A Model for Time Series Analysis

Transcription

1 Aled Mahemaal Senes, Vol. 6, 0, no. 5, A Model for Tme Seres Analyss me A. H. Poo Sunway Unversy Busness Shool Sunway Unversy Bandar Sunway, Malaysa ahhn@sunway.edu.my Absra Consder a me seres model n whh he resonse r a a shor ahead of he resen me deends on he resen resonse r and l oher resonses before me va a ondonal dsrbuon wh arameer ) x Δ deends on he resen arameer x( ) and x(,and he arameer m oher arameers before me va anoher ondonal dsrbuon wh arameer θ. We roose a mehod for esmang he above wo yes of ondonal dsrbuons. A daa se on neres rae s used o omare he erformane of he roosed model and he Chan, Karoly, Longsaff and Sanders (CKLS) model. The omarson shows ha he redon nervals derved from he roosed model have overage robables and exeed lenghs whh are omarable o hose of he redon nervals based on he CKLS model. Mahemas Subje Classfaon: 6M0 Keywords: Tme seres model, ondonal dsrbuons, redon nervals. Inroduon Gven he observed resonse varables y, y,..., y, one may use he ondonal robably densy funon (df) f y y, y,..., ) o sefy a ( y me seres model for he nex resonse y. Some examles of moran me seres model are as follows:

2 5736 A. H. Poo () The ARMA (, q) model (Box and Jenns,970) gven by y y q where and he and are onsans, and he error erms are assumed o be ndeenden and denally normally dsrbued random varables wh mean zero and varane. () The Auo-Regressve Condonal Heerosedasy (ARCH) model (Engle, 98) for he me seres s gven by z where z s a srong whe nose roess, and he seres s modeled by where he are onsans. (3) The Generalzed ARCH (GARCH) model (Bollerslev, 986). Ths model s smlar o he ARCH model exe ha he seres s now modeled by q 0 where he are onsans. (4) Sae sae model (Anderson and Moore,979; Lews,986) An examle of he sae sae model s x f ( x,, ) x y y has df g (. x, ) where x d R s he veor of he unobserved sae varables, ha of he observed resonse varable, unnown fxed arameers, s a veor of s a veor of random varables, he

3 A model for me seres analyss 5737 beng ndeenden of eah oher, f s a nown Borel measurable funon, g s an absoluely onnuous robably dsrbuon funon (df) wh bounded densy. Among he above four examles, alhough only he las examle gves he ondonal df of he nex resonse drely, he equaons relang he nex resonse o he observed resonses enable us o fnd he ondonal df for he nex resonses n he frs hree examles. Presenly, an alernave me seres model s roosed. In he roosed model, we assume ha he resonse r a a shor me ahead of he r and l oher resen me may deend on he resen resonse resonses r r,..., r ( l ) dsrbuon wh arameer x. We also assume ha he arameer x Δ deend on he resen arameer x and m arameers x Δ, x,..., ( m ), before me va a ondonal may oher x before me va anoher ondonal dsrbuon wh arameer θ. We may refer o he arameer x as a Tye arameer, andθ a Tye arameer. The deendene of he arameer x on s nororaed n he model o ae are of he suaon when he me seres formed by he resonses s nonsaonary. Furhermore, when he resen Tye arameer x and m- oher Tye arameers before me are gven, a ondonal dsrbuon s mosed on he fuure Tye arameer x Δ o desrbe he varaon of he fuure Tye arameer as nreases. The above ondonal dsrbuons wh resevely Tye and Tye arameers rovde a mehansm for generang he fuure resonses r, r,.... The roosed model and he Chan, Karoly, Longsaff and Sanders (CKLS) model are omared usng a se of neres rae daa. I s found ha he redon nervals derved from he roosed model have a erformane whh s omarable o ha of he redon nervals derved from he CKLS model n erms of he ably of he redon nervals o over he fuure neres raes, and he average lengh of he redon nervals.

4 5738 A. H. Poo A Tme Seres Model Le r ( ) [ r( ( l ) ),..., r( ), r( )] be a veor formed by he resen resonse r() and l- oher resonses before me. We assume ha he resonse r( ) a a shor me ahead of me deends on r va he ondonal df f ( r( ) r ( ) ) x() wh Tye arameer Nex le x [ x( ( m ) ),..., x( ), x( )] x. be a veor formed by he Tye arameer x a he resen me and m- oher Tye arameers before me. We assume ha he Tye arameer x Δa a shor me ahead of me deends on x va he ondonal df wh Tye arameerθ whh does no deend on. f x θ x ( ) ( ( ) ) The above wo yes of ondonal df are suffen o sefy a model for he resonses. For a gven neger, he fuure resonses r( ), r( ),..., r( J ) may be generaed from he gven values of x and () Generae r( ) r usng he followng roedure: df f ( r( ) r ( ) ) x() () For j =,,, J from r usng he ondonal () Generae j x ( j ) [ x ( j ) ( m ),..., x ( j ), x ( j ) ] usng he ondonal df x from he value () Generae r( ( j ) ) from he value r j [ r ( j ( l )),..., r ( j ), r j] usng he ondonal df f ( r( ( j ) ) r ( j )) x( j ) We may esmae he above ondonal dsrbuons usng a ye of mulvarae non-normal dsrbuon alled he mulvarae ower-normal

5 A model for me seres analyss 5739 dsrbuon whh wll be desrbed n Seon 3. Seons 4 and 5 wll deal wh he esmaon of he ondonal dsrbuons. 3 Condonal df Derved from Mulvarae Power-normal Dsrbuon Yeo and Johnson (000) onsdered he followng ower ransformaon z ] z 0, 0 logz z 0, 0 z ] z 0, 0 log z z 0, 0 [ ~,, z (3.) [ If z has he sandard normal dsrbuon, hen ~ has a non-normal dsrbuon whh s derved by a ye of ower ransformaon of a random varable wh normal dsrbuon. We may say ha ~ has a ower-normal dsrbuon. We may now use he unvarae ower-normal dsrbuon o oban he mulvarae ower-normal dsrbuon. Frs le y be a veor onssng of orrelaed random varables. The veor y s sad o have a -dmensonal ower- - normal dsrbuon wh arameers μ, H, λ,,, f y μ Hε (3.) where y μ E, H s an orhogonal marx,,,..., are unorrelaed, ~ ~ var ~ E > 0 s a onsan, and ~ has a ower-normal dsrbuon wh arameers and., (3.3) When he values of y, y,..., y are gven, we may fnd an aroxmaon for he ondonal df of y by usng he followng numeral roedure: () Sele a large neger N 0 and omue N, where y and λ y y h, y are suh ha y y y P s lose

6 5740 A. H. Poo h N o, and y () Form he veor suh ha y y y, y,..., y, y and fnd he value of y μ Hε (3) Relae, n Equaon (3.) by, and fnd z suh ha. Le he answer of z be denoed by z. T ε (4) Comue f Ex z d dz ( ) z z (5) Esmae he ondonal df (evaluaed a y ) of y by N f f ' ' 4 Esmaon of Condonal df wh Tye I Parameer Suose he observed resonses are r(j ) where j The followng s a roedure for esmang he ondonal df of r( ) when he values of r( ( l ) ),..., r( ) and r () are gven: T () Oban ~ ( n [ ~ ) r r ~ n l,, r ~ ( ) ( n ), r ( n ) ] where ~ r( n ) n N and N 0 s a hosen neger. r n, ( The value ~ n ) r may be vewed as he n-h observed value of a eran ( l ) veor ~ r of random varables. () Comue ~ ( n ) r r, N N n and,, j l ; 0,.

7 A model for me seres analyss 574 (3) Comue he l egenveors of he varane-ovarane marx (,) (,0) (0,) { mj mj mj } and form he marx H of whh he -h olumn s he h egenveor. (4) Comue. N ( ) ( n ) (5) Comue m [ s ], l ; N n, 3, 4. (6) Fnd (, ( ) ) and suh ha E( ) m, where s defned n Equaon(3.3) and l ;, 3, 4. (7) From he mulvarae ower-normal dsrbuon of ~ r H ε, we r esmae he ondonal df of r( ) usng he ondonal df of ~ r gven ~ r r( ( l ) ),..., ~ r l r( ) and ~ r l r( ). l We noe ha he veor formed by () he values n he uer rangle of he varane-ovarane marx n (3) () he omonens of r (), and, l may be vewed as an esmae of he Tye arameer x () whh has a oal of n ( l )( l ) 4( l ) omonens. 5 Esmaon of ondonal df wh Tye Parameer The followng roedure may be used o esmae he ondonal dsrbuon of x when he values of x m,, x x are gven: () Oban ~ x x ( N ~ x n l m n ), n and [ ~ x,, ~ x, ~ x ] where n m n n n N N N m l.

8 574 A. H. Poo The value may be vewed as he n -h observed value of a eran veor ~ x onssng of (m+) n random varables. ~ n () Oban x [ q] as a veor onssng of he nal m n omonens of ~ n x and q nal omonens of ~ x n, q n. ~ n x [ q] may be vewed as he The value n -h observed value of a eran veor ~ x [ q] onssng of mn q random varables. (3) For aly he mehod n Seon 4 o fnd a mulvarae ower-normal dsrbuon for ~ x [ q] and use he resulng dsrbuon o generae a value x q x m,, x, x, x, x,, for he q h omonen of x when x q are gven. (4) Reea (3) o generae m values of x, x, x and use he mehod n Seon 4, n o fnd a mulvarae ower-normal dsrbuon as an esmae of he ondonal dsrbuon of x. 6 Performane of Model We may onsru redon nerval for he fuure resonse and redon regon for he fuure Tye arameer from he ondonal df wh Tye and Tye arameers resevely, and assess he erformane of he model by usng he exeed szes of he redon nerval and redon regon, and he ably of he nerval and regon o over he reseve fuure values. From he ondonal df of r( ) when he values of r( ( l ) ),, r( ) and r() are gven, we may use he 00( )% on L and 00( )% on U of he ondonal df o form a nomnally 00( )% redon nerval L, U ] for he fuure value r( ). The moran haraerss of he [ redon nerval L, U ] are s [

9 A model for me seres analyss 5743 (A) overage robably P whh s defned as he robably ha L, U ] wll over he fuure observaon r( ) and [ (B) exeed lengh L whh s defned as he exeed value of he lengh U L Esmaes of P and L may be obaned va he followng roedure. Suose here are N observed values r,, r, r N of he resonse. For, we use he mehod n Seon 4 o oban, from he values rn, rn,, rn N l, an esmaed ondonal df, and onsru he redon nerval w w w based on he esmaed ondonal df. The redon nerval fuure observaon may or may no over he observed rn w N l. The rooron of mes (ou of w N mes) he redon nerval overs he observed fuure observaon r N l s n w hen an esmae of he overage robably P. Furhermore he average value of over nw N w s an esmae of he exeed lengh L. We noe ha when he values of x( ( m ) ),, x( ) and x() are gven,he random varable x( ) may be wren as x( ) r x H x ε x of whh he rgh sde has a sruure whh s smlar o ha of he rgh sde of Equaon (3.). Thus le n Equaon (3.3), x s a funon of a random varable (denoed as z x ) havng a sandard normal dsrbuon. A nomnally 00( )% redon regon R for x( ) an now be formed from he values of x ( ) of whh he orresondng z, x, z, x zxn have a sum of squares D n z x a h square dsrbuon wh haraerss of he redon regon whh s less han or equal o he 00( )% on n, n degrees of freedom. The moran R are s of

10 5744 A. H. Poo (A) overage robably P whh s defned as he robably ha over he fuure value x ( ) and R wll (B) exeed sze L whh s defned as he exeed value of he sze of he n -dmensonal sae. R n he A measure of he sze of R an be obaned as follows. We frs ransform n -dmensonal sae for x( ) o an n -dmensonal sheral olar oordnae sysem wh ener r x. We nex hoose a large number ( N, say) of olar angles whh are of he same sze, and fnd he average of he radal dsanes of he orresondng N ons x ( ) of whh D average radal dsane s hen a measure of he sze of R. n,. The Esmaes of P and L may be obaned n a way smlar o ha used for esmang P and L. For a redon nerval (or regon) o be lassfed as sasfaory, should a leas have a overage robably whh s no oo far from he arge value. Among wo redon nervals (or regons) of whh he overage robables are lose o he arge value, he one wh a smaller exeed sze s deemed o be a beer redon nerval (or regon). The model would be onsdered as sasfaory f he relaed redon nerval and redon regon are sasfaory. 7 Numeral Examles The fluuaon of neres rae s very moran n he deson of nvesmen and rs managemen n he fnanal mares. One-faor models are a oular lass of model for desrbng he fluuaon of neres rae. In a one- r a me may be sefed va he sohas faor model, he neres rae dfferenal equaon dr, r( ) d, r( ) dw( )

11 A model for me seres analyss 5745 where and are resevely he drf and dffuson erm of he neres rae roess, and W s a Brownan moon. An moran examle of one-faor model s he Chan, Karoly, Longsaff and Sanders (CKLS) model: dr where,, 0, 0 are onsans. r( ) d r( ) dw( ), r0 r0 The above one-faor model may be used for deermnng he res of bonds, bond oons, swas, as, floors, e. In wha follows, he CKLS model wll be used o f a real daase on neres raes. Consder he 6 monh reasury Bll Raes daa obaned from he ln h: researh.slousfed.org/fred/aegores/6 under he fle name WTB6MS.xls. The oal number of daa on s N=566 and = 7/365 reresens he lengh of a one-wee erod. The maxmum lelhood esmaes of he arameers n he CKLS model for he daase are found o be resevely =0.080, =0.065, =0.733 and = Le l = and hoose a value of 00 for N (see Seon 4). For j j N l, N l,.., N, we use he mehod n Seon 4 o esmae, he ondonal df of r( ) when r ( l ),..., r and r are gven. From he ondonal df of r( ), we fnd a nomnally 95% redon nerval for r( ). The oal number of redon nervals whh an be obaned wll hen be N N l. From he N N l redon nervals, he esmaes of he overage robably and exeed lengh of he redon nerval are found o be P = 0.96 and L =0.006 resevely. By usng he CKLS model, we an oban a oal of N N l orresondng redon nervals whh gve and as he esmaes of he overage robably and exeed lengh resevely. Table 7. gves he esmaes of overage robably and exeed lengh for he ases when l 5 are used. The able shows ha he redon nervals

12 5746 A. H. Poo based on he roosed model have smaller overage robables and shorer exeed lenghs when omared wh ha based on he CKLS model. We also noed ha he exeed lengh of he redon nerval based on he roosed model learly dereases as l nreases from o 3 bu he derease beomes less obvous afer l = 3. Table 7. dslays he esmaes of he overage robably and exeed lengh when l s fxed a 3 bu N ranges from 00 o 500. From he able, we see ha for suable hoes of N, he redon nerval based on he roosed model an have an esmaed overage robably whh s lose o he arge value, bu ye has an exeed lengh whh s shorer han ha of he redon nerval based on he CKLS model. Table 7. Esmaed overage robably and exeed lengh of redon nerval [α = 0.05, N = 00]. l P L CKLS (0.9496) (0.0063) Table 7. Esmaed overage robably and exeed lengh of redon nerval [α = 0.05, l = 3, he values n arenheses are he esmaed overage robably and exeed lengh of redon nerval based on he CKLS model]. l P L (0.0063) (0.9496) (0.944) (0.9507) (0.965) ( ) ( ) ( )

13 A model for me seres analyss 5747 Now le l, m and N =00. For j, j N l, N l,.., N, we use he mehod n Seon 4 o esmae he Tye arameer x () for he ondonal df of r( ) when r ( l ),..., r and gven. From he esmaed values of x (),we nex oban r are ~ n n N N N m l, and use he mehod n Seon 5 o esmae he arameer θ of he ondonal df of x when [ x( )] x for x s gven. For a gven value of n, a nomnally 95% redon regon R for x (( N l m n ) ) an be found. To fnd ou wheher he observed x (( N l m n ) ) les n he redon regon R, we omue D n z x where n =8 (see Seon 6), and fnd ou wheher he omued D s less han he 95% on 8.87 of he h square dsrbuon wh 8 degrees of freedom. When n = and 000, he omued values of D are found o be.44 and 7.55 resevely, ndang ha he observed values of x (( N l m n ) ) le n he orresondng redon regons. As he above wo omued values of D are boh less han he 95% on of he h square dsrbuon, here are no good reasons o suse he valdy of he ondonal df of x. A more horough nvesgaon of he ondonal df of x would be ossble f we ould allevae he roblem of long omung me nvolved n esmang he hgh- dmensonal dsrbuons. 8 Conludng Remars The model roosed for me seres analyss feaures a farly general ondonal dsrbuon of he fuure resonse when he values of he resen and as observaons are gven. Alhough he mean of he ondonal dsrbuon s resred o be a lnear funon of he resen and as observaons, he model s farly general as he varane, sewness and uross of he ondonal dsrbuon are modeled as funons of he resen and as observaons. The generaly of he model s furher enhaned by desrbng he arameer of he ondonal dsrbuon of he fuure resonse usng a mul-dmensonal me seres.

14 5748 A. H. Poo The erformane of he ondonal dsrbuon of he fuure resonse n he roosed model has been examned and found o be omarable o he wellnown CKLS model when a real daase on neres raes s used. However he erformane of he mul-dmensonal me seres formed by he arameers of he ondonal dsrbuons of he fuure resonses n he roosed model has no been exensvely examned due o he requremen of long omung me. The roosed model may be aled o oher daase n fnane and oher areas where he underlyng dsrbuons are unmodal nonnormal, and he me seres of he resonses s non-saonary. Referenes [] B.D.O. Anderson and J.B. Moore, Omal Flerng, Prene-Hall In., Englewood Clffs, New Jersey,979. [] T. Bollerslev, Generalzed Auoregressve Condonal Heerosedasy, Journal of Eonomers, 3 (986), [3] G.Box and G.Jenns,Tme seres analyss: Foreasng and onrol, San Franso: Holden-Day, 970. [4] R. Engle, Auoregressve Condonal Heerosedasy wh Esmaes of Uned Kngdom Inflaon, Eonomera, 50 (98), [5] F.L.Lews, Omal esmaon, John Wley & Sons, New Yor, 986. [6] I.K.Yeo and R.A. Johnson, A new famly of ower ransformaons o mrove normaly or symmery, Bomera, 87 no. 4 (000), Reeved: June, 0