DYNAMIC ECONOMETRIC MODELS Vol. 4 Nicholas Copernicus University Toruń Jacek Kwiatkowski Nicholas Copernicus University in Toruń

Transcription

1 DYNAMIC ECONOMETRIC MODELS Vol. 4 Nicholas Copernicus Universiy Toruń 000 Jacek Kwiakowski Nicholas Copernicus Universiy in Toruń Bayesian analysis of long memory and persisence using ARFIMA models wih an applicaion o Polish sock marke Inroducion This aricle describes Bayesian analysis of auoregressive fracionally inegraed moving average (ARFIMA) models and provides an applicaion o he weekly reurns of he rus fund Pioneer 1 on Polish sock marke. This paper also discusses basic conceps of long memory and persisence, associaed wih ARFIMA models. The origin of ineres in long memory processes has come from hydrology and climaology. The mos known example is aricle by Hurs (1951). In economerics, long memory models have really been used since around 1980 (Granger (1980), Granger, Joyeux (1980) and Hosking (1981)). ARFIMA models have been used, for insance, in sudying real oupu (Diebold and Rudebush (1989); Sowell (199b)), foreign-exchange raes (Cheung (1993) and Peers (1991)), inflaion (Delgado and Robinson (1994)). Bayesian analysis of ARFIMA models has been discussed by Koop, Ley, Osiewalski, Seel (1997) and Pai and Ravinshanker (1996). The paper is organised as follows: The secion describes long memory processes and ARFIMA models. Secion also describes impulse response funcions. Secion inroduces Bayesian analysis of ARFIMA models. Secion 4 presens applicaion o he weekly reurns of he rus fund Pioneer 1 on Polish sock marke. Secion 5 concludes. 1 This work was suppored by he Polish Commiee for Scienific Research (KBN); gran no. 1-H0B I would like o hank Professor Jacek Osiewalski for helpful commens and suggesions and Aleksander Słomiński for compuing assisance.

2 Jacek Kwiakowski. Long memory processes and ARFIMA models. Mos of he work in ime series analysis has been concerned wih series having he propery ha auocorrelaion funcion decays exponenially o zero (e.g., ARMA processes), and hen k () k ρ <, (.1) where ρ() k is auocorrelaion funcion. Consider, for example, firs-order auoregressive process y = 1 + ε, a ( 1,1), (.) ay whereε is a sequence of independen normal variables wih mean zero and variance σ, () k k= k= a k ρ is finie and equal o a = 1 + <. (.3) 1 a I urns ou however ha for some daa ses, condiion.1 does no hold. These processes imply auocovariances ha decay hyperbolically as he lag increases, and a specral densiy f ( ω ) has a pole a zero. Therefore auocorrelaion funcion akes he form () ρ α ρ k c k, (.4) where k ends o infiniy and ρ his implies ha c is a finie posiive consan. As ( ) α, k= () k = ρ (.5)

3 Bayesian analysis of long memory and persisence using ARFIMA models wih an applicaion o Polish sock marke. 3 Processes saisfied condiion.5 are called long memory processes. 1, 1 0,8 0,6 ARFIMA (0, 0,3, 0) AR(1), a = 0.8 0,4 0, Fig. 1. Auocorrelaion funcion of long memory process (ARFIMA (0, 0,3, 0) and shor memory process (AR(1), a = 0.8) An example of a long memory process is so-called fracional noise (ARFIMA (0, d, 0)) wih auocovariances { } d + 1 d d k + k + k 1 γ = 0,5γ, for k = 1,,... (.6) k For d from inerval (, 0.5) auocorrelaion funcion is given by d 1 γ ck, as k k (.7) There are also processes wih inermediae memory forα > 1and hence ρ () k <. k=

4 4 Jacek Kwiakowski where c saisfying 0 < c <. Therefore expression.7 saisfies condiions of long memory (equaions.4 and.5). When 0 < d < 0. 5, he ARFIMA (0, d, 0) process is a process wih long memory. The correlaions and parial correlaions decay monoonically and hyperbolically o zero as he lag increases. The specral densiy is concenraed a low frequencies. When 0.5 < d < 0, he ARFIMA (0, d, 0) process has a inermediae memory. When d = 0, ARFIMA (0, d, 0) is whie noise. A wider class of parameric models are he auoregressive fracionally inegraed moving averages (ARFIMA (p, d, q)), given by d ( B)( B) ( x µ ) θ ( B) ε φ 1 =, (.8) where d > 1, ε is whie noise, B is he backward shif operaor, φ () and θ () are polynomials of degrees p and q respecively. Therefore ARFIMA (p, d, q) models are a generalisaion of Box and Jenkins (1976) ARIMA models. The reason for choosing his class of processes is ha he effec of he auoregressive and moving average componens decay exponenially, while he efec of he parameer d decays hypebolically. Thus ARFIMA (p, d, q) models describe he long-, medium-, and shor-erm behaviour. Based on Koop, Ley, Osiewalski, Seel (1997) we can consider ARFIMA models for deviaions from a deerminisic rend, mainly φ where δ ( B)( B) z = θ ( B) ε 1 (.9) ( B)( y µ α ) = y µ = z 1, (.10) υ = y µ α. (.11) Using.10 o subsiue for z in.9 we obain d ( B)( B) ( y µ α ) θ ( B) ε φ 1 =, (.1) where d = δ + 1. When

5 Bayesian analysis of long memory and persisence using ARFIMA models wih an applicaion o Polish sock marke. 5 ( 1, 0,5) δ he process y is a rend saionary process wih long memory, δ ( 0,5, 0,5) he process y is a saionary process wih long memory for δ > 0 and wih inermediae memory for δ < 0, δ = 0 he process y corresponds o he sandard ARIMA (p, 1, q) model. The impulse response funcion (Campbell and Mankiw (1987); Diebold and Rudebusch (1989); Hauser, Pöcher, Reschnhofer (199)) of he esimaed ARFIMA models is one measure of he persisence in various macroeconomic ime series. Le assume ha y is a process wih saionary firs differences ( B) ε = µ + ( 1 + α1 B + α B + ) ε y y = µ + A... (.13) Impulse responses for a saionary process are he coefficiens of is moving average represenaion (Koop e al. (1997)). An impulse response funcion, I() n can be inerpreed as he effec of a shock of size one a ime on y + n, I n = 1+ α α. () n When he process y is saionary or saionary around a deerminisic linear rend he effec of any shock is ransiory, in case of he uni roo, he effec of any shock is permanen and equal o () 1 / φ() 1 I = θ (.14) + For insance, for a random walk y = y 1 + ε he effec of any shock is permanen and equal o one. δ For ARFIMA models φ( B)( 1 B) ( y µ ) = θ ( B) ε he impulse response funcion akes he form y δ ( 1 B) θ ( B) / φ( B) ε = (.15) Depending on parameerδ he impulse response funcion saifies for δ > 0 I + = θ () 1 / φ() 1 for δ = 0. (.16) 0 for δ < 0

6 6 Jacek Kwiakowski Noe ha δ = 0 corresponds o he sandard ARIMA (p, 1, q) model for y and o he ARMA (p, q) model for y. Addiionally, insead of examining he impulse response funcion a infiniy I ( ), a shor-run, medium-run and long-run effecs of a shock can be considered, (e.g., n = 4,1, 40 ) 3. Bayesian analysis of ARFIMA models. Following Koop, Ley, Osiewalski, Seel (1997) prior densiy is given by p ( ω, µ, σ ) = p( ω) p( µ ) p( σ ) σ p( ω) where ( ω) (3.1) p is a uniform densiy runcaed o he saionariy and inveribiliy region Ω. The likelihood funcion based on observaions is p ( w /, µ, σ ) f ( w / µ l, σ V ) ω = (3.) N where f N is he -variae Normal densiy funcion w = ( y,..., y ) ω ' = ( δ, Θ', Φ' )' Ω, Φ = ( φ 1,..., φ p )' C p, Θ = ( 1,..., θ q )' Cq Ω = ( 1, 0, 5) C q C p, µ R, σ R+, l - 1 wih elemens = σ γ ( i j) i, j = 1,..., ; γ () s v i j ' 1, θ, vecor of ones, V -marix, for T is he auocovariance funcion given in Sowell (199a). Afer combining 3.1 wih 3. and inegraing ou µ, σ, he marginal poserior pdf is where p 1 ( ω dane) K V 1/ ' 1 ( l V l ) 1/ SSE ( 1) / / p( ω) Ω 1/ =, (3.3) ' 1 1/ ( ) ( 1 l V l SSE )/ p( ) K = V ω dω, ' 1 ( w µ ˆl ) V ( w ˆ ) SSE = µ l ' 1 1 ' 1 ( l V l ) l V w ˆ µ =.,

7 Bayesian analysis of long memory and persisence using ARFIMA models wih an applicaion o Polish sock marke Applicaion o he Polish sock marke This secion examines evidence for fracional inegraion in Polish sock marke. We invesigae he long memory and persisence of he weekly reurns of he rus fund Pioneer 1 from 1995 o 1998 (06 observaions). We consider 18 differen models for he firs differences y corresponding o all possible ARFIMA ( p,δ,q) and ARMA ( p, q) for p, q. All models have equal prior probabiliies. Table 1. Poserior model probabiliies for ARFIMA ( p, δ, q) and ARMA ( p, q) models p, q ARFIMA ( p, q), δ ARMA ( p, q) 0,0 0, 1,0 1,1 1,,0,1, Toal * Compued by he auhor. The op wo models are he ARMA ( 0,0) wih poserior probabiliy and ARFIMA ( 0, δ,0) wih poserior probabiliy 498. In addiion, all ARMA models receive poserior probabiliy and ARFIMA models receive The poserior odds of uni roo case versus fracionalδ is hree o one. Therefore in case of he weekly reurns of he rus fund Pioneer 1 nonsaionary models wih uni roo are more likely. The resuls confirm he efficien marke hypohesis for he weekly reurns. (poserior probabiliy 0.47, while prior probabiliy 0.056) However hese facs do no negae he fracal behaviour of he reurns. (poserior probabiliy 0.15, while prior probabiliy 0.056) I is much beer resuls hen for he oher models ARMA (0, 1), (1, 0) (1, 1). Table repors poserior means and sandard deviaion forδ for all he ARIFMA models.

8 8 Jacek Kwiakowski Table. Poserior mean and sandard deviaion of δ for ARFIMA ( p, q), δ models. p, q Mean Sandard deviaion 0,0 0, 1,0 1,1 1,,0,1, * Compued by he auhor Figure plos he poserior pdf of δ mixed over he 9 ARIFMA models ,08 0,06 0,04 0,0 0-0,99-0,89-0,79-0,69-0,59-0,49-0,39-0,9-9 -0,09 0,01 1 0,31 0,41 Fig.. Poserior densiy of δ This plo indicaes ha he poserior pdf of δ is fairly Normal. The esimaes presened in Table are no very differen from zero. The peak in he densiy a suggess some sligh evidence of long memory. The regions of P δ > 0 receive 0,8889 poserior mass he parameer space which imply ha ( ) ( P ( δ > 0 ) = ), while poserior probabiliy ha δ is less han zero is

9 Bayesian analysis of long memory and persisence using ARFIMA models wih an applicaion o Polish sock marke Therefore, here is srong evidence ha I ( ) =. Noe however, ha he poserior probabiliy of δ > 0 is downweighed by he daa. (ARFIMA models receive poserior mass; Table 1). The ARFIMA models have been criicised by Hauser, Pöcher, Reschnhofer (199) for he purpose of esimaing persisence. Finding he poin esimaes of δ o be exacly equal o zero is very unlikely. Therefore, hey argue ha use of ARFIMA model corresponds o a prior informaion ha he persisence measure is zero or. In heir opinion ARFIMA models are no proper for he measuremen of persisence. Following Koop, Ley, Osiewalski, Seel (1997) here are wo responses o his criicism. Firs, apar from I ( ) also shor- run, medium-run and long-run persisence measure have been repored (Campbell and Mankiw (1987); Diebold, Rudebusch (1989)). Therefore, persisence can be esimaed eiher for I ( ) or for much shorer horizons (e.g.; I () 4, I( 1), I( 40) ). Secondly, by puing some prior mass a = 0 I conains δ, he poserior for ( ) wo poin masses a 0 and, bu is coninuous on (,+ ) 0. However, no momens exiss for such poserior (Koop, Osiewalski, Seel (1994). Table 3 repors impulse responses esimaes for n = 4,1, 40 for ARFIMA ( 0, d,0), ARMA (,0) 0, and for he overall model which averages all 18 individual ARMA and ARFIMA models. Addiionally, models which are a mixure of all 9 individual ARMA or ARFIMA models are also repored. Table 3. Poserior means of impulse responses n = 4, 1, 40. n ARFIMA 0, d,0 ( ) ARMA 0,0 ( ) ARFIMA ARMA model (0.1399) (0.0000) (0.1484) (0.0846) (0.1483) (0.363) (0.0000) (0.737) 1,0671 (0.1014) (0.13) (0.3769) * Compued by he auhor (0.0000) (0.4893) (0.1074) (0.3381) For he random walk specificaion (ARMA (0, 0) for y ) effec of any shock is exacly permanen and equal o one. Generally, all poserior means of impulse responses are very close o one for ARMA models. The resuls of applying he ARFIMA models are repored in column and 4. The evidence from P δ > 0 is more hen 0.5. There- Table 3 confirm, ha poserior probabiliy ( )

10 10 Jacek Kwiakowski fore, we can expec ha for ARFIMA models poserior mean of impulse responses will end o infiniy as n increases. Figure 3-6 plos he poserior densiy of impulse responses for n = 4,1, and 40. 0,8 0,7 0,6 0,5 0,4 0,3 ARFIMA ARMA 0, 0 Fig. 3. Poserior pdf of I () 4 0,9 0,8 0,7 0,6 0,5 0,4 0,3 ARFIMA ARMA 0, 0 0,05 0,60 1,14 1,68,,76 3,30 3,85 4,39 4,93 5,47 6,01 0,03 0,37 0,70 1,03 1,37 1,70,03,37,70 3,03 3,37 3,70 Fig. 4. Poserior pdf of I ( 1)

11 Bayesian analysis of long memory and persisence using ARFIMA models wih an applicaion o Polish sock marke. 11 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0, ARFIMA ARMA 0 0,06 0,69 1,31 1,94,56 3,19 3,81 4,44 5,06 5,69 6,31 6,94 Fig. 5. Poserior pdf of I ( 40) The poseriors of impulse responses are similar across all compared models. Moreover, since he ARMA models have received ¾ of he poserior mass, poserior densiies of impulse responses for ARMA and overall models have a similar shape. The shape of hese poseriors is unimodal wih peaks a one. The poserior densiies of impulse responses corresponding o he ARFIMA models are skewed wih mean ending o infiniy. Figure 6 graphs impulse responses I ( ) for ARMA and he overall model which is a mixure of 18 models. The impulse responses I ( ) corresponding o he overall model has a poin mass of 0.07 a zero and a poin mass a infiniy 0, +. equal o 0.18 I also has coninuous par on ( )

12 1 Jacek Kwiakowski 0,9 0,8 0,7 0,6 0,5 0,4 0,3 ARMA 0, 0 0,05 0,35 0,65 0,95 1,5 1,55 1,85,15,45,75 3,05 3,35 3,65 3,95 Figure 6. Poserior pdf of I ( ) 5. Summary and conclusions The ARFIMA models are naural exension of ARMA represenaion. Allowing any d > 1 fracional inegraion provides parsimonious modeling of long memory processes. These processes have abiliy o display significan dependence beween wo poins in ime as he disance beween hese poins increases. Unlike shor memory processes, heir auocovariances decay monoonically and hyperbolically o zero as he lag increases. Following Koop, Ley, Osiewalski, Seel (1997) Bayesian analysis of ARFIMA models wih implemenaion o measuremen of persisense has been shown. Also in his paper long memory and persisense in he weekly reurns of he rus fund Pioneer 1 on Polish sock marke has been discussed. The resuls for ARFIMA ( 0, δ,0) wih poin esimae of sugges some sligh evidence of long memory.

13 Bayesian analysis of long memory and persisence using ARFIMA models wih an applicaion o Polish sock marke. 13 References Box, G.E.P., Jenkins G.M., (1976), Time series analysis: forecasing and conrol, Holden Day, San Francisco. Campell, J.Y., Mankiw N.G., (1987), Are oupu flucuaions ransiory?, Quarerly Journal of Economics, 10, p Cheung, Y.-W., (1993), Long memory in foreign exchange raes, Journal of Business and Economic Saisics, 11, p Delgado, M.A., Robinson P.M., (1994), New mehods for he analysis of long-memory ime series: Applicaion o Spanish inflaion, Journal of Forecasing, 13, p Diebold, F.X., Rudebusch G.D., (1989), Long memory and persisence in aggregae oupu, Journal of Moneary Economics, 4, p Granger, C.W.J., Joyeoux R., (1980), An inroducion o long memory ime series models and fracional differencing, Journal of Time Series Analysis, 1, p Granger, C.W.J., (1980), Long memory relaionships and he aggregaion of dynamic models, Journal of Economerics, 14, p Hauser, M.A., Pöscher B.M., Reschenhofer E., (199), Measuring persisence in aggregae oupu: ARMA models, fracionally inegraed ARMA models i nonparameric procedures, manuscrip. Hosking, J.R.M., (1981), Fracional differencing, Biomerika, 68, p Hurs, H.E., (1951), Long erm sorage capaciy of reservoirs, Transacions of American Sociey of Civil Engineers, 116, p Koop, G., Ley E., Osiewalski J., Seel M.F.J., (1997), Bayesian analysis of long memory and persisence using ARFIMA models, Journal of Economerics, 76, p Koop, G., J. Osiewalski, Seel M.F.J., (1994), Poserior properies of long-run impulse responses, Journal of Business and Economic Saisics, 1, p (Correcion, Journal of Business and Economic Saisics, 14, p. 57). Pai, J.S., Ravishanker N., (1996), Bayesian modelling of ARFIMA processes by Markov Chain Mone Carlo mehods, Journal of Forecasing, 15, p Peers, E.E., (1991), Chaos and order in he capial markes, New York, John Wiley & Sons. Sowell, F., (199a), Maximum likelihood esimaion of saionary univariae fracionally inegraed models, Journal of Economerics, 53, p Sowell, F., (199b), Modelling long-run behaviour wih he fracional ARIMA model, Journal of Moneary Economics, 9, p