DYNAMIC ECONOMETRIC MODELS Vol. 8 Nicolaus Copernicus University Toruń 2008

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1 DYNAMIC ECONOMETRIC MODELS Vol. 8 Ncolaus Coperncus Unversy Toruń 008 Monka Kośko The Unversy of Compuer Scence and Economcs n Olszyn Mchał Perzak Ncolaus Coperncus Unversy Modelng Fnancal Tme Seres Volaly wh Markov Swchng Models. Inroducon An analyss of fnancal me seres volaly s an mporan ssue n makng many economc decsons. The volaly of hgh frequency fnancal seres changes over me and he perods of he hgh volaly are cluserng. Many auhors use GARCH models, nroduced by Bollerslev (986), o capure hese dependences. GARCH models descrbe he condonal varance cluserng effec bu her forecass are ofen oversaed (Anderson and Bollerslev, 998). An applcaon of he Markov swchng specfcaon o GARCH models can ouperform forecass of he sandard GARCH srucure. The frs Markov swchng model was used by Hamlon (989) n he analyss of he busness cycle. The ARCH model wh Markov swchng (SWARCH) was he frs specfcaon n hs class of models (Hamlon, Susmel, 994). Nex he SWARCH srucure was exended o GARCH parameers, gvng he MSGARCH model. The Markov swchng GARCH model was characerzed by Davdson (994), Klassen (00) and Gray (996), and each of hem defned an equaon of he condonal varance n a dfferen way. The condonal varance equaon s hen exploed n he esmaon of MSGARCH model parameers. The man purpose of hs arcle s o check wheher, a beer qualy volaly predcons can be obaned from MSARGARCH han from ARGARCH models. A frs he esmaon of hose ypes of models has been carred ou for Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House In he case of Mchał Perzak he publcaon was funded by he Ncolaus Coperncus Unversy whn he confnes of he 404 E gran.

2 56 Monka Kośko, Mchał Perzak he nvesgaed seres. Nex, havng wellesmaed models, oneday predcons have been poned ou for he fuure hry sessons. The followng sep of he research was calculang he ex pos predcon error, n he form of RMSE, n order o compare he predcon properes of he boh analyzed ypes of models.. MSARGARCH Model Frs of all, ARGARCH and MSARGARCH models dffer beween hemselves n he concepon of a volaly cluserng explanaon. The volaly n ARGARCH models s descrbed by a consderaon of prevous volaly levels (). Therefore, hs knd of a specfcaon characerzes he volaly cluserng effec que well, when perods of a low varance follow long perods of a hgh varance. In case of MSARGARCH modelng he volaly cluserng s dscussed as a resul of sayng n a one sae for some me and hen volen swchng o anoher sae. Because of he r saes occurrence, hereare r equaons of he condonal varance, so s called a mxure of dsrbuons. Ths propery allows o characerze a volaly cluserng n fnancal me seres as well as ARGARCH models. However he Markov swchng srucure should provde a beer forecas ables n comparson o ARGARCH model. The Markov swchng MSARGARCH model s gven by: y dla s = ( ) y Y, s, θ ~ M, () r y dla s = r where y = μ + ε, () ε = u h, u ~ IID( 0,), (3) y he emprcal value of he process n he h momen, Y process nformaon from he pas o he ()h momen, [ ] () ( ) ( ) ( ) ( ) θ = p, β 0, β, γ, df he esmaed parameers vecor n sae, s = he sae of he process n he h momen, {,,..., r}. The condonal mean can be descrbed by he auoregressve AR(p) process n he h momen and for h sae. AR(p) process can be wren as: μ = α ) y, (4) 0 ( s = ) + α ( s = ) y α p ( s = Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House p The GARCH and he MSGARCH models were expanded, respecvely o AR ARCH and MSARGARCH specfcaons wh auoregressve process AR(p), wch enable o descrbe an auocorrelaon n me seres.

3 Modelng Fnancal Tme Seres Volaly wh Markov Swchng Models 57 and he condonal varance n he h momen for h sae s gven by: p q 0 ( s = ) + βl ( s = ) ε l + γ k ( s = l= k h = β ) h. (5) k 3. MSARGARCH Model Esmaon An esmaon of he MSARGARCH model based on Equaon () s a compuaonally que dffcul. When you need o calculae he condonal varance h, for wo saes and GARCH(,) process, you have o consder wo equaons of he condonal varance h dependng on s =. Then for he each of varance equaons h you have o ake no accoun wo equaons of condonal varance h for he sake of he wo saes s =. Ths paern s done unl he = momen. So s seen ha he number of needed saes ncreases wh a number of me seres observaons. Therefore he esmaon of hs knd of equaon becomes unworkable. There are hree approaches o solve hs problem n he leraure. The frs approach was nroduced by Davdson (004), where he GARCH srucure s an auoregressve process wh nfne number of lags ARCH( ): h = β ( s β 0 ( s = ) = )... β ( s q + = ) k = δ ( s = ) ε Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House k. (6) An advanage of hs soluon s ha he condonal varance h depends only on sae s h. The second approach s a proposon of Gray (996), where lag of condonal varance s he expeced value of condonal varances for each sae. Hence n he case of GARCH model he condonal varance s gven by: h = 0 ( + β s = ) + β ( s = ) ε E( h ), (7) ( h ) = P( s = ) h + P( s = ) h E, (8) P s a probably ha he process n h momen s n h sae and he nformaon abou he process unl ()h momen s known. The hrd approach was proposed by Klassen (00), where he values of he condonal varance h, h are needed. These values n Klassen s approach are calculaed from equaon: where ( s = ) h = 0 ( + ( h β s = ) + β ( s = ) ε E( h ), (9) where an expeced value E ) s calculaed by an analogy from Equaon (8).

4 58 Monka Kośko, Mchał Perzak In hs paper he Davdson soluon was used o defne an equaon of condonal varance. The parameers of he research model were obaned by he maxmzaon of he loglkelhood funcon,whch can be wren as : where f l = T [ P( s = ) f ( y Y ) ( ( )) ( )], s = + P s = f y Y, s = = ln, (0) ( s = ) = P( s = ) p + P( s = ) p f ( y Y, s = ) Ps ( ) ( = ) = = f ( y Y, s = ) Ps ( = ) + f ( y Y, s = ) Ps ( = ) P, () Ps, () ( y Y s ), = Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House he densy funcon of a dsrbuon, ( s ) P = probably of he process n h momen beng n h sae and nformaon abou he process unl h momen s known, pj he condonal probably of he process swchng from he h sae o he jh sae (he ranson probably). The reurn rae forecass and he condonal varance forecass for k perods were calculaed from equaons gven by: P T + k = P T + k = () ( s = T ) y μ, (3) T + k () ( s = T ) h P T + k = P T + k = h, (4) T + k where he condonal probables P( s T ) Equaon (). T k = 4. The Fnancal Tme Seres Resuls + are receved recursvely from The emprcal analyss refer o he daly reurn raes of he companes quoed on he Warsaw Sock Exchange, whch creae WIG0 ndex 3. The values The loglkelhood funcon n (0) was consruced for wo saes bu can be done for any number of saes. 3 To esmae he ARGARCH and he MSARGARCH models, from 0 companes of he WIG0 ndex were chosen. I has been he consequence of he assumpon, ha a leas 000 observaons of a me seres should be aken no consderaon. The analyzed seres come from he perod from November 7, 000 (he dae of mplemenng WARSET sysem) o March 30, 007. The logarhmc raes of reurn have been mulpled by 00.

5 Modelng Fnancal Tme Seres Volaly wh Markov Swchng Models 59 of dsrbuon s characerscs and some ess have been presened n Table. These resuls verfy a me seres properes for he seleced sock marke reurns (Agora S.A., Telekomunkacja Polska S.A., KGHM Polska Medź S.A., PKN Orlen S.A. 4 ). All seres have ncreased kuross (more han 3) and he hypohess of he normal dsrbuon n accordance wh JarqueBera's es resuls s rejeced. The ARCH effec occurs n every me seres, wha s shown by he LjungBox es for he squares of reurn raes. Accordng o LjungBox es n he case of Agora S.A. and KGHM S.A., he auocorrelaon phenomenon appears. Table. Dsrbuon characerscs of raes of reurn for he chosen seres Dsrbuon s characerscs AGORA TP KGHM PKN Orlen Sandard davaon Assymery Kuross JarqueBera es LjungBox es (auocorelaon) Q( 0) [0.03] [0.477] [0.0] [0.40] LjungBox es (effek ARCH) Q( 0) Source: Calculaons n TSM programme, pvalues have been presened n brackes. The resuls 5 of he esmaon for he AR(p)GARCH(p,q) models are presened n Table. The esmaed parameers are sascally sgnfcan for all me seres. Whle analysng he resduals s worh o pay an aenon on he ncreased kuross n resdual processes and on he fac, ha accordng o JarqueBera's es for null hpohess, normaly of resdual dsrbuon s rejeced. Ths s he resul of he condonal Suden dsrbuon of resduals, whch wh low level degree of freedom have hgher cuross n comparson o he normal dsrbuon. The ARCH effec and he auocorrelaon phenomenon have been successfully elmnaed n he case of he all seres. The Schwarz nformaon creron and RMSE values are presened n he las wo rows of Table. The RMSE values were calculaed on he bass of he obaned predcons. Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House 4 Because of he lmed spare, only he resuls of he chosen 4 seres from he acqured resuls for companes have been presened. 5 Boh he ARGARCH and he MSARGARCH models were esmaed wh he assumpon of he normal dsrbuon or of he Suden dsrbuon of resduals.

6 60 Monka Kośko, Mchał Perzak Table. Resuls of he AR(r)GARCH(p,q) esmaon Parameers AGORA TPSA KGHM PKNOrlen α [0.03] α [0.054] β β [0.007] [0.00] γ df Assymery (resdual) Kuross (resdual) JarqueBera es (resdual) LjungaBox es (auocorelaonresdual) Q(0) 3.05 [0.868].998 [0.88] 8.63 [0.095] 7.46 [0.63] LjungBox es (effec Archresdual)Q(0) [0.64] [0.755] 5.59 [0.74] 0. [0.966] LL SC RMSE error Source: Calculaons n TSM programme, pvalues have been presened n brackes. Table 3 presens resuls of he MSARGARCH model esmaon where he parameers are sascally sgnfcan. The obaned resduals have smlar characerscs o he ARGARCH models. In he case of he all companes he ARCH effec has been elmnaed. The Schwarz nformaon creron and RMSE values are presened n he las wo rows of Table Conclusons Analyzng 6 he fng of he ARGARCH and MSARGARCH models one can easly noce he comparable Schwarz nformaon creron values. Ths means ha boh models have been fed smlarly o emprcal daa. Nex, comparng he values of he RMSE for he examned me seres one can asceran ha he errors for boh ypes of models are also comparable. I s esfed by he fac ha on he bass of analyzed companes ncluded n he WIG0 ndex, canno be asceraned ha he MSARGARCH model acqures beer volaly predcon properes han he ARGARCH model. Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House 6 The summary presens he resuls ganed from all me seres chosen from 0 companes of he WIG0 ndex.

7 Modelng Fnancal Tme Seres Volaly wh Markov Swchng Models 6 Table 3. Resuls of he MSAR(r)GARCH(p,q) swchng models esmaon Parameers AGORA TPSA KGHM PKN Orlen α [0.06] α [0.00] [0.055] p / p p / p ( ) β MSGARCH ~Suden 0 ( ) β () β [0.05] [0.0] [0.008] [0.368] () γ ( df ) ( df ).53 Assymery (resdual) Kuross (resdual) JarqueBera es (resdual) 4.05 [0.7] LjungaBox es (auocorelaonresdual) [0.99] [0.89] [0.03] [0.653] Q(0) LjungBox es (effec Archresdual)Q(0) 5.45 [0.85] 5.34 [0.756] 7.05 [0.65] 0.46 [0.959] LL SC RMSE error Source: Calculaons n TSM programme, pvalues have been presened n brackes. I may be concluded ha n he suaon of smlar models' qualy wh respec o he descrpon of emprcal daa, as well as o he volaly predcons of fnancal me seres, would be worh o choose he ARGARCH model as less complcaed model. Wha s also mporan s he usage of he MSAR GARCH model enables ganng addonal nformaon on he ranson mechansm and dynamcs of he process n he each sae. Snce all models are hghly sable n regmes, he average me o each regme and he average me of process duraon can be fxed, wha addonally ncreases he predcon properes of Markov model. Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House

8 6 Monka Kośko, Mchał Perzak References Anderson, T. G., Bollerslev, T. (998), Answerng he Skepcs: Yes, Sandard Volaly Models Do Provde Accurae Forecass, Inernaonal Economc Revew, vol. 39. Bollerslev, T. (986), Generalzed Auoregressve Condonal Heeroskedascy, Journal of Economercs, vol. 3. Ca, J. (994), A Markov Model of Uncondonal Varance n ARCH, Journal of Busness and Economc Sascs, vol.. Davdson, J. (994), Forecasng MarkovSwchng Dynamc, Condonally Heeroscedasc Proceses, Sascs and Probably Leers Dempser, A. P., Lard, N. M., Rubn, D. B. (977), Maxmum Lkelhood from Incomplee Daa va he EM Algorhm, Journal of he Royal Sascal Socey, vol. 39. Doman, R. (004), Forecasng he Polsh Fnancal Marke Volaly wh Markov Swchng Models, Macromodels 003, Wydawncwo Unwersyeu Łódzkego, Łódź 004. Doman, M., Doman, R. (004), Economerc Modelng of he Polsh Fnancal Marke Dynamc, Wydawncwo Akadem Ekonomcznej, Poznań. Gray, S. (996), Modelng he Condonal Dsrbuon of Ineres Raes As a Regme Swchng Process, Journal of Fnancal Economcs, vol. 4. Hamlon, J.D. (989), A New Approach o he Economc Analyss of Nonsaonary Tme Seres and he Busness Cycle, Economercs,vol. 57. Hamlon, J. D., Susmel, R. (994), Auoregressve Condonal Heeroskedascy and Changes n Regme, Journal of Economercs, vol 64. Km, C.J. (994), Dynamc Lnear Models wh MarkovSwchng, Journal of Economercs, vol. 60. Klassen, F. (00), Improvng GARCH Volaly Forecass Emprcal, Economcs, vol. 7. Marcucc, J. (003), Forecasng Sock Marke Volaly wh RegmeSwchng GARCH Models, Sudes n Nonlnear Dynamcs and Economercs, vol. 9. Copyrgh by The Ncolaus Coperncus Unversy Scenfc Publshng House