LONG RUN AND CYCLICAL DYNAMICS

Transcription

1 LONG RUN AND CYCLICAL DYNAMICS IN THE US STOCK MARKET GUGLIELMO MARIA CAPORALE LUIS A. GIL-ALANA CESIFO WORKING PAPER NO. 46 CATEGORY 1: EMPIRICAL AND THEORETICAL METHODS JULY 7 An elecronic version of he paper may be downloaded from he SSRN websie: from he RePEc websie: from he CESifo websie: Twww.CESifo-group.deT

2 CESifo Working Paper No. 46 LONG RUN AND CYCLICAL DYNAMICS IN THE US STOCK MARKET Absrac This paper examines he long-run dynamics and he cyclical srucure of various series relaed o he US sock marke using fracional inegraion. We implemen a procedure which enables one o consider uni roos wih possibly fracional orders of inegraion boh a he zero (longrun) and he cyclical frequencies. We examine he following series: 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 or zero 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, he series appear o be saionary bu o exhibi long memory wih respec o boh componens in almos all cases. The excepion is he price/dividend raio, whose order of inegraion is higher han.5 bu smaller han 1 for he long-run frequency, and is beween and.5 for he cyclical componen. Also, mean reversion occurs in all cases. Finally, we use six differen crieria o compare he forecasing performance of he fracional (a boh zero and cyclical frequencies) models wih ohers based on fracional and ineger differeniaion only a he zero frequency. The resuls show ha he former ouperforms he ohers in a number of cases. JEL Code: C, G1, G14. Keywords: sock marke, fracional cycles, long memory, Gegenbauer processes. Guglielmo Maria Caporale Cenre for Empirical Finance Brunel Universiy, Uxbridge Middlesex UB8 3PH Unied Kingdom Guglielmo-Maria.Caporale@brunel.ac.uk Luis A. Gil-Alana Universiy of Navarra Faculy of Economics 318 Pamplona Spain alana@unav.es June 7 Commens of wo anonymous referees are graefully acknowledged. We are also graeful o Torben Andersen and o paricipans in he 4 European Meeing of he Economeric Sociey, Madrid, Spain, 19-4 Augus 4, and he V Encuenro Inernacional de Finanzas, Saniago, Chile, 19-1 January 5, for useful commens and suggesions. The second named auhor also acknowledges financial suppor from he Miniserio de Ciencia y Tecnologia (SEJ5-7657, Spain).

3 The Efficien Marke Hypohesis (EMH) in is weak form saes ha i is no possible o rade profiably on he basis of hisorical sock marke prices and/or reurn informaion (see Fama, 197). This proposiion has been esed in numerous empirical sudies by rying o esablish wheher sock prices are I(1) and consequenly sock marke reurns I() series. This is based on he idea ha if sock prices fully reflec available informaion hey should follow a random walk process, which implies unpredicable reurns, and rules ou sysemaic profis over and above ransacion coss and risk premia. Therefore, a finding of mean reversion in reurns is seen as inconsisen wih equilibrium asse pricing models (see he survey by Forbes, 1996). Noe, however, ha if risk facors change sysemaically over he business cycle, expeced reurns should also be ime-varying. Similarly, allowing for business cycle variaion and shorrange dependence migh also resul in rejecing long memory in sock prices (see Lo, 1991). In general, as sressed in Caporale and Gil-Alana (), he uni roo ess normally employed impose oo resricive assumpions on he behaviour of he series of ineres, in addiion o having low power. Tha sudy suggess insead using ess which allow for fracional alernaives, and finds 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 (a he long run or zero frequency) 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, from a pure ime series viewpoin, i has been argued ha cycles should be modelled as an addiional componen o he rend and he seasonal srucure of he series 1

4 (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). 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, which enables one o es simulaneously for roos wih possibly fracional orders of inegraion a boh zero and he cyclical frequencies. This approach, due o Robinson (1994), has several disinguishing feaures compared wih oher mehods, he mos noiceable one being is sandard null and local limi disribuions. 1 Moreover, i does no require Gaussianiy (a condiion rarely saisfied in financial ime series), a momen condiion only of order wo being sufficien. Also, modelling simulaneously he zero and he cyclical frequencies 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 ess used for he empirical analysis. Secion 3 discusses an applicaion

5 o annual daa on he US sock marke. 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. 1. The saisical model Le us suppose ha {y, = 1,,, n} is he ime series we observe, which is generaed by he model: d d ( 1 L) 1 (1 cos w L + L ) y = u, = 1,,.., (1) where L is he lag operaor (Ly = y -1 ), w is a given real number, u is I() 3 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, because of he srong associaion beween observaions widely separaed in ime. 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 mean-revering, 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-mean-reversion. I is herefore crucial o examine if d 1 is smaller han or equal o or higher han 1. 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 some ime 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 varianceraio ess and found evidence of mean reversion in sock reurns. On he conrary, Lo (1991) used a generalised form of rescaled range (R/S) saisic and found no evidence agains he random walk hypohesis for he sock indices. Oher papers examining he persisence of 3

6 shocks in financial ime series are Lee and Robinson (1996), Fiorenini and Senana (1998) and May (1999). 4 Le us now consider he case of d 1 = and d >. The process is hen said o exhibi long memory a he cyclical frequency. This model was inroduced by Andel (1986) and has been sudied, among ohers, 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. 5 They also showed ha he second polynomial in (1) can be expressed in erms of he Gegenbauer polynomial C j, d, such ha, defining µ = cos w, (1 L L d µ + ) C j,d j= = µ ( )L j, () for all d, where C j,d ( µ ) = [ j/ ] ( 1) k (d ) ( ) j k j k µ Γ(d j) ; (d ) + j =, k = k!(j k)! Γ(d) where Γ(x) represens he Gamma funcion. For a formal reamen of Gegenbauer polynomials, see, for example, Szego (1975). Lildhold () shows ha his model can resul from crosssecional aggregaion of cerain AR() processes, while Bierens (1) concludes ha US real GDP can be well characerised as a model of his form wih d = 1. These 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). Modelling periodiciy in sock marke reurns has been sudied by Andersen and Bollerslev (1997). They found evidence of srong inraday periodiciy in reurn volailiy in foreign exchange and equiy model markes. To model his kind of phenomenon hey noed ha he lag-j auocovariance was proporional o cos(λj) d-1 as j, which has he long memory propery of non-summabiliy. However, hese auocovariances also oscillae, changing sign every π/λ lags, a propery ha is saisfied by he Gegenbauer processes described above. The 4

7 economic implicaions in () are similar o he previous case of long memory a he zero frequency. Thus, if d < 1 and µ < 1, or if d <.5 and µ = 1, shocks affecing he cyclical par will be mean revering (see Gray e al., 1989; Smallwood and Norrbin, 6), while d 1 (wih µ < 1) implies an infinie degree of persisence of he shocks. This ype of model for he cyclical componen has no been much used for financial ime series, (some recen examples are he papers of Bisaglia e al., 3, and Smallwood and Norrbin, 6), hough Robinson (1, pp. 1-13) suggess is adopion in he conex of complicaed auocovariance srucures. Finally, noe ha he model in (1) (wih w ) encompasses many specificaions ha have been used in financial ime series including ARMA, ARIMA and long memory fracional models. Noe ha he auocovariances no only decay a a hyperbolic rae ypical of ARFIMA models, bu also exhibi periodic behaviour associaed wih he cosine funcion. This is an imporan feaure of he presen model, since unlike fracional or ARIMA models, i can capure srong cyclical characerisics ha have been observed in he auocorrelaion funcions of economic and financial daa.. The esing procedure Robinson (1994) adops he following model: y = β' z + x = 1,,..., (3) where y is he observed 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,) T ); β is a (kx1) vecor of unknown parameers; and he regression errors x are such ha: ρ ( L ; θ ) x = u = 1,,..., (4) 5

8 where ρ is a given funcion which depends on L, and he (px1) parameer vecor θ, adoping he form: ρ (L ; θ) = (1 p 1 d L) (1 L s ) d s s 1 + θ1 d (1 cos w L + L j j + θ + θ ) j=, (5) for real given numbers d 1, d s, d, d p-1, ineger p, and where u is I(), and hus i can be specified as whie noise or any ype of weak auocorrelaed srucure. 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: H o : θ = (6) (5) becomes: ρ (L ; θ = ) = ρ (L) = (1 L) d1 s d s (1 L ) p 1 d j (1 cos w L + L ) j=. (7) This is a very general specificaion ha makes i possible o consider differen models under he null. 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: d ( L ; ) (1 L) 1 + θ1 (1 cos w L L d ) + θ ρ θ = +, (8) and, similarly, (7) becomes: d ( L) (1 L) 1 d ρ = (1 cos w L+ L ). (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. The funcional form of he es saisic, (denoed by Rˆ ) is described in Appendix 1. Based on H o (6), Robinson (1994) esablished ha, under cerain regulariy condiions: 6 Rˆ d χ, as n, (1) 6

9 where n is he sample size and d means convergence in disribuion. Thus, as shown by Robinson (1994), unlike in oher procedures, we are in a classical large-sample esing siuaion, and furhermore he ess are efficien in he Piman sense agains local deparures from he null. 7 Because Rˆ involves a raio of quadraic forms, is exac null disribuion could have been calculaed under Gaussianiy via Imhof s algorihm. However, a simple es is approximaely valid under much wider disribuional assumpions: a es of (6) will rejec H o agains he alernaive H a : θ if Rˆ >, α χ, where Prob ( χ > χ, α ) = α. A similar version of Robinson s (1994) 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 ouperforms he laer in a number of cases. Oher versions of his ess have been 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 esing simulaneously for he roos a zero and he cyclical frequencies, a saisical approach which is shown in he presen paper o represen a convenien alernaive o he more convenional ARIMA (ARFIMA) specificaions used for he parameric modelling of many ime series. 3. An empirical applicaion for he US sock marke Our daase includes annual series for 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). 8 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 7

10 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. 8

11 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 reurns Correlogram * Periodogram,6,4,4,3,,3, -,3 -,,1 -,6 -, Equiy premium Correlogram * Periodogram,6,4,4,3,,3, -,3 -,,1 -,6 -,4 1 T/ Price / Dividend raio Correlogram * Periodogram 4 3,8,5, ,1 -, * The bold lines in he correlograms refers o he Barle 95% confidence bands 1 T/ 9

12 FIGURE Firs differenced ime series, wih heir corresponding correlograms and periodograms Inflaion rae Correlogram * Periodogram,4,4,1, -, -,4, -,,8,6,4, -,6 -, T/ Real risk-free rae Correlogram * Periodogram,6,4,1,4, -,, -,,8,6,4, -,4 -, T/ Real sock reurns Correlogram * Periodogram 1,,4,6,8,4,,4 -,4 -, -,4, -,8 -, T/ Equiy premium Correlogram * Periodogram,8,4,8,4 -,4, -, -,4,6,4, -,8 -,6 1 T/ Price / Dividend raio Correlogram * Periodogram 1 8 4,4, , , * The bold lines in he correlograms refers o he Barle 95% confidence bands T/ 1

13 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. As a firs sep, we focus exclusively on he long-run frequency and implemen a simple version of Robinson s (1994) es, which is based on a model given by (3) and (4), wih z = (1,) T, 1, (,) T oherwise, and ρ(l; θ) = (1 L) d+θ. Thus, under H o (6), we es he model: y = β + β1 + x, = 1,,... (11) ( 1 L) d x = u, = 1,,..., (1) for values d =,, (.1),,, ha is, we es from d = o d = wih.1 incremens, and use differen ypes of disurbances. In such a case, he es saisic grealy simplifies, aking he form given by Rˆ in Appendix 1, wih ψ(λ s ) being exclusively defined by ψ 1 (λ s ) and û = (1 L) d y β ˆ' w. The null limi disribuion will hen be a χ 1 disribuion. However, if ρ(l; θ) = (1 L) d+θ, hen p = 1, and herefore we can consider one-sided ess based on rˆ = Rˆ, wih a sandard N(,1) disribuion. Noe ha esing he null hypohesis wih d = 1 means ha his becomes a classical uni-roo model of he same form as hose proposed by Dickey and Fuller (1979) and ohers. However, insead of using auoregressive (AR) srucures of he form: (1 (1+θ)L)x = u, we use fracional alernaives. Moreover, he use of AR alernaives resuls in a dramaic change in he asympoic behaviour of he ess: 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 (1), he behaviour of x is smooh across d, his being he inuiive reason for is sandard asympoic behaviour. 9 11

14 Table 1 displays he es resuls. Noe ha Robinson s (1994) parameric approach does no require preliminary differencing; hus, i allows us o es any real value d, encompassing boh saionary and nonsaionary hypoheses. The numbers in parenheses are he esimaes of d obained wih he While funcion. We also repor he 95% confidence bands for he nonrejecions of d. We examine separaely he cases of β = β 1 = a priori (i.e., wih no regressors in he undifferenced model (11)); 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. For example, suppose ha u in (1) is whie noise. Then, esing H o (6) in (11) and (1) wih d = 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. 1 The resuls differ subsanially from one series o anoher. For insance, for inflaion and he real risk-free rae 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 for which H o (6) canno be rejeced oscillae widely around, ranging beween.18 (equiy premium wih a linear rend) and.14 (sock reurns wih no regressors). Finally, for he price/dividend raio all he non-rejecion values are higher han.5, implying nonsaionariy wih respec o he zero frequency. 1

15 TABLE 1 Confidence inervals of he non-rejecion values of d using Rˆ in Appendix 1 wih ρ(l; θ) = (1 L) d+θ and whie noise u Time Series No regressors An inercep A linear rend INFLATION RATE [.1 (.5).45] [.13 (.5).46] [.7 (.).44] R. RISK-FREE RATE [.19 (.31).49] [.17 (.3).47] [.15 (.9).47] R. STOCK RETURN [-.9 (.).14] [-.1 (.).13] [-.1 (.).13] EQUITY PREMIUM [-.1 (-.4).1] [-.14 (-.4).1] [-.18 (-.7).8] PRICE / DIVIDEND [.7 (.83) 1.] [.58 (.73).9] [.59 (.73).9] We es he null hypohesis: d = d o in he model (1-L) d x = ε. In parenheses, he While esimaes for d. TABLE Confidence inervals of he non-rejecion values of d using Rˆ in Appendix 1 wih ρ(l; θ) = (1 L) d+θ and AR(1) u Time Series No regressors An inercep A linear rend INFLATION RATE [-.13 (-.7).19] [-.18 (-.8).] [-.44 (-.18).11] R. RISK-FREE RATE [-.11 (.4).33] [-.8 (.4).8] [-.14 (-.6).7] R. STOCK RETURN [-.17 (-.4).] [-.5 (-.4).18] [-.6 (-.5).18] EQUITY PREMIUM [-. (-.11).] [-.3 (-.1).] [-.41 (-.19) -.4] PRICE / DIVIDEND [.4 (.7).83] [.15 (.55).58] [.13 (.48).6] We es he null hypohesis: d = d o in he model (1-L) d x = u ; u = τu -1 + ε. 13

16 The significance of he resuls in Table 1 may be parly 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 Table wih hose of Table 1, one can see ha he orders of inegraion are smaller by abou. when auocorrelaion is allowed for. This migh reflec 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. 11 We also examined d, independenly of he way of modelling he I() disurbances, a he same zero frequency. For his purpose, we used wo semiparameric mehods: an approximae local While approach (Robinson, 1995), and an exac local While esimaor recenly proposed by Phillips and Shimosu (5). In he wo cases he conclusions were very similar: for inflaion and he real risk-free rae: some esimaes are wihin he I() inerval, especially if he bandwidh parameer is small; however, for mos of values of ha parameer, hey are no. For real sock reurns and he equiy premium almos all values are wihin he I() confidence inervals, bu no so for he price/dividend raio. 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. Therefore, 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 he equiy premium, some evidence of long memory for inflaion and he real risk-free rae, and srong evidence of fracional inegraion for he price/dividend raio. Of course, saionariy of sock reurns and 14

17 equiy premium is no a surprising resul, as he absence of long memory in hese wo series is a well-esablished fac in he lieraure (Lo, 1991; Cheung and Lai, 1995, ec.) The above approach o invesigaing he long-run behaviour of ime series consiss in esing a parameric model for he series and esimaing wo semiparameric ones, 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 issue 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 (1994) ha enables us simulaneously o consider roos a zero and he cyclical frequencies. 1 Before discussing he es resuls we describe a small Mone Carlo experimen we have carried ou o examine he power properies of he procedure employed below. We suppose ha he rue model is given by equaion (1) wih d 1 =.7; d =.1 and w = w r = π/6, implying long memory and nonsaionariy a he long run frequency, saionary long memory behaviour of he cyclical componen, and cycles wih a periodiciy of abou 6 periods. We also assume ha u is whie noise, hough similar conclusions were obained under weak auocorrelaion for he error erm. We perform he procedure described in Secion 3, esing he null hypohesis for d 1o - values equal o,.1,,, and d o = -.5, -.4,, 1.5, and r = 6, for sample sizes T = 1, 4, 36, 48 and 96 observaions. We generaed Gaussian series using he rouines GASDEV and RAN3 of Press, Flannery, Teukolsky and Veerling (1986), wih 1, replicaions in each case. 15

18 Table 3 repors he rejecion frequencies of he esing procedure a he 5% significance level. Thus, he values corresponding o d 1o =.7 and d o =.1 indicae he size of he es. TABLE 3 Rejecion frequencies of Robinson s (1994) procedure described in Secion 3 d 1 d T = 1 T = 4 T = 36 T = 48 T = One can see ha if he sample size is small (e.g. T = 1) he size of he es is slighly above is nominal value, hough i approximaes he 5% level wih T. Looking a local 16

19 deparures from he null (e.g., d 1 =.6 & d =.1, and d 1 =.8 & d =.1), one finds ha he rejecion frequencies wih T = 1 are. and.7 respecively. For T = 4 he corresponding values are.39 and.334, and, for T = 48 or 96, hey are higher han.9 in all cases. For he remaining deparures from he null, he rejecion probabiliies are higher han.9 in pracically all cases, even for small sample sizes. Similar conclusions were reached wih oher values of d 1 and d. 13 The procedure is hen applied o he five series under examinaion. For his purpose, le us consider now he model given by (3) and (4), wih ρ(l; θ) as in (8) and z = (1,) T. Thus, under H o (6), he model becomes: y = β + β1 + x, = 1,,... (13) d d ( 1 L) 1 (1 cos w L + L ) x = u, = 1,,..., (14) 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. 17

20 TABLE 4 Tesing H o (6) in (14), (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 * * * * * * , * * * * * * * * * , * * * * * * * * The non-rejecion values of he null hypohesis a he 5% significance level are in bold and wih an aserisk. 18

21 We firs compued he saisic Rˆ given in Appendix 1 for values of d 1 and d = -.5,, (.1),,, and r =,, (1),, n/, 14 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 4 and 9 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 4 he non-rejecion cases a he 5% level only for he case of an inercep and r = 6. The resuls for he case of a linear ime rend were very similar, and he coefficien corresponding o he 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 he real risk-free 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 he equiy premium, in some cases even being 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

22 FIGURE 3 Non-rejecion values of d 1 and d in (14), (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 RETURNS EQUITY PREMIUM,5,5,5,5 d d -,5 -,5 -,5 -,5 -,5 -,5,5,5 -,5 -,5,5,5 d1 d1 PRICE / DIVIDEND RATIO,75,5 d,5,5,5,75 1 1,5 d1 d 1 represens he order of inegraion a he zero frequency while d is he cyclical one.

23 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 3 displays he regions of (d 1, d ) values where H o canno be rejeced a he 5% level. I shows ha he combinaion of non-rejecion (d 1o, d o )-values form clusers, hough here are also some values away from he clusers in four of he five series examined. These values of he saisics are in fac close o he criical values of he χ - disribuion. Essenially, he series can be grouped ino hree caegories: inflaion and he real risk-free rae; real sock reurns and he equiy premium; finally, he price/dividend raio. Saring wih he firs group (inflaion and he real risk-free rae), we observe ha he values of d 1 range beween.1 and.5 while d seems o lie beween and.3. Thus, here appears o be 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. This procedure was also applied 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. In he AR(1) case, he AR parameer was no significanly differen from zero for mos series. The only excepion was he price/dividend raio, for which values of d 1 close o zero are obained for an AR parameer close o one, suggesing once more ha he order of inegraion a he zero frequency and he AR parameer are in compeiion. When using AR() disurbances he resuls were again very similar, hough wih larger regions for he (d 1, d )- non-rejecion values. 1

24 4. 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 in he model in (13) and (14), we use he following sraegy: afer compuing he values of he es saisic for d 1, d = -.5,, (.1),, and r =,, (1),.., n/, for he hree cases of no regressors, an inercep and an inercep wih a linear ime rend, we discriminae beween hese hree cases on he basis of he significance of he esimaed coefficiens in (13), and choose he values of d 1, d and r which produce he lowes saisic. Noe ha, for each r, he values of d 1 and d producing he lowes saisic should be an approximaion o he maximum likelihood esimaes since he procedure employed in he paper is based on he LM principle and uses he While funcion, which is an approximaion o he likelihood funcion. The seleced model for each ime series is repored in he second column of Table 5. We find ha, for he inflaion rae and he real riskfree rae, boh orders of inegraion are beween.1 and.3, he order of inegraion a zero being slighly higher han he cyclical one; for real sock reurns and he equiy premium, he values of he d s are close o zero, being slighly negaive for he zero frequency; finally, he price-dividend raio appears o be nonsaionary a he long-run frequency (d 1 =.68), and saionary wih d close o zero (d =.9) for he cyclical componen. Noe ha in his case all models are based on whie noise disurbances, he reason being ha, as menioned in he previous secion, he inclusion of auocorrelaed disurbances did no aler he conclusions excep for he price/dividend raio - for his series he associaed AR coefficien was very close o one, hus making he esimae of d 1 invalid. Moreover, he cyclical fracional polynomial can be considered as an alernaive o he ARMA specificaion when describing he shor-run dynamics of he series.

25 TABLE 5 Seleced models for each ime series Models / Series Fracional and cyclical differencing (FCD) Fracional differencing (FD) Ineger differencing (ID) Inflaion rae (1 L).17 y =.18 + x ; (.6) (1 cos w L + L ) 7.14 x = ε y =.17 + x ; (.9).19 (1 L) x = u u =.1u 1.11u +ε y =. + x ; (.9) x =.38 x 1.6 x + ε +.91ε 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).4 x =.35u 1 = u + ε y =.16 + x ; (.7) x =.381x 1 + ε Real sock reurns (1 L) y =.97 + x ; (.56).5. 5 (1 cos w5l + L ) x = ε 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 = ε y = x ; (.3) (1 L). 4 x = ε y =.574+ x; x = ε + (.11).176ε 1. 39ε Price Dividend raio (1 L).68 y = x ; (6.679) (1 cos w L + L ) 6.9 x = ε y = x ; (6.13) (1 L).73 x = ε (1 L) y = x ; (.18) x = ε +.78ε 1. 34ε Sandard errors are in parenheses. The hird column of he able repors he seleced models aking ino accoun only 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 3

26 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 (.4 for he real risk-free rae;.1 for real sock reurns, and.4 for he equiy premium). Here we have followed he same sraegy as in he fracional cyclical case, i.e., esing sequenially for a grid of values of d 1, and hen choosing he value ha produces he lowes saisic in absolue value. 16,17 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. For he shor-run componens we use ARMA(p, q) models, wih p, q 3, and choose he bes specificaion using boh LR ess and likelihood crieria (AIC, BIC). We see ha, for mos of he series, he shorrun srucure can be described by simple MA models, he only excepions being he real riskfree rae where an AR(1) process is imposed, and he inflaion rae (ARMA(,1)). 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). These measures are described in Appendix. 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 following 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 6, and is based on he average value of he forecass for each crierion. We observe ha for inflaion and he real risk-free rae he FCD model ouperforms FD and ID according o all he crieria. For real sock reurns and he equiy premium, he ID 4

27 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, RMPSE 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 6 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 sands for Fracional and Cyclical Differeniaion, FD for Fracional Differeniaion, and ID for Ineger Differeniaion. Five ou-of-sample observaions were considered in each case and he ranking was compued on he basis of he average value of he forecass for each crierion. 5

28 In Table 7 we focus on he forecass for inflaion and he price/dividend raio over a longer ime-horizon. The reason for focusing on hese wo series is ha hey are he wo ha clearly exhibi non-zero (and fracional) degrees of inegraion. We consider he forecasing performance of he hree ypes of models discussed above (FCD, FD and ID) over he period , based on specifying and esimaing he models over he ime period The new seleced models are displayed in Table 7 and we observe ha hey are very similar o hose presened in Table 5. TABLE 7 Seleced models for Inflaion and Price/Dividend raio ( ) FCD FD ID Inflaion Price/ Dividend raio (1 L) (1 L).14 y.66 =.18 + x ; (.6) (1 cos w L + L ) y 7 = x ; (6.7) (1 cos w L + L ) x x = ε = ε y =.16 + x ; (.11).4 (1 L) x = u u =.14u 1.1u + ε y = x ; (6.55).81 (1 L) x = u u =.1u 1.33u + ε y =.18 + x ; (.9) x =.36 x 1. 7 x + ε (1 L) y =.151+ x ; (.18) x = ε ε 1 6

29 TABLE 8 MSE forecass for inflaion and price/dividend raio a) inflaion 1 period 3 period 6 period 9 period 1 period 15 period FCD * * 1.771* FD 1.165* ID a) price/dividend raio 1 period 3 period 6 period 9 period 1 period 15 period FCD * * * 3.935* FD ID Table 8 repors he MSE forecass for he wo series, using he ime horizons h = 1, 3, 6, 9, 1 and 15. We observe ha for he wo series in many cases he lowes MSEs are obained wih he fracional cyclical models. However, he MSE measure used for comparing he relaive forecasing performance of our models is a purely descripive device. There exis several saisical ess for comparing differen forecasing models. One of hese ess, widely employed in he ime series lieraure, is he asympoic es for a zero expeced loss differenial of Diebold and Mariano (1995). 18 However, Harvey, Leybourne and Newbold (1997) noe ha he Diebold-Mariano es saisic could be seriously over-sized as he predicion horizon increases, and herefore provide a modified Diebold-Mariano es saisic given by: M DM = DM n + 1 h + n h (h 1) / n, where DM is he original Diebold-Mariano saisic, h is he predicion horizon and n is he ime span for he predicions. Harvey e al. (1997) and Clark and McCracken (1) show ha his modified es saisic performs beer han he DM es saisic (hough sill poorly in finie 7

30 samples), and also ha he power of he es is improved when p-values are compued wih a Suden disribuion. Using he M-DM es saisic, we furher evaluae he relaive forecas performance of he differen models by making pairwise comparisons. In Table 8 we indicae wih an aserisk, for each predicion-horizon, he rejecions of he null hypohesis ha he forecas performance of model i and j is equal in favour of he one-sided alernaive ha model i s performance is superior a he 5% significance level. 19 Given he fac ha we have hree poenial models for each predicion and we make pairwise comparisons, only he preferred model - when here is consisency for all hree specificaions - is indicaed wih an aserisk, cases no being chracerised by consisency being lef ou. We noe here ha over long horizons he fracional cyclical model produces for boh series significanly superior forecass. Similar resuls were obained when using oher ses of forecass based on rolling window saisics. 5. Conclusions In his paper we have examined he ime series behaviour of five series relaed o he US sock marke by means of saisical echniques based on long memory processes. Specifically, we have used a procedure ha has enabled us o es for uni roos wih ineger or fracional orders of inegraion, 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. Iniially, we focused only on he long-run or zero frequency, applying a suiable version of Robinson s (1994) parameric ess along wih various semiparameric esimaion procedures. 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 by mos financial series), bu only a momen condiion of order wo. Addiionally, hey have sandard null limi disribuions, which 8

31 is anoher advanage of hese ess compared o oher procedures based on AR alernaives. The order of inegraion esimaed using hese mehods varies considerably from one series o anoher, 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 he zero and he cyclical frequencies. 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 he real risk-free rae, wih he order of inegraion a he zero frequency flucuaing beween and.5 and d (cyclical inegraion) beween and.3; real sock reurns and he 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 he real risk-free rae; I() saionariy for sock reurns and he equiy premium; and nonsaionary long memory a he zero frequency bu saionariy 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 hem. A criicism ha could be made of his ype of model 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 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, 9

32 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 a 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. Clearly, for he sample period examined in his sudy, srucural breaks could also be an issue. Noe ha fracional inegraion and srucural break are issues which are inimaely relaed (see Bos e al., 1999; Diebold and Inoue, 1; Granger and Hyung, 4; Gil-Alana, 7). However, a heoreical framework for srucural breaks and fracional inegraion a boh he zero and he cyclical frequencies has ye o be developed. 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; Robinson, 1995; Tanaka, 1999; Phillips and Shimosu, 5; Mayoral, 7 ec.). For he cyclical par, here are fewer conribuions such as Areche and Robinson (), Areche () and Dalla and Hidalgo (5) and no likelihood esimaion mehods have been proposed for he join esimaion of he wo orders of inegraion. However, he goal of his paper is o show ha a model wih fracional orders of inegraion a boh he zero and he cyclical frequencies can be a credible alernaive o he convenional ARIMA (ARFIMA) specificaions. In fac, our approach produces unambiguous resuls, 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. 3

33 Furher research could be carried ou using his framework. For insance, he ess can be exended o allow for more han one cyclical componen. 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. Noe ha he periodograms displayed in Figures 1 and show in some cases muliple peaks a he cyclical frequencies. However, for real sock reurns and equiy premium, he esimaed order of inegraion a he zero frequency is exremely close o and he periodograms of he original daa (in Figure 1) exhibi a single clear significan peak. Moreover, for he remaining hree series he orders of inegraion a he long run frequency range beween and 1, and he periodograms of he firs differenced daa (in Figure ) clearly show ha ype of behaviour along wih a single peak a he non-zero frequency. 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 (1994) 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 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 31

34 forecasing performance of our preferred model. In order o increase he number of ou-ofsample observaions and gain power, a rolling design (e.g. McCracken, ) wih larger samples could be used. Daa mining is an addiional relevan issue worh exploring. 3

35 Appendix 1 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 π < λ 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 (9) are: û = d d (1 L) 1 (1 cos w L + L ) y β ˆ's, where 1 β n n d d = s + s ' s (1 L) (1 cos w L L ) y = 1 = 1 s = d d (1 L) 1 (1 cos w L + L ) z. 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 τˆ = arg min σ ( τ), where T * is a suiable compac subse of R q Euclidean space, and τ T * n 1 π σ ( τ) = g( ; ) 1 λs τ Iû ( λs ), n s = 1 1/ n = 1 iλ πs n wih I ( λ ) = ( πn) û e s ; λ =. û s s The es saisic, which is derived hrough he Lagrange Muliplier (LM) principle, akes he form: n Rˆ = rˆ'rˆ; rˆ = Â 1/ â, ˆ σ where n is he sample size, and π * = ψ λ λ τ 1 π n 1 â ( s ) g( s; ˆ) I( λs) ; σ = σ ˆ (ˆ) τ = g( λ τ 1 s; ˆ) I( λs ), n s n s = 1 33