DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń An Application of Markov-Switching Model to Stock Returns Analysis

Transcription

1 DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Coernicus Universiy Toruń 006 The Universiy of Comuer Science and Economics in Olszyn An Alicaion of Markov-Swiching Model o Sock Reurns Analysis. Inroducion The main characerisics of financial ime series is volailiy clusering for high and low aciviy eriods. Emirical researches ofen confirm occurrence of some framework ha divides a eriod ino subsaces wih differen arameers. The way o describe such a relaionshi is a model wih simulaneous swiching of exlained variable and arameers beween subsaces. A Markov-swiching models MS have his roery, because boh variable and arameers describe a rocess dynamics beween saes. Iniially, an economeric dynamic model wih Markov ye swiching was inroduced by J. Hamilon (989, 994), as a ool which characerizes inner srucure of changes beween regimes of business cycle flucuaions. In ha aer Hamilon considered a wo sae chain, for exansion and recession resecively, whereas he mean of reurn rae is secified. The coninuaion of his research was roosed by Clemens and Krolzig (000), hen models wih swiching in variance or boh variance and mean (Turner, Sarz, Nelson 989, Yin 003), Markov-swiching VAR models (Linne 00, Krolzig 00) and Markov-swiching ARCH models SWARCH (Hamilon, Susmel 993). Yin (003) in his aricle used he S&P500 monhly marke reurns ( ). The samle eriod was divided ino 4 grous and for each subsace here were models esimaed wih boh mean reurn and variance as a subjec o change in regimes. The resuls imlied ha sock marke could swich beween wo saes wih exremely differen means and variances. The good sae is characerized by abou 4,5 imes higher mean reurn and abou imes lower variance in comarison o he bad sae. Furhermore he good sae urned ou o be exremely ersisen ( =0.9999) and he bad Coyrigh by The Nicolaus Coernicus Universiy Scienific Publishing House

2 60 sae very ransiory ( = ), which was exlained by quick coming back rocess o he good sae. The conclusions hereof can be aralleled wih early emirical findings abou asymmeric volailiy of sock markes rocess, namely ha sudden increase in volailiy ends o be associaed more ofen wih large negaive reurns. I seems o be unreasonable inuiively, since aking more risks is execed o bring a higher reurn. The urose of Linne s aer (00) was o examine he conagion effecs on several emerging sock markes in Cenral and Easern Euroe as resul of currency crises in he Czech Reublic in May 997, in Asia in Summer 997 and in Russia in Augus 998. Weekly sock reurns of seven Cenral and Eas Euroean markes were used in his sudy. The research counries were he Czech Reublic, Esonia, Hungary, Poland, Russia, he Slovak Reublic and Slovenia. Linne considered a Markov-swiching vecor auoregressive models MS()-VAR(), in which auoregressive coefficiens addiionally deermined he influence of shocks on aricular markes. There were hree alernaive secificaions of model examined, in which eiher he mean, he variance, or boh differed beween wo regimes. The conagion marke resuls in higher rading aciviies, higher rice volailiy and falling sock rices. The aer is an aem o answer he following quesions: are he shifs in reurns relaed o currency crisis and are he sock reurns feaures similar across differen markes? The resuls showed an occurrence of wo saes, he calm sae was characerized by low variance and a osiive mean; he crisis sae had a higher variance and a negaive mean. The robabiliies of remaining in he same sae for boh regimes were large, and momens in which he high robabiliy of he crisis sae aeared reflec he crisis eisodes during he samle eriod (i is he mos aaren in MSMV() VAR()). The resuls imlied ha swiching model is able o caure he sock reurns volailiy common for all markes. Thereby, he model rovided an exlanaion for he volailiy clusering resen in he sock rice daa. The residuals of swiching models were esed for he resence of ARCH effec, following he F es suggesed by Garcia and Pierre (996). The resuls showed ha null hyohesis of no ARCH effecs canno be rejeced for five of seven sock reurns. The urose of his aer is inroducion o auoregressive Markovswiching model MS wih differen kinds of swiching. In emirical research weekly and daily daa, from Polish sock marke were used. The esimaed models are comared wih ARMA(,q) model and esed for ARCH effec. Coyrigh by The Nicolaus Coernicus Universiy Scienific Publishing House. Building and Esimaion of MS Model A fac ha arameers of he auoregression can change beween regimes as he resul of a firs-order Markov rocess is he main characerisic of

3 An Alicaion of Markov-Swiching Model o Sock Reurns Analysis 6 he Markov-swiching model. In his rocess, curren sae of a variable deends only on revious sae, wha may be wrien as: P( s = j s0 = i0, s = ii,..., s = i) = P( s = j s = i) = ij ( ) () (i means ha he robabiliy of he rocess a momen relies on he sae of his rocess a momen - and is defined as a robabiliy of ransiion from sae i ino j). The esimaed model is given by: ( r s ) = ( r s ) + e, ~ N ( 0, ) ( ) e () where he subjec of change is a mean or a variance, ha can be wrien in he form: s s =, s, s = s =, s = = Coyrigh by The Nicolaus Coernicus Universiy Scienific Publishing House s dla s =,. (3) dla s = The ransiion robabiliies beween a wo-sae chain are given by: P P ( s = s = ) = ( s = s = ) = and hey form he ransiion marix ha is characerized by: P =. (4) In his model, he mean ( ), variance ( e ), arameer of auoregression ( ) and he ransiion robabiliies (, ) are subjec o esimae. Moreover, hey are included in an esimaed arameer vecor ( θ ), for he following models wih: shifs in he mean MSM(): θ =,,,,, ], [ e shifs in he variance (he heeroskedasic model) MSV(): θ =,,,,, ], [ 0 e e shifs boh he mean and he variance MSMV(): θ =,,,,,, ]. [ e e The Markov-swiching model can be esimaed by deermining arameer esimaes of vecor (θ ) a maximizaion of likelihood funcion given by : Kim, Nelson (999),. 60.

4 6 and where: T L r = ( θ) = f ( ψ ) i ln (5) =, 0 f ( r ψ ) = f ( i s= s( ) = r, s, s ψ ), (6) Coyrigh by The Nicolaus Coernicus Universiy Scienific Publishing House ( r s, s ψ ) = f ( r s, s, ψ ) ( s, s ψ ) f P, (7) ψ, refers o informaion u o ime -, [,, ] θ =. (8) i ei i To sar an ieraion a ime =, iniial values of robabiliies given in he following form are assumed: π = P( s0 = ψ0 ) =, (9) P ( s = ψ ) π = 0 0 =. (0) The execed (average) reurn o regim e i is given by: m ( i ) =. ii The execed duraion of regime i can be wrien as: d( i ) =. ii 3. Emirical resuls () () In his aer daily and weekly sock marke reurns ( ) of he WIG index, TPSA, Prokom and Comarch socks were analyzed. The Markov-swiching models MSM(), MSV(), MSMV() are

5 An Alicaion of Markov-Swiching Model o Sock Reurns Analysis 63 comared wih ARMA(,q) model. Furhermore an Akaike informaion crierion and he log likelihood funcion were alied. Esimaion resuls of some seleced ime series (he WIG and TPSA socks) are shown in ables 4. Considering AIC and LL crieria, resuls achieved from MSV() and MSMV() models are insignificanly differen wih MSV() leading. Emirical resuls from MSV() and MSMV() are beer hen boh MSM() and he ARMA(,q) models. A deerminaion coefficien for all ime series. R urned ou o be higher for ARMA(,q) model Table. Esimaes of a Markov-swiching models of he WIG index (weekly; ) Parameers MSM() MSV() MSMV() (0. 034) 0.00* (0.005) (0.005) (0.054) (0.004) * (0.07) 0.68 (0.0680) 0.788* (0.0854) e e m() m() d() d() AIC LL R TR (ARCH) ARMA(0,) 0 β AIC LL R * (0.0005) ( ) Noe: resuls come from Ox 3, sandards errors are in aren heses, (*) means signific ance a he 5% level. Weekly reurns (ables ): MSV() and MSMV() models idenified wo s ignificanly differen low and high aciviy regimes for he WIG and TPSA reurns. Boh of hem are characerized by high robabiliy of remaining in regime, for he WIG index in regime : (MSV) = 0.94, (MSMV) = 0.90 and regime : (MSV) = 0.97, (MSMV) = 0.9. The regime denoed a lower value of sandard deviaion, so i is named low aciviy regime and he average imes of ersising in regime for boh models are 5 and 0 weeks accordingly. In he -nd regime, characerized by a higher value of sandard deviaion, he rocess remains for 35 and weeks on average (accordingly for Coyrigh by The Nicolaus Coernicus Universiy Scienific Publishing House Engle (98). 3

6 64 MSV and MSMV models). In case of boh regimes, sysem shifs beween regimes every week on average (see Fig. ). The MSMV () model allows o esimae means in regimes, oo. The lower mean relaes o he high aciviy sae, wha can be exlained by negaive values of sock reurn in his regime. However he boh means urned ou o be insignifican. Furhermore, he robabiliies o remain in sae for TPSA sock reurns are: (MSV) = 0.98, (MSMV) = Sae urns ou o be a low aciviy one wih a long ime of duraion in regime, wih 43 weeks for MSV and 63 weeks for MSMV model. The second sae is a high aciviy regime and he sysem remains on average 7 and 38 weeks in i (for MSV and MSMV models resecively). The ARCH effec urned ou o be insignifican in all weekly sock reurns. Table. Esimaes of a Markov-swiching models of TPSA socks (weekly; ) Parameers MSM() MSV() MSMV() (0.6) (0.004) (0.0043) (0.0590) (0.0085) (0.0739) 0.39 (0.0676) 0.4 (0.069) e e ,984 0, ,065 0,9735 m() m() ,0,03 d() d() ,3 37,69 AIC LL R TR (ARC H) ARMA(0,) 0 β AIC LL R * ( ) ( ) Noe: resuls come from Ox, sandards errors are in arenh eses, (*) means significa nce a he 5% level. Daily reurns (able 3-4): There were idenified wo saes wih low and high ac iviy, for boh WIG and TPSA sock reurns. The differences beween values of sandard deviaions in d aily series are larger hen in weekly series. Boh considered saes are reresened by high ransiion robabiliies (, ), herefore inferences of execed duraion d(i) indicae a long ime of saying in saes. For he WIG index robabiliy values in boh saes are (MSV) = = (MSMV) = 0.989, (MSV) = (MSMV) = (see Fig. ). In he low aciviy regime, he considered rocess remains 88 days on average (for boh MSV and MSMV models), whereas i says 70 days in he high aciviy regime Coyrigh by The Nicolaus Coernicus Universiy Scienific Publishing House

7 An Alicaion of Markov-Swiching Model o Sock Reurns Analysis 65 on average, which corresonds wih aroriae weekly reurns conclusions. The ransiion robabiliies of TPSA sock rocess are: (MSV) = 0.993, (MSMV) =.00, and (MSV) = , (MSMV) = The low aciviy sae is characerized by osiive mean and he high aciviy sae has a negaive mean, which can be a confirmaion of emirical research in he sock markes. The average rocess duraion in he -s sae is 43 days for MSV and days for MSMV model (he high value of d() is caused by high robabiliy of remaining in he sae =.00, which may be an oucome of incorrec assumion of iniial values (π, π ), beginning he ieraion of he EM algorihm). The average rocess duraions in he -s sae, which is characerized by -imes higher sandard deviaion (he sae of enhanced aciviy) are 8 and 53 of boh models resecively. The ARCH effec in boh Markov-swiching models for daily reurns is relevan. The obained resuls of esimaed arameers for models wih shif in he variance and model wih shif in boh he mean and he variance are similar. Hence he conclusion ha addiional swiching in he mean does no reresen an increase in efficiency. Besides all indicaed comarison crieria sugges a choice of Markov-swiching model wih shif in variance MSV. Table 3. Esimaes of a Markov-swiching models of he WIG index (daily; ) Parameers MSM() MSV() MSMV() (0.003) * (0.0003) * ( ) (0.004) ( ) * (0.0490) (0.066) * (0.0306) e e , , m() m(). 33, d() d() , AIC LL R TR (ARCH) ARMA(,0) 0 β AIC LL R * ( ) (0.0873) Noe: re suls come from Ox, sandards errors are in are nheses, (*) means significance a he 5% level. Coyrigh by The Nicolaus Coernicus Universiy Scienific Publishing House

8 66 Table. Esimaes of a Markov-swiching models of TPSA sock (weekly; ) Parameers MSM() MSV() MSMV() (0.0039) ( ) (0.0007) (0.0066) (0.00) 0.0 (0.0373) ( ) (0.0304) e e ,00 0, , ,998 m() m() ,00,00 d() d() ,0 53,35 AIC LL R TR (ARCH) ARMA(.0) 0 β AIC LL R ( ) (0.0479) Noe: resuls come from Ox, sandards errors are in arenheses, (*) means signifi cance a he 5% level. 4. Conclusions The urose of his aer was o show he alicaion of auoregressive Markov-swiching model MS in sock marke reurns analysis, hen research of he roeries of his model and is comarison wih ARMA(,q) model one of he mos oular in ime series analysis. The emirical resuls indicae he MSV m odel, boh for weekly and daily daa, as he mos roer one for descriion of differen arameers srucure and relaionshis beween hem. All examined saes were exremely ersisen, wha can be a roof for he occurrence of some srucure wih differen arameers (he mean and he variance of residuals), ha deends on curren change in rocess of exlained variable. No ARCH effec in weekly reurns imlies beer roeries of MS models for lower frequency daa, in which he ARCH effec is weaker. In he daily daa models, he ARCH effec was no eliminaed. Therefore, i is said ha MS models do no exlain he volailiy clusering resen in he daily sock reurns daa, wha can be reface o furher research aiming a consrucion a Markov-swiching ARCH model (SWARCH). Coyrigh by The Nicolaus Coernicus Universiy Scienific Publishing House

9 An Alicaion of Markov-Swiching Model o Sock Reurns Analysis 67 Aendix Fig.. The WIG reurns and robabiliies of remaining in regimes for MSV() and MSMV() models (weekly; ) Noe: resuls come from Ox. Coyrigh by The Nicolaus Coernicus Universiy Scienific Publishing House Fig.. The WIG reurns and robabiliies of remaining in regimes for MSV() and MSMV() models (daily; ) Noe: resuls come from Ox.

10 68 References Brzeszczyński, J., Kalm R. (00), Ekonomeryczne modele rynków finansowych, (Economeric models of financial markes), WIG-Press, Warszawa. Engle, R. F. (98), Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica, vol. 50. Garcia, R., Pierre, P. (996), An Analysis of he Real Ineres Rae under Regime Shif, Review of Economics and Saisics, vol. 78. Hamilon, J. D. (989), A New Aroach o he Economic Analysis of Nonsaionary Time Series and he Busines Cycle, Economerica,vol. 57. Kim, C. J., Nelson, C.R. (999), Sae-Sace Models wih Regime Swiching, The MIT Press, London. Koskinen, L., Pukkila, T. (995), An Alicaion of he Vecor Auoregressive Model wih a Markov Regime o Inflaion Raes, źródła inerneowe. Krolzig, H. M. (998), Economeric Modelling of Markov_Swiching Vecor Auoregressions using MSVAR for Ox, Insiue for Economics and Saisics, Oxford. Linne, T. (00), A Markov Swiching Model of Sock Reurns: An Alicaion o he Emerging Markes in Cenral and Easern Euroe, Eas Euroean Transiion and EU Englargemen A Quaniaive Aroach, Berlin. Podgórska M. (00), Łańcuch Markowa w eorii i zasosowaniach, (A Markov chain heory and alicaion), SGH, Warszawa. Sawicki, J. (004), Wykorzysanie łańcuchów Markowa w analizie rynków kaiałowych, (A Markov chain alicaion o caial marke analysis), Wydawnicwo UMK, Toruń. Yin, P. (003), Markov Swiching in he Sock Marke, Economics, vol. 43, www. missouri.edu/~econrm/ec43f0/yin. Coyrigh by The Nicolaus Coernicus Universiy Scienific Publishing House