Long-run and Cyclical Dynamics in the US Stock Market

Transcription

1 155 Reihe Ökonomie Economics Series Long-run and Cyclical Dynamics in he US Sock Marke Guglielmo Maria Caporale, Luis A. Gil-Alana

2 155 Reihe Ökonomie Economics Series Long-run and Cyclical Dynamics in he US Sock Marke Guglielmo Maria Caporale, Luis A. Gil-Alana May 4 Insiu für Höhere Sudien (IHS), Wien Insiue for Advanced Sudies, Vienna

3 Conac: Guglielmo Maria Caporale London Souh Bank Universiy 13 Borough Road London SE1 OAA, Unied Kingdom : +44// fax +44// g.m.caporale@lsbu.ac.uk Luis A. Gil-Alana Deparmen of Economics Universiy of Navarra Campus de Arrosadia 316 Pamplona, Spain alana@unav.es Founded in 1963 by wo prominen Ausrians living in exile he sociologis Paul F. Lazarsfeld and he economis Oskar Morgensern wih he financial suppor from he Ford Foundaion, he Ausrian Federal Minisry of Educaion and he Ciy of Vienna, he Insiue for Advanced Sudies (IHS) is he firs insiuion for posgraduae educaion and research in economics and he social sciences in Ausria. The Economics Series presens research done a he Deparmen of Economics and Finance and aims o share work in progress in a imely way before formal publicaion. As usual, auhors bear full responsibiliy for he conen of heir conribuions. Das Insiu für Höhere Sudien (IHS) wurde im Jahr 1963 von zwei prominenen Exilöserreichern dem Soziologen Paul F. Lazarsfeld und dem Ökonomen Oskar Morgensern mi Hilfe der Ford- Sifung, des Öserreichischen Bundesminiseriums für Unerrich und der Sad Wien gegründe und is somi die erse nachuniversiäre Lehr- und Forschungssäe für die Sozial- und Wirschafswissenschafen in Öserreich. Die Reihe Ökonomie biee Einblick in die Forschungsarbei der Abeilung für Ökonomie und Finanzwirschaf und verfolg das Ziel, abeilungsinerne Diskussionsbeiräge einer breieren fachinernen Öffenlichkei zugänglich zu machen. Die inhalliche Veranworung für die veröffenlichen Beiräge lieg bei den Auoren und Auorinnen.

4 Absrac This paper examines he long-run dynamics and he cyclical srucure of he US sock marke using fracional inegraion echniques, specifically a version of he ess of Robinson (1994a) which allows for uni (or fracional) roos boh a he zero (long-run) and a he cyclical frequencies. We consider inflaion, real risk-free rae, real sock reurns, equiy premium and price/dividend raio, annually from 1871 o When focusing exclusively on he long-run frequency, he esimaed order of inegraion varies considerably, bu nonsaionariy is found only for he price/dividend raio. When he cyclical componen is also aken ino accoun, mos series appear o be saionary and o exhibi long memory. Furher, mean reversion occurs. Finally, he fracional (a zero and cyclical) models are shown o forecas more accuraely han rival ones based on fracional and ineger differeniaion exclusively a he zero frequency. Keywords Sock marke, fracional cycles, long memory, Gegenbauer processes JEL Classificaion C, G1, G14

5 Commens We are very graeful o Pok-sang Lam for kindly supplying he daa. The second named auhor graefully acknowledges financial suppor from he Minserio de Ciencia y Tecnologia (SEC-1839, Spain).

6 Conens 1. Inroducion 1. The saisical model 3. The esing procedure 4 4. An empirical applicaion o he US sock marke 7 5. Forecasing and comparisons wih oher models Conclusions References 4

7 I H S Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke 1 1. Inroducion The empirical lieraure analysing sock markes ypically ess wheher he series of ineres are I(1) (sock prices), or I() (sock marke reurns). This is because, according o he Efficien Marke Hypohesis (EMH), i should no be possible o make sysemaic profis above ransacion coss and risk premia, and herefore sock prices are characerised as an enirely unpredicable random walk process, which implies ha sock reurns should be I(). Mean reversion is seen as inconsisen wih equilibrium asse pricing models (see he survey by Forbes, 1996). Caporale and Gil-Alana (), hough, sress ha he uni roo ess normally employed impose oo resricive assumpions on he behaviour of he series of ineres, in addiion o having low power. They sugges insead using ess which allow for fracional alernaives (see Robinson, 1994a, 1995a,b), and find ha US real sock reurns are close o being I() (which raises he furher quesion wheher he shocks are auocorrelaed, wih he implicaion ha markes are no efficien). Fracional inegraion models have also been used for inflaion and ineres raes (see, e.g., Shea, 1991; Backus and Zhin, 1993; Hassler and Wolers, 1995; Baillie e al., 1996, ec.). However, i has become increasingly clear ha he cyclical componen of economic and financial series is also very imporan. This has been widely documened, especially in he case of business cycles, for which non-linear (Beaudry and Koop, 1993, Pesaran and Poer, 1997) or fracionally ARIMA (ARFIMA) models (see Candelon and Gil-Alana, 4) have been proposed. Furhermore, i has been argued ha cycles should be modelled as an addiional componen o he rend and he seasonal srucure of he series (see Harvey, 1985, Gray e al., 1989). The available evidence suggess ha he periodiciy of he series ranges beween five and en years, in mos cases a periodiciy of abou six years being esimaed (see, e.g., Baxer and King, 1999; Canova, 1998; King and Rebelo, 1999; Caporale and Gil-Alana, 3). In view of hese findings, he presen paper exends he earlier work by Caporale and Gil- Alana () by adoping a modelling approach which, insead of considering exclusively he componen affecing he long-run or zero frequency, also akes ino accoun he cyclical srucure. Furhermore, he analysis is carried ou for he US inflaion rae, real risk-free rae, equiy premium and price/dividend raio, in addiion o real sock reurns. More precisely, we use a procedure due o Robinson (1994a), which enables one o es simulaneously for uni and fracional roos a boh zero and he cyclical frequencies. This approach has several disinguishing feaures compared wih oher mehods, he mos noiceable one being is sandard null and local limi disribuions. 1 Moreeover, i does no require Gaussianiy (a condiion rarely saisfied in financial ime series), wih a momen condiion only of order wo required. Addiionally, using a large srucure ha involves simulaneously he zero and he 1 Noe ha, for example, mos of he classical uni roo ess (i.e., Dickey and Fuller, 1979; Kwiakowski e al., 199; ec.) are non-sandard, in he sense ha he criical values have o be calculaed numerically on a case by case simulaion sudy.

8 Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke I H S cyclical frequencies, we can solve a leas o some exen he problem of misspecificaion ha migh arise wih respec o hese wo frequencies. We are able o show ha our proposed mehod represens an appealing alernaive o he increasingly popular ARIMA (ARFIMA) specificaions found in he lieraure. I is also consisen wih he widely adoped pracice of modelling many economic series as wo separae componens, namely a secular or growh componen and a cyclical one. The former, assumed in mos cases o be nonsaionary, is hough o be driven by growh facors, such as capial accumulaion, populaion growh and echnology improvemens, whils he laer, assumed o be covariance saionary, is generally associaed wih fundamenal facors which are he primary cause of movemens in he series. The srucure of he paper is as follows. Secion 1 briefly describes he saisical model. Secion inroduces he version of he Robinson s (1994a) ess used for he empirical analysis. Secion 3 discusses an applicaion o annual daa on several US sock marke series for he ime period Secion 4 is concerned wih model selecion for each ime series, and he preferred specificaions are compared wih oher more classical represenaions. Secion 5 conains some concluding commens.. The saisical model Le us suppose ha {y, = 1,,, n} is he ime series we observe, which is generaed by he model: d ( 1 L) 1 (1 cos w L + L d ) y = u, = 1,,.., (1) where L is he lag operaor (Ly = y -1 ), w is a given real number, u is I() and d 1 and d can be real numbers. Le us firs consider he case of d =. Then, if d 1 >, he process is said o be long memory a he long-run or zero frequency, also ermed srong dependen, sonamed because of he srong associaion beween observaions widely separaed in ime. Noe ha he firs polynomial in (1) can be expressed in erms of is Binomial expansion, such ha for all real d 1 : d (1 ) = d L j= ( 1) d ( d1 1) L 1 1 j j 1 L = 1 d 1 L + j.... These processes were iniially inroduced by Granger (198, 1981) and Hosking (1981), and were heoreically jusified in erms of aggregaion by Robinson (1978), Granger (198): cross secion aggregaion of a large number of AR(1) processes wih heerogeneous AR For he purposes of he presen paper, we define an I() process as a covariance saionary process wih specral densiy funcion ha is posiive and finie a any frequency on he specrum.

9 I H S Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke 3 coefficiens may creae long memory. Parke (1999) uses a closely relaed discree ime error duraion model, while Diebold and Inoue (1) relae fracional inegraion wih regime swiching models. 33 The differencing parameer d 1 plays a crucial role from boh economic and saisical viewpoins. Thus, if d 1 (,.5), he series is covariance saionary and meanrevering, wih shocks disappearing in he long run; if d 1 [.5, 1), he series is no longer saionary bu sill mean-revering, while d 1 1 means nonsaionariy and non-meanreversion. I is herefore crucial o examine if d 1 is smaller han or equal o or higher han 1. Thus, for example, if d 1 < 1, here is less need for policy acion han if d 1 1, since he series will reurn o is original level someime in he fuure. On he conrary, if d 1 1, shocks will be permanen, and acive policies are required o bring he variable back o is original long erm projecion. In fac, his is one of he mos holy debaed opics in empirical finance. Lo and MacKinlay (1988) and Poerba and Summers (1988) used variance-raio ess and found evidence of mean reversion in sock reurns. On he conrary, Lo (1991) used a generalized form of rescaled range (R/S) saisic and found no evidence agains he random walk hypohesis for he sock indices, conradicing his earlier finding using variance raio ess. Oher papers examining he persisence of shocks in financial ime series are Lee and Robinson (1996), Fiorenini and Senana (1998) and May (1999). Le us now consider he case of d 1 = and d >. The process is hen said o exhibi long memory a he cyclical frequency. I was examined by Gray e al. (1989, 1994), who showed ha he series is saionary if cos w < 1 and d <.5 or if cos w = 1 and d <.5. They also showed ha he second polynomial in (1) can be expressed in erms of he Gegenbauer polynomial C, such ha, calling µ = cos w, j, d d j (1 µ L L ) + = C j, d ( µ ) L, () j = for all d, where C j, d ( µ ) = ( 1) ( d ) (µ) [ j / ] k j k j k ; ( d ) j = k = k!( j k)! Γ( d + j), Γ( d ) where Γ(x) represens he Gamma funcion and a runcaion will be required in () o make he polynomial operaional. Of paricular ineres is he case of d = 1, i.e. when he process conains uni roo cycles; is performance in he conex of macroeconomic ime series was examined, for example, by Bierens (1). 4 Such processes, for which he crucial issue is o have a specral densiy wih a peak a (, π], were laer exended o he case of a finie number of peaks by Giraiis and Leipus (1995) and Woodward e al. (1998) (see also Gray e 3 Crozcek-Georges and Mandelbro (1995), Taqqu e al. (1997), Chambers (1998) and Lippi and Zaffaroni (1999) also use aggregaion o moivae long memory processes. 4 Uni roo cycles were also examined by Ahola and Tiao (1987), Chan and Wei (1988) and Gregoir (1999a, b).

10 4 Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke I H S al. (1989) and Robinson (1994a)). The economic implicaions in () are similar o he previous case of long memory a he zero frequency. Thus, if d < 1, shocks affecing he cyclical par will be mean revering, while d 1 implies an infinie degree of persisence of he shocks. This ype of model for he cyclical componen has no been previously used for financial ime series, hough Robinson (1, pp. 1-13) suggess is adopion in he conex of complicaed auocovariance srucures. 3. The esing procedure Following Bhargava (1986), Schmid and Phillips (199) and ohers in he parameerisaion of uni-roo models, Robinson (1994a) considers he regression model: y = β ' z + x = 1,,..., (3) where y is a given raw ime series; z is a (kx1) vecor of deerminisic regressors ha may include, for example, an inercep, (e.g., z 1), or an inercep and a linear ime rend (in he case of z = (1,) ); β is a (kx1) vecor of unknown parameers; and he regression errors x are such ha: ρ ( L ; θ ) x = u = 1,,..., (4) where ρ is a given funcion which depends on L, and he (px1) parameer vecor θ, adoping he form: s s p d s d d θ + θ θ j = 1 j j ρ ( L ; θ ) = ( 1 L ) ( 1 L ) ( 1 cos w L + L ), (5) for real given numbers d 1, d s, d, d p-1, ineger p, and where u is I(). Noe ha he second polynomial in (5) refers o he case of seasonaliy (i.e. s = 4 in case of quarerly daa, and s = 1 wih monhly observaions). Under he null hypohesis, defined by: Ho: θ = (6) (5) becomes: d 1 j ρ ( L ; θ = ) = ρ ( L ) = ( 1 L ) ( 1 L ) ( 1 cos w L + L ). (7) s s d p 1 j = d This is a very general specificaion ha makes i possible o consider differen models under he null. For example, if d 1 = 1 and d s, d j = for j, we have he classical uni-roo models

11 I H S Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke 5 (Dickey and Fuller, 1979, Phillips, 1987; Phillips and Perron, 1988, Kwiakowski e al., 199, ec.), and, if d 1 is a real value, he fracional models examined in Diebold and Rudebusch (1989), Baillie (1996) and ohers. Similarly, if d s = 1 and d j = for all j, we have he seasonal uni-roo model (Dickey, Hasza and Fuller, 1984, Hyllerberg e al., 199, ec.) and, if d s is real, he seasonal fracional model analysed in Porer-Hudak (199). If d 3 = 1 and d s, d j = for j 3, he model becomes he uni roo cycles of Ahola and Tiao (1987) and Bierens (1), and if d 3 is real, he Gegenbauer processes examined by Gray e al. (1989, 1994), Ferrara and Guegan (1), ec. In his paper we are concerned wih boh he long run and he cyclical srucure of he series, and hus we assume ha d s = and p = 3. In such a case (5) can be expressed as: d1 + θ d + θ 1 ρ ( L ; θ ) = (1 L) (1 cos w L + L ), (8) and, similarly, (7) becomes: ρ ( L ) = (1 L) d 1 (1 cos wl+ L ) d. (9) Here, d 1 represens he degree of inegraion a he long run or zero frequency (i.e., he sochasic rend), while d affecs he cyclical componen of he series. We nex describe he es saisic. We observe {(y, z ), = 1,, n}, and suppose ha he I() u in (4) have parameric specral densiy given by: σ f ( λ ; τ ) = g( λ ; τ ), π < λ π, π where he scalar σ is known and g is a funcion of known form, which depends on frequency λ and he unknown (qx1) vecor τ. Based on H o (6), he residuals in (3), (4) and (8) are: d1 d = (1 L) (1 cos w L + L ) y ˆ' β s, (1) ˆ u where 1 n n d ˆ 1 β = s s' s( 1 L ) ( 1 cos w L = 1 = 1 d1 d s = ( 1 L) (1 cos wl + L ) z. + L ) d y,

12 6 Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke I H S Unless g is a compleely known funcion (e.g., g 1, as when u is whie noise), we need o esimae he nuisance parameer τ, for example by ˆ τ = arg min T σ ( τ ), where T is a suiable subse of R q Euclidean space, and n π σ ( τ ) 1 1 = g( λs ; τ ) I û( λs ), n s = 1 ( π n) 1 / I ( λ ) = û e û s n = 1 wih i λs ; λs = π s. n τ The es saisic, which is derived hrough he Lagrange Muliplier (LM) principle, hen akes he form: ˆ 1/ R = rˆ ' rˆ; rˆ n = Aˆ ˆ σ aˆ, (11) where n is he sample size, and aˆ = π n * s ψ ( λ ) g( λ ; ˆ) τ s s 1 n I( λs ) ; 1 π 1 ˆ σ = σ ( ˆ) τ = g( λs; ˆ) τ I( λs ), n s = 1 Aˆ * = * * 1 * ψ ( λs ) ψ ( λs )' ψ ( λs ) ˆ( ε λs )' ˆ( ε λs ) ˆ( ε λs )' ˆ( ε λs ) ψ ( λs )' n s s s s ψ ( λs )' = [ ψ 1( λs ), ψ ( λs )]; ˆ ε( λ ) log ( λ ; ˆ s = g s τ ) ; τ ψ = ψ ( λ ) = log ( cosλ cos ). 1 ( λ s ) log sin λ s ; s s w Based on H o (6), Robinson (1994a) esablished ha, under cerain regulariy condiions: 5 Rˆ d χ, as n. (1) Thus, as shown by Robinson (1994a), unlike in oher procedures, we are in a classical largesample esing siuaion, and furhermore he ess are efficien in he Piman sense agains local deparures from he null. 66 Because Rˆ involves a raio of quadraic forms, is exac null 5 These condiions are very mild and concern echnical assumpions o be saisfied by ψ 1 (λ) and ψ (λ). 6 In oher words, if he ess are implemened agains local deparures of he form: H a : θ = δn -1/, for δ, he limi disribuion is a χ ( v ) wih a non-cenraliy parameer v, which is opimal under Gaussianiy of u.

13 I H S Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke 7 disribuion can be calculaed under Gaussianiy via Imhof s algorihm. However, a simple es is approximaely valid under much wider disribuional assumpions: a es of (6) will χ, α χ, α rejec H o agains he alernaive H a : d d o if Rˆ >, where Prob ( > ) = α. A similar version of Robinson s (1994a) ess (wih d 1 = ) was examined in Gil-Alana (1), where is performance in he conex of uni-roo cycles was compared wih ha of he Ahola and Tiao s (1987) ess, he resuls showing ha he former ouperform he laer in a number of cases. Oher versions of his ess have been successfully applied o raw ime series in Gil- Alana and Robinson (1997, 1) o es for I(d) processes wih he roos occurring a zero and he seasonal frequencies respecively. However, his is he firs empirical finance applicaion, which ess simulaneously he roos a zero and he cyclical frequencies, a saisical approach which is shown in he presen paper o represen a credible alernaive o he more convenional ARIMA (ARFIMA) specificaions used for he parameric modelling of many ime series. χ 4. An empirical applicaion o he US sock marke The daase includes annual daa on US inflaion, real risk-free rae, real sock reurns, equiy premium and price/dividend raio from 1871 o 1993, and is a slighly updaed version of he daase used in Cecchei e al. (199) (see ha paper for furher deails on sources and definiions). Figure 1 conains plos of he original series wih heir corresponding correlograms and periodograms. All of hem, wih he excepion of he price/dividend raio, appear o be saionary. However, deeper inspecion of he correlograms shows ha here are significan values even a some lags relaively disan from zero, along wih slow decay and/or cyclical oscillaion in some cases, which could indicae no only fracional inegraion a he zero frequency bu also cyclical dependence. Similarly, he periodograms also have peaks a frequencies oher han zero. For he price/dividend raio, he slow decay in he correlogram clearly suggess ha he series is no I() saionary. Figure displays similar plos for he firs differenced daa. The correlograms and periodograms now srongly sugges ha all series are overdifferenced wih respec o he frequency. On he oher hand, here are significan peaks in he periodograms a frequencies differen from zero. In view of his, i migh be of ineres o examine more in deph he behaviour of hese series using a fracional model a boh he zero and he cyclical frequencies.

14 8 Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke I H S FIGURE 1 Raw ime series, wih heir corresponding correlograms and periodograms Inflaion rae Correlogram * Periodogram,4,4,6,,,4 -, -,, -, , T/ Real risk free rae Correlogram * Periodogram, 4,6,8, -,,4, -,,6,4, -,4 -, Real sock reurn Correlogram * Periodogram, 8,3,4,4 -,4,,1 -,1,3,,1 -, , Equiy premium Correlogram * Periodogram,8,,4,4 -,4,1 -,1 -,,3,,1 -,8 -, T/ 4 Price / Dividend raio Correlogram * Periodogram X 1, 8 3,8 6, , T/ * The large sample sandard error under he null hypohesis of no auocorrelaion is 1/ n or roughly.9 for series of lengh considered here.

15 I H S Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke 9 FIGURE Firs differenced ime series, wih heir corresponding correlograms and periodograms Inflaion rae Correlogram * Periodogram,4,4,1, -, -,4, -,,8,6,4, -,4 -, T/ Real risk free rae Correlogram * Periodogram, 6, 4,1,4,,8,,6,4 -, -,, -, , T/ Real sock reurn Correlogram * Periodogram 1,, 4,6,8,4,,4 -,4 -, -,4, -,8 -, T/ Equiy premium Correlogram * Periodogram,8,4,8,4 -,4, -, -,4,6,4, -, , T/ Price / Dividend raio Correlogram * Periodogram ,4, -, , T/ * The large sample sandard error under he null hypohesis of no auocorrelaion is 1/ n or roughly.9 for series of lengh considered here.

16 1 Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke I H S As a firs sep, we focus on he long run or zero frequency and implemen a simple version of Robinson s (1994a) es, which is based on a model given by (3) and (4), wih z = (1,), 1, (,) oherwise, and ρ(l; θ) = (1 L) d+θ. Thus, under H o (6), we es he model: y = β + β + x, 1,,... (13) 1 = d ( 1 L) x = u, = 1,,..., (14) for values d =, (.1),, and differen ypes of disurbances. In such a case, he es saisic grealy simplifies, aking he form given by (11), wih ψ(λ s ) being exclusively defined d by ψ 1 (λ s ) and uˆ = (1 L) y ˆ' β w. The null limi disribuion will hen be a disribuion. However, if ρ(l; θ) = (1 L) d+θ, hen p = 1, and herefore we can consider onesided ess based on r ˆ = Rˆ, wih a sandard N(,1) disribuion: an approximae one-sided 1α% level es of H o (6) agains he alernaive: H a : θ > (θ < ) will be given by he rule: Rejec H o if rˆ > z α (rˆ < - z α ), where he probabiliy ha a sandard normal variae exceeds z α is α. Noe ha by esing he null hypohesis wih d = 1, his becomes a classical uni-roo ess of he same form as hose proposed by Dickey and Fuller (1979) and ohers. However, insead of using auoregressive (AR) alernaives of he form: (1 (1+θ)L)x = u, we use fracional alernaives. Moreover, he use of AR alernaives involves a dramaic change in he asympoic behaviour of he ess. Thus, if θ <, x is saionary; i conains a uni roo if θ =, and i becomes nonsaionary and explosive for θ >. On he conrary, under fracional alernaives of he form as in (14), he behaviour of x is smooh across d, his being he inuiive reason for is sandard asympoic behaviour. χ 1 The resuls presened in Table 1 correspond o he 95%-confidence inervals of hose values of d where H o (6) canno be rejeced, using whie noise disurbances. 7 We examine separaely he cases of β = β 1 = a priori (i.e., wih no regressors in he undifferenced model (13)); β unknown and β 1 = (wih an inercep); and β and β 1 unknown (an inercep and a linear ime rend). The inclusion of a linear ime rend may appear unrealisic in he case of financial ime series. However, i should be noed ha in he conex of fracional (or ineger) differences, he ime rend disappears in he long run. Thus, for example, suppose ha u in (14) is whie noise. Then, esing H o (6) in (13) and (14) wih d o = 1, he series becomes, for > 1, a pure random walk process if β 1 =, and a random walk wih an inercep if boh β and β 1 are unknown. The resuls vary subsanially from one series o anoher. For insance, for inflaion and real risk-free raes, he values are always higher han bu smaller han.5, oscillaing beween.7 (inflaion rae wih a linear rend) and.49 (real risk-free rae wih no regressors). For real sock reurns and equiy premium, he values of d where H o (6) canno be rejeced widely oscillaes around, ranging beween.18 (equiy premium wih a linear rend) and.14 (sock reurns wih no regressors). Finally, for 7 The confidence inervals were buil up according o he following sraegy. Firs, choose a value of d from a grid. Then, form he es saisic esing he null for his value. If he null is rejeced a he 95% level, discard his value of d. Oherwise, keep i. An inerval is hen obained afer considering all he values of d in he grid.

17 I H S Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke 11 he price/dividend raio, all he non-rejecion values are higher han.5, implying nonsaionariy wih respec o he zero frequency. TABLE 1 Confidence inervals of he non-rejecion values of d using Rˆ in (11) wih ρ(l; θ) = (1 L) d+θ and whie noise u Time Series No regressors An inercep A linear rend INFLATION RATE [ ] [ ] [ ] REAL RISK FREE RATE [ ] [ ] [ ] REAL STOCK RETURN [ ] [ ] [ ] EQUITY PREMIUM [ ] [ ] [ ] PRICE / DIVIDEND RATIO [.7-1.] [ ] [ ] We es he null hypohesis: d = d o in a model given by (1-L) d x = ε. TABLE Confidence inervals of he non-rejecion values of d using Rˆ in (11) wih ρ(l; θ) = (1 L) d+θ and AR(1) u Time Series No regressors An inercep A linear rend INFLATION RATE [ ] [ ] [ ] REAL RISK FREE RATE [ ] [ ] [ ] REAL STOCK RETURN [ ] [ ] [ ] EQUITY PREMIUM [-. -.] [-.3 -.] [ ] PRICE / DIVIDEND RATIO [ ] [ ] [ ] We es he null hypohesis: d = d o in a model given by (1-L) d x = u ; u = τu -1 + ε. The significan resuls in Table 1 may be in par due o he fac ha I() auocorrelaion in u has no been aken ino accoun. Thus, we also performed he ess imposing AR(1) disurbances (see Table ). Higher AR orders were also ried and he resuls were very similar. For all series, excep he price/dividend raio, he values oscillae around, implying ha he series may be I() saionary. However, for he price/dividend raio, he values are sill above, ranging from.13 (wih a linear ime rend) o.83 (in he case of no regressors). Comparing he resuls of his able wih hose of Table 1 (whie noise u ), we are lef wih he impression ha he orders of inegraion are smaller by abou. when auocorrelaion is allowed for. This may be relaed o he fac ha he esimaes of he AR coefficiens are Yule-Walker, which enails AR roos ha, alhough auomaically less han one in absolue value can be arbirarily close o one. Hence, hey migh compee wih he order of inegraion a he zero frequency when describing he behaviour a such a frequency. I may also be of ineres o examine d, independenly of he way of modelling he I() disurbances, a he same zero frequency. For his purpose, we use a semiparameric

18 1 Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke I H S procedure due o Robinson (1995a), which we now describe. The Quasi Maximum Likelihood Esimaor (QMLE) of Robinson (1995a) is basically a While esimaor in he frequency domain, considering a band of frequencies ha degeneraes o zero. The esimaor is implicily defined by: ˆ m 1 d = arg min log ( ) log d C d d λ s, (15) m s = 1 C( d ) 1 m = I( λ ) λ s m s = 1 π s, n m n d s, λs =, where I(λ s ) is he periodogram of he raw ime series, x, given by: I( λ ) s = 1 π n n = 1 x e i λ s, and d (-.5,.5). 8 Robinson (1995a) proved ha: Under finieness of he fourh momen and oher mild condiions, m ( dˆ d ) N(, 1/ 4) as n, o d where d o is he rue value of d, wih he only addiional requiremen ha m slower han n. Robinson (1995a) showed ha m mus be smaller han n/ o avoid aliasing effecs. A mulivariae exension of his esimaion procedure can be found in Lobao (1999). There also exis oher semiparameric procedures for esimaing he fracional differencing parameer, for example, he log-periodogram regression esimaor (LPE), iniially proposed by Geweke and Porer-Hudak (1983) and modified laer by Künsch (1986) and Robinson (1995b), and he averaged periodogram esimaor (APE) of Robinson (1994b). However, we have chosen o use here he QMLE, primarily because of is compuaional simpliciy. Noe ha, when using he QMLE, one does no need o employ any addiional user-chosen numbers in he esimaion (as in he case of he LPE and he APE). Also, here is no need o assume Gaussianiy in order o obain an asympoic normal disribuion, he QMLE being more efficien han he LPE. 8 Velasco (1999a, b) has recenly showed ha he fracionally differencing parameer can also be consisenly semiparamerically esimaed in nonsaionary conexs by means of apering.

19 I H S Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke 13 FIGURE 3 Semiparameric esimaes of d based on he QMLE (Robinson, 1995a) INFLATION RATE REAL RISK FREE RATE 1 1,5,5 -,5 -,5-1 1 T/ -1 1 T/ REAL STOCK RETURN EQUITY PREMIUM 1 1,5,5 -,5 -,5-1 1 T/ -1 1 T/ PRICE / DIVIDEND RATIO PRICE / DIVIDEND RATIO (FIRST DIFF) 1,5 1 1,5,5 1 -,5,5-1 1 T / 3 T/ The horizonal axes corresponds o he bandwidh parameer number m, while he verical one refers o he order of inegraion.

20 14 Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke I H S Figure 3 repors he resuls based on he QMLE of Robinson (1995a), i.e., dˆ given by (15) for a range of values of m from 1 o n/. 9 9 I also displays he confidence inervals corresponding o he I() hypohesis for all series and he uni roo for he price/dividend raio. We see ha, for inflaion and he real risk-free rae, here are some esimaes ha are wihin he I() inerval, especially if m is small; however, for mos of he values of m, he esimaes are higher han hose corresponding o he confidence inerval. For real sock reurns and equiy premium, almos all values are wihin such inervals, while for he price/dividend raio hey are clearly no. Also, for he laer series, he values are lower han hose wihin he uni roo inerval, clearly suggesing ha d is greaer han bu smaller han 1. Consequenly, he findings are he same as wih he parameric procedure, namely here is srong evidence in favour of I() saionariy for real sock reurns and equiy premium, some evidence of long memory for inflaion and real risk-free raes, and srong evidence of fracional inegraion for he price/dividend raio. The above approach o invesigaing he long-run behaviour of a ime series consiss of esing a parameric model for he series and esimaing a semiparameric one, relying on he long run-implicaions of he esimaed models. The advanage of he firs procedure is he precision gained by providing all he informaion abou he series hrough he parameer esimaes. A drawback is ha hese esimaes are sensiive o he class of models considered, and may be misleading because of misspecificaion. I is well known ha he possibiliy of misspecificaion can never be seled conclusively in he case of parameric (or even semiparameric) models. However, he problem can be parly addressed by considering a larger class of models. This is he approach used in wha follows, where we employ anoher version of he ess of Robinson (1994a) ha enables us simulaneously o consider roos a zero and he cyclical frequencies. For his purpose, le us consider now he model given by (3) and (4), wih ρ(l; θ) as in (8) and z = (1,). Thus, under H o (6), he model becomes: y = β + β + x, 1,,... (16) 1 = d ( 1 L) (1 cos w L + L ) 1 d x = u, = 1,,..., (17) and, if d =, he model reduces o he case previously sudied of long memory exclusively a he long-run or zero frequency. We assume ha w = w r = πj/n, j = n/r, and r indicaing he number of ime periods per cycle. 9 In he case of he price/dividend raio, and in order o ensure saionariy, he esimaes were based on he firs differenced daa, adding hen one o he esimaed values of d o ge he proper orders of inegraion.

21 I H S Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke 15 TABLE 3 Tesing H o (6) in (16), (4) and (8) wih z 1, w = w r, r = 6 and whie noise u D 1 d INFLATION RISK RATE STOCK RT PREMIUM PRICE / DIV *.69 * *.9 * * * 4.35 * *.54 * * 1.81 * * * 5.49 * *.74 * * * 5.11 * * * * * 4.5 * 5.18 * 5.48 * *.5 * * 1.78 * * 5.3 * * * 3.6 * *.4 * * 3.19 * * * *.7 * * * * * * , * * * * * * * * * , * * * * * * * * In bold and wih *, he non-rejecion values of he null hypohesis a he 5% significance level.

22 16 Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke I H S We compued he saisic Rˆ given by (11) for values of d 1 and d = -.5, (.1),, and r =,, n/, 1 assuming ha u is whie noise. For breviy, we do no repor he resuls for all saisics. In brief, he null hypohesis (6) was rejeced for all values of d 1 and d if r was smaller han 4 or higher han 9, implying ha, if a cyclical componen is presen, is periodiciy is consrained o be beween hese wo years. This is consisen wih he empirical finding in Canova (1998), Burnside (1998), King and Rebelo (1999) and ohers ha cycles have a periodiciy beween five and en years. We repor in Table 3 he non-rejecion cases a he 5% level, wih an inercep and wih r = 6. The reason for giving he resuls only for he case of an inercep is ha hose based on a linear ime rend were very similar, ogeher wih he fac ha he coefficien corresponding o he linear ime rend was found o be insignificanly differen from zero in virually all cases. Noe ha he es saisic is obained from he null differenced model, which is assumed o be I(), and herefore sandard -ess apply. Furher, we focus on r = 6 since he non-rejecion values wih r = 4, 5, 7, 8 and 9 formed a proper subse of hose non-rejecions obained wih r = 6. We see ha for inflaion and real risk rae he non-rejecion values oscillae beween.1 and.4 for d 1, and beween and.3 for d. They are slighly smaller for d in he case of sock reurns and equiy premium, in some cases being even negaive. Finally, for he price/dividend raio, he values of d 1 range beween.5 and 1, while d seems o be consrained beween and.5. In order o have a more precise view abou he non-rejecion values of d 1 and d, we recompued he ess bu his ime for a shorer grid, wih d 1, d = -.5, (.1),. Figure 4 displays he regions of (d 1, d ) values where H o canno be rejeced a he 5% level. Essenially, he series can be grouped ino hree caegories: inflaion rae and real risk-free rae; real sock reurns and equiy premium; finally price/dividend raio. Saring wih he firs group (inflaion and real risk-free raes), we observe ha he values of d 1 range beween.1 and.5 while d seems o be consrained beween and.3. Thus, we observe a slighly higher degree of inegraion a he long run or zero frequency compared o he cyclical one. For real sock reurns and equiy premium, he values of boh orders of inegraion oscillae around. Finally, for he price/dividend raio he values of d 1 range beween.5 and 1, while d is beween and.5, implying nonsaionariy wih respec o he zero frequency bu saionariy wih respec o he cyclical componen, and mean reversion wih respec o boh. Consequenly, shocks o he laer series will disappear in he long run, wih hose affecing he cyclical par ending o disappear faser han hose affecing is long-run or rending behaviour Noe ha in he case of r = 1, he model reduces o he case previously sudied of long memory exclusively a he long run or zero frequency. 11 This procedure was also conduced in he conex of auocorrelaed (AR(1) and AR()) disurbances and he resuls did no subsanially differ from hose repored here based on whie noise u.

23 I H S Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke 17 FIGURE 4 Non-rejecion values of d 1 and d in (16), (4) and (8) wih r = 6 and whie noise u INFLATION RATE REAL RISK FREE RATE,75,75,5,5 d d,5,5,5,5,75 d1,5,5,75 d1 REAL STOCK RETURN EQUITY PREMIUM,5,5,5,5 d d -,5 -,5 -,5 -,5 -,5,5,5 d1 -,5 -,5 -,5,5,5 d1 PRICE / DIVIDEND RATIO PRICE / DIVIDEND RATIO (FIRST DIFF),75,5 d,5,5,5,75 1 1,5 d1

24 18 Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke I H S 5. Forecasing and comparisons wih oher models In his secion, we ry firs o deermine he bes model specificaion for each ime series. Then, we compare he seleced models wih oher approaches based on I() and I(1) hypoheses. Given he lack of efficien procedures for esimaing he parameers involved in he model given by (16) and (17), we have decided o use he following sraegy: firs, we recompue he values of he es saisic for d 1o, d o = -.5, (.1), and r =,, n/, for he hree cases of no regressors, an inercep and an inercep wih a linear ime rend. Then, we discriminae beween he hree cases according o he -values of he esimaed coefficiens in (16), and choose he values of d 1o, d o and r which produce he lowes saisic in absolue value. The seleced model for each ime series is repored in he second column in Table 4. We observe ha for inflaion rae and real risk-free rae, boh orders of inegraion are consrained o be beween.1 and.3, he order of inegraion a zero being slighly higher han he cyclical one; for real sock reurns and equiy premium, he values of he d s are close o zero, being slighly negaive for he zero frequency; finally, for he price-dividend raio we see ha i is nonsaionary a he long-run frequency (d 1 =.68), and saionary wih d close o zero for he cyclical componen. The hird column of he able repors he seleced models aking ino accoun exclusively he componen affecing he long run or zero frequency, while he fourh refers o he case of ineger differeniaion wih respec o such a frequency. In boh cases, we model he cyclical srucure using ARMA specificaions. Saring wih he case of fracional inegraion, we observe ha he highes degree of inegraion is obained for he price/dividend raio (d =.73), followed by inflaion (d =.19). For he remaining hree series, he values are pracically zero (.3 for real risk-free rae;.1 for real sock reurns, and.4 for equiy premium). Imposing ineger orders of inegraion, for he firs four variables, we use d = while for he price-dividend raio we ry boh d = and 1. Wih respec o he shor-run componens we use ARMA(p, q) models, wih p, q 3, and choose he bes model specificaion using boh LR ess and likelihood crieria (AIC, BIC). We see ha, for mos of he series, he shor-run srucure can be described by simple MA models, he only excepion being he real risk-free rae where an AR(1) process is imposed.

25 I H S Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke 19 TABLE 4 Models / Series Inflaion rae Fracional and cyclical differencing (FCD) (1 L).17 y (1 cos w L + L ) Seleced models for each ime series =.18 + x ; (.6) 7.14 x = ε Fracional differencing (FD) y (1 L) =.17 + x ; (.9).19 x = ε Ineger differencing (ID) x y = ε =. + x ; (.9) +.396ε 1 Real risk free rae (1 L).5 y = x ; (.15) (1 cos w L + L ) 6.1 x = ε y u (1 L) r =.1 + x ; (.11).3 x =.35u = u 1 + ε x y =.16 + x ; (.7) =.381u 1 + ε Real sock reurns (1 L) y =.97 + x ; (.56).5. 5 (1 cos w5l + L ) x = y u (1 L) r = x ; (.19).1 x = u =.1 u 1 + ε y =.97 + ε ; (.16) Equiy premium (1 L) y =.58 + x ; (.4).6. 3 (1 cos w6 L + L ) x = y = x ; (.3) (1 L). 4 x = ε y =.574+ x; x = ε + (.11).176ε 1. 39ε y = x ; (1 L) y = x ; Price (6.679) (6.13) (.18) Dividend x = ε + (1 L) (1 cos w6 L + L ) x = ε (1 L) x = ε raio.78ε 1. 34ε y = x ; Sandard errors are in parenhesis. Nex, we compare he various models in erms of heir forecasing performance. Sandard measures of forecas accuracy are he following: Theil s U, he mean absolue percenage error (MAPE), he mean-squared error (MSE), he roo-mean-squared error (RMSE), he roo-mean-percenage-squared error (RMPSE) and mean absolue deviaion (MAD) (Wi and Wi, 199). Le y be he acual value in period ; f he forecas value in period, and n he number of periods used in he calculaion. Then: a) Theil s U: ( y f f ) ( x x 1 ) ;

26 Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke I H S b) Mean absolue percenage error (MAPE): ( x f ) n / x ; c) Mean squared error (MSE): ( x f ) n ; d) Roo-mean-percenage-squared error (RMSP): ( x f ) n / f ; e) Roo-mean-squared error (RMSE): ( x f ) ; n x f f) Mean absolue deviaion (MAD):. n The firs ype of evaluaion crieria measures he spread or dispersion of he forecas value from is mean. The MAD belongs o his caegory. I measures he magniude of he forecas errors. Is principal advanages are ease of inerpreaion and he fac ha each error erm is assigned he same weigh. However, by using he absolue value of he error erm, i ignores he imporance of over or underesimaion. The second ype of accuracy measure is based on he forecas error, which is he difference beween he observaion, x, and he forecas, f. This caegory includes MSE, RMSE and RMSPE. MSE is simply he average of squared errors for all forecass. I is suiable when more weigh is o be given o big errors, bu i has he drawback of being overly sensiive o a single large error. Furher, jus like MAD, i is no informaive abou wheher a model is overor under-esimaing compared o he rue values. RMSE is he square roo of MSE and is used o preserve unis. RMSPE differs from RMSE in ha i evaluaes he magniude of he error by comparing i wih he average size of he variable of ineres. The main limiaion of all hese saisics is ha hey are absolue measures relaed o a specific series, and hence do no allow comparisons across differen ime series and for differen ime inervals. By conras, his is possible using he hird ype of accuracy measure, such as MAPE, which is based on he relaive or percenage error. This is paricularly useful when he unis of measuremen of x are relaively large. However, MAPE also fails o ake over or under esimaion ino consideraion. Unlike he measures menioned above, Theil s U is a relaive measure, allowing comparisons wih he naïve (x = x -1 ) or random walk model, where a U = 1 indicaes ha he naïve mehod is as good as he forecasing echnique, whils U < 1 means ha he chosen forecasing mehod ouperforms he naïve model. The smaller he U-saisic, he beer he performance of he forecasing echnique relaive o he naïve alernaive. Despie some

27 I H S Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke 1 aracive properies, he U-saisic has he disadvanage of no being as easily inerpreable as MAPE; furher, i does no have an upper bound, and herefore is no robus o large values. The hree seleced ime series models (fracional and cyclical differencing, FCD; fracional differencing, FD; and ineger differencing, ID) for each of he series were used o generae he 5-year-ahead ou-of-sample forecass. Each forecas value was calculaed and compared wih he acual value of he series. Then, he above six crieria were used o rank he hree forecasing models for each series. The ranking in erms of forecasing performance is given in Table 5. We observe ha for inflaion and real risk-free rae he FCD model ouperforms FD and ID for all he crieria. For real sock reurns and equiy premium, he ID specificaion seems o be he mos adequae, while for he price/dividend raio he resuls are mixed. Therefore, on he basis of he MAPE, MSE, RMSP and RMSE crieria, he fracional and cyclical (FCD) model emerges as he bes specificaion, while he oher wo crieria, MAD and Theil s U, sugges ha he simple fracional model (wih d =.73) is he mos adequae one. TABLE 5 Overall ranking of forecasing performance using differen crieria Series Model Theil s U MAPE MSE RMSD RMSE MAD FCD Inflaion rae FD 1 3 ID Real risk free rae Real sock reurn Equiy premium Price Dividend raio FCD FD ID 3 FCD FD 3 ID FCD FD 1 1 ID FCD FD 1 1 ID FCD means Fracional and Cyclical Differeniaion; FD is Fracional Differeniaion and ID Ineger Differeniaion.

28 Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke I H S 6. Conclusions In his paper we have examined he ime series behaviour of he US sock marke for he ime period by means of new saisical echniques based on long memory processes. Specifically, we have used a procedure due o Robinson (1994a) ha has enabled us o es for uni and fracional roos no only a zero bu also a he cyclical frequencies. These ess have sandard null and local limi disribuions and can easily be applied o raw ime series. 1 We iniially focused exclusively on he long run or zero frequency, performing a suiable version of Robinson s (1994a) parameric ess along wih a semiparameric esimaion procedure. We used hese mehods because of he disinguishing feaures ha make hem paricularly relevan in he conex of financial ime series. Specifically, hey do no require Gaussianiy (which is an assumpion ha is no saisfied in mos financial daa), bu only a momen condiion of order. Addiionally, hey have sandard null limi disribuions, which is anoher advanage of hese ess compared o oher procedures based on AR alernaives. The order of inegraion esimaed using hese mehods varies considerably, bu nonsaionariy is found only in he case of he price/dividend raio. However, he non-rejecion values obained a he zero frequency could be parly due o he fac ha aenion has no been paid o oher possible (cyclical) frequencies of he process. Thus, we adoped a mehod suiable for simulaneously esing for he presence of roos a zero and he cyclical frequencies, as in Robinson (1994a). For he laer frequencies, he model is based on Gegenbauer processes. The resuls sugges ha he periodiciy of he series ranges beween 5 and 1 years, which is consisen wih mos of he empirical lieraure on cycles finding a periodiciy of abou six years (see, e.g., Baxer and King, 1999, Canova, 1998, and King and Rebelo, 1999). Furher, he series can be grouped ino hree differen caegories: inflaion and real risk-free raes, wih he order of inegraion a he zero frequency flucuaing beween and.5 and d (cyclical inegraion) beween and.3; real sock reurns and equiy premium, wih boh orders of inegraion flucuaing around ; and finally, he price/dividend raio, wih d 1 ranging beween.5 and 1 and d beween and.5. Thus, we found evidence of saionary long memory wih respec o boh componens for inflaion and real risk-free raes; I() saionariy for sock reurns and equiy premium; and nonsaionary long memory a he zero frequency bu saionary a he cyclical componen for he price/dividend raio. Finally, he fac ha all orders of inegraion are smaller han 1 suggess ha mean reversion akes place wih respec o boh componens for all series, hough he rae of adjusmen varies across series. An argumen ha could be employed agains his ype of models for he cyclical componen is ha, unlike seasonal cycles, business cycles are ypically weak and irregular and are spread evenly over a range of frequencies raher han peaking a a specific value. A srong 1 A diskee conaining he FORTRAN codes for he programs is available from he auhors upon reques.

29 I H S Caporale, Gil-Alana / Long-run and Cyclical Dynamics in he US Sock Marke 3 counerargumen is ha, in spie of he fixed frequencies used in his specificaion, flexibiliy can be achieved hrough he firs differenced polynomial, he ARMA componens and he error erm. In fac, Bierens (1) uses a model of his kind (wih d = 1) o es for he presence of business cycles in he annual change of monhly unemploymen in he UK. Our analysis also yields clear-cu resuls, which are consisen wih earlier findings on he periodiciy of cycles. The seleced models for each ime series were hen compared wih oher approaches based on fracional and ineger differeniaion wih respec o he zero frequency. Six forecasing crieria were employed and he resuls showed ha he fracional cyclical model ouperforms he ohers in a number of cases. I would also be worhwhile o obain poin esimaes of he fracional differencing parameers in his conex of rends and cyclical models. For he rending componen he lieraure is vas (see, e.g., Fox and Taqqu, 1986; Dahlhaus, 1989; Sowell, 199; Tanaka, 1999, ec.). For he cyclical par, here are fewer conribuions such as Areche and Robinson () and Areche (). However, he goal of his paper is o show ha a fracional model wih he roos simulaneously occurring a he zero and he cyclical frequencies can be a credible alernaive o he convenional ARIMA (ARFIMA) specificaions. In fac, our approach leads us o some unambiguous conclusions, wih he periodiciy ranging beween 4 and 1 years and mos of he orders of inegraion wihin he inervals (,.5) and (.5, 1) depending on he series and he componen under sudy. Furher research could be carried ou in his conex. For insance, he ess of Robinson (1994a) can be exended o allow for more han one cyclical componen underlying he process. The exisence of muliple cycles in financial series has no ye been examined empirically, and migh be of ineres in he conex of various laen variaes. Furher, daily daa could also be used o examine inraday periodiciy, e.g. in he volailiy of asse reurns. As an alernaive o he cyclical fracional approach, Andersen and Bollerslev (1997) modelled periodiciy in reurns by means of deerminisic weighs. The inclusion of deerminisic componens is possible in Robinson s (1994a) se-up, and is significance can be esed by means of a join es of he deerminisic regressors and of he order of inegraion. The univariae naure of he presen sudy is also a limiaion in erms of heorising, policy-making or forecasing. Theoreical models and policy-making involve relaionships beween many variables, and forecas performance can be improved hrough he use of many variables (e.g., facor based forecass based on daa involving hundreds of ime series bea univariae forecass, as shown, e.g., in Sock and Wason, ). However, he univariae approach aken in he presen paper is useful, as i enables one o decompose he series ino a long run and a cyclical componen. Moreover, heoreical economeric models for boh long run and cyclical fracional srucures in a mulivariae framework are no ye available. In his respec, he presen sudy can be seen as a preliminary sep in he analysis of financial daa from a differen ime series perspecive. Of paricular ineres in fuure work would be a more exensive sudy of he ou-of-sample forecasing performance