix is no longer consan over ime. Afer such model has been esimaed, i is always meaningful o ge o undersand he mechanism ha how he fuure series can be generaed and wheher hey fi well wih he real series. Forecasing by he BEKK-GARCH model In he condiional covariance equaion of he BEKK-GARCH model H CC A A BH B, (19) H is a funcion of he pas informaion, i.e., H -1 and 1. For his reason, he parameer esimaion of MGARCH models can be used o predic he fuure covariance marix. Forecasing by he DCC-GARCH model The forecas of he covariance marix of he DCC model is implemened in a wo-sep procedure. The predicion of he diagonal marix of he ime-varying sandard 12

17 variaion hrough he univariae GARCH models and he forecas of he condiional correlaion marix of he sandardized residuals are deal wih separaely. Under he assumpion ha he volailiy a ime is known, wha is is forecas value a ime +k? In a hree-variable case, he answer when k = 1 is given below, 2 h ii i, h, 1 ii, (20) where i = 1, 2, 3. To obain he forecas h ii,+k a ime +k, one jus need o repea he subsiuion successively. Cied from he definiion formula of he DCC-GARCH model, he srucure of he condiional correlaion marix is he equaion (11). Under he assumpion ha R Q and R +i = Q +i for i = 1,, k, a successive calculaion as before can be performed o derive R +k. MGARCH models can be used for forecasing. However, by analyzing he relaive forecasing accuracy of he wo formulaions BEKK and DCC, i can be deduced ha he forecasing performance of he MGARCH models is no always saisfacory. Many sudies, e.g. see Andersen and Bollerslev (1998), reveals ha he apparen poor forecasing effec of he MGARCH models is due o using he squared shocks as an approximae value for he rue condiional volailiy. 13

18 3. Consrucion of Mulivariae GARCH Models 3.1 Daa Descripion The original daa is provided by he European Cenral Bank (ECB) websie 2. I conains daily zero raes of AAA-raed euro area cenral governmen bonds, from 01/01/2007 o 30/04/2010. The following able gives a fracion of he daa, for example, on 2-Jan-07, he zero rae wih mauriy 6 monh is 3.61% in coninuous compounding. Table 3.1 he Zero Rae Daa from ECB 6m 1y 2y 2-Jan Jan Jan Jan Jan Jan Jan Wih given ZR i, he zero rae a ime, and mauriy T, he zero coupon bond price p i is calculaed as p i ZRi T S e, i = 1, 2, 3. (21) where S is he par value, in our case aking he value 100. The daily log reurn r is calculaed as follows: pi, r i, ln( ), i = 1, 2, 3. (22) p i, 1 Their associaed line graphs are ploed in he following Figure 3.1. Three variables (var1/var2/var3) correspond o hree daily reurns wih differen mauriies (6m/1y/2y). One may see ha during he second half of year 2008, he daily reurns exhibis high 2 hp:// 14

19 volailiy, reflecing a financial crisis. Besides, heir descripive saisics are given in Table 3.2. Moreover, he resul of ARCH effec [Waler, 2009] es proposed by Engle of each reurn series is given in Table 3.3, where H being 1 indicaes rejecing of null hypohesis ha here is no ARCH effec. One may see ha each variable/reurn has significan ARCH effec. Figure 3.1 Daily Log-Reurn of Bonds wih Differen Mauriies - 6m, 1y and 2y from op o boom Table 3.2 Descripive Saisics of Reurn Series wih differen mauriies (6m/1y/2y) Mean Median Max Min Sd.Dev. Skewness Kurosis Jarque-Bera Prob var var var

20 Table 3.3 GARCH Effec Tesing of Reurn Series var1 var2 var3 Lag H pvalue H pvalue H pvalue One can deec from Table 3.2 or Figure 3.1 ha he bonds wih he longer mauriy are much more volaile han hose wih a shorer mauriy. Addiionally, he financial daa here exhibis feaures like: Volailiy clusering Volailiy does no keep consan. I is quie common ha high reurns end o be followed by high reurns and low reurns end o be close wih low reurns. Lepokurosis effec By viewing he value of kurosis, one can conclude ha he reurn series can show he feaure of fa ails relaive o he normal disribuion as high kurosis indicaes a larger possibiliy of exreme movemens. Leverage effec Volailiy increases more afer low reurns han afer high reurns. A simple explanaion for his is ha negaive reurns imply a larger proporion of deb which leads o a high volailiy afer smaller changes. Skewness All of hree variables show evidence of some degree of skewness. The effec of skewness may be posiive or negaive, which describes heir deparure from symmery. Long-run memory effec The exisence of his effec reflecs persisence emporal dependence even beween disan observaions. In addiion, he Jarque-Bera saisics rejec he null hypohesis ha he log reurn series are normally disribued as he probabiliy of BJ es are all equal o zero. 16

21 3.2 Mulivariae-GARCH modeling The daa of 2007, 2008 and firs half of 2009, oally 635 observaions, is used o esimae MGARCH models, and he res daa, i.e., from Jul/2009 is used o evaluae model forecasing. As he BEKK-GARCH and DCC-GARCH models are he wo mos widely used mulivariae GARCH models, we will resric o model he volailiy and co-volailiy of he hree variables by using BEKK and DCC forms. Nex we presen he esimaed model, and heir diagnosics and forecasing are provided in following subsecions Model Esimaion As saed before, MGARCH models are esimaed by maximum likelihood echniques. In our case, he process was performed by he economerics sofware package RATS 7.0 (Regression Analysis of Time Series) which is used worldwide for analyzing ime series, developing or esimaing economeric models and forecasing. Because of he flexible maximum likelihood esimaion capabiliies of RATS [Esima, 2007a; 2007b], i has advanages over many oher sofware packages on esimaing sandard mulivariae-arch and mulivariae-garch models. RATS suppors differen forms of MGARCH models, including general MGARCH, BEKK, diagonal, VECH, CCC (Consan Condiional Correlaion), DCC (Dynamic Condiional Correlaions), and EWMA (Exponenially Weighed Moving Average) models. In his hesis, only wo widely used MGARCH forms, BEKK form and DCC form are esimaed. The opimizaion algorihm used for he maximum likelihood esimaion is BFGS proposed independenly by Broyden (1970), Flecher (1970), Goldfarb (1970) and Shanno (1970). As a numerical opimizaion algorihm, i uses ieraion rouines o obain he coefficien esimaion. Convergence is assumed o occur if he change in he coefficiens o be esimaed, 17

22 i.e. min( 2 1 / 1, 2 1 ), is less han he convergence crierion opion cvcri specified. The convergence crierion opion cvcri used in his hesis was chosen as he defaul value BEKK models As illusraed before, he BEKK form [Engle and Kroner, 1995] of MGARCH akes he following form: H CC A A BH B (23) Noe ha an advanage of BEKK form over VECH form is ha posiive-definieness is auomaically ensured. Parameer esimaion of BEKK form is provided in Table

23 Table 3.4 he Esimaion of BEKK-GARCH Model Parameers Variable Coeff Sd Error T-Sa Signif Mean(1) Mean(2) Mean(3) C(1,1) C(2,1) C(2,2) C(3,1) C(3,2) C(3,3) A(1,1) A(1,2) A(1,3) A(2,1) A(2,2) A(2,3) A(3,1) A(3,2) A(3,3) B(1,1) B(1,2) B(1,3) B(2,1) B(2,2) B(2,3) B(3,1) B(3,2) B(3,3) We can see from Table 3.4 ha mos of variables esimaed here are saisically significan. The esimaed BEKK-GARCH model can be obained by subsiuing he following marices ino equaion (23) A

24 B C DCC models As reviewed in previous chaper, he DCC model has he following form: H D R D, 2 2 where D diag ( h 1/,, h 1/ ), each h ii is a univariae GARCH model, and 11 NN R 1/ 2 1/ 2 1/ 2 1/ 2 diag ( q,, q ) Q diag ( q,, q ). 11 NN 11 NN The marix Q = (q ij ) is he N N symmeric posiive definie marix updaed by he following: Q. ( 1 ) Q u 1u 1 Q 1 where u / h. i i ii Parameer esimaion of DCC model from RATS is provided in Table 3.5. Table 3.5 he Esimaion of DCC-GARCH Model Parameers Variable Coeff Sd Error T-Sa Signif Mean(1) e e Mean(2) e e Mean(3) e e C(1) e e C(2) e e C(3) e e A(1) A(2) A(3) B(1) B(2) B(3) DCC(1) DCC(2)

25 Excep for he consan erms, all he oher esimaed variables are saisically significan. Then he esimaed DCC model is as following, where Q is he 3 3 uncondiional covariance marix of u : h h h , 1 2, h 3, Q h h Q ( ) Q u 1/ 2 1/ 2 1/ 2 1/ 2 R diag( q, q, q ) Q diag( q, q 11 1 u 1/ , 1 1, q 22, 1 33, Q 1/ 2 33 ) 1 where u / h. i i ii 3.3 Model Diagnosics Diagnosics of BEKK models The empirical measure of logarihmic daily reurn variabiliy is called he realized volailiy. I is compued from high-frequency logarihmic reurns [Hull, 2005]. I is calculaed using he subsequen 10 observaions on he log-reurns in our case. In conras realized volailiy consruced from high-frequency reurns wih he resricive parameric mulivariae GARCH models, links beween realized volailiy and he diagonal elemens of he condiional covariance marix have been esablished [Andersen e al., 2003]. 21

26 (a) (b) (c) Figure 3.2 Esimaed and Realized Volailiy of he BEKK Model 22

27 Basically, he esimaed volailiy follows he dynamic of he realized volailiy. And he graph reveals wo of he financial daa s feaures, he volailiy clusering and he relaion beween mauriy and volailiy, ha is, longer mauriy corresponds o higher volailiy as indicaed in Fig. 3.2 (c). On average, here exiss a horizonal lag beween hese wo lines for he reason ha we calculaed he realized volailiy by using he nex en observaions. The realized correlaion over a horizon of T days is approximaed by a consisen, empirical esimae. In our case, he realized correlaion beween he log daily reurn of var i and var j a ime over a T-day horizon is calculaed as T (var i )(var j ) k i k j k 1 (var i, var j), T (24) T T 2 2 (var ik i ) (var jk j ) k 1 k 1 where i and j are he corresponding sample means over he T-day period. 23

28 (a) (b) (c) Figure 3.3 Esimaed and Realized Correlaion of he BEKK Model 24

29 The comparison beween he esimaed and realized correlaion is shown above. I can be seen ha here is a huge decline on esimaed and realized correlaion during he second half of year Wih regard o oher ime periods, he value of correlaion beween var1 and var2/var3 is around 0.8 and he value of correlaion beween var1 and var3 is even above 0.9. As for he performance of fi, he esimaed correlaion more or less follows he dynamics of he realized correlaion excep here is also a horizonal lag beween hem. The model esimaion employed here is he Gaussian quasi MLE mehod. One of is assumpions is ha he residuals have a Gaussian disribuion. Hence, o es wheher he esimaions of he model parameers are robus, we can check wheher he residuals of he esimaed process are whie noise. Calculaed by he formula zˆ Figure 3.4 Sandardized Residuals of he BEKK Models 1/ 2 H ˆ, he sandardized residuals are shown in Fig 25

30 3.4 for he hree variables. I indicaes ha no disinc difference exiss among he disribuions of he hree residuals. They all look like whie noise on a cerain degree. Table 3.6 shows he esing resul of GARCH effec on he sandardized residuals of he BEKK model. H = 0 represens he accepance of he null hypohesis ha no GARCH effecs exis. In conras wih Table 3.3, we can conclude ha GARCH effec has eliminaed quie a lo. The Ljung-Box es based on he auocorrelaion plo ess he randomness a each disinc lag. H = 0 means ha we end o accep he null hypohesis ha he series is random. Table 3.6 GARCH Effec Tesing of each Sandardized Residuals (BEKK) var1 var2 var3 Lag H pvalue H pvalue H pvalue Table 3.7 LBQ Tes of each Sandardized Residuals of he BEKK Model var1 var2 var3 Lag H pvalue H pvalue H pvalue Resuls of he sample auocorrelaion and he sample cross-correlaion of he sandardized residuals and he squared sandardized residuals are presened here o examine he adequacy of he MGARCH model. Figure 3.5 shows he sample auocorrelaion funcion of he sandardized residual of he BEKK model (b) and compare i o he sample auocorrelaion of he reurns ahead of modeling (a). For mos of lags, he sample ACFs and XCFs are wihin he disance beween posiive and negaive 2 imes sandard deviaion lines a 95% confidence level. A 26

31 comparison beween he ACFs of he premodel daa and he sandardized residual indicaes ha GARCH effec has been removed quie a lo. In Fig 3.8, he conras beween he XCFs of he corresponding squared erms before and afer he BEKK model also proves ha he GARCH effec has been diminished a lo. 27

32 (a) (b) Figure 3.5 ACFs of Premodel Daa and Sandardized Residual of he BEKK Model 28

33 (a) (b) Figure 3.6 ACFs of he Squared Premodel Daa and he Squared Sandardized Residual of he BEKK Model 29

34 (a) (b) Figure 3.7 XCFs of Premodel Daa and Sandardized Residuals of he BEKK Model 30

35 (a) (b) Figure 3.8 XCFs of he Squared Premodel Daa and he Squared Sandardized Residuals of he BEKK Model 31

36 3.3.2 Diagnosics of DCC models Volailiy clusering is also presened in Fig 3.9. The esimaed volailiy on he whole changes along wih he realized volailiy. Also, here exiss a horizonal lag beween hese wo lines for he same reason explained before. The fiing performance of he DCC model is shown in Fig 3.9 and Fig 3.10 no such saisfying as ha of he BEKK model shown in Fig 3.2 and Fig

37 (a) (b) (c) Figure 3.9 Esimaed and Realized Volailiy of he DCC Model 33

38 (a) (b) (c) Figure 3.10 Esimaed and Realized Correlaion of he DCC Model 34

39 Figure 3.11 Sandardized Residuals of he DCC Models Table 3.8 shows he esing resul of GARCH effec on he sandardized residuals of he DCC model. In conras wih Table 3.3, we can also conclude ha GARCH effec has eliminaed quie a lo. The Ljung-Box es based on he auocorrelaion plo ess he randomness a each disinc lag. 35

40 Table 3.8 GARCH Effec Tesing of each Sandardized Residuals (DCC) var1 var2 var3 Lag H pvalue H pvalue H pvalue Table 3.9 LBQ Tes of each Sandardized Residuals of he DCC Model var1 var2 var3 Lag H pvalue H pvalue H pvalue A comparison, see in fig. 3.12, beween he ACF of he premodel daa and he sandardized residual indicaes ha GARCH effec has been erased much. For mos of lags, he sample ACFs and XCFs are wihin he disance beween posiive and negaive 2 imes sandard deviaion lines a 95% confidence level. ACFs of he squared daa before and afer he modeling show ha hey are serially uncorrelaed. The cross-correlaions of he squared pre-model daa and he squared sandardized residuals of he DCC model also reflecs ha less GARCH effec exiss in he squared sandardized residuals afer modeling. 36

41 (a) (b) Figure 3.12 ACFs of Premodel Daa and Sandardized Residual of he DCC Model 37

42 (a) (b) Figure 3.13 ACFs of he Squared Premodel Daa and he Squared Sandardized Residual of he DCC model 38

43 (a) (b) Figure 3.14 XCFs of Premodel Daa and Sandardized Residuals of he DCC Model 39

44 (a) (b) Figure 3.15 XCFs of he Squared Premodel Daa and he Squared Sandardized Residuals of he DCC model 40

45 3.3.3 Comparison of BEKK and DCC models The mean absolue error (MAE) [Engle, 2000] can measure how close he esimaed variables are o he realized values. I is also called he mean average error. In our case MAE is calculaed by 1 MAEvi n n k1 ˆ (25) ik ik for volailiy where n is he oal number of observaions or MAEij 1 n n k1 ˆ (26) ijk ijk for correlaion where i, j = 1, 2, 3. Table 3.10 MAE in correlaion and volailiy of he BEKK model Average error in correlaion Average error in volailiy MAE MAEv MAE MAEv MAE MAEv Table 3.11 MAE in correlaion and volailiy of he DCC model Average error in correlaion Average error in volailiy MAE MAEv MAE MAEv MAE MAEv The values of he measure absolue error beween hese models sugges ha he parameer esimaion of he BEKK model is more accurae han ha given by he DCC model even hrough he magniude of he difference beween heir corresponding MAEs is no enough. 3.4 Forecasing We spli our sample ino wo pars, 2.5-year esimaion period and he subsequen half-year forecas periods. The dynamic characerisics of he logarihmic daily reurns of he zero-coupon bonds have been simulaed during he esimaion period by he mulivariae GARCH models. Afer he parameers of he model are esimaed, he deerminaion of he predicion on he condiional covariance marix H +k a ime +k can be aained. 41

46 As for BEKK-GARCH models, he ieraion formula for he purpose of forecas is H CC A A BH B Wih regards o DCC-GARCH models, he ieraion formula for he purpose of forecas is H k D k R k D k where D +k and R +k can be compued separaely and 1/ 2 D k diag ( h, k,, h 1/ , k 1/ 2 1/ 2 1/ 2 R k diag ( q, k,, q33, k ) Q kdiag ( q11, k,, q ), each h ii, +k is a univariae GARCH model, and 1/ , k The marix Q +k = (q ij,+k ) is he 3 3 symmeric posiive definie marix updaed by following: ). Q k ( 1 ) Q uk 1u k1 Q k1. where u i, k i, k / hii, k. 42

47 (a) (b) (c) Figure 3.16 he Esimaed and Forecased Volailiies-BEKK Fig shows he performance of he predicion is beer as he mauriy ges longer. Especially in he hird plo (c), a horizonal lag is presened clearly. Wha is 43

48 needed o pay aenion is ha a very sparse observaion appears a he very beginning of year Tha is why a peak suddenly emerges. Bu he endency is ha he forecas volailiy hen goes back o is normal dynamics exponenially. Fig 3.17 presens he poor performance of he BEKK-GARCH model on forecasing hrough he comparison of he realized correlaions and he forecas correlaions. On he lef of he verical line in Fig 3.17 presens he comparison beween he realized correlaion and he esimaed correlaion by he BEKK form. On he righ side of he verical line, i shows he poorer performance on forecasing he correlaion among each pair of variables in he subsequen half year. The forecasing performance of DCC-GARCH models looks beer han ha of he BEKK-GARCH model. The forecas volailiy generally follows he dynamics of he realized volailiy. For he same reason ha here is a very sparse observaion, a peak also appears in he following figure. The forecased correlaions by he DCC form in Fig 3.19 also fi beer wih he realized ones. 44

49 (a) (b) (c) Figure 3.17 he Esimaed and Forecased Correlaions-BEKK 45

50 (a) (b) (c) Figure 3.18 he Esimaed and Forecased Volailiies-DCC 46

51 (a) (b) (c) Figure 3.19 he Esimaed and Forecased Correlaions-DCC 47

52 The relaively beer predicion performance of DCC-GARCH models can also be presened in he comparison of he esimaed and forecas correlaions. One of he reasons for his disincion is ha he number of parameers esimaed in he BEKK-GARCH models is more han ha of DCC-GARCH models so ha he summaion of he error accumulaed by each parameer of he BEKK-GARCH models ends o be larger han ha of he DCC-GARCH models. 48

53 4. Conclusions and Fuure Work This hesis focuses on he consrucion and he diagnosics of wo formulaions of mulivariae GARCH models he BEKK and DCC forms. The esimaion process is fulfilled in he sofware package RATS 7.0 hrough he maximum likelihood mehod. Afer he parameers of hese models are esimaed, he forecas of he condiional covariance marix is conduced by he ieraion process. All our implemenaions are realized under he assumpion ha he residual erms are followed by a Gaussian disribuion. Therefore, he diagnosics in evaluaing he adequacy of modeling are operaed by checking wheher such assumpion is credible enough. By comparing he goodness of fi hrough he mean absolue error, we find ha he fiing performance of he BEKK GARCH form is beer han DCC GARCH form in our case. This difference may due o he number of parameers of he BEKK GARCH model is comparaively more; so ha BEKK GARCH model has a beer capabiliy in explaining he informaion hidden in he hisory daa. In he opposie, he DCC GARCH model has an advanage over he BEKK GARCH model in he area of forecasing as he DCC GARCH model is more parsimonious han he BEKK GARCH model. In his sense, i is crucially imporan o balance parsimony and flexibiliy when modeling mulivariae GARCH models. Regarding he diagnosic ess applied o mulivariae GARCH models, our work is inadequae because of he fac ha few ess are applicable o mulivariae cases and also due o he difficuly in implemening hose exended forms of he ess for deecing he univariae GARCH effec. 49

54 References Alexander C A primer on he orhogonal GARCH model. Unpublished manuscrip. ISMA Cenre, Universiy of Reading, UK. Andersen T., Bollerslev T., Diebold F.X. and Labys P Modeling and forecasing realized volailiy. Economerica 71: Andersen T. and Bollerslev T Answering he skepics: Yes, sandard volailiy models do provide accurae forecass. Inernaional Economic Review 39: Annasiina S. and Timo T Mulivariae GARCH models. SSE/EFI Working Paper Series in Economics and Finance No Baba Y., Engle R.F., Kraf D. and Kroner K Mulivariae simulaneous generalized ARCH, unpublished manuscrip, Universiy of California, San Diego. Bauwens L., Lauren S., and Rombous J.V.K Mulivariae GARCH models: A survey. Journal of Applied Economerics 21: Bollerslev T Generalized auoregressive condiional heeroscedasiciy. Journal of Economerics 31: Bollerslev T Modeling he coherence in shor-run nominal exchange raes: a mulivariae generalized ARCH model. Review of Economics and Saisics 72: Bollerslev T., Engle R.F., Wooldridge J.M A capial asse pricing model wih ime varying covariances. Journal of Poliical Economy 96: Box G.E.P., Pierce D.A Disribuion of he auocorrelaions in auoregressive moving average ime series models. Journal of American Saisical Associaion 65: Broyden C.G The convergence of a class of double-rank minimizaion algorihms. Journal of he Insiue of Mahemaics and Is Applicaions 6: Engle R.F Auoregressive condiional heeroskedasiciy wih esimaes of he variance of he Unied Kingdom inflaion. Economerica 50: Engle R.F Dynamic condiional correlaion A simple class of mulivariae GARCH models. Journal of Business and Economic Saisics 20(3): Engle R., Kroner F.K Mulivariae simulaneous generalized ARCH. Economeric 50

55 Theory 11: Engle R.F., Ng V.K Measuring and esing he impac of news on volailiy. Journal of Finance 48(5): Esima, 2007a. RATS User s Guide. Unied Saes of America. Esima, 2007b. RATS Reference Manual. Unied Saes of America. Flecher R A new approach o variable meric algorihms. Compuer Journal 13: Glosen L., Jagannahan R. and Runkle D Relaionship beween he expeced value and he volailiy of he nominal excess reurn on socks. Journal of Finance 48: Goldfarb D A family of variable meric updaes derived by variaional means. Mahemaics of Compuaion 24: Henschel, Ludger, All in he family nesing symmeric and asymmeric GARCH models. Journal of Financial Economics 39(1): Hosking J.R.M The mulivariae pormaneau saisic. Journal of American Saisical Associaion 75: Hull J. C Opions, fuures and oher derivaives. Prenice Hall: New York. Nelson D.B Condiional heeroskedasiciy in asse reurns: a new approach. Economerica 59: Paul R.K Auoregressive condiional heeroscedasic (ARCH) family of models for describing volailiy. Seminar Paper. Indian Agriculural Saisics Research Insiue. Shanno D.F Condiioning of quasi-newon mehods for funcion minimizaion. Mahemaics of Compuaion 24: Waler E Applied economeric ime series. John Wiley & Sons, Inc. The hird ediion. 51