RELATIONSHIP BETWEEN VOLATILITY AND TRADING VOLUME: THE CASE OF HSI STOCK RETURNS DATA Mchaela Chocholaá Unversy of Economcs Braslava, Slovaka
Inroducon (1) one of he characersc feaures of sock reurns s he me-varyng volaly he poneerng work n he area of modellng volaly was presened by Engle [1982] nowadays a large number of modfcaons of he sandard ARCH and GARCH models have been developed hough he ARCH/GARCH class models allow he volaly shocks o perss over me, hey ddn provde he economc explanaon for hs phenomenon
Inroducon (2) he paper of Lamoureux and Lasrapes [1990] offers he explanaon for volaly perssence her approach has been appled n varous sudes o boh ndvdual socks (sock-level analyss) and sock marke ndces (marke-level analyss) hey proved ha he daly radng volume has a sgnfcan explanaory power regardng he varance of daly reurns
The am of he presenaon: o analyse he relaonshp beween he radng volume and he daly volaly of he Hong Kong HSI sock reurns daa usng he GJR-GARCH models and applyng he approach of Lamoureux and Lasrapes [1990]
Daa and Mehodology he whole analyss was done on logarhmc ransformaon of daly ndex reurns and daly radng volume he logarhmc sock reurns are calculaed as he logarhmc frs dfference of he daly closng values of he analyzed sock ndex,.e. P r P d(ln( P)) ln P where s he closng value of he sock ndex a me and r denoes logarhm of he correspondng sock reurn 1
Closng values of he HSI sock ndex and descrpve sascs of he logarhmc reurn seres 32000 28000 24000 20000 16000 12000 8000 2005 2006 2007 2008 2009 2010 CLOSE 500 400 300 200 100 0-0.10-0.05-0.00 0.05 0.10 Seres: DLCLOSE Sample 1/03/2005 3/31/2011 Observaons 1535 Mean 0.000328 Medan 0.000947 Maxmum 0.134068 Mnmum -0.135820 Sd. Dev. 0.018115 Skewness 0.081094 Kuross 11.53212 Jarque-Bera 4657.657 Probably 0.000000
Mehodology condonal mean equaon he logarhmc sock reurns equaon,.e. he condonal mean equaon, can be n general wren as a Box-Jenkns ARMA(m,n) model of he form: r m n 0 + φ jr j + θkε k j 1 k 1 ω + ε where ω0 s unknown consan, φ ( j j 1,2,...m ) and θ k ( k 1,2,...n ) are he parameers of he approprae ARMA(m,n) model, s a dsurbance erm. ε
Mehodology condonal varance equaon he condonal varance equaon n case of a GJR-GARCH(p,q) model can be specfed as: where from, s clear he dfferen mpac of he posve shocks and negave shocks on he condonal varance o examne he effec of radng volume on sock reurns volaly, he followng modfcaon of he condonal varance equaon s used + + + q p q I h h 1 2 1 1 2 0 ε γ β α ε α > < 0 0, 0 1, f f I ε ε > 0 ε 0 < ε q p q V I h h δ ε γ β α ε α + + + + 1 2 1 1 2 0 V
Emprcal resuls he analyss was done n wo seps whou and wh radng volume ncluded no he condonal volaly equaon he approprae ARMA (m,n) model for logarhmc sock reurns was esmaed (able 1) he esmaon resuls of condonal varance equaons whou and wh he radng volume ncluded usng he GJR-GARCH(1,2) model are n able 2 and able 3, respecvely
Table 1 Dependen Varable: D(LOG(CLOSE)) Mehod: Leas Squares Dae: 11/20/11 Tme: 18:59 Sample (adjused): 1/04/2005 3/31/2011 Included observaons: 1534 afer adjusmens Convergence acheved afer 5 eraons Backcas: 12/21/2004 1/03/2005 Varable Coeffcen Sd. Error -Sasc Prob. C 0.000328 0.000429 0.762914 0.4456 MA(10) -0.070451 0.025503-2.762422 0.0058 R-squared 0.005097 Mean dependen var 0.000327 Adjused R-squared 0.004447 S.D. dependen var 0.018121 S.E. of regresson 0.018080 Akake nfo creron -5.186683 Sum squared resd 0.500808 Schwarz creron -5.179726 Log lkelhood 3980.186 F-sasc 7.848098 Durbn-Wason sa 2.075733 Prob(F-sasc) 0.005151 Invered MA Roos.77.62+.45.62-.45.24-.73.24+.73 -.24-.73 -.24+.73 -.62-.45 -.62+.45 -.77
Table 2 Dependen Varable: D(LOG(CLOSE)) Mehod: ML - ARCH (Marquard) - Normal dsrbuon Dae: 11/20/11 Tme: 19:03 Sample (adjused): 1/04/2005 3/31/2011 Included observaons: 1534 afer adjusmens Convergence acheved afer 14 eraons MA backcas: 12/21/2004 1/03/2005, Varance backcas: ON GARCH C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-1)^2*(RESID(-1)<0) + C(6)*RESID(-2)^2 + C(7)*GARCH(-1) Coeffcen Sd. Error z-sasc Prob. C 0.000376 0.000277 1.357983 0.1745 MA(10) -0.014608 0.025503-0.572803 0.5668 Varance Equaon C 3.07E-06 6.99E-07 4.398775 0.0000 RESID(-1)^2-0.064064 0.016873-3.796743 0.0001 RESID(-1)^2*(RESID(-1)<0) 0.109838 0.016381 6.705210 0.0000 RESID(-2)^2 0.125669 0.025270 4.973077 0.0000 GARCH(-1) 0.871804 0.015290 57.01663 0.0000 R-squared 0.001888 Mean dependen var 0.000327 Adjused R-squared -0.002034 S.D. dependen var 0.018121 S.E. of regresson 0.018139 Akake nfo creron -5.772294 Sum squared resd 0.502423 Schwarz creron -5.747947 Log lkelhood 4434.350 F-sasc 0.481319 Durbn-Wason sa 2.069974 Prob(F-sasc) 0.822649 Invered MA Roos.66.53+.39.53-.39.20+.62.20-.62 -.20+.62 -.20-.62 -.53-.39 -.53+.39 -.66
Table 3 Dependen Varable: D(LOG(CLOSE)) Mehod: ML - ARCH (Marquard) - Normal dsrbuon Dae: 11/20/11 Tme: 19:04 Sample (adjused): 1/04/2005 3/31/2011 Included observaons: 1534 afer adjusmens Convergence acheved afer 25 eraons MA backcas: 12/21/2004 1/03/2005, Varance backcas: ON GARCH C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-1)^2*(RESID(-1)<0) + C(6)*RESID(-2)^2 + C(7)*GARCH(-1) + C(8)*LOG(VOLUME) Coeffcen Sd. Error z-sasc Prob. C 0.000409 0.000268 1.522045 0.1280 MA(10) -0.020115 0.023929-0.840626 0.4006 Varance Equaon C -0.000119 2.59E-05-4.617316 0.0000 RESID(-1)^2-0.068117 0.014880-4.577800 0.0000 RESID(-1)^2*(RESID(-1)<0) 0.160458 0.022972 6.984846 0.0000 RESID(-2)^2 0.117919 0.023677 4.980276 0.0000 GARCH(-1) 0.816716 0.021827 37.41748 0.0000 LOG(VOLUME) 6.31E-06 1.34E-06 4.713631 0.0000 R-squared 0.002474 Mean dependen var 0.000327 Adjused R-squared -0.002101 S.D. dependen var 0.018121 S.E. of regresson 0.018140 Akake nfo creron -5.787474 Sum squared resd 0.502128 Schwarz creron -5.759648 Log lkelhood 4446.992 F-sasc 0.540768 Durbn-Wason sa 2.070523 Prob(F-sasc) 0.803971 Invered MA Roos.68.55-.40.55+.40.21+.64.21-.64 -.21+.64 -.21-.64 -.55-.40 -.55+.40 -.68
he receved resuls show que hgh degree of he q p volaly perssence, snce he sum ˆα + s 1 ˆβ 1 hgh n model whou radng volume varable akes value of 0,933409 and besdes hs fac also he exsence of he leverage effec was proved (snce he correspondng parameer s sascally sgnfcan and posve) n model wh radng volume varable he volaly perssence slowly declned o 0,866518
The dagnosc check sascs of he sandardzed resduals n order o have he nformaon abou adequacy of he presened esmaes, we esed he sandardzed resduals he uncorrelaedness of he sandardzed resduals and squared sandardzed resduals was proved usng he Ljung Box Q sascs and Q 2 sascs, respecvely he normaly was no confrmed (Jarque Bera es)
Condonal varance whou and wh he radng volume ncluded.007.007.006.006.005.005.004.004.003.003.002.002.001.001.000 2005 2006 2007 2008 2009 2010.000 2005 2006 2007 2008 2009 2010 Condonal varance whou he radng volume Condonal varance wh he radng volume ncluded
Concludng remarks he logarhm of he radng volume was ncluded no he condonal volaly equaon n order o nvesgae f s a good proxy for nformaon arrval akng no accoun some oher papers (e.g. [Grard and Bswas 2007], [Gursoy e al. 2008], [Sharma e al. 1996]), he resuls of our analyss concde wh hers,.e. ha he radng volume can be n general consdered (n case of he marke-level analyss) o be only a poor proxy for nformaon flow
References (Exrac) Engle R.F. (1982) Auoregressve Condonal Heeroscedascy wh Esmaes of he Varance of Uned Kngdom Inflaon, Economerca, 50, No. 4. Grard E., Bswas R. (2007) Tradng Volume and Marke Volaly: Developed versus Emergng Sock Markes, The Fnancal Revew, 42, pp. 429 459. Gursoy G., Yuksel A., Yuksel A. (2008) Tradng volume and sock marke volaly: evdence from emergng sock markes, Invesmen Managemen and Fnancal Innovaons, Vol. 5, Issue 4, pp. 200 210. Lamoureux C., Lasrapes N. (1990) Heeroscedascy n sock reurn daa: volume versus GARCH Effecs, The Journal of Fnance, Vol. XLV, No. 1, pp. 221-229. Sharma J.L., Mougoue M., Kamah R. (1996) Heeroscedascy n sock marke ndcaor reurn daa: volume versus GARCH effecs, Appled Fnancal Economcs, Vol. 6, pp. 337-342.