Singapore Centre for Applied and Policy Economics

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1 Sngapore Centre for Appled and Polcy Economcs Volatlty Dynamcs n Foregn Exchange Rates: Further Evdence from the Malaysan Rnggt and Sngapore Dollar by Kn-Yp Ho and Albert K Tsu Department of Economcs SCAPE Workng Paper Seres Paper No. 2008/05 - August

2 Volatlty Dynamcs n Foregn Exchange Rates: Further Evdence from the Malaysan Rnggt and Sngapore Dollar Kn-Yp Ho Department of Economcs Cornell Unversty Ithaca, NY , USA and Albert K Tsu* Department of Economcs Natonal Unversty of Sngapore AS2 Level 6, 1 Arts Lnk Sngapore Abstract The evoluton of volatlty and correlaton patterns of the Malaysan rnggt and the Sngapore dollar are analyzed n ths paper. Our approach can smultaneously capture the emprcal regulartes of persstent and asymmetrc effects n volatlty and tmevaryng correlatons of fnancal tme seres. Consstent wth the results of Tse and Tsu (1997), there s only some weak support for asymmetrc volatlty n the case of the Malaysan rnggt when the two currences are measured aganst the US dollar. However, there s strong evdence that deprecaton shocks have a greater mpact on future volatlty levels compared wth apprecaton shocks of the same magntude when both currences measured aganst the yen. Moreover, evdence of tme-varyng correlaton s hghly sgnfcant when both currences are measured aganst the yen. Regardless of the choce of the numerare currency and the volatlty models, shocks to exchange rate volatlty are found to be sgnfcantly persstent. Keywords: Constant correlatons; Exchange rate volatlty; Fractonal ntegraton; Long memory; Bvarate asymmetrc GARCH; Varyng correlatons JEL Classfcaton: C12; G15 *Correspondng emals: kh267@cornell.edu and ecsatsu@nus.edu.sg 1

3 1. Introducton Followng the semnal work of Engle (1982) and Bollerslev (1986), modellng the tme-varyng condtonal volatlty of fnancal tme seres by the generalzed autoregressve condtonal heteroscedastcty (GARCH) type of models has generated tremendous applcatons n the past two decades. The GARCH-type models are mportant as they can successfully capture the stylzed fact n fnance that large changes n many asset prces tend to be followed by other large changes, whle small changes are usually followed by other small changes. Ths phenomenon s now commonly known as volatlty clusterng. A volumnous lterature has developed on the estmaton and forecastng of volatlty wth GARCH-type models, and ther specfc applcatons n emprcal asset prcng, fnancal rsk management, and opton prcng and hedgng. Among others, some mportant papers nclude Bollerslev et al. (1988), Campbell and Hentschel (1992), Chrstoffersen and Debold (2000), Duan (1995, 1997), French et al. (1987), Glosten et al. (1993), Maheu and McCurdy (2004), Pagan and Schwert (1990), and Schwert (1989). Moreover, there are many survey artcles that provde detaled dscusson of GARCH-type models and ther scope of research. See, for example, Bollerslev et al. (1992), Bera and Hggns (1993), Campbell et al. (1997), Engle (2002), L et al. (2002), and more recently Bauwens et al. (2006). Other than asset prcng and rsk management, GARCH-type models are also employed to analyze the volatlty dynamcs of foregn exchange rates. Among others, see Hseh (1989a, 1989b, 1993), Bollerslev (1990), Balle and Bollerslev (1994), Balle, Bollerslev, and Mkkelsen (1996), Tse and Tsu (1997), Tse (1998) and Tsu and Ho (2004), respectvely. Several well-establshed emprcal regulartes may be hghlghted: [a] evdence of volatlty clusterng s detected n exchange rate returns; [b] unlke stock market volatlty, asymmetrc responses to postve versus negatve shocks of the same 2

4 magntude n exchange rate volatlty are not common; and [c] exchange rate volatlty may dsplay sgnfcant persstence and dependence between observatons, a phenomenon commonly descrbed as long-range dependence or long memory. In partcular, Tse and Tsu (1997) examne the condtonal volatlty of the exchange rates of the Malaysan rnggt and the Sngapore dollar aganst the US dollar usng the unvarate asymmetrc power ARCH model proposed by Dng, Granger, and Engle (1993). They fnd evdence of negatve asymmetrc effects n the Malaysan rnggt. More recently, Tsu and Ho (2004) detect evdence of asymmetrc volatlty n the Malaysan rnggt and the Sngapore dollar usng unvarate fractonally ntegrated GARCH-type models. However, such establshed fndngs are based on unvarate GARCH-type models and on the blateral USD rates. Lttle work has been done on foregn exchange rates wth dfferent choce of numerare currency and GARCH-type structures n a multvarate framework. Whle the unvarate GARCH-type models may appear reasonably adequate for capturng the volatlty dynamcs n exchange rates, they are not talored to accommodate co-movements n foregn exchange volatlty. Several academcs and practtoners have noted that exchange rates are sgnfcantly correlated and these correlatons can nfluence currency tradng strateges (see Len (2005, 2006), Jang, Ma, and Ca (2007), Munandy and Unng (2006), and Mzuno et al. (2006)). Therefore, t s mperatve to understand how dfferent pars of currences move wth one other over tme. By keepng track of these co-movements, traders can understand ther exposure to exchange rate rsk. An example gven n Len (2006) llustrates that an effectve foregn exchange trader should understand hs/her overall portfolo's senstvty to market volatlty. The man reason s that currences are prced n pars, no sngle par trades completely ndependently of the others. Once a trader knows about ther correlatons 3

5 and the pattern of changes over tme, he/she can take advantage of them to control the portfolo's exposure. As such, t s mportant to analyze currency volatlty n a multvarate framework n order to accommodate the potental nterdependences between the exchange rates. Moreover, t has been frequently argued that nformaton transmsson from one foregn exchange market to another can nfluence currency volatlty. In partcular, Engle, Ito and Ln (1990) argue that volatlty n one foregn exchange market s transmtted to other markets lke a meteor shower, whle Ross (1989) shows that volatlty n asset returns depends upon the rate of nformaton flow. Snce the rate of nformaton flow and the tme used n processng that nformaton vares wth each ndvdual market (sector), one should expect dfferent volatlty patterns across markets (sectors). The ncreasng ntegraton of major fnancal markets has generated strong nterest n understandng the volatlty spllover effects from one market to another. These volatlty spllovers are usually attrbuted to cross-market hedgng and change n common nformaton, whch may smultaneously alter expectatons across markets. Apparently, a multvarate framework to capture such features s more approprate. Emprcally there are lmted studes on the volatlty dynamcs of exchange rates by the MGARCH-type models. The man obstacle s due to the computatonal dffcultes n estmatng the ncreased number of parameters and there s no guarantee of the postve-defnteness for the condtonal varance-covarance matrx durng optmzaton. Bollerslev (1990) proposes the constant-correlatons (CC)-MGARCH model, whch automatcally guarantees the postve-defnteness of the varance-covarance matrx once convergence s acheved. However, ths approach s qute restrctve and has not been successful n several studes (see, for example, Tsu and Yu (1999), Tse (2000), 4

6 Engle and Sheppard (2001), Bera and Km (2002), and McAleer et al. (2008)). A multvarate verson that s dfferent from the CC-MGARCH model s prevously proposed by Engle, Granger, and Kraft (1984), who derve the necessary condtons for the matrx of the model to be postve-defnte; however, generalzng ths model to hgher dmensons s rather ntractable. Alternatvely, Bollerslev, Engle, and Wooldrdge (1988) propose the vech-representaton, whch s an extenson of the unvarate GARCH representaton to the vectorzed condtonal varance-covarance matrx. However, condtons that guarantee the postve-defnteness of the varance-covarance matrx are dffcult to sustan durng optmzaton. Moreover, Engle and Kroner (1995) ntroduce the Baba-Engle-Kraft-Kroner (BEKK) model, whch automatcally guarantees the postvedefnteness of the varance-covarance matrx once parameter estmates are obtaned. Another approach looks nto the condtonal volatlty of dfferent assets as a factor model; see Debold and Nerlove (1989), Engel and Rodrgues (1989) and Engle, Ng, and Rothschld (1990) for detals. However, the shortcomngs of the BEKK and factor models are that the parameters cannot be easly estmated and nterpreted, and ther net mpact on the future varance and covarance are not readly observed. In addton, the ncreasng number of parameters to be estmated under the BEKK and factor-garch models exacerbates the dffcultes of achevng convergence n parameter estmaton. For example, Bera et al. (1997) report that the BEKK model does not perform well n the estmaton of the optmal hedge ratos, and Len et al. (2002) report dffcultes n obtanng meanngful estmates of the BEKK model durng optmzaton. For a detaled comparson of these MGARCH-type models, see Bauwens et al. (2006). In ths paper, we follow up on the study of asymmetrc volatlty of two currences n the Asa-Pacfc markets by Tse and Tsu (1997), namely the Malaysan rnggt and the Sngapore dollar. To ensure consstency n comparson, we confne our nvestgaton to 5

7 the GARCH-type models. Instead of usng unvarate APARCH models by Tse and Tsu (1997), we employ the MGARCH framework of Tse and Tsu (2002) to create a famly of bvarate MGARCH models whch can concurrently capture the stylzed features of volatlty asymmetry, long-range persstence n volatlty, and tme-varyng correlatons. The proposed models automatcally ensure the postve defnteness of the condtonal varance-covarance matrx once convergence s obtaned. One added advantage of the Tse and Tsu approach s that the parameter estmates are relatvely easy to nterpret, as the unvarate GARCH-type equatons are retaned. Unlke Bollerslev s (1990) constant correlaton MGARCH model, the ncluded tme-varyng condtonal correlatons n the proposed models can map out the tme-path of condtonal correlatons between the two currences. We also nvestgate the behavour of long-memory persstence n volatlty of the Malaysan rnggt and the Sngapore dollar usng fractonally ntegrated GARCH-type models. The fractonally ntegrated models help to dstngush between long persstence and exponental decay n the mpacts of exchange rate volatltes. Furthermore, we examne the robustness of the volatlty dynamcs of the two currences aganst the Japanese yen as alternatve numerare currency besdes the US dollar. We are motvated by several studes on the senstvty of alternatve numerare currency. See, for example, Papell and Theodords (2001), Zambrano (2005), and Norrbn and Ppatchapoom (2007), respectvely. In partcular, Papell and Theodords (2001) demonstrate that choce of dfferent numerare currency can have sgnfcant mpacts on purchasng power party (PPP). They fnd that PPP s stronger for European than for non-european base currences, and the volatlty of the exchange rate s one of the key determnants of such a fndng. In addton, Schlossberg (2007) provdes several examples n tradng n cross rates where the Japanese yen s used as numerare for the 6

8 Canadan dollar, the New Zealand dollar and the Brtsh pound, respectvely. He argues that tradng n currency crosses can provde a multtude of proftable opportuntes. The rest of the paper s organzed as follows. In Secton 2, we present the methodology of syntheszng features of volatlty asymmetry, long-memory and tmevaryng correlatons n a bvarate GARCH framework. Secton 3 brefly descrbes the data sets used and the estmaton results. Secton 4 provdes some concludng remarks. 2. Methodology In what follows we frst brefly descrbe the gst of the bvarate GARCH(1,1) model wth tme-varyng condtonal correlatons (VC-GARCH) proposed by Tse and Tsu (2002). We then ncorporate two dfferent structures of asymmetrc volatlty and long memory nto the condtonal varance equatons so as to synthesze the bvarate GARCH-type models. Let y t = (y 1t, y 2t ) be the bvarate vector of varables wth tme-varyng varancecovarance matrx H t, and let μ t (ξ ) be the arbtrary condtonal mean functons whch depend on ξ, a column vector of parameters. A typcal bvarate GARCH(1,1) model can be specfed as follows: yt = μ t ( ξ ) + εt, = 1,2 (1) where ε, ε )' Φ ~ ( O, H ) (2) ( 1t 2t t 1 t Note that Φ t s the σ-algebra generated by all the avalable nformaton up to tme t. The random dsturbance terms ε t and the condtonal varance equatons h t are modelled as follows: 7

9 ε = h e, where e t ~ N(0,1) (3) t t t h t 2 = + α εt 1 + βht 1 η (4) where (4) s the standard Bollerslev s (1986) symmetrc GARCH(1,1) model. Denote the j-th element (, j = 1, 2) n H t by h jt. The condtonal correlaton coeffcents can be defned as ρ jt = h h t jt h jjt. Essentally, Tse and Tsu (2002) assume that the tme-varyng condtonal correlaton matrx Γ = { } followng recurson = 12 ( ) + + t π π ρ12 π 1 ρ12, t 1 π 2ψ 12, t 1 t ρ jt s generated by the ρ (5) 2 2 e a = 1 1, t a e 2, t a ψ 12, t 1 = (6) ( ) e = 1 1, )( a t a e a= 1 2, t a e1 t e2t 2ρ12te1 te lt ( θ ) = + log h log(1 12 ) 1 t ρ t 2 2 = 2 2(1 ρ ) 12t 2t (7) The condtonal correlaton equaton n (5) nherts the prototype property of GARCH(1,1) structure, and t nests Bollerslev s (1990) constant-correlatons GARCH (CC-GARCH) structure when π 1 = π 2 = 0. Hence, the lkelhood rato test can be readly appled to compare the performance of both models. Owng to computatonal dffcultes, there are few emprcal studes on long-range temporal dependence. See Tse and Tsu (1997), Teyssere (1997, 1998), and Brunett and Glbert (2000), among others. These studes have manly appled the multvarate verson of the fractonally ntegrated symmetrc GARCH (FIGARCH) model of Balle, Bollerslev, and Mkkelsen (1996) to stock market and exchange rate data. However, they 8

10 often exclude the ssue of asymmetrc condtonal volatlty, and for convenence, they adopt the assumpton of constant condtonal correlatons n the volatlty structure. In what follows, we ncorporate the structures of asymmetrc volatlty and long memory dynamcs nto the VC-GARCH model by modfyng the symmetrc condtonal varance equaton n (4). To mantan consstency n comparson, we choose two well-establshed asymmetrc structures among the GARCH-type models. They nclude: the asymmetrc GARCH (1,1) (AGARCH (1,1)) model proposed by Engle (1990) and the asymmetrc power ARCH (1,1) (APARCH (1,1)) model of Dng, Granger, and Engle (1993), respectvely. Indeed, Tse and Tsu (1997) use the APARCH (1,1) model to capture the possbly asymmetrc effects of exchange shocks on future volatltes. In addton, these asymmetrc GARCH-type models are less restrctve n assumptons and are more flexble to accommodate alternatve varatons. Ther man features are brefly summarzed as follows: [a] Engle s (1990) asymmetrc GARCH(1,1) (AGARCH(1,1)) model: h t 2 = + α( εt 1 γ ) + β ht 1 ω (8) where γ s the asymmetrc coeffcent. When γ = 0, (8) becomes the GARCH(1,1) model and when β = 0, t becomes the prototype ARCH(1) model. [b] Dng, Granger, and Engle s (1993) asymmetrc power ARCH(1,1) (APARCH (1,1)) model. h δ 2 δ 2 t ( t 1 t 1 ) h δ = η + α ε γ ε + β t 1 (9) where γ s the asymmetrc coeffcent. When δ = 1, equaton (9) becomes the threshold GARCH(1,1) model, whch ncludes an asymmetrc verson of the Taylor/Schwert (1986/1989) model and Zakoan s (1994) threshold ARCH model. When δ = 2, t becomes the leveraged GARCH model, whch nests the GJR model of Glosten, 9

11 Jaganathan and Runkle (1993). When δ approaches 0, Dng, Granger, and Engle (1993) show that (9) becomes the logarthmc GARCH(1,1) model, thereby ncorporatng an asymmetrc verson of the Geweke/Pantula (1986) model. We note n passng that although the APARCH structure nests 7 models together (see Dng, Granger, and Engle (1993) for detals), t does not nest the AGARCH model. Turnng to the structure of long-memory dynamcs n volatlty, we may transform the condtonal varance equatons n (4), (8) and (9) so that they are fractonally ntegrated. We follow the methodology by Balle et al. (BBM) (1996). Below s a summary of the condtonal varance equatons for three fractonally ntegrated (FI) GARCH-type models obtaned by the BBM approach. [a]fractonally ntegrated GARCH(1,1) model (FIGARCH(1,d,1)) h t η 2 = + λ ( L) ε t (10) 1 β a 1 d where λ ( L ) = λ L = 1 (1 β L) (1 φ L)(1 L). a= 1 a [b] Fractonally ntegrated asymmetrc GARCH(1,1) model ((FIAGARCH)(1,d,)) model h t ω 2 = + λ ( L)( εt γ ) (11) 1 β where λ (L) s defned as n (10). Note that (11) s smlar to the FIGARCH(1,d,1) model n (10), except that t allows past return shocks to have asymmetrc effects on the condtonal volatlty. [c] Fractonally ntegrated APARCH(1,1) model (FIAPARCH(1,d,1)) 10

12 η 1 β δ 2 δ h t = + λ ( L)( ε γ ε ) (12) t t where λ (L) s defned as n (10). Smlar to the FIAGARCH(1,d,1) model n (11), (12) allows past shocks to have asymmetrc effects on the condtonal volatlty. Detals of the dervatons are gven n Tsu and Ho (2004). The parameters of these bvarate fractonally ntegrated GARCH-type models can be estmated usng Bollerslev-Wooldrdge s (1992) quas-maxmum lkelhood estmaton (QMLE) approach. Approprate assumptons for the start-up condtons are made to facltate convergence of the QMLE optmzaton process. These nclude the computaton of λ (L), the number of lags, and the ntal values. For nstance, the response coeffcents for each of the fractonally ntegrated GARCH-type models a 1 d λ ( L ) = λal = 1 (1 βl) (1 φl)(1 L) are computed by adoptng the = a 1 followng nfnte recursons gven n Bollerslev and Mkkelsen (1996): λ = φ β + d, 1 λb = βλb 1 + [( b 1 d) / b φ ] ζ b 1, b = 2,..., (13) where ζ b = ζ b 1( b 1 d) / b, wth ζ 1 = d As can be observed from (13), when b approaches nfnty, an adequate fnte truncaton s necessary to secure the long-memory dynamcs. We have sampled the 1000 and 2000 lags, respectvely; and found that the parameter estmates trmmed at 1000 lags are reasonably close to those trmmed at 2000 lags. To save the computatonal tme, we truncate λ (L) after the frst 1000 lags. 11

13 On the choce of ntal values, we set the pre-sample observatons ε 2 t to the uncondtonal sample varance for the FIGARCH(1,d,1) model. As for the bvarate FIAGARCH(1,d,1) model, the pre-sample observatons of g 2 2 ( εt ) = ( ε t γ ) are equated to the sample mean of ˆ ε ˆ γ ) ( t 2, where γˆ s the estmate of γ based on the unvarate FIAGARCH(1,d,1) model. In the case of the bvarate FIAPARCH(1,d,1) model, the pre-sample observatons of δ g( ε ) ) t δ = ( ε γ ε are equated to the t t sample mean of ˆ ˆ ˆ δ ( εt γ ε t ), where ˆ γˆ and δˆ are the estmates of γ and δ based on the unvarate FIAPARCH(1,d,1) model. We shall nvestgate 6 dfferent model specfcatons, ncludng 3 basc symmetrc and asymmetrc GARCH-type models and ther extensons to the correspondng fractonally ntegrated GARCH-type models. We then apply these unvarate models ndvdually to the Malaysan rnggt and the Sngapore dollar aganst the dollar or the yen, thereby obtanng 6 bvarate CC-MGARCH-type models and 6 bvarate VC- MGARCH-type models, respectvely. 3. Data and Estmaton Results Our data sets consst of 2998 daly observatons of the Malaysan rnggt (MYR) and the Sngapore dollar (SGD), coverng the perod from 2 January 1986 to 30 June More recent observatons are excluded to avod the possble dstortons caused by the outbreak of the 2-year Asan fnancal crss snce July The exchange rates aganst the US dollar (USD) are obtaned drectly from DataStream Internatonal and detals of these seres are dscussed n Tsu and Ho (2004). Owng to the non- 12

14 avalablty of the blateral Japanese yen (JPY) exchange rates for the perod under study, we utlze the mpled cross rates nstead. They are obtaned by dvdng the exchange rate of a naton s currency aganst the US dollar wth the Japanese yen-us dollar (JPY/USD) exchange rate. The daly nomnal exchange rate returns expressed n percentage are computed on a contnuously compoundng bass as: St y t = log( ) 100 (14) S t 1 where S t s the daly exchange rate. We assume that the condtonal mean equaton s captured by a lower-order autoregressve flter wth lag order p: yt p = ξ 0 + ξ at yt a + ε t, = 1,2 a = 1 (15) Table 1 provdes a summary of the descrptve statstcs of y t for the two currences measured aganst the dollar or the yen. For a standard normal dstrbuton, the skewness and kurtoss have values of 0 and 3, respectvely. As can be observed from Panel A of Table 1, all dfferenced logarthmc seres have kurtoss greater than 3. In partcular, the MYR and SGD exhbt much hgher kurtoss when they are measured aganst the dollar. Fgures 1-2 present the plots of the exchange rates of the two currences and ther returns seres aganst the dollar and the yen, respectvely. It can be observed that the return seres are centred about zero and the ampltude of the returns s changng. The magntude of the changes s sometmes large (small) followng the prevous large (small) ones over the sample perod, thereby reflectng the stylzed fact of volatlty clusterng. 13

15 Fgure 1. Malaysan Rnggt (MYR) and Sngapore Dollar (SGD) aganst the US dollar (USD) Fgure 2. Malaysan Rnggt (MYR) and Sngapore Dollar (SGD) aganst the Japanese Yen (JPY) 14

16 Table 1. Summary Statstcs of Exchange Rates aganst the Japanese Yen and the US Dollar Varable MYR/JPY SGD/JPY MYR/USD SGD/USD Mean Medan Maxmum Mnmum Std. Dev Skewness Kurtoss Observatons Q 1 (5) Q 1 (10) Q 2 (5) Q 2 (10) Q 3 (5) Q 3 (10) BDS(e=3,l=1.5) BDS(e=4,l=1.5) BDS(e=5,l=1.5) BDS(e=3,l=1.0) BDS(e=4,l=1.0) BDS(e=5,l=1.0) R R R Notes: 1. JPY = Japanese Yen, MYR = Malaysan rnggt, SGD = Sngapore dollar, USD = US dollar 2. Q(m) refers to the Ljung-Box Q-statstc wth m degrees of freedom. Q for = 1, 2, 3 denote the seres y t, y t, and y 2 t respectvely. 3. For the BDS Test, e represents the embeddng dmenson whereas l represents the dstance between pars of consecutve observatons, measured as a multple of the standard devaton of the seres. Under the null hypothess of ndependence, the test statstc s asymptotcally dstrbuted as standard normal. 4. For the Runs Test, R for = 1, 2, 3 denote the runs tests of the seres y t, y t, and y 2 t respectvely. Under the null hypothess that successve observatons n the seres are ndependent, the test statstc s asymptotcally standard normal. Table 2. Unt Root Tests Exchange Rate ADF Model ADF Test Statstc Q-statstc (20 lags) PP Statstc MYR/JPY Case 3 (20) SGD/JPY Case 3 (20) MYR/USD Case 3 (20) SGD/USD Case 3 (20) Notes: 1. ADF Model: Case 1 refers to the regresson equaton wthout any determnstc regressors; Case 2 refers to the equaton wth ntercept; Case 3 refers to the equaton wth both the ntercept and the determnstc tme trend. The fgure n parenthess hghlghts the number of lagged dfference terms. 2. For the PP test, both ntercept and tme trend are ncluded and 8 truncaton lags are chosen. It s found that the results are robust to dfferent lag lengths. 3. Q statstc refers to the Ljung-Box Q-statstc wth 20 degrees of freedom. 15

17 As dsplayed n Table 2, the augmented Dckey-Fuller and Phllps-Perron tests are all nsgnfcant at the 5% level, thereby ndcatng that the return seres of the MYR and SGD are statonary. However, the Ljung-Box Q-statstcs and the BDS test statstcs (Brock, Dechert, and Schenkman (1996)) suggest that both foregn exchange seres are not ndependently and dentcally dstrbuted. In addton, as can be seen from Table 1, the hghly sgnfcant Lung-Box Q statstcs and the runs tests consstently ndcate the presence of condtonal heteroscedastcty n the return seres. As such, the GARCHtype modellng of the volatlty structures may be approprate. We shall estmate the condtonal mean, varance and correlaton components of the proposed bvarate GARCH-type models smultaneously usng Bollerslev and Wooldrdge s (1992) quas maxmum-lkelhood estmaton (QMLE) procedure coded n Gauss verson 5.0. The QMLE approach provdes consstent estmators even for nonnormal errors wth a thck-taled dstrbuton. For the mean equaton, we fnd that the parsmonous AR(1) model s a reasonably adequate flter, takng nto consderaton of the log-lkelhood values and the resdual checks. To save space, we report only estmates of the condtonal varance and correlaton equatons from the followng models: the VC-GARCH, VC-AGARCH, VC-APARCH, VC-FIGARCH, VC-FIAGARCH and the VC-FIAPARCH, respectvely. Except for the correlaton coeffcents and the loglkelhood values, most of the parameter estmates from the constant-correlaton models are omtted. The complete set of estmaton results s avalable upon request. Tables 3-8 summarze the QMLE of the parameters of the bvarate VC-GARCH, VC-APARCH, VC-AGARCH, VC-FIGARCH, VC-FIAPARCH and VC-FIAGARCH models, respectvely. We frst dscuss the evdence of asymmetrc volatlty. For the currences aganst the dollar, only the Malaysan rnggt exhbts asymmetrc volatlty 16

18 under the VC-FIAPGARCH model, whereas there s no evdence of asymmetrc effects for the SGD. Our results are consstent wth the fndngs by Tse and Tsu (1997), and Tsu and Ho (2004), respectvely. In partcular, Tse and Tsu (1997) report that the deprecaton shocks of MYR/USD generate greater future volatltes compared to apprecaton shocks of the same magntude. In contrast, when the yen s used as the numerare currency, we detect sgnfcant evdence of negatve asymmetrc volatlty for the SGD based on all of the GARCH-type models. As for the MYR, except for the VC- AGARCH model, we do not detect evdence of asymmetrc volatlty. Apparently, the support of volatlty asymmetry s senstve to the specfcaton of the condtonal volatlty and to the choce of numerare currency. The estmated values of fractonal dfferencng parameter (d) of varous models are reported n Tables 6-8. Two nterestng results are n order. Frst, all the estmates are statstcally sgnfcantly dfferent from 0 and 1, ndcatng that the mpact of shocks to the condtonal volatlty dsplays a hyperbolc rather than exponental rate of decay. Ths result s robust to the choce of the numerare currency and the models. Second, most of the fractonal dfferencng parameters for the MYR and the SGD are smlar across the GARCH-type models for a gven numerare currency. For example, when the dollar s used as the numerare, the estmated values of d for MYR and SGD are and respectvely for the symmetrc VC-FIGARCH model; and for the asymmetrc VC-FIAPARCH model; and and for the asymmetrc VC- FIAGARCH model, respectvely. Smlarly, when the yen s used as the numerare currency, the MYR and the SGD have consstently lower estmated values for d wthn the range of than that of the correspondng GARCH-type counterparts under the dollar. Moreover, the lkelhood rato test statstcs reported n Table 9 are all 17

19 sgnfcant at the 5% level, thereby ndcatng that the fractonally ntegrated models are more adequate than those wthout the long memory structure. To assess the correlaton dynamcs of the two currences, we apply the lkelhood rato (LR) test to the null hypothess of π 1 = π 2 = 0 (see equaton (5)). Under the null hypothess, the LR test statstcs follows an asymptotc ch-squared dstrbuton wth two degrees of freedom. Also, the sgnfcance of the estmated values of π 1 and π 2 are examned ndvdually. As shown n columns 8-13 of Tables 3-8, all the LR tests ndcate that the null hypothess of constant condtonal correlatons s rejected at the 5% level of sgnfcance, thereby suggestng that the condtonal correlatons are tmevaryng. Such fndngs are robust across models. In contrast, almost all of the ndvdual estmates of π 1 and π 2 are statstcally nsgnfcant when the MYR and SGD are measured aganst the dollar; and all ndvdual estmates are sgnfcant at the 5% level when ther exchange rates are based on the yen. Ths mples that the evdence of tmevaryng correlatons between MYR/USD and SGD/USD s relatvely weaker, and t s consstent wth Tse s (2000) concluson that the hypothess of constant condtonal correlaton cannot be rejected for the MYR and SGD. However, we detect strong support of tme-varyng correlatons between MYR and SGD when the Japanese yen s used as the numerare currency. The reason as to why the asymmetrc effects are not robust to exchange rates under dfferent numerare currency s stll unknown to researchers. Apparently t s a challengng topc for future researchers. 18

20 Table 3. Estmaton Results of Bvarate VC-GARCH(1,1) Model: h t = η + αε 2 t-1 + βh t-1 ; Γ t = (1 π 1 π 2 )Γ + π 1 Γ t-1 + π 2 Ψ t-1 Varable η β α Γ π 1 π 2 LL (VC) Corr (CC) LL (CC) LR MYR/USD (0.0012) (0.0453) (0.0349) (0.0263) (0.7034) (0.0744) (0.0229) SGD/USD (0.0016) (0.0562) (0.0381) MYR/JPY (0.0236) (0.0952) (0.0502) (0.0087) (0.0363) (0.0077) (0.9044) SGD/JPY (0.0133) (0.0586) (0.0293) Notes: 1. All standard errors (n parenthess) are the heteroskedastc-consstent Bollerslev-Wooldrdge standard errors computed based on the Quas-Maxmum Lkelhood Estmaton (QMLE) technque. 2. Log-lkelhood value (VC) and Log-lkelhood value (CC) refer to the lkelhood values obtaned from the VC-GARCH(1,1) and CC-GARCH(1,1) models respectvely. 3. Correlatons (CC) refer to the condtonal correlaton coeffcent obtaned from the CC-GARCH(1,1) model. 4. LR s the lkelhood rato statstc for H 0 : π 1 = π 2 = 0 n the VC-GARCH(1,1) model. It s dstrbuted as ch-squared wth 2 degrees of freedom under H 0. Table 4. Estmaton Results of Bvarate VC-APARCH(1,1) Model: h δ/2 t = η + α( ε t-1 - γε t-1 ) δ + βh δ/2 t-1; Γ t = (1 π 1 π 2 )Γ + π 1 Γ t-1 + π 2 Ψ t-1 Varable η β α γ δ Γ π 1 π 2 LL (VC) Corr (CC) LL (CC) LR MYR/USD (0.0019) (0.0470) (0.0358) (0.0656) (0.2646) (0.0263) (0.9150) (0.1008) (0.0228) SGD/USD (0.0054) (0.0512) (0.0359) (0.0921) (0.3099) MYR/JPY (0.0249) (0.0659) (0.0377) (0.1117) (0.2053) (0.0089) (0.0383) (0.0080) (0.0065) SGD/JPY (0.0163) (0.0425) (0.0235) (0.1132) (0.1713) Notes: 1. All standard errors (n parenthess) are the heteroskedastc-consstent Bollerslev-Wooldrdge standard errors computed based on the Quas-Maxmum Lkelhood Estmaton (QMLE) technque. 2. Log-lkelhood value (VC) and Log-lkelhood value (CC) refer to the lkelhood values obtaned from the VC-APARCH(1,1) and CC-APARCH(1,1) models respectvely. 3. Correlatons (CC) refer to the condtonal correlaton coeffcent obtaned from the CC-APARCH(1,1) model. 4. LR s the lkelhood rato statstc for H 0 : π 1 = π 2 = 0 n the VC-APARCH(1,1) model. It s dstrbuted as ch-squared wth 2 degrees of freedom under H 0. 19

21 Table 5. Estmaton Results of VC-AGARCH(1,1) Model: h t = η + α(ε t-1 - γ) 2 + βh t-1 ; Γ t = (1 π 1 π 2 )Γ + π 1 Γ t-1 + π 2 Ψ t-1 Varable η β α γ Γ π 1 π 2 LL (VC) Corr (CC) LL (CC) LR MYR/USD (0.0013) (0.0459) (0.0355) (0.0236) (0.0258) (0.6992) (0.0734) (0.0227) SGD/USD (0.0015) (0.0535) (0.0367) (0.0306) MYR/JPY (0.0154) (0.0653) (0.0341) (0.0770) (0.0089) (0.0360) (0.0079) (0.0070) SGD/JPY (0.0097) (0.0436) (0.0219) (0.0774) Notes: 1. All standard errors (n parenthess) are the heteroskedastc-consstent Bollerslev-Wooldrdge standard errors computed based on the Quas-Maxmum Lkelhood Estmaton (QMLE) technque. 2. Log-lkelhood value (VC) and Log-lkelhood value (CC) refer to the lkelhood values obtaned from the VC-AGARCH(1,1) and CC-AGARCH(1,1) models respectvely. 3. Correlatons (CC) refer to the condtonal correlaton coeffcent obtaned from the CC-AGARCH(1,1) model. 4. LR s the lkelhood rato statstc for H 0 : π 1 = π 2 = 0 n the VC-AGARCH(1,1) model. It s dstrbuted as ch-squared wth 2 degrees of freedom under H 0. Table 6. Estmaton Results of Bvarate VC-FIGARCH(1,d,1) Model Varable η φ β D Γ π 1 π 2 LL (VC) Corr (CC) LL (CC) LR MYR/USD (0.0012) (0.1215) (0.1090) (0.1481) (0.0257) (0.3147) (0.0409) (0.0236) SGD/USD (0.0010) (0.1357) (0.1093) (0.2290) MYR/JPY (0.0365) (0.2298) (0.2411) (0.0439) (0.0080) (0.0349) (0.0079) (0.0063) SGD/JPY (0.0143) (0.1093) (0.0993) (0.0574) Notes: 1. All standard errors (n parenthess) are the heteroskedastc-consstent Bollerslev-Wooldrdge standard errors computed based on the Quas-Maxmum Lkelhood Estmaton (QMLE) technque. 2. Log-lkelhood value (VC) and Log-lkelhood value (CC) refer to the lkelhood values obtaned from the VC-FIGARCH(1,d,1) and CC-FIGARCH(1,d,1) models respectvely. 3. Correlatons (CC) refer to the condtonal correlaton coeffcent obtaned from the CC-FIGARCH(1,d,1) model. 4. LR s the lkelhood rato statstc for H 0 : π 1 = π 2 = 0 n the VC-FIGARCH(1,d,1) model. It s dstrbuted as ch-squared wth 2 degrees of freedom under H 0. 20

22 Table 7. Estmaton Results of Bvarate VC-FIAPARCH(1,d,1) Model Varable η φ γ δ β D Γ π 1 π 2 LL (VC) Corr (CC) LL (CC) LR MYR/USD (0.0023) (0.1167) (0.0636) (0.1685) (0.1080) (0.1332) (0.0254) (0.3452) (0.0406) (0.0232) SGD/USD (0.0036) (0.1662) (0.0814) (0.1803) (0.1102) (0.2592) MYR/JPY (0.0720) (0.2247) (0.1004) (0.1731) (0.2517) (0.0532) (0.0080) (0.0402) (0.0087) (0.0063) SGD/JPY (0.0295) (0.0802) (0.1012) (0.1549) (0.1004) (0.0755) Notes: 1. All standard errors (n parenthess) are the heteroskedastc-consstent Bollerslev-Wooldrdge standard errors computed based on the Quas-Maxmum Lkelhood Estmaton (QMLE) technque. 2. Log-lkelhood value (VC) and Log-lkelhood value (CC) refer to the lkelhood values obtaned from the VC-FIAPARCH(1,d,1) and CC-FIAPARCH(1,d,1) models respectvely. 3. Correlatons (CC) refer to the condtonal correlaton coeffcent obtaned from the CC-FIAPARCH(1,d,1) model. 4. LR s the lkelhood rato statstc for H 0 : π 1 = π 2 = 0 n the VC-FIAPARCH(1,d,1) model. It s dstrbuted as ch-squared wth 2 degrees of freedom under H 0. Table 8. Estmaton Results of Bvarate VC-FIAGARCH(1,d,1) Model Varable η φ γ β d Γ π 1 π 2 LL (VC) Corr (CC) LL (CC) LR MYR/USD (0.0013) (0.1215) (0.0238) (0.1118) (0.1538) (0.0257) (0.3162) (0.0411) (0.0236) SGD/USD (0.0011) (0.1392) (0.0312) (0.1190) (0.2368) MYR/JPY (0.0316) (0.2027) (0.0703) (0.2070) (0.0387) (0.0081) (0.0388) (0.0084) (0.0063) SGD/JPY (0.0151) (0.1340) (0.0692) (0.1218) (0.0489) Notes: 1. All standard errors (n parenthess) are the heteroskedastc-consstent Bollerslev-Wooldrdge standard errors computed based on the Quas-Maxmum Lkelhood Estmaton (QMLE) technque. 2. Log-lkelhood value (VC) and Log-lkelhood value (CC) refer to the lkelhood values obtaned from the VC-FIAGARCH(1,d,1) and CC-FIAGARCH(1,d,1) models respectvely. 3. Correlatons (CC) refer to the condtonal correlaton coeffcent obtaned from the CC-FIAGARCH(1,d,1) model. 4. LR s the lkelhood rato statstc for H 0 : π 1 = π 2 = 0 n the VC-FIAGARCH(1,d,1) model. It s dstrbuted as ch-squared wth 2 degrees of freedom under H 0. 21

23 Table 9. Lkelhood Rato Test: Bvarate VC and VC-FI Models Varable VC-GARCH VC-FIGARCH LR VC-APARCH VC-FIAPARCH LR VC-AGARCH VC-FIAGARCH LR MYR/USD SGD/USD MYR/JPY SGD/JPY Note: LR s the lkelhood rato test statstcs Another noteworthy fndng s that the magntude of the tme-varant component of the correlaton equaton = { } Γ s much hgher when the Japanese yen s the ρ j numerare. For example, the estmated correlatons of the MYR/USD and SGD/USD (MYR/JPY and SGD/JPY) based on the VC-GARCH, VC-APARCH, VC-AGARCH, VC- FIGARCH, VC-FIAPARCH and VC-FIAGARCH models are (0.9408), (0.9408), (0.9400), (0.9409), (0.9398) and (0.9394); respectvely. Ths also apples to the correspondng estmates for the CC-GARCH, CC- APARCH, CC-AGARCH, CC-FIGARCH, CC-FIAPARCH and CC-FIAGARCH models. They are: (0.9044), (0.9045), (0.9041), (0.9058), (0.9056) and (0.9053), respectvely. Moreover, t can be seen that all estmates of the constant components of the condtonal correlatons are sgnfcant at the 5% level. However, for the same par of exchange rates, the magntude of correlaton based on the tme-varyng correlaton models s consstently hgher than that under the constant correlaton models. Ths s consstent wth the fndng of Tse and Tsu (2002). Furthermore, the tme-varyng models are able to keep track of the tme path of the condtonal correlaton between the two currences across models. Fnally, we perform resdual dagnostcs for all the models. Most of the Ljung-Box Q-statstcs and McLeod-L test statstcs of the standardzed resduals are nsgnfcant at the 5% levels. However, the BDS test statstcs for the bvarate VC-APARCH model are stll sgnfcant at the 5% level, suggestng that dependences are stll present for the 22

24 MYR/JPY and SGD/JPY seres. But the BDS tests are less sgnfcant for resduals of the same seres n the VC-FIAPARCH model. Apparently, the fractonally ntegrated model acts as a better varance flter than those wthout such a structure. We also apply the dagnostc tests to the cross-product of the standardsed resduals. Under the null hypothess of constant correlatons, these resduals should be serally uncorrelated (Bollerslev (1990)). Indeed, most of the Ljung-Box Q-statstcs based on the cross product of the standardsed resduals are nsgnfcant at the 5% level, thereby suggestng the absence of seral correlaton. Ths s corroborated by the BDS test results. However, the tme-varyng models are preferred to the constant-correlaton models as there s less evdence of seral correlaton n the cross product of the standardsed resduals. The complete test results are avalable from the authors upon request. Indeed, the relatonshp between currency hedgng and exchange rate volatlty has been extensvely dscussed by many researchers, such as Grammatkos and Saunders (1983), Kroner and Sultan (1993), Glen and Joron (1993), Tong (1996), Jong, de Roon, and Veld (1997), Gagnon et al. (1998), Bos et al. (2000), Brooks and Chong (2001), and Bollen and Rasel (2003). In partcular, Kroner and Sultan (1993) argue that neglectng tme-varyng volatlty and the condtonal dstrbutons of the currency returns affects the performance of currency hedgng strateges. They estmate the rskmnmzng futures hedge ratos for several currences usng a symmetrc GARCH framework wth constant correlatons, and fnd evdence of greater rsk reducton n the GARCH model than those of the conventonal models. Moreover, Bollen and Rasel (2003) compare the opton valuaton model based on the GARCH framework wth the standard smle model and note that the symmetrc GARCH model outperforms the standard model n terms of hedgng. Consstent wth prevous fndngs, our results have 23

25 mplcatons for currency hedgng n three ways. Frst, most of the prevous research on currency hedgng wth GARCH-type models assumes that shocks to volatlty do not have asymmetrc effects. Snce t s possble for currency volatlty to be asymmetrc under dfferent numerare currences, a currency hedgng model that does not ncorporate asymmetres can potentally be based. Second, as the condtonal correlaton of exchange rate volatlty can be sgnfcantly tme-varyng, the optmal hedge rato wll most lkely requre frequent updatng. The assumpton of condtonal correlatons n prevous research s clearly nadequate. Thrd, wth the sgnfcant presence of hgh persstence n exchange rate volatlty, a dynamc hedgng strategy presumng that shocks to volatlty subsde n a relatvely short perod can underestmate the optmal hedge rato over tme. As noted by Balle, Bollerslev and Mkkelsen (1996), optmal hedgng decsons must take nto account any such long-run dependences. Regardng the mplcatons for nternatonal portfolos, t s wdely accepted that the ssue of nternatonal dversfcaton of portfolos cannot be separated from foregn exchange rsk. De Sants and Gerard (1997) test the condtonal captal asset prcng model (CAPM) for the world s eght largest equty markets by usng a parsmonous GARCH parameterzaton. They show that the expected gans from nternatonal dversfcaton for a US nvestor average 2.11 percent per year and have not sgnfcantly declned over the last two decades. However, ther results are predcated on the assumpton that the numerare currency s the US dollar and nvestors do not cover ther exposure to exchange rate volatlty. Analyzng the case of ncorporatng exchange rate volatlty n nternatonal portfolo dversfcaton s beyond the scope of ths paper, but our results may provde some prelmnary evdence that the benefts of nternatonal dversfcaton could be overstated f exchange rate volatlty were gnored. In addton, nvestors n mutual funds based on foregn frms need to determne the rsks of ther 24

26 foregn exchange. However, most emprcal regulartes of exchange rate volatlty and correlaton are derved from the US dollar exchange rates. Our fndngs ndcate that such a relance on the US dollar as the numerare currency could be rather restrcted as the volatlty and correlaton propertes of foregn exchange are dependent on the choce of the numerare currency. 4. Concludng Remarks We have followed up the study of Tse and Tsu (1997) to examne the emprcal evdence of asymmetrc volatlty and long memory of the Malaysan rnggt and the Sngapore dollar n the Asa-Pacfc markets usng a famly of bvarate GARCH-type models. The proposed models can concurrently capture the stylzed features of longrange persstence, asymmetrc condtonal volatlty and tme-varyng correlatons assocated wth the exchange rate returns. Besdes the possble gans n effcency n jont estmaton of parameters, the bvarate approach s capable of trackng down the tme path of condtonal correlatons between the two currences. Consstent wth prevous studes by Hseh (1993), Tse and Tsu (1997), and Tsu and Ho (2004), we fnd that n general the returns of the Malaysan rnggt and the Sngapore dollar aganst the dollar do not exhbt asymmetrc effects n ther condtonal volatltes. In contrast, we detect strong evdence of negatve asymmetrc volatlty when the Sngapore dollar s measured aganst the yen. Ths may mply an unbalanced degree of uncertanty nduced by deprecaton and apprecaton of the Sngapore dollar aganst the yen n the market. In addton, we detect evdence of long-range temporal dependence n volatlty n the two currences, regardless of the choce of the numerare currency. It seems that the mpacts of exchange rate shocks dsplay much longer 25

27 persstence than the standard exponental decay. By comparng the log-lkelhood values, we fnd that the bvarate fractonally ntegrated models generally outperform those models wthout the long-range dependent structures n the condtonal varance. Moreover, we fnd sgnfcant evdence of tme-varyng condtonal correlatons n the two currences aganst the yen. In contrast, the evdence of tme-varyng correlatons among the blateral USD rates s much weaker. The tme-varyng models help to map out nterestng tme paths of the correlaton between the Malaysan rnggt and the Sngapore dollar. Overall, ths study has shown that the choce of numerare currency (ether the US dollar or the yen) for both the Sngapore dollar and Malaysan rnggt can affect the sgnfcance of volatlty asymmetry and tme-varyng correlatons. In addton, we have dscussed the mplcatons for currency hedgng strateges and nternatonal nvestment portfolos. Our fndngs may also be useful for emprcal researchers n several areas, ncludng the computaton of VaR (Value at Rsk) as a way to measure the rsks of portfolos nvolvng multple currences; the prcng of optons based on the GARCH framework; the hedgng of portfolos nvolvng dervatve securtes; and the analyss of the mpact of foregn exchange nterventon on currency volatlty. As noted by Tse and Tsu (1997), n ther study of the Malaysan rnggt and the Sngapore dollar aganst the US dollar, the rnggt exhbts asymmetrc volatlty whereas the Sngapore dollar does not. They argue that the outcome probably depends on the partcular market mcrostructure of each currency. They further suggest that, durng the perod of analyss, heterogeneous expectatons and central bank nterventon probably contrbute to the sgnfcance of persstence and asymmetrc effects n condtonal volatlty of the MYR/USD. More recently, McKenze (2002) also fnds evdence of 26

28 volatlty asymmetry n the Australan dollar aganst the US dollar, and suggests that ths may have to do wth foregn exchange nterventon operatons conducted by the central bank. Moreover, Ramchander and Sant (2002) note that Fed nterventon s assocated wth negatve changes n the US dollar/japanese yen volatlty durng the perod from An nterestng topc for future research would be to nvestgate whether central bank nterventon does consstently lead to sgnfcant asymmetres n the volatlty of varous currences of developed and developng countres. Acknowledgement The second author wants to acknowledge the support by SCAPE of the Department of Economcs and the unted research grants provded n 2006/07 by the Natonal Unversty of Sngapore. 27

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