ARFIMA PROCESS: TESTS AND APPLICATIONS AT A WHITE NOISE PROCESS, A RANDOM WALK PROCESS AND THE STOCK EXCHANGE INDEX CAC 40

Transcription

1 Professor Régis BOURBONNAIS, PhD LEDa, Uiversité Paris-Dauphie, Frace Mara Magda MAFTEI, PhD The Bucharest Academy of Ecoomic Studies ARFIMA PROCESS: TESTS AND APPLICATIONS AT A WHITE NOISE PROCESS, A RANDOM WALK PROCESS AND THE STOCK EXCHANGE INDEX CAC 4 Abstract. The assumptio of liearity is implicitly accepted i the process which geerates a time series coditio submitted to a ARIMA. That is why, i this paper, we shall discuss the research of log memory i the processes: the fractioal ARIMA models, deoted as ARFIMA, where d ad D, the degree of differetiatio of the filters is ot iteger. After presetig the characteristics of the ARFIMA process, we shall discuss the log-memory tests (statistics rescaled Rage Lo ad R/S* Moody ad Wu). Fially three examples ad tests o a white oise process, a radom walk model ad the stock idex of Paris Stock Exchage (CAC4) will illustrate the method. Key-words: log-memory test, o statioary processes, ARIMA process, ARFIAM process. JEL Classificatio: C, C3, C3.. The ARFIMA process The ARMA processes are processes of short memory i the sese where the shock at a give momet is ot sustaiable ad does ot affect the future evolutio of time series. Ifiite memory processes such as DS (Differece Statioary) processes have a opposite behaviour: the effect of a shock is permaet ad affects all future values of the time series (R. Bourboais, M. Terraza, ). This dichotomy is iadequate to accout for log-term pheomea as show by the works of Hurst (956) i the field of hydrology. The log memory process, but ot ifiite, is a itermediary case, i that the effect of a shock has lastig cosequeces for future values of the time series, but it will fid its "atural" equilibrium level (Migo V. 997). This type of behaviour has bee formalized by Madelbrot ad Wallis (968) ad Madelbrot ad Va Ness (968) startig from fractioal Browia motios ad from fractioal Gaussia oises. From these studies Grager ad Joyeux (98) ad Hoskis (98) defie the fractioal ARIMA process as ARFIMA. More recetly

2 Régis Bourboais, Mara Magda Maftei these processes have bee exteded to seasoal cases (Ray 993, Porter-Hudak 99) ad are oted as SARFIMA process... Defiitios Let us remember that a real process x t from Wold : xt ψ at with ψ, ψ R ad a t is i.i.d.(, σ a ) is statioary uder the coditio that ψ <. The statioary process x t is a log memory if ψ. Let us cosider a process cetred o x t, t,,. We say that x t is a statioary itegrated process, oted ARFIMA (p, d, q) if it is writte: d φ ( B )( B) x θ ( B) a with: p t q t φ (B) ad θ (B) are respectively polyomial operators i B of parties AR(p) p ad MA(q) of the process, a t is i.i.d.(, σ ), d R. q a ( B) d is called fractioal differece operator ad is writte startig from the time series expasio: d d d( d) d( d) L( d ) ( B) C ( B) db B L B L! π B d)! With π,, ad Γ is the Euleria fuctio. + ) d) d Be it the process ARFIMA (, d, ) : ( B ) x a also called process FI(d). It is this process that cotais the log-term compoets, the party ARMA brigs together the short-term compoets. Whe d < ½, the process is statioary ad it has a ifiite movig average represetatio. t t

3 ARFIMA Process: Tests ad Applicatios at a White Noise Process, A Radom d d+ ) x t ( B) at ψ a t ψ ( B) at with ψ or the d) + ) fuctio h) is as such : h t t e dt if h> O h) ( h )! if h + h) if h< h ad /) π / Whe d > ½, the process is ivert ad has ifiite autoregressive represetatio: d d) ( B ) xt π ( B) xt π x t at with π d) + ) The asymptotic value of the coefficiets ψ et π : d ad Limψ d ) d decrease with a hyperbolic rhythm at a rate which is lower tha Limπ π ( d ) the expoetial rate of the process ARMA. The FAC has the same type of behaviour which allows to characterize the process FI(d). Fially if (Hoskis 98): < d < ½, the process FI(d), is a log-memory process d <, the process FI(d) is a ati persistet process, ½ < d <, the process FI(d) is ot of log-memory, but it does ot have the behaviour of ARMA. This itermediate case called ati-persistet by Madelbrot correspods to alteratios of icreases ad decreases i the process. This behaviour is also called the "Joseph effect" by referece to the Bible. The process FI(d), thus statioary is ivert for ½ < d < ½.

4 Régis Bourboais, Mara Magda Maftei.. Log-memory tests a) The "Rescaled Rage" statistics The statistics R/S was itroduced i 95 i a study related to the debits of the Nile by the hydrologist Harold Edwi Hurst. His purpose is to fid the itesity of a aperiodic cyclical compoet i a time series cosidered oe of the aspect of the log-term depedece (log memory) developed by Madelbrot. Be it x t a time series producig a statioary radom process with t,, ad t the cumulated time series. The statistics R/S oted Q is the * x t x u u extet R of partial sums of stadard deviatios of the series from its mea divided by its stadard deviatio S: Q R S max k k k ( x x) mi ( x x) k ( x x) The first term of the umerator is the maximum k of partial sums of the k i stadard deviatio of x from its average. This term (max) is always positive or zero. By defiitio the secod term (i mi) is always egative or ull. Therefore R is always positive or ull. The statistics Q ~ is always o-egative. The statistics H of Hurst applied to a time series x t is based o the divisio of time ito itervals of legth d, for give d we obtai (T + ) sectios of time. The statistics H- is calculated o each sectio (Madelbrot) usig the previous method of Hurst takig ito accout the gap operated o the time scale. I this case: R ( t, d) max ( u) mi ( u) (u) is the liear iterpolatio of * x t x s t s u d / [ ] [ ] where [ ] u d * * u * * betwee t et t + d ; be it u) [ x x ] [ x x ] ( t u t t+ d t k the expressio brought at the differece d of ( x x ) + it is about d used i order to calculate R. The stadard deviatio is the writte: S ( t, d) xt+ u xt+ u. d u d d u d We ca calculate R /S for each of (T + ) sectio of d legth but also their arithmetic average. We ca also demostrate (Madelbrot Wallis) that the We may cosult for this paragraph Migo (997).

5 ARFIMA Process: Tests ad Applicatios at a White Noise Process, A Radom itesity of the log-term depedece is give by the coefficiet H situated betwee zero ad oe i the relatioship: R H Q cd S. Be it log(r /S ) log c + H log d, d H is the estimator of OLS (Ordiary Least Square) i this relatio. I real life, we build M fictioary samples ad we choose M arbitrary startig ( ) poits of the time series. This startig poit is give by : : t + ad the M ( M ) legth of the sample is: l. M The liear adustmet of the cloud obtaied leads to the estimatio of the expoet Hurst. The r² of the cloud depeds o the iitial differece obtaied. We remember (Herard, Moullard, Strauss Kha, 978, 979) for H the oe which gives the maximum r² maximum for a iitial differece d. The iterpretatio of the H values is the followig: If < H < ½ ati-persistet process, If H ½ a simply radom process or ARMA process. There is a log term depedece absece. If ½ < H < log-term process, the depedece is eve stroger as H teds towards. b) The statistics of Lo The statistic of the expoet Hurst ca ot be tested because it is too sesitive to the short term depedece. Lo (99) shows that the aalysis proposed by Madelbrot ca be cocluded towards the presece of log memory, while the time series has oly a short-term depedece. Ideed, i this case, the expoet Hurst by aalysig R/S is biased upward. Lo proposes a ew modified statistics ~ R R/S oted: Q ˆ σ ( q) ~ Q x x max k k k ( x x) mi ( x x) k q + ω (q) x i + x x i x / This statistics is differet from the previous oe Q by its deomiator, which takes ito accout ot oly the variaces of idividual terms but also the autocovariace weighted accordigly to differeces of q as related to: For which the first poits are removed (trasitory phase).

6 Régis Bourboais, Mara Magda Maftei ω ( q) where q <. q+ Adrews Lo (99) proposed the followig rule for q: /3 / 3 3 ˆ ρ q [k ] whole party of k k ρˆ is the estimatio of ˆ ρ the autocorrelatio coefficiet of order ad i this case ω. k Lo proves that uder the hypothesis H : x t i.i.d.(, σ x ) ad for which teds towards the ifiity, the asymptotic distributio of Q ~ coverges step by ~ step towards V : Q V where V is the rak of a Browia bridge, a process with idepedet Gaussia icreases costraied to uity ad for which H ½. The distributio of the radom variable V is give by Keedy (976) ad k Siddiqui (976): F ( v) + ( 4k v ) V e ( k v) The critical values of this symmetrical distributio the most commoly used are: P(V<ν) ν π / The calculatio of H is doe as above ad Lo aalyzes the behaviour of uder alterative log-term depedecy. He the shows that: V ~ Q P pour H pour H [.5;] [ ;.5] Uder the hypothesis of H, there is a short memory i the time series ( H [,5;] ). For a acceptace threshold at 5% H is accepted if v [,89 ;,86]. He cocludes that: For the values of H betwee,5 ad the acceptace threshold of log memory at % is ν >,6. For the values of H betwee,5 ad the acceptace threshold of the ati-persistet hypothesis at % is ν >,86. We ca verify that there is a relatio betwee the values d ad the ARFIMA processes ad H of the expoet Hurst (d H,5). Q ~

7 ARFIMA Process: Tests ad Applicatios at a White Noise Process, A Radom c) The statistics R/S* modified by Moody ad Wu (996) The statistics Lo fix some flaws of the R/S traditioal statistics of Hurst whe the umber of observatios is too low. Moody ad Wu show o the applicatio of exchage rates: There is a error i estimatig the extet of R related to short-term depedecies i the series. The value of the traditioal R/S statistics led to acceptig the existece of a log-term compoet abset i the geeratig process. Whe the umber of observatios is importat, the statistics of Lo corrects this error. For a small umber of observatios, the statistics of Lo ad the expoet Hurst are poorly estimated. For a umber of importat observatios, the right lie correspodig to the Lo statistic is idepedet of q: the traditioal statistics ad those of Lo have the same expoet Hurst. Moody ad Wu suggest itroducig a differet deomiator S* i the statistics: q * q ˆ ( ) S + ω (q) + σ ω (q) x x x x i i + or: ˆ σ ( ) x x is the estimatio of the variace. ω (q) are weights such as q < so that the deomiator of the q+ statistics be positive. For q the statistics of Moody ad of Wu lead either to the traditioal statistics or to that of Lo. Applicatios: simulatio ad calculatio of the statistics of Hurst, Lo ad of Moody ad Wu for a white oise, for a radom walk model ad for CAC4 (idex stock exchage Paris)..3. Applicatio for a white oise of observatios. We have simulated a white oise of observatios icluded betwee ( ) ad (+) ad we have calculated the statistics of Hurst, Lo ad of Moody ad Wu. For the statistics of Hurst we have i mid: a iitial gap of, samples ad a threshold poit of, which allows to iterpolate H o the thirty most sigificat values, be it 36% observatios. /

8 Régis Bourboais, Mara Magda Maftei Results Threshold poit 5 3 H R² For the statistics of Lo we have 3 samples ad the iterpolatio is realised o 5% estimatios. Results Threshold poit 5 5 H R² ν Whatever the method, the most reliable values of H are those for which R² teds to oe. This is the case for the gap betwee ad 3 (Hurst) ad 5 ad (Lo). We ote that H teds to.5 i accordace with the theory. For values of H tedig towards.5, the variable ν must be betwee.86 ad.6 to accept the hypothesis of zero memory: this is the case with this exercise. Hurst's method evaluates the memory aroud about times the periodicity, whereas that of Lo estimates it at ust 5. Simulatio results (Tests of Lo ad Moody Wu) White oise ( ) : expoet Hurst ad statistics R/S modified (Lo) q H V White oise ( ): expoet Hurst ad statistics R/S modified (Moody Wu) q H V

9 ARFIMA Process: Tests ad Applicatios at a White Noise Process, A Radom Radom walk ( ): expoet Hurst ad statistics R/S modified (Lo) q H V Radom walk ( ): expoet Hurst ad statistics R/S modified (Moody Wu) q H V Applicatio to a radom walk of observatios. We have simulated a radom market x t x + a i with a t i.i.d.(, ) ad x o observatios ad we calculated the statistics of Hurst ad of Lo. For the statistics of Hurst we have chose a iitial differece of 5 ad of samples. For iterpolatio, we used gaps superiors at or 7 estimatios o (3%). Results Threshold poit 5 7 H R² The calculatio of the statistics of Lo is doe uder the same coditios but with a uique sample: Results Threshold poit 5 5 H R² ν The most reliable value of H by the method of Hurst is.889, which ca therefore coclude towards the presece of a log memory. The oe give by Lo is made for a gap betwee ad 5, which is about H.9.The value ν is superior to.6 ad it cofirms the structure of log-term depedecy. These calculatios obtaied from a o-statioary series show the misuderstadig that ca be made with these tests. i

10 Régis Bourboais, Mara Magda Maftei.5. The statistics of Hurst ad of Lo o the data of CAC4 kow for 9 days Fially, o a series of CAC4 (idex of Paris Stock Exchage), we have used the method GPH ad the maximum of likelihood i order to estimate the order d of the geeratig process FI(d) of the raw time series ad of first differeces. The geeratig process of CAC4 cotais a uitary root. Statistics H of Hurst calculated o the raw series are of aroud.9 for a threshold poit betwee 5 ad 6 ad have a value equivalet to a gap as by comparig to the statistics of Lo (ν 7.4 superior to.6). We could ifer the existece of a positive depedece betwee 4 ad 7 values. I fact whe the geeratig process is statioary by the trasitio to first differeces, the statistics of Hurst ad Lo are respectively of.46 ad of.45 ad the value of the coefficiet ν is of.94 less tha.6. We ca the coclude that there is o log-term depedecy i the CAC4 series i first differeces ad that the results issued from the raw series are ot cosistet with the assumptios of applyig tests. The calculatios are made with the software Gauss ad TSM uder Gauss. The results are the followig: Estimated GPH Raw series Differetial series Stadard deviatio (Differetial series) No widow Rectagular Bartlett Daiell Tukey Parze Bartlett Priestley Estimatio by maximum likelihood series i level Number of observatios 9 Number of estimated parameters: Value of the likelihood fuctio 5.85 Parameter Estimatio Stadard deviatio t statistics Prob. d Sigma

11 ARFIMA Process: Tests ad Applicatios at a White Noise Process, A Radom Series i first differeces Number of observatios 8 Number of estimated parameters: Value of the likelihood fuctio Parameter Estimatio Stadard deviatio t statistics Prob. d Sigma These results show that the time series has a uit root, as with or without widow, the GPH estimator is close to as well as the oe of the maximum likelihood. Whe the time series is differetiated i order to become statioary, accordig to the theory, the hypothesis H of ullity of the coefficiet of fractioal itegratio is accepted i both cases: d gph estimatio GPH <.96. Std error maximum de likelihood (cf. the critical probabilities). Fially, the relatioship H -.5 (H Hurst statistic), ca help us to verify that it leads to a result cotradictory to the raw series (d.4 by the relatio ad d by calculatio). For the differetiated series, we obtai d -.37 from H of Hurst ad H.5 by the statistics of Lo. These results are accordig to the estimatios. REFERENCES []Bourboais R., Terraza M. (), Aalyse des séries temporelles e écoomie, Duod, 3 ème Edt.; []Che G., Abraham B., Peiris M.S. (994), Lag Widow Estimatio of the Degree of Differecig i Fractioally Itegrated Time Series; Joural of Time Series Aalysis, Vol. 5; [3]Chug C. F. (996), A Geeralized Fractioally Itegrated Autotregressive Movig Average Process; Joural of Time Series Aalysis, Vol. 7; [4]Geweke J., Porter-Hudak S. (983), The Estimatio ad Applicatio of Log Memory Time Series Models; Joural of Time Series Aalysis, Vol. 4; [5]Grager C.W.J., Joyeux R. (98), A Itroductio to Log Memory Time Series ad Fractioal Differecig ; Joural of Time Series Aalysis, Vol. ; [6]Gray H. L., Zag N. F., Woodward W. (989), A Geeralized Fractioal Process; Joural of Time Series Aalysis, Vol. ; [7]Hassler U. (993), Regressio of Spectral Estimators with Fractioally Itegrated Time Series ; Joural of Time Series Aalysis, Vol. 4;

12 Régis Bourboais, Mara Magda Maftei [8]Hassler U. (994), Specificatio of Log Memory i Seasoal Time Series; Joural of Time Series Aalysis, Vol. 5; [9]Herad D., Mouillard M., Strauss-Kah D. (978), Du bo usage de R/S, Revue de Statistique Appliquée, Vol. 4; []Herad D., Mouillard M., Strauss-Kah D. (979), Forme typique du spectre et dépedace temporelle e écoomie, Revue de Statistique Appliquée, Vol. 4; []Hoskig J. R. M. (98), Fractioal Differecig; Biometrika, Vol. 68; []Hurst H. (95), Log Term Storage Capacity of Reservoirs; Trasactio of the America society of civil egieers, Vol. 6; [3]Hurvich C., Ray B. (995), Estimatio of the Memory Parameter of Nostatioary or Noivertible Fractioally Itegrated Processes; Joural of Time Series Aalysis, Vol. 6; [4]Lo A.W. (99), Log Term Memory i Stock Market Prices; Ecoometrica, Vol. 59; [5]Madelbrot B., Va Ness J.W. (968), Fractioal Browia Motio, Fractioal Noises ad Applicatio; SIAM Review, Vol.; [6]Madelbrot B., Wallis J. (968), Noah, Joseph, ad Operatioal Hydrology, Water resources research, Vol. 4; [7]Migo V. (997), Marchés fiaciers et modélisatio des retabilités boursières; Ecoomica Publishig House, Bucharest; [8]Moody J., Wu L. (996), Improved Estimates for the Relaxed Rage ad Hurst Exposat; Neural Network i Egeeerig, World Scietific; [9]Porter-Hudak S. (99), A Applicatio of the Seasoal Fractioally Differeced Model to the Moetary Aggregates; Joural of the America Statistical associatio, Vol. 85; []Ray B. (993), Log Rage Forecastig of IBM Product Reveues Usig a Seasoal Fractioally Differeced ARMA Model; Iteratioal Joural of Forecastig, Vol. 9.