Regime-Switching Stochastic Volatility and Short- Term Interest Rates

Transcription

1 Regime-Swichig Sochaic Volailiy ad Shor- Term Iere Rae Madhu Kalimipalli School of Buie ad Ecoomic Wilfrid Laurier Uiveriy Waerloo, Oario NL 3C5 Tel: Ex.: 87 ad Raul Sumel C.T. Bauer College of Buie Uiveriy of Houo, Houo,Tx , USA (Verio: May Abrac I hi paper, we iroduce regime-wichig i a wo-facor ochaic volailiy model o explai he behavior of hor-erm iere rae. The regime-wichig ochaic volailiy (RSV proce for iere rae i able o capure all poible exogeou hock ha could be eiher dicree, a occurrig from poible chage i he uderlyig regime, or coiuou i he form of `marke-ew' eve. We eimae he model uig a Gibb Samplig baed Markov Chai Moe Carlo algorihm ha i robu o complex olieariie i he likelihood fucio. We compare he performace of our RSV model wih he performace of oher GARCH ad ochaic volailiy wo-facor model. We evaluae all model wih everal i-ample ad ou-of-ample meaure. Overall, our reul how a uperior performace of he RSV wo-facor model. Key Word: Shor-erm iere rae, ochaic volailiy, regime wichig, MCMC mehod. JEL Claificaio: G We ackowledge he comme of Arhur Warga, ad emiar paricipa a he Uiveriy of Houo, McGill Uiveriy ad he NFA Meeig i Waerloo. We hak Siddharha Chib for providig u wih very helpful compuaioal ip.

2 Regime -Swichig Sochaic Volailiy ad Shor-Term Iere Rae I hi paper, we iroduce regime-wichig i a wo-facor ochaic volailiy model o explai he behavior of hor-erm iere rae. The regime-wichig ochaic volailiy (RSV proce for iere rae i able o capure all poible exogeou hock ha could be eiher dicree, a occurrig from poible chage i he uderlyig regime, or coiuou i he form of `marke-ew' eve. We eimae he model uig a Gibb Samplig baed Markov Chai Moe Carlo algorihm ha i robu o complex olieariie i he likelihood fucio. We compare he performace of our RSV model wih he performace of oher GARCH ad ochaic volailiy wo-facor model. We evaluae all model wih everal i-ample ad ou-of-ample meaure. Overall, our reul how a uperior performace of he RSV wo-facor model. Key Word: Shor-erm iere rae, ochaic volailiy, regime wichig, MCMC mehod. JEL Claificaio: G

3 I. Iroducio The volailiy of hor-erm iere rae play a crucial role i may popular wofacor model of he erm rucure. The level ad he volailiy of he hor rae are commoly ued a ae variable i wo-facor model. For example, Logaff ad Schwarz (99 derive a wo-facor geeral equilibrium model, wih he hor rae level ad he hor rae codiioal volailiy a facor. They how ha a wo-facor model improve upo a igle facor model, which oly ue he level of he hor rae. They fid ha he codiioal volailiy facor provide addiioal iformaio abou he erm rucure ha may be ueful i pricig iere rae opio ad hedgig iere rae rik. Similarly, Breer e al. (996 iclude a level effec ad a GARCH effec io heir iere rae model. They fid ha model wih boh level ad GARCH effec ouperform hoe ha exclude oe of hem. Noe ha a GARCH model diplay a igle coiuou iformaio hock; while i a ochaic volailiy (SV model here are wo coiuou iformaio hock. Followig hi more geeral formulaio for he codiioal variace, Adero ad Lud (997 ad Ball ad Torou (999 iclude a level facor ad a ochaic volailiy facor io he iere rae mea pecificaio. They fid a wo-facor model wih ochaic volailiy perform beer ha he more radiioal wo-facor model wih GARCH volailiy. The iformaio hock i boh GARCH ad SV model are coiuou. Ball ad Torou (995 build a wo-facor model, bu iroducig dicree hock from a uderlyig ae variable ha follow a wo-ae Markov proce. I Ball ad Torou (995, he codiioal volailiy diplay Hamilo (989 regimewichig. Iroducig regime-wichig i he volailiy proce of he hor-erm iere rae i coie wih previou udie ha docume a rog evidece for regimewichig i hor-erm iere rae (ee Hamilo (988, Driffill (99 ad Gray (996. Regime-wichig i he volailiy proce of he hor rae ha impora implicaio for he dyamic of he yield curve ad immuizaio raegie. A poied ou by Lierma, Scheikma ad Wei (99, he volailiy of he hor rae (for example, hree-moh T-Bill rae affec he curvaure of yield curve. I See alo Brow ad Schaefer (995. 3

4 paricular, he volailiy of he hor rae ha wo coueracig effec o he yield curve. Fir, higher volailiy of he hor rae iduce higher expeced rae for he loger mauriie (premium effec. Secod, higher volailiy of he hor-erm iere rae icreae he covexiy of he dicou facor fucio ad, herefore, reduce he yield for loger mauriie (covexiy effec. The premium effec domiae a he hor ed of he yield curve, while he covexiy effec domiae a he log ed makig he yield curve more humped. Whe regime wichig i o coidered, volailiy hock ed o be very perie ad, herefore, he covexiy effec ad he hump i he yield curve could be more proouced ha hey ough o be. Gray (996 oe ha here i evidece for mea reverig high-volailiy ae wih low volailiy periece, ad o-mea reverig low-volailiy ae wih high volailiy periece i oe-moh U.S. T-Bill yield. Thi implie ha he hape of he yield curve deped upo he dyamic of he hor rae, i volailiy ad he lae volailiy ae. Regime-wichig i he volailiy proce alo ha impora implicaio for hedgig. A rader hould accou for boh coiuou ad dicree hock o volailiy i compuig dyamic hedge raio. While coiuou hock refer o marke-ew eve, dicree hock could refer o he high or low volailiy ae of he marke, high or low liquidiy i he marke or high or low eime i he marke. I hi paper, we follow Ball ad Torou (995 ad Adero ad Lud (997. We iroduce regime-wichig i a wo-facor model, where volailiy follow a SV proce. We model he volailiy of hor-erm iere rae a a ochaic proce whoe mea i ubjec o hif i regime. Tha i, our wichig ochaic volailiy for iere rae capure all poible exogeou hock ha could be eiher dicree, a occurrig from poible chage i uderlyig regime, or coiuou, i he form of marke ew eve. We eimae our wo-facor regime-wichig ochaic volailiy model for horerm iere rae uig a Gibb Samplig baed Markov Chai Moe Carlo algorihm. We coduc a exeive i-ample ad ou-of-ample evaluaio of our wo-facor model agai oher wo-facor model. I-ample, our model perform ubaially beer ha he GARCH baed wo-facor model ad he igle-ae ochaic volailiy wo-facor model. Ou-of-ample, he regime-wichig ochaic volailiy model ed o 4

5 ouperform he oher model. The ou-of-ample foreca from he regime-wichig ochaic volailiy model, however, are o ha differe from he igle-ae ochaic volailiy model. The re of he paper i rucured a follow. Secio II iroduce our regimewichig ochaic volailiy (RSV wo-facor model. Secio III examie he daa e ued i hi paper. Secio IV dicue he reul from eimaio. Secio V pree he i-ample ad ou-of-ample comparaive performace of he RSV model. Secio VI ummarize ad pree our cocluio. II. Two-Facor Model ad Regime Swichig A commo empirical fidig i wo-facor model i he high periece i he codiioal variace. For example, Breer e al. (996 eimae he periece parameer i he codiioal variace equaio o be.8 uig weekly hree-moh U.S. T-Bill daa. Ball ad Torou (999 repor periece parameer o be.98 uig mohly oe-moh U.S. T-Bill daa, while Adero ad Lud (997 repor volailiy periece o be.98 for weekly hree-moh U.S. T-Bill daa. High periece i he codiioal variace implie ha hock o he codiioal variace do o die ou quickly -i.e., curre iformaio ha a igifica effec o he codiioal variace for fuure horizo. Lamoreux ad Larape (99 how ha high periece could be relaed o poible rucural chage ha have occurred durig he ample period i he variace proce. They fid ha a igle-regime GARCH pecificaio lead o puriou high periece uder he preece of rucural break. By allowig for poible regime wichig i he daa, high periece oberved i he igle regime model eem o loger valid. Similar reul have bee documeed by Hamilo ad Sumel (994, Cai (994 ad So, Lam ad Li (998. Hamilo (988 ad Driffill (99, amog oher, fid rog evidece for regime-wichig i he U.S. hor-erm iere rae. Variou macro-ecoomic eve were repoible for regime wichig i he U.S. iere rae. Thee eve iclude he OPEC oil crii, he Federal Reerve experime of 979-8, he Ocober 987 crah ad war ivolvig U.S. ad re of he world. Whe hor rae could wich radomly bewee differe regime i.e., where each regime i aociaed wih i ow 5

6 mea ad variace-, we may fid high periece i he daa whe we average daa from differe regime. I i he poibiliy of a hif i he uderlyig regime ha we explicily icorporae i our hor-rae proce. Nex, we iroduce a wo-facor model ha e level ad ochaic volailiy effec. Coider he hor-erm iere rae proce decribed below: r r r ( l (h µ φ ( l( h µ h r ε, > µ x' β ( I model (, r i he hor rae ad h i he codiioal variace of he hor rae, capure he level effec i he model, µ i he aioary mea of he log codiioal, φ meaure he degree of periece of codiioal variace, ad ε ad η repree hock o he mea ad o he volailiy, repecively. Boh hock are whie oie error, which are aumed o be diribued idepedely of each oher. We call model ( he Sigleae Sochaic Volailiy (SSV model. Thi model i ued i Ball ad Torou (999 ad i Adero ad Lud (997. The eimaio of SSV model ivolve he eimaio of mea parameer {, } ad variace parameer {, β, σ η, φ }. Noe ha model ( wih a GARCH pecificaio iead of a SV pecificaio for he codiioal volailiy become he Breer e al. (996 model. Now, le µ be a fucio of he lae ae, which follow a k-ae ergodic dicree fir-order Markov proce a i Hamilo (988. Tha i, a a give poi i ime, he mea of he log volailiy belog o oe of he k ae. A k-ae aioary raiio probabiliy marix gover he dyamic of he raiio from oe ae o he ex ae. Thi implie ha he lae volailiy, h, i drive by a coiuou hock, η, a i ( above ad alo by a dicree hock ha ake o dicree ieger value {,,..., k}. We ca alo hik of our lae volailiy a a mixure of k deiie, where each deiy correpod o a igle ae. The lae volailiy a a give ime come from a igle deiy, which i decided by a uderlyig k-ae Markov proce. Tha i, our Regime-wichig Sochaic Volailiy (RSV model i give by: r r ( l (h µ φ ( l( h µ µ β γ r γ > h r ε, σ η η.5 σ η η {,,...k} ( 6

7 where µ ( refer o he ae depede mea of codiioal volailiy. The parameer γ meaure he eiiviy of he mea wih repec o he uderlyig ae ad i coraied o be poiive. The uderlyig ae ca aume k poible ae, i.e. oe of {,,...,k} where higher value of lead o higher iercep erm i he log variace equaio. A a ideificaio codiio, we require each regime o correpod o a lea oe ime poi. I addiio, ad maily for coveiece, we e he level parameer.5, which ha bee ued i may previou paper. Thi aumpio alo avoid poeial o-aioary problem aociaed wih >, a how i Gray (996 ad Bli ad Smih (998. The eimaio of he RSV model ivolve eimaio of mea parameer {, }, variace parameer {β, γ, σ η, φ }, ad he raiio probabiliy parameer {p, p }, where p ij repree he raiio probabiliy of goig from ae i o j. I ummary, he RSV model pecificaio combie a level effec ad a codiioal volailiy proce ha capure all poible exogeou hock. Noe ha he RSV model reduce o he SSV model whe here i o regime hif i he daa -i.e., whe γ i rericed o zero. A cloely relaed paper i So, Lam ad Li (998, which ue a wichig ochaic volailiy model o explai he periece i he log volailiy for S&P 5 idex weekly reur. Uig a hree-ae model (wih high, medium ad low volailiy ae, So e al. (998 fid ha he volailiy ae i le perie, while he low volailiy ae i more perie. Our model lighly differ from So e al. (998. I our RSV model, he drif erm of he codiioal variace i a fucio of boh curre ad la period ae, while i So e al. (998 he codiioal variace i a fucio oly of he curre period ae. Thi differece i our volailiy pecificaio alo lead o differece i our likelihood fucio ad, hece, i our poerior deiie. Eimaio of he RSV model ivolve eimaig wo lae variable -i.e., h ad i addiio o he model parameer. I he preece of wo lae variable, he likelihood fucio for he model eed o be iegraed over all he poible ae of he wo lae variable. Jacquier, Polo ad Roi (993 how ha maximum likelihood baed mehod ed o fail uder complex pecificaio of he likelihood fucio. Coequely, we reor o Moe Carlo Markov Chai (MCMC mehod o eimae he For example, Cox, Igerol ad Ro (985 ad Adero ad Lud (997. 7

8 RSV model. 3 Noe ha Model 3 ca eaily icorporae more complicaed dyamic. For example, we ca make σ η a fucio of a lae ae, or we ca pecify a ARMA rucure for he codiioal mea equaio (wih parameer drive by he lae ae, or we ca coider muliple regime. Our eimaio echique ca accommodae all hee exeio. III. Daa The daa coi of aualized yield baed o weekly obervaio of hreemoh U.S. T-bill daa for he period /6/6 o 6/3/98 (3 weekly obervaio ad i obaied from he Chicago Federal Reerve daabae. Wededay rae are ued ad if Wededay i a radig holiday, he, Tueday' rae are ued. Similar daa e have bee previouly ued by Adero ad Lud (997 ad Gray (996. Figure plo he aualized yield ad alo he fir differece i yield baed o weekly obervaio of hree-moh T-Bill daa. There are everal epiode of large flucuaio i omial iere rae: he oil hock durig ad , he Federal Reerve moeary experime of 978-8, ad he marke crah of 987. Thi obervaio ugge more ha a igle regime i he daa. For iace, we ca hik of a high volailiy regime durig he period cied above ad a low volailiy regime durig re of he period. We could allow for a addiioal regime, a i Hamilo ad Sumel (994, o capure oulier i he daa. Followig he exiig wichig lieraure, however, we limi ourelve o wo regime for he uderlyig volailiy. The weekly hree-moh T-Bill rae (r i o-aioary 4. Fir differecig make he daa aioary. Uig Box-Jeki mehod, a ARIMA(,, model eem o provide a aifacory fi for he auocorrelaio i he yield. Followig Paga ad Schwer (99 ad Ball ad Torou (995, 999, we fi he RSV model ( o he reidual (RES from regreig r o a coa ad r -. For he purpoe of eimaio ad compario o aleraive volailiy model, we wrie our mea adjued verio of he RSV model a 3 Appedix A give he deail of our MCMC eimaio ad Appedix B pree he reul of a imulaio experime uig our eimaio mehod. 4 ADF e could o rejec he ull of a ui roo i he yield erie. 8

9 r ( ˆ RES ˆ r h r ( l (h µ φ ( l( h µ µ β γ ε, RES γ >.5 σ η η {, } (3 where all he aumpio o he error erm made i ( ill hold. Agai, oe ha whe we e γ o zero, he RSV model reduce o he SSV model. Table pree he ummary aiic of he daa. Chage i yield, r, eem o be lef kewed idicaig ha period of high T-Bill reur were le commo compared o period wih low reur. There i alo a rog evidece of kuroi i he reur erie. The Ljug-Box aiic ugge ha here i a high degree of auocorrelaio for he raw yield (r. There i a very high periece i he raw erie. O he oher had, r erie eem o be much le perie ad i characerized by low auocorrelaio. High auocorrelaio i he wo erie ( r ad log( r ugge he kid of o-lieariy i he daa ha ca be explaied by a SV model. Table pree he reul from GARCH e o he daa. We repor he Ljug-Box aiic for he quared reidual (RES a variou lag. The ull of o GARCH effec i rogly rejeced by he daa. IV. Reul from he Sochaic Volailiy model To bechmark our reul, fir, we igore he poibiliy of regime-wichig i he daa. The reul from he MCMC eimaio of he SSV model are preeed i Table 3, where he parameer e i θ {β, φ, σ η }. The periece parameer φ i very high idicaig ha he half-life of a volailiy hock, meaured a -l(/l(φ, i abou fouree week. Sadard error for he parameer are mall idicaig ha parameer are highly igifica. Figure plo he poerior deiie of he parameer. All he parameer have ymmeric deiie while half-life deiy i righ kewed idicaig ha half-live loger ha fouree week are more commo. We, ex, eimae he RSV model for our weekly iere daa e. Table 5 pree he prior ad poerior parameer eimae of he parameer e θ i our model, where θ {β, γ, φ, σ η, p, p }. Sadard error for he parameer are mall a before. 9

10 The periece parameer, φ, drop igificaly o.68 from.95 i he SSV model. Thi implie ha a wich i he lae regime creae a high periece i volailiy ad cofirm he earlier reul of Hamilo ad Sumel (994, Cai (994 ad So, Lam ad Li (998. The diribuio of φ i lef kewed wih a media.647 (ee Figure 3, implyig ha eve lower periece ha.68 i commo. The raiio probabiliie, p ad p, are eimaed a.994 ad.966. Thee eimae are comparable o.9896 ad.9739 repecively repored i Gray (996 ad.9878 ad.94 repecively repored i Cai (994. Our reul imply ha he effec of a volailiy hock i much more perie i he low volailiy ae ha i he high volailiy ae. A volailiy hock la o average of, a lea, week i he low volailiy ae compared o abou 3 week i he high volailiy ae, where duraio of he hock i obaied a (-p ii -. Figure 3 plo he Gauia kerel deiie for he poerior parameer eimae. The poerior deiie eem o be ymmeric for β ad γ ad righ kewed for σ η (wih a media.897. Figure 4 plo he Gibb parameer eimae from ru (deail i appedix A. Gibb ru idicae o auocorrealio i ucceive draw. Figure 5 plo auocorrelaio for he parameer. The auocorrelaio become iigifica a very early lag implyig ha he Gibb draw are draw a radom. Table 4 ad 6 pree he correlaio bewee he parameer. Boh Table 4 ad 6 repor rog egaive correlaio bewee β ad φ, ad φ ad σ η. Table 6 alo repor rog poiive correlaio bewee β ad γ, ad β ad σ η. Togeher, hee reul imply ha a he variace periece decreae he ucodiioal variace -i.e., he log-ru mea of l(h - icreae, large volailiy hock are o a perie a mall volailiy hock ad 3 large volailiy hock ed o be aociaed wih higher log-ru mea compared o mall volailiy hock. The fir wo pael of Figure 6 plo he T-Bill yield ad he reidual from a regreio of r o a coa ad r -, repecively. The hird pael plo he uderlyig aualized volailiy (geeraed by a muli-move imulaio mooher, ad he fourh pael plo he imulaed mooher probabiliie of beig i high volailiy ae, i.e., Prob(. Followig Hamilo (988, we coider a obervaio a belogig o ae oe if he moohed probabiliy i higher ha.5. The imulaio mooher how period of high volailiy durig he oil hock of 969 ad 973, he Federal Reerve

11 moeary experime, ad he marke crah of 987. The mooher probabiliie idicae ha here i a large probabiliy ha he T-Bill yield durig 969, 973, 979-8, ad belog o a high volailiy regime. Thi daig i i agreeme wih he daig repored by Cai (994 ad Gray (996. V. Performace of he RSV Model We coduc a exeive evaluaio of he i-ample ad ou-of-ample performace of he SV wo-facor model ad oher wo-facor model, baed o he GARCH family of model. We coider hree popular GARCH model: GARCH(, model, GARCH(,-L model -i.e., GARCH(, wih a aymmery effec of egaive lagged error, o capure he leverage effec- ad EGARCH(, model. The fir GARCH model i he formulaio ued by Logaff ad Schwarz (99. The ecod ad hird GARCH model reai a leverage effec, a i Breer e al. (996. All he GARCH model are pecified o iclude a level effec (for pecificaio ee Table 7. The MLE reul for he hree GARCH model are preeed i Table 7. There i evidece for a leverage effec baed o he igifica -aiic for κ i he GARCH(,-L model ad he igifica -aiic for δ i he EGARCH(, model. The leverage effec, however, i mall relaive o he uual ize foud i equiy reur. All he eimae i he codiioal variace equaio are igifica for he hree model. Noe ha he eimae how he uual high periece i he codiioal variace. We exrac oe-week(ep-ahead i-ample foreca variace from he igle ae ad regime-wichig wo-facor model ad compare hem o oher model. I addiio o he full ample period, /6/6-6/3/98, we coider hree ub-ample period. The hree ub-ample period are: ( /6/6-3//78, ( /6/6-3//8, ad (3 /6/6-3//9. The fir ample iclude he oil hock, he ecod ample iclude he Fed moeari experime of 979-8, ad he hird ample iclude he Ocober987 ock marke crah. Baed o he eimae for he hree ub-ample, we eimae ou-of-ample foreca uil he ed of he ample. We alo coider horer ample ad horer ou-of-ample foreca period. A a example, we iclude a fourh ample //76-3//87, which allow a evaluaio of he performace of he model i

12 a horer daa e. Thi forh ample ha wo well defied pell of high volailiy: he Fed experime ad he Ocober 987 ock marke crah. Figure 7 ad 8 how he i-ample (aualized codiioal volailiie implied by all he model. The codiioal volailiie from wo-facor model are relaively le mooh compared o hoe from he GARCH ype model. Thi i becaue he wo-facor model are more eiive o hock. For example, he RSV wo-facor model pick up a oulier i lae 98, which goe udeeced by he oher model. Table 8 how he likelihood fucio for all he model. The RSV ha he bigge likelihood. Uforuaely, he GARCH model ad he SSV model are o eed. Therefore, adard likelihood raio e are o correc. I addiio, adard likelihood raio cao be ued for he SSV model ad he RSV model ice here are uideified parameer uder he ull hypohei of o-wichig -ee Hae (99. Therefore, Table 8 how four differe i-ample evaluaio crieria for he differe model. The ochaic volailiy model perform beer ha all he GARCH model ad he SSV model. I paricular, he RSV model ha a higher likelihood fucio, higher adjued R, ad higher AIC/SBC value relaive o he GARCH model ad he SV model. Table 8 alo repor poerior odd raio of he compeig model wih repec o he coa variace model ee Kim ad Ko (994. If he odd raio i poiive, he he compeig model i more likely o have geeraed he daa ha he coa variace model. The model wih he highe value of poerior odd raio repree he mo likely compeig model pecificaio. The ochaic volailiy model have higher odd raio ha GARCH model. I paricular, he RSV model ha a odd raio a lea 56% higher ha he oher compeig model. Amog he wo-facor GARCH model, wih he excepio of he Adjued R crieria, he E-GARCH(, perform beer ha he oher model for all he evaluaio meaure. The E-GARCH(, i alo he model ued by Ball ad Torou (999 o evaluae he i-ample evaluaio of he SSV model. Baed o hee coideraio, he E-GARCH(, i he GARCH model we elec o evaluae he ou-ofample performace of he SV model. Table 9 pree i-ample ad ou-of-ample oe-ep ahead foreca for all he model for he four differe ub-ample. We pree he mea quared error (MSE ad mea abolue error (MAE for he SSV model, he RSV model, a coa volailiy

13 model, ad for he be performig GARCH model, he E-GARCH(,. We keep a coa volailiy model i our ou-of-ample compario, give he reul i Figlewki (997, where he coa volailiy perform well relaive o GARCH model. Table 9 how ha he RSV model ed o ouperform he GARCH ad SSV model. Coie wih he i-ample reul of Table 8, he RSV model alway bea i-ample he oher formulaio. Ou-of-ample, he RSV ed o do beer ha he EGARCH ad SSV model. The ou-of-ample performace of he RSV model, however, i imilar o he ouof-ample performace of he SSV model. Coie wih Figlewki (997, he coa variace model how a good ou-of-ample performace, epecially i he MSE meric. Noe ha he coa variace model i he fir ub-ample bea all he oher model. The E-GARCH model ever perform beer ha he SV model. The la ub-ample pree a hor period of ou-of-ample foreca, oly oe year. Agai, he RSV model i he domiaig model. 5 VI. Cocluio I hi paper, we iroduce regime-wichig i a ochaic volailiy model o explai he behavior of hor-erm iere rae. The regime-wichig ochaic volailiy proce for iere rae capure all poible exogeou hock ha are eiher coiuou i he form of `marke-ew' eve or dicree a occurrig from poible chage i uderlyig regime. We iroduce he regime-wichig ochaic volailiy proce i a wo-facor model for he hor-erm iere rae. We eimae he wo-facor model uig a Gibb Samplig baed Markov Chai Moe Carlo algorihm ha i robu o he uual o-lieariie i he likelihood fucio. We fid ha he uual high volailiy periece i ubaially reduced by he iroducio of regimewichig. We coduc a exeive i-ample ad ou-of-ample evaluaio of everal wo-facor model. We ue everal ub-ample ad differe evaluaio crieria o compare he RSV model wih oher GARCH model ad igle-ae ochaic volailiy model. Overall, our reul are very upporive of our RSV wo-facor model. 5 For he fourh ample, we alo calculae (o repored ou-of-ample foreca for a wo-year period ad a e-year period. Overall, he reul are imilar, alhough a he ou-of-ample forecaig period i exeded, he performace of he SSV model become very imilar o he performace of he RSV model. 3

14 I-ample ad ou-of-ample, he RSV model ed o ouperform all he oher wofacor model. 4

15 Appedix A: The Gibb Algorihm for Eimaig RSV Model I he RSV model (, we eed o eimae he parameer vecor θ {β, γ, σ η, φ, p, p } alog wih he wo lae variable H {h,...,h } ad S {,., }. The parameer e herefore coi of ω {H, S, θ} for all. We ue Baye heorem o decompoe he joi poerior deiy a follow. f ( H, S, θ f ( Y H f ( H S, θ f ( S θ f ( θ We ex draw he margial f(h Y, S, θ, f(s Y,H, θ ad f(θ Y, H S, uig he Gibb amplig algorihm decribed below: Sep : Specify iiial value θ ( {β (, γ (, σ η, (,φ ( (, p, p ( }. Se i. Sep : Draw he uderlyig volailiy uig he mulimove imulaio ampler of De Jog ad Shephard (995, baed o parameer value from ep. The uderlyig volailiy vecor for all he daa poi i obaied a a fucio of uderlyig diurbace ha are draw a a block uig a imulaio mooher. Coider he RSV model (3, reproduced below: r ( ˆ ˆ r RES RES h r ε, ( l (h µ φ ( l( h µ µ β γ γ >.5 {, } σ η η (3 The codiioal mea equaio ca be wrie a, l( RES l( h l( r l( ε (A - The erm l(ε ca be approximaed by a mixure of eve ormal variae (Chib, Shephard, ad Kim (998. l (e z f(z 7 i f N ( z m 74.,v i {,,... 7} (A - i i i 5

16 Now, (A- ca be wrie a l( RES l( h l( r [ z k i] (A - 3 where k i oe of he eve uderlyig deiie ha geerae z. Oce he uderlyig deiie k, for all, are kow, (A-3 become a deermiiic liear equaio ad alog wih he RSV model (3 ca be repreeed i a liear ae pace model. Nex, apply he De Jog ad Shephard (995 imulaio mooher o exrac he uderlyig log volailiy from he oberved daa. Sep 3: Baed he o oupu from ep ad, he uderlyig k i (A-3 i ampled from ormal diribuio a follow -ee Chib, Shephard ad Kim (998: f [ z i l( y,l( h ] qi f N ( zi l( h mi.74, vi i k (A - 4 For every obervaio, we draw he ormal deiy from each of he eve ormal diribuio {k,,..,7}. The, we elec a k baed o draw from uiform diribuio. Sep 4: Baed o he oupu from ep, ad 3, we draw he uderlyig Markov-ae followig Carer ad Koh (994. We ue he mooher for he above ae-pace model (3, o derive he vecor of uderlyig ae variable,,,..., Sep 5: Cycle hrough he codiioal of parameer vecor θ {β, γ, σ η, φ, p, p } for he volailiy equaio uig Chib (993, uig oupu from ep -4. Aumig ha f (θ ca be decompoed a: f ( θ Y, H, S f ( β Y f ( φ Y, H, H, S, S, θ, θ φ β f ( γ Y f ( p, p, H, S Y, H, θ γ, S f ( σ, θ p ij Y, H, S, θ σ (A -5 where θ -j refer o he θ parameer excludig he jh parameer. The repecive codiioal diribuio (ormal for β, γ ad φ, ivere gamma for σ ad bea for p ij are 6

17 decribed i Chib (993. The parameer γ i draw uig a ivere CDF wih he rericio ha i i poiive. The prior mea ad adard deviaio are pecified i Table 3 ad 5. Sep 6: Go o ep. Eimaio of SSV model ( ha he ame ep a i RSV model (3, excep ha we do o have o draw he lae ae ad raiio probabiliie. For he Gibb eimaio, we leave ou he fir 4 draw (i.e., bur i ieraio are 4 ad ample from he ex 6 draw. We chooe every fifh obervaio o miimize, ad if poible elimiae, ay poible correlaio i he draw. Our effecive umber of draw herefore drop o (i.e., effecive e ieraio are. We coruc 95% cofidece ierval for he parameer, baed o draw. We coruc he adard error for he parameer uig he bach-mea mehod -ee Chib (993. We eimae he deiy fucio for he parameer uig he Gauia kerel eimaor -ee Silverma (986. 7

18 Appedix B: Moe Carlo Experime wih he Gibb Algorihm We perform a Moe Carlo experime of he RSV model (3, wihou level effec. Tha i: RES h ε ( (h µ φ ( l( h µ l σ η η µ β γ γ > {, } Uig he raiio probabiliie p ad p, we geerae a ae vecor (wih value or of ize. Uig he ae vecor ad he rue parameer β, γ, σ η, ad φ, we geerae ochaic volailiy, l(h. Baed o he above model, he ochaic volailiy erie i ued i o geerae he reidual vecor, RES. (All he rue parameer value ued i he imulaio are lied below. The, akig RES a give, we eimae he parameer e θ {β, γ, σ η, φ, p, p } uig he MCMC algorihm a explaied i appedix A. We e he umber of bur i ieraio equal o 4 ad he umber of effecive e ieraio equal o. Thu, we coruc he 95% cofidece ierval for he parameer baed o draw. We coruc he adard error for he parameer uig he bach-mea mehod -ee, Chib (993. The reul are repored i Table B.. Table B. Reul from a Moe Carlo experime T ( ample ize : parameer True value Prior Value Poerior Value Mea Sd. deviaio Mea ( d. Error Sd. deviaio 95% Cofidece Ierval β (.3. ( γ (.8.9 ( φ.4.39 (.5.85 (.-.54 σ (.5.9 ( p...6. (..7 ( p (.3.36 ( *Prior diribuio of σ (ivere gamma i improper We fid ha he poerior mea of parameer are quie cloe o he rue value. The adard error are mall, idicaig a high preciio of he poerior mea. For he variace ad he raiio probabiliy p, he poerior mea are lighly higher ha rue value. However, hey clearly lie wihi he 95% cofidece boud. 8

19 Figure B. how he lae volailiy ad ae. The op pael coi of imulaed reidual RES. The ecod pael pree boh he rue ad lae volailiy, he laer obaied uig he imulaio mooher. The hird pael pree he rue ae i.e., eiher or - ad mooher probabiliie of beig i he high volailiy ae. From he ecod ad hird pael, we ee ha he mooher volailiy ad probabiliie cloely approximae heir rue couerpar. Figure B.. Simulaed Yield ad Correpodig Lae Volailiy ad Sae 9

20 Referece: Alber, J. H. ad S. Chib (993, Baye Iferece via Gibb Samplig of Auoregreive Time Serie Subjec o Regime Shif, Joural of Buie Saiic ad Ecoomic,, -5. Adero, T. ad J. Lud (997, Eimaig Coiuou Time Sochaic Volailiy Model of he Shor-erm Iere Rae, Joural of Ecoomeric, 77, Ball, C. ad Torou, W. N. (995, Regime Shif i Shor Term Rikle Iere Rae Workig Paper, #5-95, The Adero School a UCLA Ball, C. ad Torou, W. N. (999, The Sochaic Volailiy of Shor-erm Iere Rae: Some Ieraioal Evidece, Joural of Fiace, 56, Bli, R. R. (997, Moveme i he Term Srucure of Iere Rae, Federal Reerve Bak of Alaa Ecoomic Review, Bli, R. R. ad Smih, D. C. (998, The Elaiciy of Iere Rae Volailiy: Cha, Karolyi, Logaff, ad Sader Reviied, Workig paper 97-3-a, Federal Reerve Bak of Alaa. Breer, R. J., R. Harje ad K. Kroer (996, Aoher Look a Model of Shorerm Iere Rae, Joural of Fiacial ad Quaiaive Aalyi, 3, Brow, R. H. ad S. M.Schaeffer, (995, Iere Rae Volailiy ad he Shape of he Term Srucure, Mahemaical Model i Fiace edied by S. D. Howio, F. P. Kelly ad P. Wilmo. Chapma ad Hall, Lodo. Cai, J. (994, A Markov Model of Swichig-Regime ARCH, Joural of Buie ad Ecoomic Saiic,, Carer, C. K. ad R. Koh (994 O Gibb Samplig for Sae Space Model, Biomerika, 8,3, Chib, S. (993, Baye Eimaio of Regreio wih Auoregreive Error: A Gibb Samplig Approach, Joural of Ecoomeric, 58, Chib, S. (996, Calculaig Poerior Diribuio ad Modal Eimae i Markov Mixure Model, Joural of Ecoomeric, 75, Chib, S. ad E. Greeberg (995, Uderadig Meropoli-Haig algorihm, Joural of America aiicia, 49, Cox, J. C., J. E. Igeroll ad S. A. Ro, (985, Theory of Term Srucure of Iere Rae, Ecoomerica, Vol. 53,,

21 De Jog, P. ad N. Shephard (995, Simulaio Smooher for Time Serie Model, Biomerika 8, Driffill, J. (99, Chage i Regime ad Term Srucure, Joural of Ecoomic Dyamic ad Corol, 6, Figlewki, S. (997, Forecaig Volailiy, Fiacial Marke, Iiuio ad Irume, 6, 997. Gray, S. (996, Modelig he Codiioal Diribuio of Iere Rae a a Regime Swichig Proce, Joural of Fiacial Ecoomic, 4, 7-6 Hamilo, J. D. (988, Raioal Expecaio Ecoomeric Aalyi of Chage i Regime, Joural of Ecoomic Dyamic ad Corol,, Hamilo, J. D. ad R. Sumel (994, ARCH ad Chage i Regime, Joural of Ecoomeric, 64, Hae, B.E. (99, "The Likelihood Raio Te uder No-adard Codiio: Teig he Markov Tred Model of GNP," Joural of Applied Ecoomeric, 7, S6-S8. Jacquier, E., N. G. Polo ad P. E. Roi (995, Bayeia Aalyi of Sochaic Volailiy Model, Joural of Buie ad Ecoomic Saiic,, Kim. D ad S. J. Ko (994, Aleraive model for he codiioal heerocedaiciy of ock reur, Joural of Buie, 67, No.4, Kim. S, N. Shephard ad S. Chib (998, Sochaic Volailiy: Likelihood Iferece ad Compario wih ARCH Model, Review of Ecoomic Sudie, 65, Lamoureux, C. ad B. Larape (99, Periece i Variace, Srucural Chage, ad he GARCH Model, Joural of Buie ad Ecoomic Saiic, 8, Lierma, R., J. A. Scheikma ad L. Wei (99, Volailiy ad Yield Curve, Joural of Fixed Icome, Logaff, F. A. ad E. Schwarz (99, Iere Rae Volailiy ad Term Srucure: A Two-facor Geeral Equilibrium Model, Joural of Fiace, 4, Paga, A. R. ad G.W. Schwer (99, Aleraive Model for Codiioal Sock volailiy, Joural of Ecoomeric, 45, So, M., K. Lam ad W. Li (998, A Sochaic Volailiy Model Wih Markov Swichig, Joural of Buie ad Ecoomic Saiic, 6,

22 Silverma, B. W. (986, Deiy Eimaio for Saiic ad Daa Aalyi. New York: Chapma ad Hall.

23 Table Summary aiic for weekly iere rae o 3-moh T-Bill for he period /6/6 o 6/3/98 r r ( r log ( r Mea Sadard error Variace Sadard error Skewe Sadard error Kuroi Sadard error Ljug-Box ( Noe: Ljug-Box (4. Ljug-Box aiic calculaed wih 4 lag. χ (4 criical value for a 95% cofidece level i Table Te for GARCH effec i weekly iere rae o 3-moh T-Bill for he period /6/6 o 6/3/98 Lag Ljug-Box Saiic χ (lag aiic ( 95% cofidece level Noe: We obai he reidual (RES from regreig r o a coa ad r - ad repor he Ljug- Box aiic for he quared reidual a differe lag. The Ljug-Box aiic for quared reidual i highly igifica a all lag. 3

24 Table 3 Reul from he MCMC eimaio of he Sigle-Sae Sochaic Volailiy (SSV model uig weekly 3-moh T-Bill yield for he period /6/6 o 6/3/98 parameer Prior Value Poerior Value Mea Sadard Mea ( d. error Sadard 95% Cofidece Ierval Deviaio Deviaio β.5.83 (..9 ( φ.95 (..9 ( σ (..3 (.5-.4 Noe: The SSV model ued i Table (Model : r ( ˆ RES ˆ r h r ( l (h µ φ ( l( h µ µ β ε, RES.5 σ η η The ample ize i 3. Prior diribuio of σ (ivere gamma i improper. Deail abou he model eimaio are i appedix A. Table 4 Correlaio marix of he parameer for he SSV model uig weekly 3-moh T-Bill yield for he period /6/6 o 6/3/98 Parameer β σ φ β.3 -. σ φ

25 Table 5 Reul from MCMC eimaio of he Regime-wichig Sochaic Volailiy (RSV model uig weekly 3-moh T-Bill yield for he period /6/6 o 6/3/98 Parameer Prior Value Poerior Value Mea Sadard Deviaio Mea ( d error Sadard Deviaio 95% Cofidece Ierval β 5.58 (..98 ( γ (..47 ( φ.68 (..46 ( σ (..3 ( p..6.6 (..3 (.-.3 p (..3 ( Noe: The RSVmodel i eimaed i Table 5 (Model 3: r ( ˆ ˆ r RES RES h r.5 ( l (h µ φ ( l( h µ µ β γ ε, γ > {, } σ The ample ize i 3. Prior diribuio of σ (ivere gamma i improper. Deail abou he model eimaio are i appedix A.. η η Table 6 Correlaio marix of he parameer for he RSV model uig weekly 3-moh T-Bill yield for he period /6/6 o 6/3/98 parameer β γ σ φ a :p B: p β γ σ φ a :p b: p

26 6 Table 7 Reul from MLE eimaio of GARCH model uig weekly 3-moh T-Bill yield for he period /6/6 o 6/3/98 κ δ δ β GARCH(, ( (7.64 (47. GARCH(,-L (5.5 ( (6.57 ( EGARCH(, (.8 - (.567 ( (8.9 Noe: -aiic are repored i parehei. Model pecificaio: GARCH(, wih level effec: GARCH(,-L wih level effec : EGARCH(, wih level effec:, ( ~ (.5, ˆ ˆ ( > Ω h h u h N r h RES RES r r β β ε ε (, ~ (.5, ˆ ˆ ( > > Ω h u if u if d h u u d h N r h RES RES r r β κ β κ ε ε l l( l( (, ~ (.5, ˆ ˆ ( > Ω (h h h h N r h RES RES r r β ε ξ β ξ δ π ξ δ ε ε

27 Table 8 I-ample compario of aleraive model for he eire ample period /6/6 o 6/3/98 (ample ize: 3 umber of parameer Log- Likelihood AIC SBC Adj R Odd raio Coa Variace GARCH(, GARCH(,-L EGARCH(, SSV model RSV model Noe: AIC: log-likelihood value le umber of parameer SBC: log-likelihood value le.5 log (T*umber of parameer where T: ample ize MSE: MAE: T [ RES h ] T T RES T h Adj R : Adjued R i calculaed for he regreio Re a bh u u ~N(, {,..N}, where RES are he reidual from regreig r agai coa ad r - ad h {,..N} are codiioal volailiy eimae Odd raio: he poerior odd raio of aleraive pecificaio relaive o he coa variace. Thi i obaied a differece of he Schwarz Bayeia Crierio (SBC of each compeig model ad he SBC of he coa variace model ee, Kim ad Ko (994. All he model ued here are decribed i Table 3-7.

28 Table 9 I-ample ad ou-of-ample compario of aleraive model for hree differe ample period Sample I-ample (T: 99 /6/6-3//78 Ou-of-ample (T: //79-6/3/98 MSE MAE Adj. R MSE MAE Co. Variace EGARCH(, SSV model RSV model Sample I-ample (T: 99 /6/6-3//8 Ou-of-ample (T:83 //83-6/3/98 MSE MAE Adj. R MSE MAE Co. Variace EGARCH(, SSV model RSV model Sample 3 I-ample (T: 668 /6/6-3//9 Ou-of-ample (T:334 //9-6/3/98 MSE MAE Adj. R MSE MAE Co. Variace EGARCH(, SSV model RSV model Sample 4 I-ample (T: 66 //76-3//87 Ou-of-ample (T:4 //88-3//89 MSE MAE Adj. R MSE MAE Co. Variace EGARCH(, SSV model RSV model Noe: T: refer o he ample ize. The be model i highlighed. MSE: MAE: T [ RES h ] T T RES T h The eimaed coefficie from he i-ample period are ued o geerae oe-week (ep ahead codiioal

29 9 volailiy eimae for he ou-of-ample period. Oe-ep ahead codiioal volailiy foreca are geeraed uig he followig equaio (baed o he GARCH model defied uder Table 7 ad he SV model ad 3: GARCH(,: GARCH(,-L: EGARCH(,: SSV model: RSV model: ( ( β σ β σ w where w w ( ( κ β σ κ β σ w where w w (l( ( l( β σ β σ w where w w (l( ( l( µ σ φ µ σ [ ] [ ] ( ( l( ( ( l( l( pr pr pr pr σ σ σ

30 Figure. Weekly 3-moh T-Bill perceage yield (Sample: /6/6 o 6/3/98 3

31 Figure. Poerior Deiy Plo for Parameer of he SSV Model Figure 3. Poerior Deiy Plo for Parameer of he RSV model 3

32 Figure 4. Gibb Ru for Parameer of he RSV Model Figure 5. Auocorrelaio Fucio for Parameer of he RSV Model 3

33 Figure 6. T-Bill Yield ad Correpodig Lae Volailiy ad Sae (Sample: /6/6 o 6/3/98 33

34 Figure 7. Compario of i-ample Codiioal Volailiie Acro Differe Model (Sample: /6/6 o 6/3/98 34