Multivariate Stochastic Volatility

Transcription

1 SMU ECONOMICS & STATISTICS WORKING PAPER SERIES Mulivariae Sochasic Volailiy Manabu Asai, Michael McAleer, Jun Yu December 2006 Paper No ANY OPINIONS EXPRESSED ARE THOSE OF THE AUTHOR(S) AND NOT NECESSARILY THOSE OF THE SCHOOL OF ECONOMICS & SOCIAL SCIENCES, SMU

2 Mulivariae Sochasic Volailiy Manabu Asai Faculy of Economics Tokyo Meropolian Universiy Michael McAleer School of Economics and Commerce Universiy of Wesern Ausralia Jun Yu School of Economics and Social Science Singapore Managemen Universiy January 2005 * The auhors wish o acknowledge helpful discussions wih Yoshi Baba, Jii Gao, Suhejla Hoi, Essie Maasoumi, Peer Phillips, Peer Robinson and Ruey Tsay. The firs auhor appreciaes he financial suppor of he Gran-in-Aid for he 2s Cenury COE program Microsrucure and Mechanism Design in Financial Markes from he Minisry of Educaion, Culure, Spors, Science and Technology of Japan, he second auhor is graeful for he financial suppor of an Ausralian Research Council Discovery Gran, and he hird auhor graefully acknowledges financial suppor from he Wharon-SMU Research Cenre a SMU.

3 Absrac The lieraure on mulivariae sochasic volailiy (MSV) models has developed significanly over he las few years. This paper reviews he subsanial lieraure on specificaion, esimaion and evaluaion of MSV models. A wide range of MSV models is presened according o various caegories, namely (i) asymmeric models; (ii) facor models; (iii) ime-varying correlaion models; and (iv) alernaive MSV specificaions, including models based on he marix exponenial ransformaion, Cholesky decomposiion, Wishar auoregressive process, and he empirical range. Alernaive mehods of esimaion, including quasi-maximum likelihood, simulaed maximum likelihood, Mone Carlo likelihood, and Markov chain Mone Carlo mehods, are discussed and compared. Various mehods of diagnosic checking and model comparison are also examined. Keywords and phrases: mulivariae sochasic volailiy, asymmery, leverage, hresholds, facor models, ime-varying correlaions, ransformaions, esimaion, diagnosic checking, model comparison. 2

4 . Inroducion A wide range of mulivariae GARCH and sochasic volailiy (SV) models has been developed, analysed and applied exensively in recen years o characerize he volailiy ha is inheren in financial ime series daa. Bauwens e al. (2004) provided a recen survey of mulivariae GARCH, or condiional volailiy, models. The GARCH lieraure has expanded considerably since he univariae ARCH process was developed by Engle (982). The univariae SV model was proposed by, among ohers, Taylor (982, 986), and he univariae SV lieraure was surveyed in Ghysels e al. (996). Alhough here have already been many pracical applicaions of mulivariae GARCH models, he heoreical lieraure on mulivariae sochasic volailiy (MSV) models has developed significanly over he las few years. Some of he more imporan exising univariae and mulivariae GARCH and SV models have been analysed in McAleer (2005). However, a comprehensive review of he imporan aspecs of exising MSV models in he lieraure does no ye seem o exis. Owing o he developmen of a wide variey of MSV models in recen years, his paper reviews he subsanial lieraure on he specificaion, esimaion and evaluaion of MSV models. There are boh economic and economeric reasons why mulivariae volailiy models are imporan. The knowledge of correlaion srucures is vial in many financial applicaions, such as opimal porfolio risk managemen and asse allocaion, so ha mulivariae volailiy models are useful for making financial decisions. Moreover, as financial volailiy moves ogeher across differen asses and markes, modelling volailiy in a mulivariae framework can lead o greaer saisical efficiency. The remainder of he paper is organized as follows. Secion 2 presens a range of MSV models according o various caegories, including asymmeric models, facor models, ime-varying correlaion models, and several alernaive specificaions, including he marix exponenial ransformaion, Cholesky decomposiion, Wishar auoregressive models, and an empirical range-based model. Secion 3 compares and discusses alernaive mehods of esimaion, including he quasi-maximum likelihood, simulaed maximum likelihood, Mone Carlo likelihood, and Markov Chain Mone Carlo echniques. Various mehods of diagnosic checking and model comparison are examined in Secion 4. Some concluding commens are given in Secion 5. 3

5 2. MSV Models This secion reviews several varians of MSV models according o four caegories, as follows: (i) asymmeric models; (ii) facor models; (iii) ime-varying correlaion models; and (iv) alernaive MSV specificaions, including models based on he marix exponenial ransformaion, Cholesky decomposiion, Wishar auoregressive process, and empirical range. In wha follows, y = ( y, K, y m )' denoes a vecor of reurns for m financial asses. For exposiional purposes, i is assumed ha he condiional mean vecor of y is zero, alhough his can easily be relaxed. Moreover, exp(.) and log(.) denoe he elemen-by-elemen exponenial and logarihmic operaors, respecively, and diag{} x = diag{ x, K, x m } denoes he m m diagonal marix wih diagonal elemens given by x = ( x, K, x m). 2. Basic Models The firs MSV model proposed in he lieraure is due o Harvey e al. (994), as follows: y = Dε, h /2 hm /2 { K } { ( ) } D = diag e, e = diag exp 0.5 h, () h = µ + φ o h + + η, (2) ε 0 Pε O ~ N,, η 0 O Ση (3) where h = ( h, K, h m) is an m vecor of unobserved log-volailiy, µ and φ are m parameer vecors, he operaor o denoes he Hadamard (or elemen-by-elemen) produc, η { ση,ij} Σ = is a posiive-definie covariance marix, and P ε = { ρ ij } is he correlaion marix, ha is, P ε is a posiive definie marix wih 4

6 ρ = and ρ < for any i j, i, j =,, m. Harvey e al. (994) also considered ii ij he mulivariae disribuion for ε as his specificaion permis greaer kurosis as compared wih he Gaussian assumpion. for The model of Harvey e al. (994) can easily be exended o a VARMA srucure h, as follows: Φ ( Lh ) = µ +Θ ( L) η, + where p i Φ ( L) = Im ΦiL, i= q j Θ ( L) = Im ΘjL, j= and L is he lag operaor. Assuming ha he off-diagonal elemens of Σ η are all equal o zero, he model corresponds o he consan condiional correlaion (CCC) model proposed by Bollerslev e al. (988) and Bollerslev (990) in he framework of mulivariae GARCH processes. In he CCC model, each condiional variance is specified as a univariae GARCH model, ha is, wih no spillovers from oher asses, while each condiional covariance is given as a consan correlaion coefficien imes he produc of he corresponding condiional sandard deviaions. If he off-diagonal elemens of Σ η are no equal o zero, hen he elemens of h are no independen. Before inroducing various MSV models, we presen he long-memory MSV model analyzed by Anderson e al. (2003). By using a common degree of fracional inegraion, d, Anderson e al. (2003) specified he mulivariae long-memory model as follows: 5

7 Φ L L h = µ + η. (4) ( )( ) d + As heir analysis depends on he realized value of volailiy, h, esimaion of his model is relaively sraighforward, as follows: (i) esimae he common d using a mulivariae exension of he GPH (Geweke and Porer-Hudak (983)) esimaor, as developed by Robinson (995); and (ii) esimae he model by applying OLS o each equaion separaely. 2.2 Asymmeric Models I has long been recognized ha he volailiy of many financial asses responds differenly o bad news and good news. This is especially rue for sock reurns. In paricular, while bad news ends o increase he fuure volailiy, good news of he same size will increase he fuure volailiy by a smaller amoun, or may even decrease he fuure volailiy. A popular explanaion for his asymmery is he leverage effec, as firs proposed by Black (976) (see also Chrisie (982)), which predics ha volailiy ends o decrease in response o good news bu increase in response o bad news. Oher forms of asymmery, such as he asymmeric V-shape, have o be explained by reasons oher han he leverage effec. Alernaive reasons for he volailiy asymmery ha has been suggesed in he lieraure include he volailiy feedback effec (Campbell and Henschel (992)). The volailiy asymmery has been examined exensively in he conex of univariae SV models. The news impac funcion (NIF) of Engle and Ng (993) is a powerful ool for analysing he volailiy asymmery for GARCH-ype models. The idea of he NIF is o examine he relaionship beween condiional volailiy in period + (defined by 2 σ + ) and he sandardized shock o reurns in period (defined by ε ) in isolaion. Yu (2004b) generalized he NIF for he relaionship beween E ε ) (lnσ 2 + and ε in isolaion, so ha he NIF is also applicable o SV models. I is now possible o review he various asymmeric MSV models according o he differen shapes of he NIF. 6

8 2.2. Leverage Effec Yu (2004a) defined he leverage effec o be a negaive relaionship beween E (ln 2 σ + ε ) and ε, holding everyhing else consan. According o his definiion, he leverage effec mus lead o a decreasing NIF. A univariae discree ime SV model wih he leverage effec was firs proposed by Harvey and Shephard (996), alhough Wiggins (987) and Chesney and Sco (989), among ohers, considered a coninuous ime model iself and discreized i for purposes of esimaion. The model of Harvey and Shephard (996) may be regarded as he Euler approximaion o he coninuous ime SV model ha is used widely in he opion price lieraure (see, for example, Hull and Whie (987), who generalized he Black-Scholes opion pricing formula o analyse SV and leverage). These papers assume he negaive correlaion beween he innovaions. Yu (2004b) showed ha, afer rewriing he model in a Gaussian non-linear sae space form wih uncorrelaed measuremen and ransiion equaion errors, he NIF is a sraigh line which slopes downwards. An alernaive discree ime SV model wih leverage effec was proposed by Jacquier e al. (2004), which differs from he specificaion in Harvey and Shephard (996) in how he correlaion of wo error processes is modelled. Yu (2004a) argued ha i is difficul o inerpre he leverage effec in he laer specificaion, whereas he inerpreaion of he leverage effec is sraighforward in he former. Danielsson (998) and Chan e al. (2003) considered a mulivariae exension of he model of Jacquier e al. (2004). The model is given by equaions () and (2), ogeher wih ε 0 Pε L ~ N,, η 0 L Ση L = { λσ }, ij η, ij (5) 7

9 where he parameer λ ij capures asymmery. In he empirical analysis, Danielsson (998) did no esimae he mulivariae DL model because he daa used in his analysis did no sugges any asymmery in he esimaed univariae models. Chan e al. (2003) employed he Bayesian Markov Chain Mone Carlo (MCMC) procedure o esimae he DL model. However, he argumen of Yu (2004a) regarding he leverage effec also applies o he model in (5). Thus, he inerpreaion of he leverage effec in (5) is unclear and, even if λ < 0, here is no guaranee ha here will, in fac, be a leverage effec. ii Asai and McAleer (2004b) considered a mulivariae exension of he model of Harvey and Shephard (996). The model is given by equaions () and (2), ogeher wih ε 0 Pε L ~ N,, η 0 L Ση L= diag{ λσ, K, λ σ }, η, m η, mm (6) where he parameer λ i, i =,,m, is expeced o be negaive. Asai and McAleer (2004b) developed an esimaion mehod for his MSV model based on he Mone Carlo likelihood (MCL) echnique proposed in Durbin and Koopman (997) Threshold Effec In he GARCH lieraure, Glosen e al. (993) proposed modelling he asymmeric responses in condiional volailiy using hresholds. In he univariae SV lieraure, So e al. (2002) proposed a hreshold SV model, in which he consan erm and he auoregressive parameer in he SV equaion changed according o he sign of he previous reurn. Alhough he mulivariae hreshold SV model has no ye been developed in he lieraure, i is sraighforward o inroduce a mulivariae hreshold SV model wih The case λ < 0 implies a leverage effec for asse i as i implies ha bad news for ii asse i a period ends o increase he volailiy of asse j a period +. 8

10 mean equaion given in (), ogeher wih he volailiy equaion, as follows: h = µ ( s) + φ( s) o h + + η, (7) where ( ) µ ( s ) = µ ( s ), K, µ ( s ), m m ( ) φ( s ) = φ ( s ), K, φ ( s ), m m and s is a sae vecor, wih elemens given by s i 0, if yi < 0, = (8), oherwise. I is sraighforward o show ha he NIF of a hreshold SV model can have a flexible shape General Asymmeric Effec Wihin he univariae framework, Danielsson (994) suggesed an alernaive asymmeric specificaion o he leverage SV model, which is similar in spiri o an exension of he univariae exponenial GARCH (EGARCH) model of Nelson (99). The EGARCH model incorporaed he absolue value funcion o capure he sign and magniude of he previous normalized reurns shocks o accommodae asymmeric behaviour. In he model suggesed by Danielsson (994), which was ermed he asymmeric leverage (AL) model in Asai and McAleer (2004a), wo addiional erms, y and y, were included in he volailiy equaion, while he correlaion beween he wo error erms was assumed o be zero. Asai and McAleer (2004a) noed ha he absolue values of he previous realized reurns were included because i was no compuaionally sraighforward o incorporae he previous normalized shocks in he framework of SV models. Yu (2004b) proposed an alernaive specificaion o Danielsson (994) and 9

11 exended he volailiy equaion of he sandard leverage SV model by including an addiional erm, namely y. Finally, Asai and McAleer (2004a) proposed a more general model which ness boh he model of Yu (2004b) and he AL model as special cases. I is sraighforward o show ha he NIF of all hree models can have a flexible shape. In he mulivariae conex, Asai and McAleer (2004b) suggesed an exension of he AL SV model of Danielsson (994) as equaion (), ogeher wih h = µ + γ o y + γ o + 2 y + φ o h + η, (9) where γ and γ 2 are p parameer vecors. Asai and McAleer (2004b) developed an esimaion mehod for his MSV model, based on he Mone Carlo likelihood (MCL) echnique. 2.3 Facor Models In an aemp o reduce he dimensionaliy of he parameer space, various facor MSV models have been proposed. The facor MSV model was originally proposed by Harvey e al. (994), and exended by Shephard (996) and Jacquier e al. (999). This model has several aracive feaures, including parsimony of he parameer space, and he abiliy o capure he common feaures in asse reurns and volailiies. Alernaive dynamic facor models are he laen facor ARCH model of Diebold and Nerlove (989), and he facor ARCH represenaion used in Engle e al. (990) Addiive Facor Models The addiive facor MSV model was firs inroduced by Harvey e al (994), and subsequenly exended in Jacquier e al (995, 999), Shephard (996), Pi and Shephard (999a), and Aguilar and Wes (2000). The basic idea is borrowed from he facor mulivariae ARCH models, where he reurns are decomposed ino wo addiive componens. The firs componen has a smaller number of facors which capures he informaion relevan o he pricing of all asses, while he oher one is idiosyncraic noise which capures he asse specific informaion (for furher deails, see Diebold and Nerlove (989)). 0

12 Denoe he K vecor of facors as f ( K < m), and D is an m K dimensional marix of facor loadings. The addiive K facor MSV model presened by Jacquier e al (995) can be wrien as y = Df + e, f = exp( h / 2) ε, i i i (0) h = µ + φh + η, i =, K, K, i, + i i i i where e, ε i and η i are assumed o be muually independen. In order o guaranee he idenificaion of D and f uniquely, he resricions D = 0 and D ii = for i=, K, m and j < i are usually adoped (see Aguilar and Wes (2000)). ij in Model (0) was exended in Pi and Shephard (999a) by allowing each elemen e o evolve according o a univariae SV model. Chib e al. (2005) furher exended he model by allowing for jumps and for idiosyncraic errors which follow he suden SV process. Jacquier e al. (999), Pi and Shephard (999a) and Aguilar and Wes (2000) proposed esimaion mehods based on single-move MCMC algorihms. Chib e al (2005) argued ha he single-move algorihms can be simulaion-inefficien and suggesed a more efficien muli-move MCMC algorihm. Based on he Efficien Imporance Sampling (EIS) mehod proposed by Richard and Zhang (2004), Liesenfeld and Richard (2004) developed an alernaive muli-move MCMC algorihm. Liesenfeld and Richard (2003) showed ha he EIS mehod can be used o approximae he likelihood funcion, so ha i can faciliae a simulaed ML approach. Yu and Meyer (2004) showed ha addiive facor models accommodae boh ime varying volailiy and ime varying correlaions. In he conex of he bivariae one-facor SV model given by:

13 y = Df + e, e ~ N(0, diag{ σ, σ }) 2 2 e e2 f = exp( h / 2) ε, ε ~ N(0,), h = µ + φh + η, η ~ N(0,), + Yu and Meyer (2004) derived he condiional correlaion coefficien beween y 2 as y and d, ( + σ exp( h))( d + σ exp( h)) e e2 where (, d )' = D. I is clear from he above expression ha he correlaion depends on he volailiy of he facor. Philipov and Glickman (2004a) proposed a high-dimensional addiive facor MSV model in which he facor covariance marices are driven by Wishar random processes, as follows: y = Df + e, e ~ N(0, Ω), f V ~ N(0, V ), V V, A, v, d ~ Wish( v, S ), /2 d /2 S = A ( V ) ( A )', v where V is a marix of facor volailiy, A is a symmeric posiive definie marix, d is a scalar persisence parameer, Wish is he Wishar disribuion, v is he degrees of freedom parameer of he Wishar disribuion. A Bayesian MCMC algorihm is developed o esimae he model. 2

14 2.3. Muliplicaive Facor Models The muliplicaive facor MSV model, also known as he sochasic discoun facor model, was considered in Quinana and Wes (987). The one-facor model from his class decomposes he reurns ino wo muliplicaive componens, a scalar common facor and a vecor of idiosyncraic noise, as follows: y = exp( h / 2) ε, h = µ + φ ( h µ ) + η. + () The firs elemen in Σ ε is assumed o be one for purposes of idenificaion. Compared wih he basic MSV model, his model has a smaller number of parameers, which makes i more convenien compuaionally. Unlike he addiive facor MSV model, however, he correlaions are now invarian wih respec o ime. Moreover, he correlaion in log-volailiies is always equal o one. Ray and Tsay (2000) exended he one-facor model o a k-facor model, in which long range dependence is accommodaed in he facor volailiy: y = exp( h ' v / 2) ε, ε ~ N(0, P ), ε d ( L) h = µ + η, where v is an ( m k ) marix of rank k, wih k < m. 2.4 Time-Varying Correlaion Models The assumpion of consan correlaions in he correlaion marix P ε in equaion (3) can be relaxed by considering he ime-varying correlaion marix, P ε { ρ ij, } =, where ρ ii, = and ρij, ρ ji, =. 3

15 Following he suggesion made by Tsay (2002) and Chrisodoulakis and Sachell (2002) in he bivariae GARCH framework, Yu and Meyer (2004) proposed ha he Fisher ransformaion of ρ 2, could be modelled in a bivariae SV framework, as follows: ρ ( v ) ( v ) = exp( ) exp( ) +, 2, v v u u N 2 + = µ v + ϕ ( µ v) +, ~ (0, σu). (2) The firs equaliy in (2) guaranees ha ρ 2, <. Yu and Meyer (2004) esimaed he bivariae model in equaions (), (2) and (2) using he Bayesian MCMC mehod. The obvious drawback wih his specificaion is he difficuly in generalizing i o a higher-dimension. In order o develop an MSV model which accommodaes ime-varying correlaion, Asai and McAleer (2004c) and Yu and Meyer (2004) suggesed wo alernaive SV exensions of he dynamic condiional correlaion (DCC) model of Engle (2002) (see he VCC model of Tse and Tsui (2002) for a relaed developmen). Suppose ha y, condiional on covariance marix is given by Σ, has a mulivariae normal disribuion, N(0, Σ ), where he Σ = DΓ D. (3) In equaion (3), he ime-varying correlaion marix is given by marix D is defined by equaions () and (2). For he DCC model, follows: Γ, while he diagonal Γ is specified as Γ = Q QQ, ( ) /2 where Q diag{ vecd ( Q) } =, by using some posiive definie marix Q. 4

16 Asai and McAleer (2004c) exended he DCC model by specifying follows: Q as Q = ( ψ) Q + + ψq +Ξ, Ξ ~ W ( ν, Λ), m (4) where W ( ν, Λ) denoes a Wishar disribuion. This dynamic correlaion MSV model m guaranees he posiive definieness of Γ under he assumpion ha Q is posiive definie and ψ <. The laer condiion also implies ha he ime-varying correlaions are mean revering. In he special case where ν =, Ξ can be expressed as he cross-produc of a mulivariae normal variae wih mean zero and covariance marix given by Λ. Yu and Meyer (2004) proposed an alernaive MSV exension of DCC by specifying Q as follows: Q = S+ Bo( Q S) + Ao( ee S) + = So( ιι A B) + BoQ + Aoee, (5) where e ~ N(0, I m). According o Ding and Engle (200) and Engle (2002), if A, B and ( ιι A B) are posiive semi-definie, hen Q will also be posiive semi-definie. Moreover, if any one of he marices is posiive definie, hen Q will also be posiive definie. 2.5 Alernaive Specificaions This sub-secion inroduces four alernaive MSV models based on he marix exponenial ransformaion, he Cholesky decomposiion, he Wishar auoregressive process, and he observed range, respecively Marix Exponenial Transformaion 5

17 Chiu e al. (996) proposed a general framework for he logarihmic covariance marix based on he marix exponenial ransformaion, which is well known in he mahemaics lieraure (see, for example, Bellman (970)). In his sub-secion, we denoe Exp(.) as he marix exponenial operaion o disinguish i from he sandard exponenial operaion. For any m m marix A, he marix exponenial ransformaion is defined by he power series expansion: S Exp( A) = (/ s!) A, (6) s= 0 0 where A reduces o he m m ideniy marix and A s denoes he sandard marix muliplicaion of A s imes. Thus, in general, he elemens of Exp( A ) do no ypically exponeniae he elemens of A. The properies of he marix exponenial and marix logarihm are summarized in Chiu e al. (996). For any real symmeric marix A, we noe he singular value decomposiion A = TDT, where he columns of he m m orhonormal marix T denoe he appropriae eigenvecors of A, and D is an m m diagonal marix, wih elemens equal o he eigenvalues of A. Therefore, Exp( A) = T Exp( D) T, where Exp( D ) is an m m diagonal marix, wih diagonal elemens equal o he exponenial of he corresponding eigenvalues of A. If i is assumed ha Σ ε = Exp( A) for any symmeric marix A, hen Σ is posiive definie. Similarly, he marix logarihmic ransformaion, Log( B ), for any m m posiive definie marix, B, is defined by using he specral decomposiion of B. Using he marix exponenial operaor, we propose he following model: y ~ N(0, Σ ), Σ = Exp( A ), (7) where α = vech( A ) is a vecor auoregressive process, as follows: 6

18 α µ φ α η = +ϒ x + + o +, η ~ N(0, Σ ), η (8) wih (, ) x = y y, n parameer vecors µ and φ, where n = 0.5m(m+), n n covariance marix Σ, and an n 2m marix of parameers ϒ. A limiaion of his η specificaion is ha i is no sraighforward o inerpre he relaionship beween he elemens of Σ and A Cholesky Decomposiion One of he mos serious difficulies in modelling mulivariae volailiy is o ensure ha he covariance marix is posiive semi-definieness (see, for example, Engle and Kroner (995)). Alhough he convenional approach is o impose suiable parameric resricions, Tsay (2002) advocaed an alernaive approach, which uses he Cholesky decomposiion. For a symmeric, posiive-definie marix Σ, he Cholesky decomposiion facors he marix Σ uniquely in he form Σ = L G L ', where L is a lower riangular marix wih uni diagonal elemens, and posiive elemens. G is a diagonal marix wih The MSV model of Tsay (2002) is given as follows: 7

19 y Σ ~ N(0, Σ ), Σ = LGL, ' L 0 L 0 q2, 0 L =, M M M M qm, qm2, L G = diag{ g, K, g } = diag{exp( h ), K,exp( h )},, mm, m h = µ + φh + η, i =, K, m, i+ i i i i q = α + β q + u, i > j. ij, + ij ij ij, ij, The elemens in G are always posiive due o he exponenial ransformaion. Consequenly, he Cholesky decomposiion guaranees he posiive semi-definieness of Σ. I can be seen ha he elemens in L and G are assumed o follow an AR() process. Moreover, i is sraighforward o derive he relaionship beween he variances and correlaions, on he one hand, and he variables in L and G, on he oher, as follows: σ i 2 ii, ik, kk, k = = q g, i =, K, m, σ j = q q g, i > j, i = 2, K, m, ij, ik, jk, kk, k = ρ ij, σ = = σ σ ij, k = j q q g ik, jk, kk, i j ii, jj, 2 2 qik, gkk, qjk, gkk, k= k=. I is clear from hese expressions ha he dynamics in g ii, and q ij, are he 8

20 driving forces underlying he ime-varying volailiy and he ime-varying correlaion. However, he dynamics underlying volailiy are no deermined separaely from hose associaed wih he correlaions, as boh are dependen on heir corresponding AR() processes. This resricion is, a leas in spiri, similar o ha associaed wih facor MSV models Wishar Auoregressive Models Gourieroux e al. (2004) proposed he Wishar auoregressive (WAR) mulivariae process of sochasic posiive semi-definie marices o develop an alogeher differen ype of dynamic MSV model. Le Σ denoe a ime-varying covariance marix of y. Gourieroux e al. (2004) defined he WAR( p ) process, as follows: Σ = K k = x k x ' k (9) where K > m and each x k follows he VAR( p ) model, given by: p x = Ax + ε, ε N(0, Σ). k i k, i k k i= By using he realized value of volailiy, Gourieroux e al. (2004) esimaed he parameers of he WAR() process using a wo-sep procedure based on nonlinear leas squares. Philipov and Glickman (2004b) suggesed an alernaive model based on Wishar processes, as follows: y Σ ~ N(0, Σ ), Σ ν, S ~ W ( ν, S ), m (20) where ν and S are he degrees of freedom and he ime-dependen scale parameer of he Wishar disribuion, respecively. Wih a ime-invarian covariance srucure, he above model may be considered as a radiional Normal-Wishar represenaion of he 9

21 behavior of mulivariae reurns. However, Philipov and Glickman (2004b) inroduced ime variaion in he scale parameer, as follows: d /2 /2 ( )( ) ( ) S = A Σ A, ν where A is a posiive definie symmeric parameer marix ha is decomposed hrough /2 /2 a Cholesky decomposiion as A ( A )( A ) =, and d is a scalar parameer. The quadraic expression ensures ha he covariance marices are symmeric posiive definie. Philipov and Glickman (2004b) esimaed he parameers of he above model using he Bayesian MCMC echnique Range-Based Model Tims and Mahieu (2003) proposed a range-based MSV model. As he range can be used as a measure of volailiy, which is observed (or realized) when he high and low prices are recorded, Tims and Mahieu (2003) suggesed a mulivariae model for volailiy direcly, as follows: log( range ) = D ' f + ε, ε ~ N(0, Σ), η, η ~ (0, ). f+ =Φ f + N Ση As he volailiy is no laen in his model, efficien esimaion of he parameers is achieved hrough he use of he Kalman filer. I is no known, however, how o use his model for purposes of asse pricing. 3. Esimaion As SV models ypically do no have a closed-form expression for he likelihood funcion, he esimaion of he parameers for he wide range of univariae and mulivariae SV models has araced significan aenion in he lieraure. An imporan concern for he choice of a paricular esimaion mehod lies in is 20

22 efficiency. In addiion o efficiency, oher imporan issues relaed o esimaion include: () esimaion of he laen volailiy; (2) deerminaion of he opimal filering, smoohing and forecasing mehods; (3) compuaional efficiency; (4) applicabiliy for flexible modelling. Broo and Ruiz (2004) provided a recen survey regarding he numerous esimaion echniques for SV models, wih an emphasis on univariae SV models and mehods. Some of hese echniques have also been applied o he esimaion of MSV models. 3. Quasi-Maximum Likelihood In order o esimae he parameers of he model ()-(3), Harvey e al. (994) proposed a Quasi-Maximum Likelihood (QML) mehod based on he propery ha he ransformed vecor 2 2 (ln,,ln m) y = y K y has a sae space form wih he measuremen equaion given by: y = h + ξ, ξ = ln ε = (ln ε, K,ln ε ) m (2) and he ransiion equaion (2). The measuremen equaion errors, ξ, are non-normal, wih mean vecor E( ξ ) =.2793ι, where ι is an m vecor of uni elemens. Harvey e al. (994) showed ha he covariance marix of { } 2 Σ ξ = ( π /2) ρ ij, where ρ ii = and ξ, denoed Σ ξ, is given by 2 ( n )! 2n ρij = ρ, 2 ij (22) π (/ 2) n n= n where ( x) n = x( x+ ) L ( x+ n ) and ρ ij is defined by (3). Treaing ξ as a Gaussian error erm, QML esimaes may be obained by applying he Kalman filer o equaions (2) and (2). Taking accoun of he non-normaliy in ξ, he asympoic sandard errors can be obained by using he resuls esablished in Dunsmuir (979). 2

23 If ρ ij can be esimaed, hen i is also possible o esimae he absolue value of ρ ij, and he cross-correlaions beween he differen values of ε i. Esimaion of he signs of ρ ij may be obained by reurning o he unransformed observaions, and noing ha he sign of each of he pairs, εiε j ( i, j =, K, m), will be he same as he corresponding pairs of observed values, yy i j. Therefore, he sign of ρ ij is esimaed as posiive if more han one-half of he pairs, yy i j, is posiive. One of he main feaures of his ransformaion is ha ξ and η are uncorrelaed even if he original ε and η are correlaed (see Harvey e al. (994)). Since he leverage effecs assume a negaive correlaion beween ε and η, as in equaion (5), he ransformaion may ignore he informaion regarding he leverage effecs. In he univariae case, Harvey and Shephard (996) recovered i in he sae space form, given he signs of he observed values. As for he mulivariae case, Asai and McAleer (2004b) derived he sae space form for he leverage effecs in he model (), (2) and (5), based on pairs of he signs of y i and y j. This represenaion enables use of he QML mehod based on he Kalman filer. However, Asai and McAleer (2004b) adoped he Mone Carlo likelihood mehod for purposes of efficien esimaion. The main advanages of he QML mehod are ha i is compuaionally convenien, and also sraighforward for purposes of filering, smoohing, and forecasing. Unforunaely, he available (hough limied) Mone Carlo experimens in he conex of he basic univariae SV model sugges ha he QML mehod is generally less efficien han he Bayesian MCMC echnique and he likelihood approach based on Mone Carlo simulaion (for furher deails, see Jacquier e al. (994) and he discussions conained herein). I is naural o believe ha inefficiency remains for he QML mehod relaive o he Bayesian MCMC echnique and he likelihood approach in he mulivariae conex, 22

24 alhough no Mone Carlo evidence is available o dae. 3.2 Simulaed Maximum Likelihood One Simulaed Maximum Likelihood (SML) mehod is he Acceleraed Gaussian Imporance Sampling (AGIS) approach, as developed in Danielsson and Richard (993). The AGIS approach is designed o esimae dynamic laen variable models, whereby Mone Carlo mehods are used o inegrae he laen variables ou of he join densiy of he laen and observable variables o obain he marginal densiies of he observable variables. Danielsson s commens on Jacquier e al. (994) show ha he finie sample propery of his SML esimaor is close o ha of he Bayesian MCMC mehod. As for MSV models, Danielsson (998) applied he AGIS approach o esimae he parameers of he MSV model in ()-(3). I seems difficul o exend he AGIS approach o accommodae more flexible SV models, as he mehod is specifically designed for models wih a laen Gaussian process. While he AGIS echnique has limied applicabiliy, he Efficien Imporance Sampling (EIS) procedure proposed by Richard and Zhang (2004), and applied by Liesenfeld and Richard (2003, 2004), is applicable o models wih more flexible classes of disribuions for he laen variables. As in he case of AGIS, EIS is a Mone Carlo echnique for he evaluaion of high-dimensional inegrals. The EIS relies on a sequence of simple low-dimensional leas squares regressions o obain a very accurae global approximaion of he inegrand. This approximaion leads o a Mone Carlo sampler, which produces highly accurae Mone Carlo esimaes of he likelihood. In order o esimae he parameers of he addiive facor model (0) wih one facor, Liesenfeld and Richard (2003) proposed an SML approach by approximaing he likelihood funcion based on EIS, while Liesenfeld and Richard (2004) developed a muli-move MCMC algorihm by sampling he laen variables based on EIS. Le λ denoe a q -dimensional vecor of laen variables, and f( Y, Λ ; θ ) be he join densiy of Y = { y } = and { λ } = T T Λ=. The likelihood funcion associaed wih he observable variables, Y, is given by he ( T q) -dimensional inegral L( θ; Y) = f( Y, Λ; θ) dλ. The likelihood funcion can be facorized as: 23

25 T T = = L ( θ; Y ) = f ( y, λ Y, Λ, θ) d Λ = g ( y λ, Y, θ) p ( λ Y, Λ, θ) d Λ, where Y = { y } = and Λ = { λ } = s s s s. I should be noed ha he second equaliy implies ha y is independen of Λ, given ( λ, Y ), which is a sandard assumpion in he analysis of SV models. Alhough i is assumed, for noaional convenience, ha he iniial values { y0, y, K } and { λ0, λ, K } are known consans, his condiion can be relaxed. A Mone Carlo esimae of L( θ ; Y ) based on he above facorizaion is given by: N T () ˆ( ; ) ( i L θ Y = g y λ ( θ ), Y, θ ) N i= = %, () i where he { λ θ } () i % ( ) T are samples drawn from he condiional densiy = p( λ Y, Λ %, θ ). This Mone Carlo esimae is inefficien in he sense ha % () i λ ( θ ) has no relaion o he acual value λ as i does no use any informaion abou y o generae % () i λ ( θ ) from p( λ Y, Λ %, θ ). () i In order o cope wih his problem, he AGIS proposed by Danielsson and Richard (993) and Danielsson (994), and he EIS suggesed by Richard and Zhang (2004), consider an auxiliary sampler, m( λ Λ, a). These mehods enable he facorizaion given above o be rewrien as: L Y f( y, λ Y, Λ, θ) m a d T T ( θ; ) = ( λ Λ, ) Λ, = m( λ Λ, a) = which yields an imporance sampling Mone Carlo esimae, as follows: N T () i () i f( y, % λ ( a) Y, Λ% ( a ), ) L% θ ( θ; Y) =, () i () i N i m( % = = λ ( a) Λ% ( a ), a) (23) 24

26 () i where he { λ a } % ( ) T are samples drawn from imporance densiy = m( λ Λ %, a ). () i The EIS mehod denoes an approximaion of he densiy f( y, λ Y, Λ, θ ) as κ( Λ ; a ), and consrucs he auxiliary densiy m( λ Λ, a ), as follows: κ( Λ; a) m( λ Λ, a) =, χ( Λ ; a ) where χ( Λ ; a) = κ( Λ; a) dλ. I should be noed ha maching f( y, λ Y, Λ, θ ) wih κ( Λ ; a) may leave χ( Λ ; a) unexplained. As χ( Λ ; a) does no depend on λ, i can be ransferred back o he period minimizaion sub-problem. Taken ogeher, EIS requires solving a simple back-recursive sequence of low-dimensional leas squares problems of he form: N { () i () i () i ( λ θ ) ( ˆ θ θ χ θ + ) aˆ = arg min log f y, % ( ) Y, Λ ( ), Λ% ( ); a a i= ( % () i θ aˆ + )} c log κ Λ ( ); for : T, wih χ ( ΛT; at+ ). The c are unknown consans o be esimaed joinly wih he unknown a. In order o obain highly efficien imporance samplers, a small number of ieraions of he EIS algorihm is required. When such ieraions converge o fixed values of he auxiliary parameers, a ˆ, his would be expeced o produce opimal imporance samplers. Finally, he EIS esimae of he likelihood funcion for a given value of θ is obained by subsiuing { a } = for { a } equaion (23). ˆ T T = in Moreover, he EIS mehod can be used o compue he filered esimaes of he laen variables. Le h( λ ) denoe a funcion such as, for example, exp( λ ), which represens he condiional reurn variance. Then he sequence of condiional expecaions of h( λ ), given Y, he pas observaions of he reurns, provides a sequence of filered esimaes of h( λ ). For he SV model, hese expecaions ake he 25

27 form of a raio of inegrals, as follows: h( λ) p( λ Λ, Y, θ) f( Y, Λ ; θ) dλ E[ h( λ) Y ] =, f( Y, Λ ; θ ) dλ in which boh he numeraor and denominaor can be esimaed by he EIS algorihm. In he applicaion of he Bayesian MCMC mehod, Liesenfeld and Richard (2004) proposed using a combinaion of he EIS-sampler wih Tierney's (994) Accepance-Rejecion Meropolis-Hasings (AR-MH) algorihm o simulae Λ Y, θ. The basis of such a procedure is he fac ha he EIS densiy for Λ provides a very close approximaion o f( Λ Y, θ ). 3.3 Mone Carlo Likelihood The Mone Carlo likelihood (MCL) approach for non-gaussian models is based on imporance sampling echniques, so ha he mehod may be classified as an SML mehod. The MCL mehod can approximae he likelihood funcion o an arbirary degree of accuracy by decomposing i ino a Gaussian par, which is consruced by he Kalman filer, and a remainder funcion, whose expecaion is evaluaed hrough simulaion. Durbin and Koopman (997) demonsraed ha he log-likelihood funcion of sae space models wih non-gaussian measuremen disurbances could be expressed simply as pξ ( ξ θ) ln Ly ( θ) = ln LG( y θ) + ln EG, pg ( ξ y, θ) (24) where y = ( y, K y T )', y = ( y, K ym)', and ξ = ( ξ, K, ξ T )' and ln LG ( y θ ) are he vecors of measuremen disurbances and he log-likelihood funcion of he approximaing Gaussian model, respecively, p ξ ( ξ θ ) is he rue densiy funcion, p (, ) G ξ y θ is he Gaussian densiy of he measuremen disurbances of he 26

28 approximaing model, and E G refers o he expecaion wih respec o he so-called imporance densiy pg ( ξ y, θ ) associaed wih he approximaing model. Equaion (24) shows ha he non-gaussian log-likelihood funcion can be expressed as he log-likelihood funcion of he Gaussian approximaing model plus a correcion for he deparures from he Gaussian assumpions relaive o he rue model. A key feaure of he MCL mehod is ha only he minor par of he likelihood funcion requires simulaions, unlike oher SML mehods. Therefore, he mehod is compuaionally efficien in he sense ha i needs only a small number of simulaions o achieve he desirable accuracy for empirical analysis. The MCL esimaes of he parameers, θ, are obained by numerical opimizaion of he unbiased esimae of equaion (24). The log-likelihood funcion of he approximaing model, ln L ( y θ ), can be used o obain he saring values. Sandmann G and Koopman (998) is he firs paper o have used his MCL approach in he SV lieraure. Asai and McAleer (2004b) developed he MCL mehod for asymmeric MSV models. As noed in Asai (2004), his MCL mehod is also able o accommodae he addiive facor MSV model. 3.4 Markov Chain Mone Carlo Markov Chain Mone Carlo (MCMC) mehods are used widely in he SV lieraure, following he developmen in Jacquier e al. (994), which has been grealy refined and simplified by Shephard and Pi (997) and Kim e al. (998). The idea behind MCMC mehods is o produce variaes from a given mulivariae densiy (he poserior densiy in Bayesian applicaions) by repeaedly sampling a Markov chain whose invarian disribuion is he arge densiy of ineres. The MCMC mehod focuses on he densiy π ( θ, h y) insead of he usual poserior densiy, π ( θ y), since he laer requires compuaion of he likelihood funcion f ( y θ) = f( y h, θ) f( h θ) dh. The MCMC procedure only requires alernaing back and forh beween drawing from f ( h θ, y) and f ( θ hy, ). This process of alernaing beween condiional disribuions produces a cyclic chain. 27

29 As for he propery of sample variaes from an MCMC algorihm, hey are a high-dimensional sample from he arge densiy of ineres. These draws can be used as he basis for drawing inferences by appealing o suiable ergodic heorems for Markov chains. For example, poserior momens and marginal densiies can be esimaed (or simulaed consisenly) by averaging he relevan funcion of ineres over he sampled variaes. The poserior mean of θ is esimaed simply as he sample mean of he simulaed θ values. These esimaes can be made arbirarily accurae by increasing he simulaion sample size. One paricularly imporan echnical advanage of he Bayesian MCMC mehod over classical inferenial echniques is ha MCMC does no need o use numerical opimizaion. This advanage becomes especially imporan when he number of parameers o be esimaed is large, as in he applicaion of MSV models o he analysis of financial daa. Jacquier e al. (999), Pi and Shephard (999a), and Aguilar and Wes (2000) have applied he MCMC procedure o esimae addiive facor MSV models, while Yu and Meyer (2004) compared various MSV models. Moreover, Yu and Meyer (2004) employed he purpose-buil Bayesian sofware package called BUGS (Bayesian Analysis Using he Gibbs Sampler). Each of hese MCMC algorihms is based on a single-move algorihm. The main drawback wih he single-move algorihm for MSV models lies in is slow convergence. This is no surprising since he componens of he laen volailiy process are highly persisen (Kim e al. (998)). In order o improve he simulaion efficiency, Chib e al. (2005) developed a new MCMC algorihm which grealy improves simulaion efficiency for a facor MSV model augmened wih jumps. Liesenfeld and Richard (2004) proposed an alernaive muli-move MCMC mehod based on EIS, which can be used o esimae SV models by maximum likelihood, as well as simulaion smoohing. Bos and Shephard (2004) modelled he Gaussian errors in he sandard Gaussian, linear sae space model as an SV process, and showed ha convenional MCMC algorihms for his class of models are ineffecive. Raher han sampling he unobserved 28

30 variance series direcly, Bos and Shephard (2004) sampled in he space of he disurbances, which decreased he correlaion in he sampler and increased he qualiy of he Markov chain. Using he reparameerized MCMC sampler, hey showed how o esimae an unobserved facor model. Smih and Pis (2005) used a bivariae SV model o measure he effecs of inervenion in sabilizaion policy. Missing observaions were accommodaed in he model and a daa-based Wishar prior for he precision marix of he errors in he ransiion equaion were suggesed. A hreshold model for he ransiion equaion was esimaed by MCMC joinly wih he bivariae SV model. 4. Diagnosic Checking and Model Comparison Pi and Shephard (999a) conduced diagnosic checking which is applicable o he MSV models. Alhough heir mehod is based on he paricle filer algorihm (Pi and Shephard (999b)), oher simulaion filering echniques, such as he EIS filer of Liesenfeld and Richard (2003) and he reprojecion echnique of Gallan and Tauchen (998), may also be applicable. By using hese filering mehods, we can obain samples from he predicion densiy, f( h + Y; θ ), where Y = ( y, K, y ). Pi and Shephard (999a) focus on four quaniies for assessing overall model fi, ouliers and observaions which have subsanial influence on he fied model. The firs quaniy is he log-likelihood for +, l+ = log f( y+ Y; θ ). As we have he following: f( y Y; θ ) = f( y h ; θ) df( h Y; θ), Mone Carlo inegraion may be used as fˆ( y Y ; ) f ( y h ; θ ), M i + θ = + + M i= 29

31 i where h + ~ f( h + Y; θ ). I is possible o evaluae he log-likelihood a he ML (MCL, SML) esimaes or a he poserior means. The second quaniy is he normalized log-likelihood, l n. Pi and Shephard (999a) used samples from z j ( j,, j = K S ), where z ~ f( y + Y; θ ), o obain i samples l + using he above mehod. Denoe he sample mean and sandard deviaion l of he samples of log-likelihood as µ + and σ l +, respecively. The normalized log-likelihood a + may be compued as ( ) l = l µ σ. If he model and n l l parameers are correc, hen his saisic should have mean zero and variance one. Large negaive values indicae ha an observaion is less likely han would be expeced from he model. The hird quaniy is he uniform residual, u+ = F( l+ Y; θ ), which may be esimaed as uˆ Fˆ ( l ) ( S) I( l l ), S j + = + = j + < = + j where he l + are consruced as above. Assuming ha he parameer vecor θ is known, under he null hypohesis ha he model is correc, i follows ha uˆ ~ (0,) + UID. as follows: Finally, he fourh quaniy is he disance measure, d, which may be compued ( ) M i + V y+ Y θ M V y i h = + + Σ = ( ; ) ( ; θ ), 30

32 i where h + ~ f( h + Y; θ ). If he condiional disribuion of y is mulivariae normal, hen he quaniy d = y Σ y is independenly disribued as χ 2 m under he null hypohesis ha he parameers and model are correc. Therefore, we may use T 2 d ~ χ = mt as a es saisic. When he MCMC procedure is used, i may require checking convergence of Markov chains and prior sensiiviies. The former can be assessed by correlograms, and he laer by using alernaive priors (for furher deails, see Kim e al. (998), Chib (200) and Chib e al. (2005)). Turning o model selecion, we may use he likelihood raio es for he nesed models, and Akaike informaion crierion (AIC) or Bayesian informaion crierion (BIC) for he non-nesed models, in he conex of he likelihood-based mehods, such as SML and MCL. In he Bayesian framework, model comparison can be conduced via he poserior odds raio or Bayes facor. For boh values, he marginal likelihood needs o be calculaed, for which esimaion is based on he procedure proposed by Chib (995) and is various exensions. The AIC is inappropriae for he MCMC mehod because, when MCMC is used o esimae he SV models, as menioned above, he parameer space is augmened. For example, in he basic univariae SV model, we include he T laen volailiies in he parameer space, wih T being he sample size. As hese volailiies are dependen, hey canno be couned as T addiional free parameers. Consequenly, AIC is no applicable for comparing SV models. Recenly, Berg e al. (2004) showed ha model selecion of alernaive univariae SV models can be performed easily using he deviance informaion crierion (DIC) proposed by Spiegelhaler e al. (2002), while Yu and Meyer (2004) compared alernaive MSV models using DIC. 3

33 5. Concluding Remarks As he lieraure on mulivariae sochasic volailiy (MSV) models has developed significanly over he las few years, his paper reviewed he subsanial lieraure on specificaion, esimaion and evaluaion of MSV models. A wide range of MSV models was presened according o various caegories, namely (i) asymmeric models; (ii) facor models; (iii) ime-varying correlaion models; and (iv) alernaive MSV specificaions, including models based on he marix exponenial ransformaion, Cholesky ecomposiion, Wishar auoregressive process, and he empirical range. Alernaive mehods of esimaion, including quasi-maximum likelihood, simulaed maximum likelihood, Mone Carlo likelihood, and Markov chain Mone Carlo mehods, were discussed and compared. Various mehods of diagnosic checking and model comparison were also examined. Relaive o he exensive heoreical and empirical mulivariae GARCH lieraure, he MSV lieraure is sill in is infancy. The majoriy of exising research in he MSV lieraure deals wih specificaions and/or esimaion echniques, which are ofen illusraed by fiing a paricular symmeric or asymmeric MSV model o financial reurns series. Few papers have direcly addressed imporan economic issues using MSV models. To our knowledge, Nardari and Scruggs (2003) and Han (2002) are wo excepions. Nardari and Scrugg used MSV models o address he resricions in he APT heory while Han examined he economic values of MSV models. Clearly, furher applicaions of MSV models are needed. Mos of he MSV models discussed in his paper have been esimaed using a mos 3 or 4 reurn series. Chib e al. (2005) is he firs paper in he lieraure where genuinely high-dimensional MSV models have been esimaed. Chan e al. (2003) and Nadari and Scruggs (2003) also esimaed high-dimensional MSV models. Wih he developmen of superior esimaion echniques and he availabiliy of greaer compuing power, he lieraure on specificaion, esimaion and evaluaion of high-dimensional MSV models will be broadened appreciably. 32

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