Econometric Modelling of. Selected Approaches. Michaela Chocholatá University of Economics Bratislava

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1 Economerc Modellng of Fnancal Tme Seres: Seleced Aroaches Mchaela Chocholaá Unversy of Economcs Braslava

2 The man am of he resenaon a bref nroducon no he economerc modellng of he fnancal me seres ars: volaly modellng usng he ARCH-class models nvesgaon of he relaonshs beween ars of me seres based on correlaon analyss, conegraon conce and Granger causaly wh llusraon for he Neherland sock exchange daa

3 Inroducon (ARCH-class Models) fnancal me seres are mosly analysed n he form of reurns (frs dfferences), he man feaure of whch s he me-varyng volaly condonal volaly aroach - Engle (98), Bollerslev (986), ec. oday - huge amoun of varous ARCH-class models from he ersecve of he raonal nvesor reresens he hegh of reurns only one ar of he decson makng rocess anoher essenal facor o ake no accoun s he amoun of rsk,.e. he volaly of reurns ogeher wh s seasonal anomales (urn-of-he-year, urn-of-he-monh, day-of-he-week effecs, ec.) neresng area s also o nvesgae he role of he radng volume n exlanaon of he volaly erssence of sock reurns

4 Daa and Mehodology he analyss s beeng usualy done on logarhmc ransformaon of reurn seres he logarhmc reurn seres are also calculaed as he logarhmc frs dfferences of he ndvdual seres measured n levels, e.g. daly closng values of he sock ndces or daly exchange raes: P r = P d(ln( P)) = ln P where s he rce (e.g. closng value of he sock ndex, exchange rae) a me and r denoes logarhm of he corresondng reurn

5 Mehodologcal asecs: he logarhmc reurns euaon,.e. he condonal mean euaon - Box-Jenkns ARMA(m,n) model condonal varance euaon - arorae ARCH-class model

6 Some ARCH-class Models Model/ auhors, year ARCH() Engle, 98 GARCH(,) Bollerslev, 986 IGARCH(,) Engle and Bollerslev, 986 EGARCH(,,r) Nelson, 99 TGARCH(,) Zakoan, 99 Mahemacal form/ arameer condons/some exlanaons h = α + αε + αε αε α >, α for =,,..., = α + αε + β = =, α, h h α for =,...,, β for =,,..., α + αε + β for α + h = h ln( = ε = = h ) = α + α + β ln( h ) = h = + = β = ε r k γ k k = h k leverage effec: γ <, asymmery n volaly: γ h k = α + αε + β h + γ ε I = = = > α, α + γ for,,..., α, =, β for =,,...,, f ε < I =, leverage effec: γ >, asymmery n, f ε > volaly: γ PARCH(,) δ δ Dng e al., 993 ( h ) = α + α( ε γ ε ) + β ( h ) = > α, δ, β for =,,...,, α a < γ < for = =,,...,, asymmery n volaly: γ δ k

7 Modellng of seasonal anomales: a GARCH(,) case sudy Condonal mean euaon r = ω + = + ε + Condonal varance euaon m φ r n k = θ ε k k + l= Κ l Y l +ΛM + 5 r= Π r D r h = α + = α ε + β h + κ lyl + λm + = l= 5 r= π D r r urn-of-he-year effec: dummy varables monhs from February o December reresenng urn-of-he-monh effec: dummy varable M akng he value of one n he monh s frs ffeen calendar days (and zero n he remanng days) day-of-he-week effec: dummy varables r ndvdual radng days wh exceon of Monday Y l ( l =,3,..., ) D ( r =,3,4,5 ) for

8 The role of he radng volume n exlanaon of he volaly erssence of sock reurns GARCH(,) case sudy o examne he effec of radng volume on sock reurns volaly, he followng modfcaon of he condonal varance euaon s used h = + αε + = = α β h + δv where V s he logarhm of he radng volume accordng o he Lamoureux e al. (99) he arameer should be osve and he volaly erssence should become neglgble δ

9 Ineracons beween he Fnancal Tme Seres e.g. muual relaonshs beween: :exchange raes and sock ndces :ars of exchange raes :ars of sock ndces varous rocedures: :grahcal analyss :correlaon rocedures :long-run relaonshs - conegraon rocedures (Engle-Granger, Johansen) :shor-run relaonshs - Granger causaly, mulse resonses analyss

10 Long-run relaonshs conegraon rocedures: Engle-Granger = conegraon aroach for a bvarae sysem (one conegrang vecor) -based on esng of resduals from conegrang euaon erformng e.g. he ADF es on resduals Johansen = more general echnue alcable also n case of more han wo varables (for N number of varables we can have u o N- conegrang vecors) -based on maxmum lkelhood mehod usng boh he λrace and λ sascs max

11 Shor-run relaonshs Granger causaly -we can say, ha he me seres x Granger-causes me seres y, f y can be redced beer by usng as values of x han by usng only he hsorcal values of he y -f hs doesn hold, we can say ha x doesn Granger-cause y -wheher he me seres y Granger-causes he me seres x can be esed n analogcal way -he corresondng VAR(k) model: k k y = α + α x + β y + ε = = k k x = α + α x + β y + ε = = -esng sascs: Wald F-sascs Imulse resonses analyss -o examne he shor-run dynamc relaons beween analysed me seres

12 Relaonshs beween he overall sock marke ndex and s secoral ndces: A Case Sudy for Neherland Grahcal analyss Q Q3 Q4 Q Q Q3 Q4 Q LAEX LBAS_MA LCON_GO LCON_SER LFIN LHC LINDU LOIG LTECH LTEL general ndex: AEX secoral ndces: Basc Maerals (BAS_MA), Consumer Goods (CON_GO), Consumer Servces (CON_SER), Fnancals (FIN), Healh Care (HC), Indusrals (INDU), Ol&Gas (OIG), Telecomuncaons (TEL), Technology (TECH) source of daa: NYSE Euronex webage Noe: refx L denoes naural logarhm

13 Descrve sascs and ADF es resuls - ndces Indces LAEX LBAS_ MA LCON_ GO LCON_ SER LFIN LHC LINDU LOIG LTECH LTEL Mean 5,7966 6,8978 7,39 6,9644 6,459 6,44 6,998 6,8397 6,8 7,64 Maxmum 5,948 7,35 7,6 7,78 6,46 6,6398 7,894 7,7 7,38 7,864 Mnmum 5,5738 6,3854 6,8797 6,769 5,79 6,33 6,647 6,645 6,5696 6,86 Sd. Dev.,793,63,48,89,8,59,43,84,8,78 Skewness -,76 -,868 -,965 -,9383 -,5488,99 -,7487 -,967,37 -,763 Kuross,868,376 3,477,6388,39,46,39,98,779,7439 Jarue-Bera 45,7 65,6 84, 78,4 35,5 57,3 58,5 6,6 9,7 46,7 ADF sa. (whou rend and whou consan) -,7 -,635 -,463 -,44 -,334,398 -,46,8,665 -,33 Noe: Symbols, **, * denoe n he whole resenaon he reecon of he null hyohess on he %, 5 %, and % sgnfcance level, resecvely.

14 Descrve sascs and ADF es resuls - reurns Reurns DLAEX DLBAS _MA DLCON _GO DLCON_ SER DLFIN DLHC DLINDU DLOIG DLTECH DLTEL Mean, -,6,, -,,4,,5,5 -,7 Maxmum,77,85,67,396,4,3,3,4,548,366 Mnmum -,457 -,84 -,393 -,463 -,886 -,997 -,76 -,493 -,536 -,86 Sd. Dev.,34,9,5,,5,89,65,5,65,6 Skewness,7,7,944 -,3599,347 7,549,479 -,46 -,6 -,377 Kuross 5,45 5,4 4,835 5,544 7,6347 3,668 7,636 4,77 3,8564 9,33 Jarue-Bera 8,8 89,4 7, 9,9 469, ADF sa. (rend+consan) -,6 -,356 -,37 -,363 -,644 35, 46, 76,9 5,7 94,3 -,3557 -,366-9,6799 -,6954 -,67

15 Correlaon coeffcens of AEX ndex wh s secoral ndces based on he grahcal analyss we can exec ha here are some relaonshs beween he overall ndex (AEX) and some of s secoral ndces whch was roved by calculaon of correlaon coeffcens: Correl. coeff. LBAS_ MA LCON_ GO LCON_ SER LFIN LHC LINDU LOIG LTECH LTEL levels,944,8484,97,9393,7535,944,3,64,656 reurns,8948,8488,8446,949,3948,883,7876,76,556

16 Engle-Granger conegraon es resduals ADF es sascs LBAS_MA LCON_GO LCON_SER LFIN LHC LINDU LOIG LTECH LTEL ADF sa. -,488-3,735 -,3387 -,895 -, ,757 -, ,8797 -,7883 was no ossble o reec he un roo hyohess,.e. no long-run relaonsh was confrmed

17 Granger causaly es resuls H: AEX ndex reurns does no Granger cause secoral ndex reurns F-sasscs Lags: Lags: Lags: 3 Lags: 4 Lags: 5 DLBAS_MA,579,6568*,36*,4*,7889 DLCON_GO,34,756,869,693,644 DLCON_SER 9,363 4,884 3,66**,554**,58* DLFIN,98,66,535,558,5566 DLHC,555,845,899,698,593 DLINDU 3,3* 3,477**,57*,78* 3,654 DLOIG,686,37,393,3748,576 DLTECH,74,553,396,36,469 DLTEL,33,8665,747,44,833

18 A Case Sudy Concluson was no ossble o reec he hyohess abou he exsence of he un roo for he overall AEX ndex as well as for all he secoral ndces; he reurns were saonary concernng he grahcal analyss and correlaon coeffcens, he sronges relaonshs (correlaon coeffcen more han,9) were ndenfed beween LAEX and followng secoral ndces: LBAS_MA, LCON_SER, LFIN, LINDU based on Granger causaly es resuls, was roved ha for ars wh confrmed shor-run relaonshs he correlaon coeffcens were hgh no long-run relaonsh was confrmed smlarly as Waścńsk e al. (9) we roved he exsence of he shor-run relaonshs n some cases (alhough he resuls are srongly deenden on he concree analysed ar of ndces), bu he long-run relaonshs of he overall AEX ndex o s secoral ndces were confrmed n neher case

19 References Bollerslev T. (986) Generalzed Auoregressve Condonal Heeroscedascy. Journal of Economercs, Vol. 3, No.3. Chocholaá M. () Relaonshs Beween General Index and s Secoral Indces: A Case Sudy for Seleced Euroean Indces. Proceedngs of he Inernaonal Scenfc Conference 3 h May s June, Braslava, Slovaka, Engle R.F. (98) Auoregressve Condonal Heeroscedascy wh Esmaes of he Varance of Uned Kngdom Inflaon, Economerca, Vol. 5, No. 4. Engle R. F., Granger C. W. J. (987) Co-negraon and Error Correcon: Reresenaon, Esmaon, and Tesng. Economerca, Vol. 55, Granger C. W. J. (969) Invesgang Causal Relaons by Economerc Models and Cross-Secral Mehods. Economerca, Vol. 37, July, Lamoureux C., Lasraes N. (99) Heeroscedascy n sock reurn daa: volume versus GARCH Effecs, The Journal of Fnance, Vol. XLV, No.,. -9. Johansen S. (988) Sascal Analyss of Conegraon Vecors. Journal of Economc Dynamcs and Conrol, Vol., Waścńsk T., Przekoa G., Sobczak, L. (9) A Long- and Shor-Term Relaonsh Beween Secoralndexes and he WIG (Warsaw Exchange Index). Polsh Journal of Envronmenal Sudes, Vol. 8, No. 5B,