A Varieties of Spurious Long Memory Process

Transcription

1 Inernaional Journal of Business and Social Science Vol. No. 3 [Special Issue - January 0] Absrac 5 A Varieies of Spurious Long Memory Process Charfeddine Lanouar Assisan Professor Universiy of Gabes Insiu Supérieur de Gesion de Gabes Deparmen Quaniaive Mehods France lanouar_charf@yahoo.fr his paper inroduces a large varieies of models which can generae a spurious long memory behavior. By using Mone Carlo simulaions have invesigaed he behavior of he, he Exac Local While and he Wavele mehods when he daa generaing process is weakly dependen wih changes in mean. he resuls show ha hese long memory esimaion mehods fail o accep he null hypohesis of shor memory. A srong size disorion have been deeced. he resuls sress he need for a es saisic which can discriminae he rue long memory behavior from he spurious behavior. In he empirical applicaions we use sock reurns of seven French companies wih he CAC 40 sock marke index. We show ha he observed long memory behavior is a rue behavior and ha presence of breaks in hese ime series do no have a maor effec on he long memory parameer. Keywords: spurious long memory, srucural change, empirical power. JEL classificaion: C3;C5,C5. Inroducion Modelling ime series using long memory processes have largely increased since he seminal papers of Granger and Joyeux (980 and Hosking (98. Acually, i is well known ha many financial ime series exhibi srong persisence behavior. his long range dependence is generally deeced in he squares or absolue values of he reurns. A he same ime, many empirical and heoreical works have shown ha several srucural changes models like he mean-plus-noise model of Chen and iao (990, he sochasic permanen break model of Engle and Smih (999, he sign model of Granger and eräsvira (999, he Markov swiching model of Hamilon (989 and he hreshold Auo-Regressive (AR model of Lim and ong (980 among ohers can generae a long memory behavior in erms of decaying auocorrelaionsand fracional value of he esimaed parameer d. Only few works have examined he link beween srucural changes models and long memory behavior. Unil now, no consensus has been reached abou his problem. Diebold and Inoue (00 argued ha regime swiching models and long memory behavior are inimaely relaed. Granger and Hyung (004 sugges ha i is difficul o discriminae beween hese wo classes of models. he exising empirical sudies in esing for long-range dependence show ha, when he daa are weakly dependen wih changes in mean, saisical ess for long-range dependence have a size disorion. hey are no able o accep he null hypohesis of shor memory wih a high power. So, i appears ha he exisence of breaks in daa ses can lead o a misspecified model selecion. his paper have hree main obecives. Firs, we propose a varieies of srucural change models ha can generae spurious long memory behavior. hen, we sudy he possibiliy of discriminaion beween spurious long memory behavior and rue behavior. o deec long memory behavior we use he following esimaion mehods: he echnique of Geweke and Poer-Hudak (983, he Exac Local While (hereafer proposed by Philips and Shimosu (005 and he wavele mehod inroduced by Lee (005. Finally, we invesigae he rue naure of he observed long memory behavior in seven absolue reurns French companies wih he CAC 40 one.he remain of he paper is organized as follows: Secion reviews he definiions of long memory models. Secion 3, presens long memory esimaion mehods. In Secion 4, we inroduces simulaneously he general form of spurious long memory models and we presen he resuls of he simulaions sudies. Secion 5 presens our empirical resuls concerning he eigh French sock reurns and Finally secion six concludes. Long memory process In his secion we presen differen alernaives definiions of long memory processes. hese definiions will be a useful ools in he analysis of spurious long memory process. Firs, we inroduce he ypical ARFIMA model proposed by Granger and Joyeux (980 and Hosking (98.

2 he Special Issue on Behavioral and Social Science Cenre for Promoing Ideas, USA hen, we presen he echnique of Geweke and Poer-Hudak (983, he Exac Local While ( mehod inroduced in 005 by Shimosu & Phillips and he wavele esimaion mehod as in Lee (005.. Long memory Definiions In ime domain, a series ( y, =,...,, has a long memory behavior if is auocorrelaion funcion decays hyperbolically o zero. In his case, he auocovariance funcion, γ k, is defined by, γ k c( k k d where d > 0 and c(k is a slowly varying funcion a infiniy. In frequency domain, long range dependence is defined in erms of raes of explosion of low-frequency specra, his means ha he specral densiy ends o infiniy for frequency near zero. he frequency domain definiion of long memory is, f ( ω as ω ends o 0 A hird possible definiion of long memory process is given by he Variance( S = O( d + where S is he parial sums of he ime series ( y, S = = ( y. he ime domain, he frequency domain and he variance of parial sums definiions of long range dependence will be a useful ools o show ha cerain simple nonlinear models can generae a long memory behavior, see Diebold and Inoue (00 for insance.. he ARFIMA model he firs long memory process inroduced in he lieraure is he popular Auo-Regressive Fracionally Inegraed Moving Average (ARFIMA model developed independenly by Granger and Joyeux (980 and Hosking (98. A process { y, wih N follows a saionary ARFIMA(p,d,q process if i akes he following form, } d Φ( B( B ( y µ = Θ ( B u where Φ (. and Θ (. are polynomials of order p and q respecively, whose roos lie ouside he unie circle and µ is unknown mean. ( u is a Gaussian srong whie noise N(0, u and B is he lag operaor. If d (-/,0 he ARFIMA(p,d,q model is an inverible saionary process wih inermediae memory. Now, if d (0,/ model ( is saionary and inverible. In he laer case, he auocorrelaion funcion ρ( k exhibis a slow decay when he lag k increases, see Beran (994 for insance. his means ha we are in presence of long memory behavior. In he following we concenrae our analysis on he model ( under he resricion p = q = 0 and 0 < d < /. In ha case he process is called a Fracionally Inegraed noise FI(d. 3 Long Memory Esimaion Mehods o es he presence of long memory inside a ime series a large varieies of esimaion mehods have been proposed in he las wo decades. Besides he mehod of Geweek Poer-Hudak (983 which is he mos widely used in applicaions, we use also he Exac Local While ( mehod of Shimosu & Phillips (005 and he wavele mehod ( recenly by Lee ( he echnique he echnique of Geweke and Poer-Hudak (983 is he mos widely used mehod in empirical applicaions. Based on his mehod he esimaed fracional long memory parameer d is obained from he following leas squares regression, { I } a d { } ln ( ω = ln 4 sin ( ω / + ( Wih, =,...,m, and ( y a frequency ω = π, where is he number of observaions. Consisency requires ha m grows wih sample size, bu a a slower rae. Diebold and Inoue (999 adap a popular rule of humb of m = p. he ordinae leas square esimaor of d is I ω is he periodogram of he process { } asympoically normal wih sandard error equal o π (6 m ( /, see Geweke and Poer-Hudak ( he Exac Local While he second semi-parameric mehod, used in his paper, is he exac local while mehod of Shimosu and Phillips (005. his mehod avoids some of he srong assumpions of he Local While (LW mehod, see Yaima (

3 Inernaional Journal of Business and Social Science Vol. No. 3 [Special Issue - January 0] Conrary o he LW mehod which is no consisen ouside he saionary region d, he mehod have a good properies when he value of d is less han 9/. In his case he esimaor is consisen and have he N(0, /4 limi disribuion for all value of d when he opimizaion covers an inerval of widh less han 9/. he esimaed value d is obained as follow: d = Arg min [, ] R( d (3 d d d where d and d are he lower and upper bounds of he admissible values of d such ha < d < d < + and, m R( d = log G( d d log( ω m (4 = m Wih G( d = I d ( ω where is he y m I = d d ( ( y = π y e d d is he periodogram of ( y = ( B y =.he esimaor saisfies he propery m( d d d N(0,/ 4, which induces a simple asympoic inference. 3.3 he Wavele mehod In recen years, he use of waveles have been exended o many areas oher han physic. In economerics, Jensen and Whicher (000 and Jensen (999 have applied waveles o esimae he fracional parameer of long memory. o sar, consider a wavele defined by ψ b a a, b = a ψ where a 0, a and b are respecively he dilaion and ranslaion parameer. ψ ( is called he moher + wavele which saisfies he condiion ψ ( d = 0. Waveles can be disinguished by heir regulariy and by heir number of vanishing momens. he number of vanishing momens of a wavele ψ ( is he larges + ineger υ which saisfies υ ψ ( d = 0, for all r = 0,..., υ. For some special values of a and b and a special choice of ψ, he wavele ψ a, b consiues an orhonormal basis for L( R. As commonly used in he lieraure, we se a =, b = k wih and k Ζ, hen ψ, ( k ψ = ( k. he dyadic orhonormal iω wavele ransform of a ime series ( y, =,..., = is defined as, ω =< y(, ψ ( >= y( ψ ( k d, (, k Z, k, k R he wavele coefficien ω, k is he deail coefficien a scale and posiion k. Our ime series ( y can be easily reconsruced from is deail coefficiens. When he scale is large, he coefficien ω, k capures low frequency or coarse scale behavior of ( y. In he oher hand, when is small, he wavele coefficien capures he high-frequency or fine scale deails of he signal ( y. he mehod proposed by Jensen is only robus for values of d lie inside (-/, /, hen when he expeced value of d is in he non-saionary range we need a more robus es. Lee (005 use a new mehod which is robus for a value of d (0,3, see also Kao and Masry (999. In his paper we use he laer mehod wih he Haar wavele and υ =. he Haar wavele ψ ( = if 0 < < and ψ ( = if < < and 0 oherwise. he specral densiy of he wavele ransform a he scale around zero frequency for d (0,3 ; ( d f ( λ = C λ g ( λ as λ 0 (8 where C = c π < is a consan erm wih g( λ g( λ = for all, as λ 0 and 0 < g(0 <.. he specral densiy f ( λ a fixed scale is given by, (5 54

4 he Special Issue on Behavioral and Social Science Cenre for Promoing Ideas, USA ( q ω ( k exp( iλ q k π k = 0 I =, q =,,..., m (9 Where λ = π q and m, m 0as. From (8 a log-linear relaionship is, q λ ln( Iq = ln( C g( ( d ln( hen we have he regression, = λ λ + for q =,..., m ln( Iq ln( C g( ( d ln( λ which can be esimaed by he ordinary leas squares yielding he d Wave esimaor consisen for d (0,3. m ( dwave d d N(0, π 4 as ( 4 Spurious Long Memory Models and Simulaions Sudies In his secion we inroduce he general form of models ha can creae spurious long memory behavior in erms of linear propery of he daa such, he auocorrelaion funcion, he specrum or using he esimaed value of he long memory parameer which lie, generally for his kinds of models, ino (0,. Consider he following model given by, y = µ + where µ is a random variable and ² is a srong whie noise N(0, σ. Assumpion : We assume ha µ follows a random walk wih drif in noise, µ = µ + g( η wih g( is a bounded funcion beween zero and one and η is a srong whie noise N (3 (0 ( (0, σ η. As shown in assumpion he occasional level shifs process µ is conrolled by wo variables, he funcion g( which describes he naure of he swiching process and he srong whie noise η which describes he size of he umps. wo rivial cases are of ineress, when g( is equal o zero he underlying process y is I(0 and when g( is equal o one he processes y is I(. In his paper we suppose ha here are a coninuous behavior of he fracional long memory parameer beween zero and one for his process. Under his assumpion he esimaed value of d is expeced o lies beween 0 and. his expeced value of d is caused by he presence of occasional shifs in he noise. Proofs. If g( = 0, and µ = µ = µ, hen y = µ + which mean ha y is I(0. Now, if g( =, y = µ + and µ =, hen y = µ + η which mean ha y = + η, y is I(. he only assumpion ha we have made abou he funcion g( is "' g( is bounded beween zero and one"'. wo large classes of g( funcions are o consider. Under he firs class he funcions g( behave smoohly which make a gradually swiching in daa se. he second class of models includes processes wih abrup changes in mean. 4. Gradually swiching process When g( is a coninuous funcion which decreases or increases smoohly beween zero and one a gradually swiching in mean occur in he series y. his gradually swiching model can be easily confused wih long memory model. Generally, when we esimae he fracional long memory parameer we obain a value of d significanly differen from zero. his value of d depends on he naure of he behavior of he funcion g(. We suppose here ha g( is a funcion of. o creae a gradually swiching process, many g( funcions can be proposed, Engel and Smih (999 proposed a smoohly ransiion funcion g( = which γ + depends on a posiive parameer γ. he naure of he behavior of he ransiion funcion depends on he parameer γ and he degree of he exponen of. 55

5 Inernaional Journal of Business and Social Science Scie A second possibiliy is o use he logisic funcion, g ( = Vol. No. 3 [Special Issue - January 0], which decreases beween 0 and. We + exp( can enriched his funcion by adding a posiive parameer γ in he denominaor. Anoher possible funcion is g ( = exp( which increases es beween 0 and. In hese differen diff cases, by varying C differen behavior, γ + exp( especially in empirical finance, can be deeced. hese ransiion funcions creae a gradually swiching process which induces a problem in n inference inferenc saisics because model (3-(A. Figure : - (A, (B and (C are respecively he raecory, he ransiion funcion and he ACF of he SOPBREAK model (3-(A wih = 000 and σ = (D, (E and (F are respecively he raecory, he ransiion funcion and he ACF of he SOPBREAK model (3-(A wih = 000 and σ = 0.5. wih g ( follows one of he ransiion funcion described above among ohers, can generae a long memory behavior. A graphical represenaions of he raecories, he ransiion funcions and he auocorrelaion funcions when we use he firs rs and he hird g ( funcions are repored in figure. o invesigae he behavior of he previous long memory esimaion mehods we conduc some experimen simulaions3. We generae 000 series of he process (3-(A (3 when g ( is he ransiion funcion proposed by Engel and Smih (999, he logisic ransiion funcion and g ( = exp(. he simulaions resuls are repored in γ + exp( able for differen eren values of he parameer γ. We use a sample size = 000 for he and he mehods and = = 048 for he wavele mehod.

6 he Special Issue on Behavioral and Social Science Cenre for Promoing Ideas, USA able : Mean esimae of he long memory parameer d when he DGP is he process (3-(A wih σ =. g( λ d g( = d γ + d d g( = d γ + exp( d d exp( g( = d γ exp( d he simulaions resuls are consisen wih he fac ha when g( = 0 he expeced value of d is close o zero and when g( = he value of d is close o one. his resul is observed in able, for example when γ ends o he process y behaves like an I(0 process. he resuls show ha all long memory esimaion mehods do no have he same behavior. he wavele mehod have he wors performance when he value of d approach zero and he bes performance when he value of d is expeced o be higher han 0.3. he mehod performs beer han he mehod. Neverheless, hese hree mehods canno accep he null hypohesis of shor memory only for large value of γ (when g( 0. he resuls repored in able confirm ha when he daa is weakly dependen wih swich in mean a spurious long memory behavior is deeced. As we have noed in previous secions he behavior creaed by he ransiion funcion g( has an imporan role in empirical applicaions. he inuiion behind his follows from he fac ha when he funcion g( varies beween he wo exremes value he expeced value of d varies, generally, beween 0 and. his means ha hese models can cap shocks varying beween ransiory o permanen shocks. 4. Abrup swiching process Now, we assume ha g( is a discree funcion which changes brually beween 0 and. Cheen and io (990 proposed he binomial disribuion as a process which governs he swich of he funcion g(, hen g( = 0 wih probabiliy (w.p p and g( = w.p p for more deails see Cheen and io (990. We can propose many ohers process for he funcion g( like he Markov swiching model developed by Hamilon (989, he hreshold Auo-Regressive model inroduced by Lim and ong (980 and he srucural change model of Quand (958 among ohers. In he case of Markov swiching model, he ransiion beween he wo regimes is governed by a Markov chain s which akes values zero and one. he probabiliies of ransiion beween he wo saes is defined by p i = (s = i s - =, hen g( = 0 when s = 0 and g( = when s =. I is also possible o allow o he process which governs g( o depend in an observed variable, generally y l,wih l is he delay parameer. For his SEAR specificaion, g( = 0 if y l r where r is he hreshold parameer and g( = if y l > r, in our simulaions we se l = and r = 0. Finally, we propose he srucural change specificaion where g( = 0 if < π and g( = oherwise, where π lies ino (0, and π is he dae of break. he process (3-(A and g( follows one of he processes described above have a same properies han long-range dependence models. his behavior can be observed in he auocorrelaion funcion which decays slowly o zero or in he esimaed long memory parameer d which is significanly differen from zero. Above we give some graphical represenaions of he raecories and he auocorrelaion funcions of models (3-(A when he funcion g( follows he binomial disribuion, he Markov swiching, he SEAR and he SCH models. Basis in he linear properies of hese models differen model can be proposed o describe his ime series. 57

7 Inernaional Journal of Business and Social Science Scie Vol. No. 3 [Special Issue - January 0] In order o invesigae he behavior of he previous long memory esimaion mehods when he da daa generaing process is (3-(A (A wih an abrup changes we conduc some empirical simulaions. We generae 000 series of he process (3-(A when g ( follows he binomial disribuion, he Markov swiching process (MSAR, he AR process and he Srucural CHange (SCH model. Figure : - (A, (C, (E and (G are he raecories ries of model (3-(A (3 wih g ( follows abrup swiching process, = 000 and σ = 0.5, γ = 5, g ( = γ + - (B, (D, (F and (H are he ACF of model (3-(A (3 wih g ( follows abrup swiching process, = 000 and σ = 0.5, γ =, g ( = γ + exp( he resuls are repored in able and 3. he wavele mehod behaves beer han he and he mehods. he esimaed value of d depends on he parameers ha governs each model, he probabiliy p for he binomial disribuion, he noise's variance ση for he AR model and he probabiliies of ransiion ( p00, p for he Markov swiching model. For he SCH model, he resuls seem do no depend on he dae of break π. In a simulaions no repored here he esimaed value of d is a posiive funcion of ση, see Charfeddine and Guégan (006.

8 he Special Issue on Behavioral and Social Science Cenre for Promoing Ideas, USA able : Mean esimae of he long memory parameer d when he DGP is he process (3-(A model wih σ =. g( p d Binomial disribuion d d g( d AR Model d d g( d SCH Model d d able 3: Mean esimae of he long memory parameer d when he DGP is he process (3-(A model wih g( follows he MS-AR model andσ =. ( p00, p (0.5, 0.95 (0.95, 0.95 (0.95, 0.99 (0.99, 0.99 (0.999, d d d From able and 3 i follows ha all esimaion mehods of long memory have a srong size disorion. he hypohesis of long memory canno be reeced wih higher power. his means ha when he daa have breaks in mean a long memory behavior can be observed. hen, by using saisical inference researchers believe ha he daa generaing process is a long memory model even if he rue DGP is models wih shifs in mean. Assumpion We assume ha µ = µ 0( h( + µ h( where h( is an indicaor funcion which akes values 0 and.now, we assume ha he daa generaing process is (3-(A. he process which governs he ransiion beween µ 0 and µ varies across models. When h( = s, wih s is an unobserved exogenous Markov chain, he process (3-(A is he Markov swiching model of Hamilon (989. In he case of hreshold Auo-Regressive model h( = 0 if y < l r and h( = if y l r, wih l is he delay parameer and r is he hreshold parameer. For he Srucural CHange model h( = 0 if he h observaion is less han h π and h( = if he observaion is higher han π wih π (0,. For hese kinds of models he separaion beween he wo regimes is more clear which means ha he changes in mean is more abrup han (3-(A.We use experimen simulaions o sudy he performance of he, he and he wavele mehods when he daa generaing process is he process (3-(A. We generae 000 series and we esimaed he mean value of he long memory parameer d. he resuls are repored in able 4 and 5. able 4: Mean esimae of he long memory parameer d when he DGP is he process (3-(A model wih h( is governed by an MS-AR model and σ =. ( p00, p (0.5, 0.95 (0.95, 0.95 (0.95, 0.99 (0.99, 0.99 (0.999, d d d

9 Inernaional Journal of Business and Social Science Vol. No. 3 [Special Issue - January 0] able 5: Mean esimae of he long memory parameer d when he DGP is he process (3-(A model wih σ =. h( is governed by σ = d AR Model d d h( is governed by π d SCH Model d d he resuls confirm ha he wavele mehod is no robus for a value of he parameer d < 0.3. When he process ( µ follows he MS-AR model, he AR model or he SCH model he performance of esimaion mehods are ambiguous. For he MS-AR model he mehod have he bes performance. In he case of AR model, he performance of long memory esimaion mehods depend on σ. For large value of noise's variance he performs beer han he mehods exceps when σ η is very small (<0.. he wavele mehod have he wors performance. When he DGP are he SCH model, he performance of he long memory esimaion mehods depend in he locaion of breaks. For a value of σ < 0.5 he have he bes performance. When σ > 0.5 he wavele performs beer han he wo ohers mehods. he resuls repored η in able 4 and 5 confirm ha he behavior generaed by a weakly dependen daa wih changes in mean can be easily confused wih a long memory behavior. In he following secion, we use absolue reurns of several French companies and he CAC 40 sock marke o invesigae he rue behavior of he deeced long memory in he absolue reurns. 5 Empirical Applicaions his secion is devoed o analyze he long range properies of absolue reurns for seven French companies wih he CAC 40 index. We use a daily daa from 0/0/994 unil 3//003 obained from Daasream Base. his daa provides us a large sample size of = 606 observaions. raecories and auocorrelaion funcions of he absolue reurns are repored respecively in figures 5 and 6. In all auocorrelaion funcions a slowly decaying are observed. For mos absolue reurns auocorrelaions remain significanly differen from zero for a lags higher han 60 ( years which means ha absolue reurns are characerized by a srong persisence behavior. able 6: Esimae values of d for he differen absolue reurns of he and mehods. m BNP 0.56 ( ( ( (7.7 LVMH 0.47 ( ( ( (6.63 PPR 0.33 ( ( ( (8.79 Sain Gobain 0.38 ( ( ( (6.0 Soc. Générale 0.44 ( ( ( (9.4 Suez 0.34 ( ( ( (8.553 F 0.6 ( ( ( (5.983 CAC ( ( ( (7.0 -sas are in parenheses. η η

10 he Special Issue on Behavioral and Social Science Cenre for Promoing Ideas, USA In he empirical applicaion we concenrae our analysis only on he and he mehods. For he mehod we use a values of m equal o and. and / and /4 for he mehod. he esimaed values of he fracional long memory parameer are repored in able 6. he resuls show ha all absolue reurns are characerized by a long range dependance behavior. he resuls show ha he mehod is sensiive o he frequency m. hese resuls are consisen wih some works ha have sressed ha a values of m higher han 0:5 are more consisen wih he hypohesis of long memory behavior. Moreover, when we use 0.75 he mehod wih m = he obained esimaors values of d are close o hose obained from he mehod. So, our empirical resuls confirm ha a larger value of he frequency m is more consisen wih he long range dependence hypohesis. Unlike he mehod, he esimaor of d is no sensiive o he frequency m. his laer mehod is more efficien han he one, for a deails reviews see Banere and Ungra (004. Figure 3: - Plo of he ime-varying long memory parameer d for he BNP, LVMH, PPR, Sain Gobain, Sociéé Générale, Suez, F and he CAC 40 absolue reurns. - he daes of breaks using Bai and Perron procedure are in verical dash lines. he quesion of he rue naure of he long memory in absolue reurns have ineresing many researchers, for insance see Lobao and Savin (998 and Granger and Hyung (004. In order o invesigae if he observed long memory is due o srucural breaks or i is a rue behavior generaed by he daa mechanisms, we propose he following sraegy. Firs, we esimae a ime varying values of d, we sar a =000 observaions and we incremen he sample size by 5 observaions ( week each ime. his sep have a wo mains obecives. Firs, i allows us o deec he daes of shifss in he long memory parameer. hen, i allows o us o compare he daes of changes on he long memory parameer wih breaks daes obained by using he sequenial procedure of Bai and Perron (998, 003. he idea behind his follows from he fac ha if he long memory in absolue reurns is due o he presence of srucural breaks hen he changes in long memory parameer should coincide wih srucural breaks daes. In figure 3 we plo he ime varying parameer d wih he daes of breaks (dash line as deeced using he sequenial procedure of Bai and Perron (998, 003. For he seven companies and he CAC 40 index he breaks daes do no coincide wih he daes of changes on he long memory parameer. his means ha for hese absolue reurns he observed long memory behavior is no due o he presence of breaks.

11 Inernaional Journal of Business and Social Science Vol. No. 3 [Special Issue - January 0] Moreover, we invesigae he possibiliy ha he long memory parameer is over esimaed because of he presence of breaks. So, we sar by filering ou he breaks and hen we re-esimae anoher one he long memory parameer. he resuls are summarized in able 7. I follows from columns -5 ha no maor difference is deeced compared wih he resuls obained from he original ime series. Only a small changes in he long memory parameer have been occurred. his means ha he presence of changes in hese ime series have a lile affecs on he esimaed value of d. his can be explained by he small size of umps. In oher word, as shown in many simulaions experimen he value of d depends on he size of umps and he daes of breaks able 7: Esimae values of d for he differen absolue reurns afer filering ou he breaks. m BNP 0.3 ( ( ( (6.883 LVMH 0.35 ( ( ( (6.336 PPR 0.09 ( ( ( (6.933 Sain Gobain 0. ( ( ( (4.83 Soc. Générale 0.8 ( ( ( (7.084 Suez 0.9 ( ( ( (6.49 F (0.43 CAC (7.00 -sas are in parenheses. 6 ( (5.890 ( (3.946 ( (6.07 Now, in order o confirm previous resuls we concenrae our analysis only in he CAC 40 sock marke reurns. We use he mehod proposed by Hsu (005 and Hsu and Kuan (000. Firs, we presen briefly he Hsu and Kuan Local while mehodology. hen, we applied i o absolue reurns and we show ha he deeced breaks has a lile impac on he esimaed long memory parameer d. o sar consider he model given by (3-(A where h( is equal o zero before he observaion 0 and afer ha. We rewrie he model as follows, µ 0 + if =,..., 0 y = (4 µ + if = 0 +,..., Hsu and Kuan (000 and Hsu (005 propose he following mehod: Firs, hey propose o esimae µ 0 and µ, τ µ 0 = y, µ = y τ = τ = τ + (5 hen, for each value of τ he residuals are y µ 0 for τ = (6 y µ for τ + And finally, o selec he opimal dae of change-poin τ, Hsu and Kuan (000 propose o ake τ = ArgMin LW ( d, τ [ τ, τ τ τ (7 τ, τ 0, and he esimaor of d is d ( τ. he auhors show by using empirical simulaions ha he where [ ] ( finie-sample disribuion of he proposed esimaor is very close o he normal one boh for he shor- and long-memory daa, see Hsu and Kuan (000. he principal advanage of his mehod lie on he oinly esimaion of he long memory parameer d and he dae of break τ. In our empirical applicaions we use he esimaor I = ( e π = iω.

12 he Special Issue on Behavioral and Social Science Cenre for Promoing Ideas, USA Figure 4: Plos of he ime-varying long memory parameer d as esimaed using Hsu and Kuan (000 mehod. We applied his mehod o he CAC 40 absolue reurns. Firs, we consider he full sample size (P and hen we consider hree subperiods, P=:000, P=750:750 and P3=500:500 observaions. We use m = / 4, τ = 0.*, τ = 0.9*. he resuls repored in figure 4 show ha he obained daes of changes in he esimaed long memory parameer coincide exacly wih Bai and Perron breaks. Moreover, hese changes are no very srong. his means ha he long memory behavior in he CAC 40 absolue reurns is a rue behavior. Neverheless, he presence of breaks in he CAC 40 absolue reurns amplifies he long memory parameer by a value around Noe ha for he second period he higher change of d is around 0.0 ( his change is no so srong o consider i as a veriably ump in he d parameer. 6 Conclusion his paper inroduces a general form of spurious long memory models. wo classes of models are proposed. hey differ by he naure in which he changes occur and by he process in which i is governed. We use he mos hree robus mehods of esimaing long memory in order o sudy heir behavior and heir robusness when he daa is weakly dependen wih changes in mean. he resuls confirm ha a spurious long memory behavior can be deeced and his behavior can be confused wih he rue one. he resuls express he need for a new es saisic which can be robus in hese cases. In he empirical applicaions we show ha he observed long memory behavior in he absolue reurns of he seven French companies wih he CAC 40 sock marke index is a rue behavior. We show also ha he deeced breaks have a lile effecs, and i amplifies he value of d only by Moreover, we show ha he fracional long memory parameer have a ime varying behavior. References [] Bai J., Perron P "Esimaing and esing linear models wih muliple srucural changes". Economerica 66: [] Beran, J. (994 "Saisics for Long-Memory Processes". Chapman and Hall, London. [3] Baneree A., and G. Urga (004. "Modelling srucural breaks, long memory and sock marke volailiy: an overview". Journal of Economerics,, forhcoming. [4] Charfeddine, L. and Guégan, D. (006. "Is i possible o discriminae beween srucural breaks model and long memory processes: Empirical evidence in he French and US Inflaion raes?". Working paper, -CES- Cachan, December. 8.

13 Inernaional Journal of Business and Social Science Vol. No. 3 [Special Issue - January 0] [5] Chen, C. and iao, G.C. (990 "Random level-shif ime series models, ARIMA approximaions, and level-shif deecion". Journal of Business and Economics Saisics, 8, [6] Diebold, F.X., Inoue, A., (00 "Long memory and regime swiching". Journals of Economerics 05, [7] Engel, R.F., Smih, A.D., (999 "Sochasic permanen breaks". Reviews of Economics and Saisics 8, [8] Geweke, J., Porer-Hudak, S., (983 "he esimaion and applicaion of long memory ime series models". Journal of ime series Analysis 4, [9] Granger, C.W.J. and Hyung, N., (004. "Occasional srucural breaks and long memory wih applicaion o he S&P500 absolue sock reurns". Journal of empirical finance,, [0] Granger, C.W.J., Joyeux, R., (980. "An inroducion o long memory ime series and fracional differencing". Journal of ime Series Analysis,, [] Granger, C.W.J., eräsvira,., (999 " A simple nonlinear ime series model wih misleading linear properies". Economics Leers 6, [] Hamilon, J.D. (989. "A new Approach o he Economic Analysis of Non saionariy imes Series and he Business Cycle," Economerica 57, [3] Hosking, J.R.M. (98. "Fracional differencing". Biomerika, 68, [4] Hsu, C.C and Kuan C.M., (000. "Long memory or srucural Change: esing mehod and empirical examinaion". [5] Jensen, J. (999. "Using waveles o obain a consisen ordinary leas squares esimaor of long memory parameer". Journal of Forecasing 8, 7-3. [6] Jensen, J. and Whicher, B. (000. "ime-varying Long-Memory in Volailiy: Deecion and Esimaion wih Waveles". echnical Repor, Deparmen of Economics, Universiy of Missouri. [7] Kao,., Masry, E., (999. "On he specral densiy of he wavele ransform of fracional Brownian moion". Journal of ime Series Analysis 0, [8] Lim, K.S. and ong, H. (980. "hreshold auoregressions, limi cycles, and daa," Journal of he Royal Saisical Sociay, B 4, 45-9 (wih discussion. [9] Lee, J., (005. "Esimaing memory parameer in he US in_aion rae". Economics Leers 87, [0] Lee, J. and Robinson, P.M., (996. "Semiparameric Exploraion of Long Memory in Sock Prices". Journal of Saisical Planning and inference 50, [] Lobao I.N., Savin N.E. (998. "Real and spurious long memory properies of sock marke daa". Journal of Business and Economic Saisics 6: [] P "Compuaion and analysis of muliple srucural change models". Journal of Applied Economerics 8: -. [3] Robinson P.M. (995 "Log-periodogram regression of ime series wih long range dependence". he annals of Saisics, 3, (995, [4] Quand, R. (958 "he Esimaion of he Parameer of a linear Regression Sysem Obeying wo Separae Regimes" Journal of he American Saisical Associaion, 53, [5] Shimosu, K. and Phillips C.B.P. (005 "Exac Local While Esimaion of Fracional Inegraion", Annals of Saisics, 33(4, pp [6] Yaima, Y. (989. "A cenral limi heorem of Fourier ransforms of srongly dependen saionary processes". Journal of ime series analysis, 0,

14 he Special Issue on Behavioral and Social Science Cenre for Promoing Ideas, USA Figure 5: raecories of he absolue sock reurn for he BNP, LVMH, PPR, Sain Gobain, Sociéé Générale, Suez and F companies and he CAC 40 sock marke index. 65

15 Inernaional Journal of Business and Social Science Vol. No. 3 [Special Issue - January 0] Figure 6: Auocorrelaion funcions of he level (dash line and he absolue (solid line reurns for he BNP, LVMH, PPR, Sain Gobain, Soc. Générale, Suez and F companies and he CAC 40 sock marke index. 66