Parameter Estimation of a Class of Hidden Markov Model with Diagnostics

Transcription

1 Joural of oder Applied Saisical ehods Volume Issue Aricle 5--3 Parameer Esimaio of a Class of Hidde arkov odel wih Diagosics E. B. Nkemole Uiversiy of Lagos, Nigeria, Africa O. Abass Uiversiy of Lagos, Nigeria, Africa, olabass@uilag.edu.g R. A. Kasumu Uiversiy of Lagos, Nigeria, Africa Follow his ad addiioal works a: hp://digialcommos.waye.edu/jmasm Par of he Applied Saisics Commos, Social ad Behavioral Scieces Commos, ad he Saisical Theory Commos Recommeded Ciaio Nkemole, E. B.; Abass, O.; ad Kasumu, R. A. (3 "Parameer Esimaio of a Class of Hidde arkov odel wih Diagosics," Joural of oder Applied Saisical ehods: Vol. : Iss., Aricle. DOI:.37/jmasm/36738 Available a: hp://digialcommos.waye.edu/jmasm/vol/iss/ This Regular Aricle is brough o you for free ad ope access by he Ope Access Jourals a DigialCommos@WayeSae. I has bee acceped for iclusio i Joural of oder Applied Saisical ehods by a auhorized edior of DigialCommos@WayeSae.

2 Joural of oder Applied Saisical ehods Copyrigh 3 JAS, Ic. ay 3, Vol., No., /3/$95. Parameer Esimaio of a Class of Hidde arkov odel wih Diagosics E. B. Nkemole O. Abass R. A. Kasumu Uiversiy of Lagos, Nigeria, Africa A sochasic volailiy (SV problem is formulaed as a sae space form of a Hidde arkov model (H. The SV model assumes ha he disribuio of asse reurs codiioal o he lae volailiy is ormal. This aricle aalyzes he SV model wih he sude- disribuio ad he geeralized error disribuio (GED ad compares hese disribuios wih a mixure of ormal disribuios from Kim ad Soffer (8. A Sequeial oe Carlo wih Expecaio aximizaio (SCE algorihm echique was used o esimae parameers for he exeded volailiy model; he Akaike Iformaio Crieria (AIC ad forecas saisics were calculaed o compare disribuio fi. Disribuio performace was assessed usig simulaio sudy ad real daa. Resuls show ha, alhough comparable o he ormal mixure SV model, he Sude- ad GED were empirically more successful. Key words: Hidde arkov odel, sequeial oe Carlo, expecaio maximizaio, Sude- disribuio, sae-space model, sochasic volailiy, likelihood, sock exchage. Iroducio The Hidde arkov odel (H, origially iroduced i 957, (acdoald & Zucchii, 997, Cappe, e al., 5 has may applicaios i fields such as sigal processig, medicie, egieerig ad maageme. The H is a doubly sochasic process, ( X, Y, wih a uderlyig sochasic process, X, ha is o direcly observable bu ca be observed hrough aoher process, Y, ha produces a sequece of idepede radom observaios.h are equivalely defied via a fucioal represeaio kow as a sae space model. The sae space model (Douce & Johase, 9 of a H is represeed by wo equaios: sae ( ad observaio ( as E. B. Nkemole is a Lecurer i he Deparme of ahemaics. her a: ekemole@uilag.edu.g. O. Abass is a Professor i he Deparme of Compuer Sciece. him a: olabass@uilag.edu.g. R. A. Kasumu is a Lecurer i he Deparme of ahemaics. him a: bkasumu@yahoo.com. x = f ( x + w ( y = g( x + v ( where f ad g are liear or oliear fucio sad w ad v are whie oise processes. odels represeed by ( ad ( are referred o as sae space models ad iclude a class of Hs wih o-liear Gaussia sae-space model such as he sochasic volailiy (SV model. The SV model (Taylor, 98, accous for ime-varyig ad persise volailiy ad he lepokurosis i fiacial reur series. The SV model has become popular for explaiig he behavior of fiacial variables, such as sock prices ad exchage raes (Durbi & Koopma, ; Douce & Tadic, 3 ad is populariy has resuled i several differe proposed approaches for esimaig model parameers. Though heoreically aracive, he SV model is empirically challegig due o he fac ha he uobserved volailiy process eers he model i a o-liear fashio which leads o he likelihood fucio depedig upo highdimesioal iegrals. Esimaio procedures, such as he Geeralized ehod of omes (G 8

3 PARAETER ESTIATION OF A HIDDEN ARKOV ODEL WITH DIAGNOSTICS (ellio & Turbull, 99 ad he Efficie ehod of omes (E (Galla, e al., 997 have bee proposed for he SV model. Oher proposed esimaio procedures iclude he mehod of momes ad he quasi maximum likelihood approach mehodology o approximae he SV model o a liear Gaussia model (Harvey, e al, 994; Ruiz, 994. Durbi ad Koopma ( used he idea of liearizaio of geeral sae-space models ad mached erms i he likelihood of a liearized model o hose of a liear Gaussia model. Several sudies (Jacquier, e al., 994; Chib, e al., ; Kim, e al., 998 adoped he Gibbs samplig scheme, ad Shephard ad Pi (997 applied he eropolis-hasigs scheme for he aalysis of he SV.Kim ad Soffer (8 showed how he icorporaio of he E algorihm ad SC (paricle filers ad smoohers forms a basic idea o hadle he parameer esimaio problem i he SV model. Esimaio ca be accomplished by applyig a filerig algorihm. (Kiagawa& Sao, combied paricle filerig mehods ad gradie algorihms. This aricle expads he scope of applicaio of SV models, by exedig SC echiques wih he E algorihm developed by Kim ad Soffer (8 o esimae SV model parameers wih he sude- disribuio. The SV model usually assumes ha he disribuio of asse reurs codiioal o he lae volailiy is ormal. However, fiacial daa ofe have heavier ails ha ca be capured by he sadard SV model: This has led o he use of o-ormal disribuios o beermodel ad o deal wih he heavy ails (Shephard, 996; Kim, e al., 998; Bai, e al., 3; Sadorsky, 5; Kim & Soffer, 8. Liesefeld ad Jug ( fi a Sude- disribuio o he error disribuio i he SV model usig he simulaed maximum likelihood mehod developed by Daielsso ad Richard (993 ad Daielsso (994. A promisig disribuio ha models boh skewess ad kurosis is he Skewed Sude- (Feradez & Seel, 998. Hece, i is ecessary o deermie he bes-fied model ou of a poeially huge class of cadidaes; i has become perie o develop efficie model selecio crieria. As his backgroud illusraes here is a evergrowig lieraure o ime-varyig fiacial marke volailiy; i is aboud wih empirical sudies i which compeig models are evaluaed ad compared o he basis of heir forecas performace (Aderse, e al., 5. Sochasic Volailiy (SV odels SV models belog o class of Hidde arkov model ad accou for volailiy of daa. The SV model ca be expressed as a auoregressive (AR process: x φ + w (3 = x x y = β exp v (4 where w~n(, τ, x ~N( μ, σ, v~n(,, { y } is he log-reur o day, ad β is he cosa scalig facor so ha { x } represes he log of volailiy of y (Taylor, 98.To esure saioariy of y i is assumed ha φ <. Squarig (4 ad akig is logarihm resuls i he liear equaio y = α + x + z. (5 Equaios (3 ad (5 form a versio of he SV model ha ca be modified i may ways; ogeher hey form a liear, o-gaussia, saespace model for which (5 is he observaio equaio ad (3 is he sae equaio. Sochasic Volailiy wih Heavy-Tailed Disribuio The sadard form of he SV modelwas show i equaios (3 ad (4; i (4 v follows a ormal disribuio. Various auhors have argued ha real daa may have heavier ails ha ca be capured by he sadard SV model. The Sochasic Volailiy odel wih ixure The observaioal oise process (Kim & Soffer, 8 is a mixure of wo ormal s wih ukow parameers give as 8

4 NKENOLE, ABASS & KASUU yk = x + z (6 wih z = Iz + ( I z, z ~ N(m,R, z ~ N(m,R, m = α μπ, m = α + ( π μ ad I ~ Beruolli ( π. I is a idicaor variable, where π is a ukow mixig probabiliy, ha is, p( I = = π = p( I ~ Beruolli( π. The likelihood of { x,, x, y,, y, I,, I } is show i Figure where R * * = I R + ( I R, μ = I q + ( I q. I he SV-ormal mixure model defied by (6, he vecor of he model parameer is deoed by { q, q, R, R, π}. These parameers are esimaed alog wih he oher parameers, { φ, τ} (see Kim & Soffer, 8 for deails. Sude- as a Observaio Noise Equaios (3 ad (5 are a exesio of he liearized versio of he SV model wherei i is assumed ha he observaioal oise process, z is a sude- disribuio. The model, firs preseed i Shumway ad Soffer (6, reais he sae equaio for he volailiy as: x = φ x + w. However, he proposed Sude- disribuio wih degrees of freedom, v, for he observaio error erm, z, effecs a chage i he observaio equaio: y = α + x + z v z ~, =,, (7 The disribuio of he error erm for his specificaio accordig o Shimada & Tsukuda (5 akes he form f(y x = v+ Γ x ye e π (v v + v Γ v+ x (8 where v represes a parameer of degree of freedom ad Γ sads for he Gamma fucio. The likelihood fucio of x, x,, x, y,, y } is { f(x,y = x μ exp πσ σ x ϕ x exp = πτ τ v+ Γ + Γ v+ x x ye e = π(v v v Figure : Likelihood of x,, x, y,, y, I,, I } { where R * = I R + ( I R, μ * = I q + (I q (x f(x,y,i = exp πσ (x ϕ x I I exp π ( π μ πτ τ = * σ = (y x μ exp * * = π R R 83

5 PARAETER ESTIATION OF A HIDDEN ARKOV ODEL WITH DIAGNOSTICS Geeralized Error Disribuio as a Observaio Noise The disribuio of he GED accordig o Bao, e al. (6 akes he form v exp (( y α v / ψ f ( y x = + v where ψ = ψ Γ v v Γ v 3 Γ v The log-likelihood fucio for he GEDmodel is: v y α v log f ( y x = log v = ψ logψ logγ v + v log. The E Algorihm The paramou parameer esimaio oolo achieve maximum likelihood esimaor is he E algorihm ad i has bee widely applied o he cases where he daa is cosidered o be icomplee i he sese ha i is o fully observable. I is comprised of he wo followig seps: E-sep: Compue he expeced likelihood, Q( θ θ (k ' ' Q ( θ θ = E (log f ( x θ y, θ ( k + -sep: Choose θ he parameer values ha (k maximize he fucio, Q ( θ θ (for deails see Baum, e al, 97; Dempser, e al., 977; Rabier, 989. v The E- ad - seps are repeaed uil some soppig crieria is me, such as ˆ + θ ˆ θ < Q, for some specified Q, obaiig suiable iiial parameers iclusive. A olie E algorihm recely proposed for discree H ca be exeded o more geeral seigs, icludig o-liear o-gaussia sae-space models ha ecessiae he use of Sequeial oe Carlo (SC filerig approximaios. Sequeial oe Carlo ehods (SC Afer is iroducio i he 96 s, SC has become a emergig mehodology for he oliear or o-gaussia sae-space models. The chief iiiaive is o represe he ieresed desiy fucio p ( x: k y: k a ime k by a se of radom samples wih associaed weighs, { x: k, w: ki =,, N} ad compue esimaes based o hese samples ad associaed weighs. As he umber of samples becomes very large, his oe Carlo characerizaio develops io a equivale represeaio o he fucioal descripio of he probabiliy desiy fucio (Arulampalam, e al.,. If { x: k, w: ki =,, N} are samples ad associaed weighs approximaig he desiy fucio p x y, wih ( w i i= : N k approximaed by p( x ( : k : k =, he he desiy fucio is : k y : k N w δ ( x i= k k x k where δ (x sigifies he Dirac dela role. The N paricle approximaio { wk, xk } i= is rasformed io a equally weighed radom sample from p ( x: k y: k by samplig, wih replaceme from he discree disribuio N { wk, xk } i=. This procedure, also called resamplig, produces a ew sample wih uiformly disribued weighs so ha ( w i = N k. 84

6 NKENOLE, ABASS & KASUU Paricle filers ad smoohers are SC mehods grouded i paricle represeaios ad are cosidered geeralizaios of Kalma filers ad smoohers for geeral sae-space models. The fudameal approach used o obai paricles from he desired desiy is based o sequeial imporace samplig (SIS ad resamplig. SIS, a oe Carlo mehod, forms he basis for mos paricle filerig mehods. To approximae he codiioal desiy of x give previous saes, x, ad pas ad prese daa, y, p ( x x, y, SIS iroduces a imporace samplig desiy, q ( x x, y where i is easier o sample from π ( x x, y ha p ( x x, y (Douce, e al.,. Paricle Filer Algorihm If a ime weighed paricles { f, w } draw from f ( x y, f is a se of paricle filer wih associaed weigh w, he his is cosidered a empirical approximaio for he desiy comprised of poi masses, f ( x y w δ ( x f. i= Kiagawa & Sao ( ad Kiagawa (996 provide a algorihm for filerig i geeral sae space model. This is a oe Carlo filerig for geeral sae-space models:. For i =,, N, geerae a radom umber ( f i x ~ p(. Repea he followig seps for =,, T. a. For i =,, N, geerae a radom ( umber w i ~ q( w. b. For i =,, N, compue p = F ( f, w c. For i =,, N, compue w = p ( y p ( d. Geerae f i, i =, N by ( N resamplig p,, p The oe Carlo filer reurs { f,i =,,N, =,,m } so ha N δ (x f f(x Y. N i= Paricle Smoohig Algorihm If { s, w } i= is a se of paricle smoohers ad associaed weighs approximaig he desiy fucio f ( x Y, he he desiy fucio is approximaed by: f ( x Y w δ ( x j= s The problem wih smoohed esimaes is degeeracy. Godsill, e al. (4 suggesed a ew smoohig mehod (paricle smooher usig backwards simulaio. The mehod assumes ha filerig has already bee performed, hus, he paricles ad associaed weighs, { } =, { w } i= f. f i ca approximae he filerig desiy, ( x Y, by w = N δ ( x i= w f. The algorihm from Godsill, e al. (4 supposes ha weighed paricles { f, w ; i =,,, } are available for =,,,. The algorihm for i =,,, is:. Choose s = f wih probabiliy ( j ( j w. For o ( j ( j ( j a. Calculae w + w f ( s+ f for each j. ( j b. Choose s = f wih probabiliy w +. ( j 85

7 PARAETER ESTIATION OF A HIDDEN ARKOV ODEL WITH DIAGNOSTICS 3. s: = ( s,, s is a approximae realizaio from p X Y. ( Sequeial oe Carlo Expecaio aximizaio (SCE Algorihm Aalysis Parameer Esimaio The SCE esimaio procedure cosiss of hree mai seps: filerig, smoohig ad esimaio. Parameer esimaio for he Sude- ad model were cosidered. A basic approach for he Sude- SV model, equaio (7, is o apply he E algorihm; wih he oupu of filerig ad a smoohig sep a approximae expeced likelihood is calculaed. Filerig Sep The algorihm for he filerig ad smoohig seps shows a sligh modificaio of Godsill, e al. (4 ad Kim ad Soffer (8. samples from f ( x, Y for each were obaied as: (. Geerae f i ~ N( μ, σ. For =,, a. Geerae a radom umber ( w i ~ N(, τ, j =,, b. Compue c. Compue w = p( y d. Geerae weighs, p + = φ f w ( j w p, e x + x y e v f by resamplig wih Smoohig Sep I he smoohig sep, paricle smoohers ha are eeded o acquire he expeced likelihood i he expecaio sep of he E algorihm were obaied. Suppose ha equally ( weighed paricles f i { }, i =, from, v+ f ( x, Y are available for =,,, from he filerig sep. ( j. Choose [ s ] = [ f ] wih probabiliy. For o calculae ( j ( j (s+ ϕ f w + f(s+ f exp τ v+ v+ Γ (j s+ s+ ye exp + π(v v v Γ for each j ( j a. Choose [ s ] = [ f ] wih probabiliy j w ( s: = {( s,, s } is he radom sample from f x,, x Y ( 4. Repea -3, for i =,, ad calculae s (s ˆx i= i= ˆx =, ˆp =, ˆ ˆ (s x (s x i= ˆp, =, v+ x ye E+ = v (v x y+ v ye (v+ y e + = v 86

8 NKENOLE, ABASS & KASUU Esimaio Sep This sep cosiss of obaiig parameer esimaes by seig he derivaive of he expeced likelihood, of he complee daa { x,, x, y,, y } give { x,, x }, wih respec o each parameer o zero ad solvig for ˆ φ, ˆ, τ ad αˆ. The complee likelihood of x, x,, x, y,, y } is { log f ( X,Y = (x μ log + log exp π σ σ (x ϕ x + log exp = πτ τ v+ (yαv Γ log e = π (v v + Γ v+ (yαv ye + v This mehod resuls i he esimaes: ˆ α = log where ˆ S φ = S S ˆ τ = S, S (v x y+ v ye (v+ y e + = v ˆ α = ( y v = S = = = + = S ( x p, S v v ( x p, = = x x, + p, Whe z follows he GED, i is o possible o represe i as equaio (3.. Hece, for he SV- GED model, he parameer v, as well as he oher parameers, ad x were sampled from heir full codiioal disribuios usig SCE echiques. ehodology The proposed mehod o compare he fi of he disribuios is illusraed usig hree simulaed daa ses ad daily exchage raes of he Nigeria Naira, Ghaa Cedi, Briish Poud ad Euro compared o he U. S. Dollar, from arch 3, 9 o arch 3,. Figures -3 show he plos ad hisograms of daa geeraed from he ormal mixure, Sude- ad model respecively ad Tables -4 show he resuls of he esimaio for he models. Simulaio Daa were geeraed from he ormal mixure SV model x =.7x + w, y =.75+ x + v where w ~ N(,.96, v ~ I N(,6 + (I N( 3.5,4 ad I ~ Beroulli(.5 wih rue parameer se ( φ, τ,, q, R, R, π q = (.7,.96, 3.5,, 4, 6,.5. The echique based o mixure ad Sude- SV was applied o his daa o examie he performace of he proposed model. To make he process saioary,, samples were geeraed ad he firs, values were discarded. Figures a ad b show he plo ad hisogram for Simulaio. Simulaio Daa were geeraed from he Sude- SV model wih rue parameer se ( φ, τ, α, v = (.8,.45,3., 8. The echique based o he mixure ad Sude- SV models was applied o his daa o examie he meri of he Sude- idea; he legh of he daa, { y }, was,. Figures 3a ad 3b show he plo ad hisogram for Simulaio. The secod daa se was used o observe he behavior of he esimaio procedure whe a deparure from he ormal mixure observaioal error assumpio exiss. 87

9 PARAETER ESTIATION OF A HIDDEN ARKOV ODEL WITH DIAGNOSTICS Figurea: Represeaio of SCE SequeceSimulaed from he ixure SV odel Figure b: Hisogram of Fial Values of Parameers of he ixure SV odel 5 5 x y Figure3a: Represeaio of SCE Sequece Simulaed from he Sude- SV odel Figure 3b: Hisogram of Fial Values of he Parameers of he Sude- SV odel x y Figure4a: Represeaio of SCE Sequece Simulaed from he ixure SV odel Figure 4b: Hisogram of Fial Values of Parameers of he ixure SV odel x y

10 NKENOLE, ABASS & KASUU Simulaio 3 Daa were geeraed from he model wih rue parameer se ( φ, τ, α, v = (.9,.6,.7. Techiques based o mixure ad he model were applied o his daa o examie he meri of he GED idea; he legh of he daa, { y }, is,. Figures 4a ad 4b show he plo ad hisogram for Simulaio 3. Resuls Usig he procedures described [.95,.79,.6794, , 4., 4.,.5] were seleced for he iiial parameers for ( φ, τ, q, q, R, R, π. Table shows fial esimaes wih heir sadard error (i parehesis for Simulaio. The fial esimaes, alog wih heir sadard deviaios (i pareheses, were: φˆ =.7568 (.786, τˆ =.3466 (.93, ˆq =.9486 (.989, ˆq = 3.76 (.869, ˆR =.369 (.936, ˆR = (.674, πˆ =.3854 (.635 where he rue parameers are (.7,.6, 3.5,, 4, 6,.5. I his approach, ˆ α = ˆ π qˆ ˆ + ( ˆ π q =.6475; ( ˆ, φ ˆ, τ ˆ α = (.7568,.3466,.6475; based o resuls, he esimaio procedure based o he ormal mixure model works well because ha he esimaes are close o he rue parameers. Based o he Sude- echique, (.95,.79,.496 were used as he iiial values for parameers ( φ, τ, α ; he process was sopped whe he value of relaive likelihood was less ha.. The fial esimaes, alog wih heir sadard deviaios (i pareheses were: φˆ =.693 (.3798, τˆ =.336 (.4839, αˆ =.99 (.45. These resuls show ha he model provides good esimaes despie he fac ha he rue observaio oise is o a ormal mixure disribuio.a similar simulaio sudy was performed usig he daa from simulaio (see Table. The iiial parameer se [.84,.3359,.783, 5.783, 4., 4.,.5] was seleced for parameers ( φ, τ, q, q, R, R, π. Table shows he resuls of he parameer esimaio procedure based o he ormal mixure. The fial esimaes, alog wih heir sadard deviaios (i pareheses were: φˆ =.6547 (.57, τˆ =.93 (.473, ˆq = 3.8 (..354, ˆq = (.445, ˆR = (.5, ˆR = (.3564, πˆ =.486 (.4338 where he rue parameers are (.8,.45, 3. for he parameers, ( ˆ ϕ, ˆ τ, ˆ α ; where ˆ α = ˆ π qˆ + ( ˆ π qˆ = Whe he daa from simulaio was fied wih he echiques based o he Sude- (.84,.3359,.83 were used as iiial parameers of ( φ, τ, α. A he h ieraio he relaive likelihood was less ha. ad he process was cosidered coverged. The fial esimaes, alog wih heir sadard deviaios were: φˆ =.8383 (.855, τˆ =.5357 (.43, αˆ =3.9 (.53. These esimaes are similar o he rue parameers (.8,.45, 3., while mixure reurs (.6547,.93,4.388 as ( ˆ, φ ˆ, τ ˆ α. The mehod based o he Sude- SV model worked well i boh cases. Whe he esimaio procedure based o he ormal mixure SV model was applied, he esimaes were disa o he rue parameer. Coversely, he applicaio of he echique based o Sude- model idicaed a beer proximiy o he rue parameers; herefore, exesio of he SV model by adopig Sude- is meaigful. Table 3 shows he resuls of he parameer esimaio procedure o echique based o he ormal mixure SV ad GED o daa geeraed from he ormal mixure model; [.8699, , , , 4., 4.,.5] were seleced for he iiial parameer for he parameers ( φ, τ, q, q, R, R, π. The fial esimaes, alog wih heir sadard deviaios (i pareheses were: φˆ =.9869 (.334, τˆ = (.5468, ˆq =

11 PARAETER ESTIATION OF A HIDDEN ARKOV ODEL WITH DIAGNOSTICS (.89, ˆq = (.639, ˆR = (.69579, ˆR = (.668, πˆ =.4895 (.3833 where he rue parameers are (.8, 3.5, 5, 8, 3, 4.,.5. I his approach, ˆ α = ˆ π qˆ ( ˆ ˆ + π q = 5.4 ; ( ˆ, φ ˆ, τ ˆ α = (.9869, 4.936, 5.4. Resuls show ha he esimaio procedure based o he ormal mixure model worked well i he sese ha he esimaes are close o he rue parameers. For he GED echique, (.8699, , , were used as he iiial values for parameers ( φ, τ, α. Table 3shows he resuls of he esimaio procedure. The process was sopped whe he value of relaive likelihood was less ha.. The fial esimaes, alog wih heir sadard deviaios (i pareheses were: φˆ =.87 (.7854, τˆ = (.65, αˆ =4.744 ( Resuls show ha he GED model gives good esimaes eve hough he rue observaio oise is o a ormal mixure disribuio. Table 4 shows resuls of he parameer esimaio procedure o echique based o he ormal mixure SV ad GED o daa geeraed from he model. The mehod based o he GED model works well i boh cases. Whe he esimaio procedure based o he ormal mixure SV model was applied, he esimaes were far from he rue parameers. By coras, he applicaio of he echique based o GED model idicaed a beer proximiy o he rue parameers. (.95,.388,.639 were used as iiial parameers ( φ, τ, α. The fial esimaes, alog wih heir sadard deviaios were: φˆ =.9749 (.6845, τˆ =.3496 (.678, αˆ =.68 (.447. These esimaes are similar o he rue parameers (.9,.6,.7 while he ormal mixure reurs (.755,.3496, as ( φ, τ, α. Thus, he mehod based o he GED works well i boh cases. Table : Parameer Esimaes ad Sadard Errors (i parehesis o Techique Based o ixure ad Sude- o Daa Geeraed from ixure odel True Parameer φ.7 τ.6 q 3.5 q - R 4 R 6 π.5 ixure SV Sude- SV ixure SV Sude- SV ixure SV Sude- SV = 5 ε =. = ε =. = ε = ( ( ( ( ( ( (.7748 α Rel. Lik.6976 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

12 NKENOLE, ABASS & KASUU Table : Parameer Esimaio o Techique Based o he ixure ad Sude-o Daa Geeraed from he Sude- odel Sude- SV Sude- SV Sude- SV ixure SV ixure SV ixure SV True Parameer = 5 ε =. = ε =. = ε =. φ.8 τ.45 q q R R π.6388 ( ( ( ( ( ( (.568 α Rel. Lik.8439 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Table 3: Parameer Esimaio o Techique Based O he ixure SV ad GED o Daa Geeraed from he ixure odel ixure SV ixure SV ixure SV True Parameer = 5 ε =. = = φ.8 τ 3.5 q 5 q 8 R 3 R 4. π ( ( ( ( ( ( (.356 α Rel. Lik.8485 ( ( ( ( ( (.64 v ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

13 PARAETER ESTIATION OF A HIDDEN ARKOV ODEL WITH DIAGNOSTICS Table 4: Parameer Esimaio o Techique Based O he ixure SV ad GED o Daa Geeraed from he GED odel True Parameer φ.9 τ.6 q q R R π ixure SV ixure SV ixure SV = 5 ε =. = =.95 ( ( ( ( ( ( (.576 α Rel. Lik.977 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Applicaio o Real Life Fiacial Daa The ormal mixure, Sude- ad GED SV model were applied o aalyze daily raes o he Naira/Dollar, Cedi/Dollar, Poud/Dollar ad Euro/Dollar exchage raes from arch 3, 9 o arch 3,. Figures 5-8 show he plos of he daily exchage raes ad log reurs of he daa.paers of behavior are evide i he secod plos i Figures 5-8: he daa experiece a small variace for some periods of ime, ad for oher periods hey show a large variace. For his reaso, i cao be assumed ha he daa have a cosa variace. Table 6 preses he esimaio resuls alog wih heir sadard deviaios for he Sude-, ormal mixure ad he models. These disribuios produce comparable maximum likelihood values, idicaig a accepable overall fi. The values (ragig from.97 o.988 sugges high persisece of he volailiy of he series idicaigha volailiy cluserig is observed i all he exchage raes reur series. The Akaike values ad he evaluaio saisics usig he all daa are show i Table 7. The AIC ad he log-likelihood values highligh he fac ha (GED Sude- disribuio beer esimaes he series ha he ormal mixure disribuio for he SV model. I fac, he loglikelihood fucio icreases, leadig o AIC crieria of.85, , ad wih he ormal mixure versus ( , ( , ( ad ( wih he o-ormal desiies, for he Naira/Dollar, Cedi/Dollar, Poud/Dollar ad Euro/Dollar rae respecively. The saisics from he volailiy forecass (Sadorsky, 5 are preseed. I erms of SE, he Sude- performs beer ha he ormal mixure for he Naira/Dollars ad he Euro/Dollar exchage rae while he opposie is rue for he Cedi/Dollars ad Poud/Dollar exchage rae. Geerally, he AE resuls are o differe from he SE resuls. I erms of APE, he Sude- SV model is preferred i hree cases ad he model oce. 9

14 NKENOLE, ABASS & KASUU Table 5: Descripive Saisics of Daily Reurs for he Exchage Rae Saisics Naira/Dollar Rae Cedi/Dollar Poud/Dollar Euro/Dollar ea Sadard Deviaio Skewess Kurosis Jarque-Bera Figure5: Naira/Dollar Daily Exchage Rae ad Log Reurs 58 NIGERIA - USD.5 Nigeria Naira - US Exchage Rae Reurs Figure 6: Cedi/Dollar Exchage Rae ad Log Reurs.6 x 4 Ghaa Cedis - USD.3 Ghaa Cedis - USD.55.. Exchage Rae.5.45 Reurs

15 PARAETER ESTIATION OF A HIDDEN ARKOV ODEL WITH DIAGNOSTICS Figure 7: Euro/Dollar Daily Exchage Rae ad Log Reurs.3 Euro - USD. Euro - USD... Reurs Exchage Rae Figure 8: Poud/Dollar Daily Exchage Rae ad Log Reurs. Briish Poud - USD.74 Briish Poud - USD Reurs Exchage Rae Coclusio A exesio of he observaio error i he SV model from ormal mixure o Sude- ad GED disribuios was preseed. A sequeial oe-carlo expecaio maximizaio experime was used o esimae parameers for he exeded SV model. Fucios provided by ATLAB eabled echiques based o he Sude- ad model o be developed alog wih a sraegy for fiig a model ha combies he E algorihm ad SC; his chage o he proposed model allowed for a more robus fi, providig a ew ool o explore he ail fi. The Sude- ad model was compared wih he ormal mixure. The E algorihm makes i possible o obai maximum likelihood esimaors. The esimaio Algorihm was compleed by applyig he Godsill, e al. (4 paricle smoohig algorihm o he SV model wih (3 ad (5 as he observaio ad sae equaios. The oucome of he simulaio ad real daa aalyses cofirm he viabiliy of he proposed mehod. Resuls show ha he proposed esimaio algorihm yields accepable resuls whe he ormal assumpio is violaed as well as whe i holds, hus wideig he rage of applicaio of he SV model.saisics were calculaed o compare he fi of disribuios. Resuls, based o daa from he Naira/Dollar, Cedi/Dollar, Poud/Dollar ad Euro/Dollar exchage raes, reveal ha he Sude- is comparable o he ormal mixure SV model bu is empirically more successful. 94

16 NKENOLE, ABASS & KASUU Table 6a: Esimaio Resuls - Disribuio Compariso Naira/Dollar Cedi/Dollar φ τ q q R R π ixure SV (SD.9759 ( ( ( ( ( ( (.769 α.73 Sude- SV (SD.9769 ( ( (.5369 (SD.9684 ( ( (.35 ixure SV (SD.983 ( ( ( ( ( ( ( Sude- SV (SD.9887 ( ( (.977 (SD.974 (..983 ( (.97 Poud/Dollar Euro/Dollar φ τ q q R R π ixure SV (SD.9895 ( ( ( ( ( ( (.8666 α Sude- SV (SD.9754 ( ( (.4 (SD.9697 ( ( ixure SV (SD.9579 ( ( ( ( ( ( ( Sude- SV (SD.93 ( ( (.768 (SD.9763 ( ( (

17 PARAETER ESTIATION OF A HIDDEN ARKOV ODEL WITH DIAGNOSTICS Table 7: Evaluaio Saisics - Disribuio Compariso AIC Log-like SE AE APE Naira/Dollar ixure SV Sude- SV Cedi/Dollar ixure SV Sude- SV Poud/Dollar ixure SV Sude- SV Euro/Dollar ixure SV Sude- SV Refereces Aderse, T. G., Bollerslev, T., & Diebold, F. X. (5. Parameric ad oparameric fiacial ecoomerics. Amserdam: Norh-Hollad. Arulampalam,. S., askell, S., Gordo, N., & Clapp, T. (. A uorial o paricle filer for o-lie o-liear/o- Gaussia Bayesia rackig. IEEE Trasacio o Sigal Processig, 5, Bai, X., Russell, J. R., & Tiao, G. C. (3. Kurosis of GARCH ad Sochasic Volailiy odels wih No-ormal Iovaios. Joural of Ecoomerics, 4, Bao, Y., Tae-Hwy, L., & Burak, S. (6. Comparig desiy forecas models. Joural of Forecasig, 6(3, 3-5. Baum, L. E., Peerie, T., Souled, G., & Weiss, N. (97. A maximizaio echique occurrig i he saisical aalysis of probabilisic fucios of arkov chais. Aals of ahemaical Saisics, 4(, Cappe, O., oulies, E., & Ryde, T. (5. Iferece i hidde arkov models. New York, NY: Spriger. Chib, S., Nardari, F., & Shephard, N. (. arkov chai oe Carlo mehods for sochasic volailiy models. Joural of Ecoomerics, 8, Daielsso, J., & Richard, J. F. (993. Acceleraed Gaussia imporace sampler wih applicaio o dyamic lae variable models. Joural of Applied Ecoomerics, 8, Daielso, J. (994. Sochasic volailiy i asse prices: Esimaio wih simulaed maximum likelihood.joural of Ecoomerics, 6, Dempser, A. P., Laird, N.., & Rubi, D. B. (977. aximum likelihood from icomplee daa via he E algorihm. Joural of he Royal Saisical Sociey, 39(, -38. Douce, A., De Freias, J. F. G., & Gordo, N. (. Sequeial oe Carlo mehods i pracice. New York, NY: Spriger- Verlag. Douce, A., & Tadic, B. B. (3. Parameer esimaio i geeral sae-space models usig paricle mehods. Aals of Isiue of Saisical ahemaics, 55,

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