Lancaster University Management School Working Paper 2007/014. Trading volume and the number of trades: A comparative study using high frequency data

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1 Lancase Unesy Managemen School Wokng Pape 007/014 Tadng olume and he numbe of ades: A compaae sudy usng hgh fequency daa Mawan Izzeldn The Depamen of Economcs Lancase Unesy Managemen School Lancase LA1 4YX UK Mawan Izzeldn All ghs eseed. Sho secons of ex, no o exceed wo paagaphs, may be quoed whou explc pemsson, poded ha full acknowledgemen s gen. The LUMS Wokng Papes sees can be accessed a hp:// LUMS home page: hp://

2 Tadng Volume and he Numbe of Tades: A Compaae Sudy Usng Hgh Fequency Daa Mawan Izzeldn Depamen of Economcs, Lancase Unesy, Lancase LA1 4YX, UK. (m.zzeldn@lancase.ac.uk) Mach 007 Absac Tadng olume and he numbe of ades ae boh used as poxes fo make acy, wh dsageemen as o whch s he bee poxy fo make acy. Ths pape nesgaes hs ssue usng hgh fequency daa fo Csco and Inel n A numbe of economec mehods ae used, ncludng GARCH augmened wh lagged adng olume and numbe of ades, ess based on momen escons, egesson analyss of olaly on olume and ades, nomaly of euns when sandadzed by olume and numbe of ades, and Coelaon analyss usng olaly geneaed fom GARCH and ealzed olaly. Ou esuls show ha he numbe of ades s he bee poxy fo make acy. Keywods: Tadng olume; numbe of ades; ealzed olaly, GARCH olaly, Mxue of dsbuon hypohess.

3 I. Inoducon The olaly-olume elaon s cenal o many models n fnance and economcs. Snce he ealy 1970s, he elaon beween adng olume and sock pces olaly has been wdely nesgaed n an mpesse body of empcal and heoecal leaue. The fs eamen of he elaon goes back o Osbone (1959) n hs aemp o model he sock pce change as a dffuson pocess wh olaly elaed o he numbe of ansacons. Ths was followed by he wok of Yng (1966) and Couch (1970), who fnd a sascally sgnfcance pose coelaon beween absolue euns and daly olumes fo boh make ndces and nddual socks. Clak (1973) fnds a pose elaon beween squaed euns and aggegaed olume usng daly daa fom he coon fuues make. Wesefeld (1977) fnds a smla elaon n a sample of euns and olumes fo a numbe of common socks, as do Tauchen and Ps (1983) usng daly daa fom he Teasuy Bll fuues make. Epps and Epps (1976) fnd a pose elaon beween he sample aances of euns a gen olume leels usng ansacons fom 0 socks. Has (1986, 1987) fnds a pose coelaon beween olume and he squae of he pce change usng daly daa. Moeoe Kapoff (1987, 1988), Lamaouex and Lasapes (1990, 1994), Lesenfeld (1998, 001), Rchadson and Smh (1994) and Tauchen and Ps (1983) hae all emphaszed he ole of olume as an acy aable. All he wok ced aboe on he olaly-olume elaon comes unde wha s known as he Mxue of Dsbuon Hypohess (MDH) model. The MDH model assumes olume and olaly ae posely coelaed, and such coelaon ases due o pose assocaon of boh olume and olaly o he

4 unfoeseen nfomaon flow pocess. The MDH also ells us ha olume s he bes poxy fo make acy; hence we expec he coelaon beween olume and any olaly poxy o be an nceasng funcon of he accuacy of he olaly measue n use. Howee, hs s no he case wh he daa we use hee whee he numbe of ades s found o show hghe coelaon wh ealzed olaly han olume, whch n un suggess a olaly-numbe of ades elaon as opposed o he olaly-olume elaon mpled by he MDH. The suppo fo he numbe of ades ha we fnd n hs pape s n lne wh a gowng leaue whch ends o emphasze he ole of he numbe of ades oe olume. Fo example, Mach and Rock (1986) fnds ha he ne numbe of ades has smla explanaoy powe as ne olume. Jones e al.(1994) ague ha adng olume has no nfomaonal conen beyond ha conaned n he numbe of ades. As a esul hey sugges he use of he numbe of ades as a subsue fo olume. Moe ecen edence can be found n he wok of Easley and O'Haa (199), Easley e al. (1997), Hasbouk (1999).and Ané and Geman (000). Ths pape adds o he exsng leaue by compang olume and he numbe of ades usng hgh fequency daa fo Csco and Inel n (1997). We consde a numbe of economec pocedues peously used o addess he oles of hese wo acy aables whch nclude: a) The dec es of he MDH model adoped by Rchadson and Smh (1996). b) The augmenng of GARCH wh olume as n Lamaouex and Lasapes (1990) bu exended o allow fo he numbe of ades. c) The sandadzaon of euns by olume and ades as se

5 ou n Has (1986) and Ané and Geman (000). Moeoe we consde coelaon analyss by whch we look a he coelaon beween olume, he numbe of ades and olaly geneaed fom a aey of commonly used GARCH models - exponenal GARCH, heshold GARCH, GARCH n-hemean, faconal GARCH, faconal EGARCH, wo componens GARCH - and ealzed olaly. The oulne of hs pape s as follows. In Secon II, we dscuss he daa and he economec pocedues o be used. In secon III, we dscuss ou fndngs. We pesen ou conclusons n secon IV. II. Daa and Mehodology We use he Csco and Inel hgh fequency daa fo 1997 as used n Ané and Geman (000). We dd no hae access o Reues - he souce used by Ané and Geman (000), so we use daa fom he Whaon Reseach Daa Seces webse. We calculae he na day (9.30 am o 4 pm) euns ( ), olume ( ) and he numbe of ades ( neals. n ) a he 10, 30 and 60 mnue and daly me We consde a numbe of economec pocedues / mehods as peously used o nesgae he olaly-olume and olaly-numbe of ades elaonshps. These mehods ae descbed below. The fs mehod daws fom he wok of Rchadson and Smh (1994),

6 Andesen (1996), and Lesenfeld (1998, 001).and s based on esng he momen escons mpled by he MDH model usng he Genealzed Mehod of Momens (GMM) J- es of oedenfyng escons. The MDH assumes ha, condonal on he nfomaon flow, euns and he obseed "make acy" a (olume, log olume, he numbe of ades ec.) ae ndependenly and nomally dsbued as: 0 µ N,, a µ a 0 a (1) The model mples a se of momen escons ha can be mposed on he daa and ealuaed usng he GMM J-es of oe-denfyng escons. We consde he momen escons se ou n equaon 4 of Rchadson and Smh (1994, p. 106) whch can be wen as follows: m = µ m 1 1 m = m + µ m 1 m = 3µ m + µ m m 3 3 = µ m a 1 a 1 m = m + µ m a a 1 a m = 3µ m + µ m m m m 3 3 a a a = µµ m a 11 a 3 3 a 1 = aµ m + aµ m3 a 1 = µ am + µ am3 () whee, m,..., 1 m 3 denoe he fs hee (cenal) momens of nfomaon, m,..., 1 m 3 denoe he fs hee uncondonal (cenal) momens of euns, a m,..., 1 m a 3 denoe he fs hee uncondonal (cenal) momens of acy and a a a 11, 1, 1 m m m denoe he co-aances beween a),(, a) and ( a, ).

7 Muphy and Izzeldn (007) pon ou, he paamees µ and ae only denfed up o scale snce s no obseed. Suppose s eplaced by κ wh κ > 0 so ha m,..., 1 m 3 become κm,..., 1 κm3n he momen condons. Then µ / κ and / κ sasfy he new momen condons. Thus µ, and m canno be denfed sepaaely. To oecome hs poblem, we nomalze he mean of 1 unobseed nfomaon flow pocess m1 o one. Followng nomalzaon, he emanng sysem consss of nne momen condons and sx paamees o be esmaed ( µ, µ,,, m, m a a 3). Ths leaes hee oe-denfyng escons whch ae ealuaed usng he J-es of oe-denfyng escons. Fo example, f J > 7.815, hen we can ejec he momen escons a he 5 % leel. The acy aable whose momen escons bes f he daa, can be aken as a good poxy of make acy. In ohe wods, we seek o esablsh whehe a olaly-olume elaon o a olaly-numbe of ades s appopae fo he olaly-acy elaon mpled by he MDH. The second mehod bulds on he wok of Lamaouex and Lasapes (1990). Lamaouex and Lasapes (1990) exends he GARCH model by augmenng he GARCH aance equaon wh adng olume. Ths augmened model bee fs he daa and accommodaes fo pessence n he GARCH olaly, a esul whch suppo he ole of olume as a poxy fo make acy. We eplcae he execse by addng lagged olume o lagged ades o he GARCH aance equaon, and check whehe lagged olume o he lagged ades bee explans he GARCH effecs.

8 We consde he basc GARCH model oulned n Lamaouex and Lasapes (1990), bu wh he mean equaon gen by = c+ u (3) and he hee aance equaons as shown below α0 α1 1 α 1 = + + (4a) = α0 + α1 1+ α 1+ α3 1 (4b) = α + α + α + α (4c) n 1 We hen selec he model whch bes fs he daa usng he Akake nfomaon ceon (AIC), he sgnfcance of he coeffcens on 1 and n 1, and he leel of pessence gen by he sum α1+ α. The hd mehod noles compang he pefomance of olume and he numbe of ades n explanng olaly changes. We consde he egessons oulned n Jones e al. (1994) and Ané and Geman (000) whch ae as follows: 1 s = a+ β + ρ s + e (5a) ˆ j j j= 1 1 s = b+ γ n + ρ s + u (5b) ˆ j j j= 1 ˆ j j j= 1 1 s = c+ β + γ n + ρ s +η (5c) whee a, b and c ae consans, and n ae he fs dffeences of olume and he numbe of ades, and sˆ s he Schwe (1990) daly olaly measue.

9 To geneae s, we un a egesson of he eun oe 1 lagged euns as ˆ shown n equaon 6 below: 1 α δ j 1 j ε = j = + + (6) We hen defne sˆ ˆ π = ˆ ε (7) s The sandadzaon π follows fom an elemenay esul on he Gaussan dsbuon whch asses ha f X N(0, ), hen ( ) E X = π. In he fouh mehod we es one of he assumpons unde he MDH whch asses ha euns sandadzed by a good poxy fo acy s nomally dsbued. Clak (1973) shows ha euns subodnaed/sandadzed usng olume s nomally dsbued. Ané and Geman (000) clams ha euns sandadzed by he numbe of ades ae nomal. In ou execse we consde euns sandadzed by olume and euns sandadzed by he numbe of ades. The bes acy poxy s he one whch achees a hghe leel of nomaly fo he sandadzed euns. Fnally, we look a he coelaon of olume and ades wh olaly geneaed fom GARCH ( gach ), exponenal GARCH ( egach ), heshold GARCH ( gach ), GARCH n-he-mean ( pgach ), faconal GARCH ( fgach ), faconal exponenal GARCH ( fegach ) and wo componens GARCH ( gach ). All hese models ae used exensely n he fnancal leaue and hae been found o pode a good f o fnancal daa. See fo example,

10 Bollesle e al. (1994), Engle (001) and Glosen (1993). We also consde he coelaon beween olume, ades and ealzed olaly. Realzed olaly s defned as he sum of he na-day squaed euns whch, n he absence of mco-sucue effecs, podes an unbased and accuae measue of olaly. See, Andesen. e al. (001) and Bandoff-Nelsen and Shephad (001) fo example. The ealzed olaly n ou case s consuced by summng 5 mnue na-day squaed euns o he daly neal. III. Resuls [ Table 1 aound hee] Table 1 epos some sascal popees fo Csco and Inel olumes, log olumes, ades and he numbe of ades. We scale olume by 1/ and he numbe of ades by 1/100 o make he esuls compaable. The hghe mean and sandad deaon of and n fo Inel oe Csco, ndcae moe acy fo Inel. Log and log n ae moe nomal elae o and n as shown by he Jaque-Bea es sasc. The able also shows he esuls fo he Auoegesse faconally negaed mong aeage, Afma (p, d, q) model appled o,log, n, and log n. The faconal dffeencng paamee d shows hghe alues fo l og n. Hgh pessence s a sylzed fac of a good olaly model. Hence follows ha a good acy poxy ha s hghly coelaed wh olaly should also possess hgh pessence. Snce he numbe of ades s moe pessen han olume ndcaes ha he numbe of ades has moe n common wh olaly han olume. The esuls fo and n should

11 no be aken as defne? Snce he cusom s o apply he Afma fo he log sees and no he leel sees. [ Table aound hee] Table epos he esmaed second and hd momens of he nfomaon flow fo Csco and Inel along wh he χ (3) sasc of he J-es of oe-denfyng escons. Fo all he me neals consdeed he baae momens wh ades achees a lowe alue fo he J-es han ha wh olume excep fo he Csco (60 mnue) case, whee he esuls show moe suppo fo olume. These esuls suppo he MDH model wh boh adng olume and he numbe of ades acng as mxng aables, bu wh geae emphass on he ole of he numbe of ades. [ Table 3 aound hee] Table 3 epos he esuls of basc and augmened GARCH models, wh lagged olume and numbe of ades. Fo all cases, GARCH augmened wh lagged numbe of ades shows has lowe AIC and lowe pessence as gen α + α. These esuls show ha he numbe of ades enhances he f of he by 1 GARCH model n a smla o bee fashon o olume. [ Table 4 aound hee]

12 Table 4 epos he esuls of egesson equaons (5a, 5b and 5c) as oulned n secon II. These pode a mehod by whch o compae he pefomance of olume and he numbe of ades n explanng olaly changes. Ou esuls show mxed suppo fo olume and he numbe of ades. Fo example, n he Csco case, he numbe of ades shows a hghe R elae o olume a all he me neals consdeed. A he 60 mnue me neal he combned pesence of olume and he numbe of ades endes olume nsgnfcan. Moeoe, egessons 5b and 5c ae no sascally dffeen fom each ohe. Ths shows ha he numbe of ades has moe explanaoy powe han olume. Volume conans no exa nfomaon o ha poded by he numbe of ades. Fo Inel, olume shows a hghe R han he numbe of ades excep fo he 60 mnue me neal, whee R s hghe fo he numbe of ades. Moeoe, and smla o he Csco 60 mnue case, he pesence of he numbe of ades and olume endes he coeffcen of olume nsgnfcan. If he Csco esuls ae aken o be moe bndng (snce hey ell he same soy acoss all me neals) we can conclude ha he numbe of ades s moe coelaed wh he Schwe (1990) olaly measue han s olume. Suppo fo he numbe of ades s conssen wh Jones e al. (1994) and Ané and Geman (000) boh of whom oban (fom a smla famewok) esuls faong he numbe of ades. [ Table 5 aound hee]

13 Table 5 epos he esuls fo esng he nomaly of euns sandadzed by olume o he numbe of ades, boh escaled o hae a mean of uny. Resuls obaned show euns sandadzed by he numbe of ades ae moe nomal han hose sandadzed usng olume, as shown by he Jaque-Bea es sasc. The mplcaon s ha he numbes of ades possess moe flaon powe han olume and hence ae able o emoe some of he facos causng eun non-nomaly. To he bes of ou knowledge, no sudy has managed o ecoe full euns nomaly usng he numbe of ades o olume as sandadzng aables. [ Table 6 aound hee] Table 6 epos he coelaon beween he GARCH models, olume and he numbe of ades a he 60 mnue me neal. We consde leel coelaons and log-coelaons. Fo Csco, leel coelaon shows ha he numbe of ades s moe closely coelaed wh GARCH models olaly han s olume. On he conay, he log-coelaon esuls show ha olume s moe coelaed wh hese models. In he Inel case, esuls ae mxed. A he leel coelaon mos GARCH models show a hghe coelaon wh he numbe of ades han wh olume, wh he excepon of fegach. Usng log coelaon, gach, gach, and pgach shows a hghe coelaon wh he numbe of ades han wh olume, wheeas egach, fgach, fegach, gach ae moe coelaed wh olume han wh he numbe of ades. Theefoe he oucome of hs execse s ambguous and depends on he funconal fom whch bes descbes he elaon beween olaly and acy.

14 [ Table 7 aound hee] Table 7 shows he coelaon beween ealzed olaly, olume and he numbe of ades. Ou esuls show ha ealzed olaly s moe coelaed wh he numbe of ades han wh olume. In conas wh he coelaon fo GARCH models, hs esul holds fo boh leel and log-coelaon. Gen ha ealzed olaly s consdeed moe accuae han GARCH geneaed olaly, he esuls n able 7 hae geae cedbly: he numbe of ades s a bee poxy fo make acy han olume. IV. Concluson A numbe of economec mehods ncludng GARCH augmened wh lagged olume o numbe of ades, Tess based on momen escons and Coelaon analyss usng olaly geneaed fom GARCH and ealzed olaly ae consdeed o decde whch s he moe appopae measue of make acy: () olume o () he numbe of ades. Ou geneal concluson confms ohe fndngs fom ecen leaue: ha he numbe of ades s a bee measue of make acy han olume. Ou esuls show ha he olaly-olume elaonshp mpled by he Mxue of Dsbuon Hypohess model could also be saed as a olaly-numbe of ades elaonshp.

15 Ou sudy can be exended n aous ways. Fs: o addess he queson of why he numbe of ades s a bee poxy han adng olume. Wok would be necessay a he mcosucue leel o denfy dffeences and smlaes. Second: o exend ou esuls usng ohe measues of olaly, such as mpled olaly and ealzed ange. Ohe measues of coelaon mgh also be examned, fo example, Copulas and fequency doman based measues of coelaon such as Coheency. Thd: o nesgae whehe foecass based on GARCH could be enhanced by usng GARCH augmened wh he numbe of ades.

16 Acknowledgmens I wsh o hank Dad Peel, Anhony Muphy, Ian Paya, Kwok Tong Soo and Gey Seele fo he helpful commens. Any eos ae he auho s esponsbly. Refeences Andesen, T. G. (1996) Reun olaly and adng olume: An nfomaon flow Inepeaon of sochasc olaly, The Jounal of Fnance, 51 (1), Andesen T. G., T. Bollesle, F. X. Debold & H. Ebens (001) The dsbuon of ealsed sock eun olaly, Jounal of Fnancal Economcs, 61, Ané, T. & H. Geman (000) Ode flow, ansacon clock, and nomaly of asse euns, The Jounal of Fnance, 55(5), Bandoff-Nelsen, O. E. & N. Shephad (00) Economec analyss of ealzed olaly and s use n esmang olaly models, Jounal of he Royal Sascal Socey Sees B 64, Bollesle, T., Chu R.Y. & K. F. Kone (1994) ARCH modelng n fnance: A selece eew of he heoy and empcal edence, Jounal of Economecs 5, Clak, P. K. (1973) A subodnaed sochasc pocess model wh fne aance fo speculae pces, Economeca, 41(1), Couch, R. L. (1970) A nonlnea es of he andom-walk hypohess, Amecan Economc Reew, 60, Easley, D. & M. O.Haa (199) Tme and he pocess of secuy pce adjusmen, The Jounal of Fnance, 47 (), Easley, D., N. Kefe & M. O.Haa (1997) One day n he lfe of a ey common sock, Reew of Fnancal Sudes, 10, Engle, R. F (001) GARCH 101: The use of ARCH/GARCH models n appled economcs, Jounal of Economc Pespeces, 15, Epps, W. & M. Epps (1976) The sochasc dependence of secuy pce changes and ansacon olumes: mplcaons fo he mxue of dsbuon hypohess, Economeca, 44 (), Gallan, A. R., J. Ross & G. Tauchen (199) Sock pces and olume, Reew of Fnancal Sudes, 5,

17 Glosen, L. R., R. Jagannahan & D. E. Runkle (1993) On he elaon beween he expeced alue and he olaly of he nomnal excess euns on socks., Jounal of Fnance, 48 (5), Has, L. (1986) Coss-secuy ess of he mxue of dsbuons hypohess, Jounal of Fnancal and Quanae Analyss, 1, Has, L. (1987) Tansacon daa of he mxue of dsbuons hypohess, Jounal of Fnancal and Quanae Analyss. (), Hasbouk, J.(1999) Tadng fas and slow: Secuy make eens n eal me, Wokng Pape, New Yok Sock Exchange. Jones, M., G. Kaul & M. Lpson (1994) Tansacons, olume and olaly, Reew of Fnancal Sudes, 7(4), Kapoff, J. M. (1987) The elaon beween pce changes and adng olume: A suey, Jounal of Fnancal and Quanae Analyss, (1), Kapoff, J. M. (1988),.Cosly sho sales and he coelaon of euns wh olume, Jounal of Fnancal Reseach, 11, Lamaouex, C. G. & D. Lasapes (1990) Heeoskedascy n sock eun daa: Volume esus GARCH effecs, Jounal of Fnance, 45, 1-9. Lamaouex, C. G. & D. Lasapes (1994) Endogenous adng olume and momenum n sock-eun olaly, Jounal of Busness and Economc Sascs, 1, Lesenfeld, R. (1998) Dynamc baae mxue models: Modelng he behao of pces and adng olume, Jounal of Busness and Economc Sascs, 16, Lesenfeld, R. (001) A genealzed baae mxue model fo sock pce olaly and adng olume, Jounal of Economecs, 104, Mash, T. A. & K. Rock (1986) The ansacons pocess and aonal sock pze dynamcs, wokng pape, Unesy of Calfona a Bekeley. Muphy, A. & M. Izzeldn (007) Recoeng he momens of nfomaon flow and he nomaly of asse Reuns, fohcomng, Fnance Lees. Osbone, M. F. M. (1959) Bownan moon n he sock make, Opeaons Reseach, 7, Rchadson, M. & T. Smh. (1994) A dec es of he mxue of dsbuon hypohess: measung he daly flow of nfomaon, The Jounal of Fnancal and Quanae Analyss, 9(1),

18 Tauchen, G. & M. Ps (1983) The pce aably-olume elaonshp on speculae makes, Economeca, 51, Wesefeld, R. (1977) The dsbuon of common sock pce changes, Jounal of Quanae Analyss, 1(5), Yng, C. (1966) Sock make pces and olume of sales, Economeca, 34,

19 Table 1. Sascal popees of olume and he numbe of ades Csco log n log n log Inel n log n 10 Mnue 10 Mnue Mean Sd Skewness Kuoss JB Afma (d) S.e (0.03) 0.05 (0.031) (0.01) (0.040) (0.01) (0.04) (0.0) BIC ADF Mnue 30 Mnue Mean Sd Skewness Kuoss JB Afma (d) S.e (0.043) (0.038) (0.00) (0.000) (0.039) (0.039) (0.038) (0.036) BIC ADF Mnue 60 Mnue Mean Sd Skewness Kuoss JB Afma(d) S.e (0.040) (0.048) (0.037) (0.080) (0.061) (0.058) (0.041) (0.058) BIC ADF Noes: 1.Vaables n alcs ae found no sgnfcan a he 5% leel.. denoes olume, n denoes he numbe of ades., Sd denoes sandad deaon.,. JB denoes Jaque-Bea es sasc,. BIC denoes Bayesan nfomaon ceon, S.e denoes sandad eo. 3. Afma (Auoegesse Faconally Inegaed Mong Aeage) and d s he faconal dffeencng paamee. 4. ADF denoes Augmened Dckey Fulle Tes. The 5% and 1% ccal alues ae -.86 and Tuncaon lags fo ADF wee chosen accodng o he AIC (Akake nfomaon ceon) and ae 37, 0 and 4 fo he 10, 30 and 60 mnues fequences.

20 Table. Esmaed momens of nfomaon and J-es of oe denfyng escons Csco Inel m m 3 J χ (3) m m 3 J χ (3) 10 Mnue 10 Mnue Baae momens wh olume (0.091) (0.5) (0.4) (0.7) (1.741) (0.031) Baae momens wh ades (0.044) (0.53) (0.590) (0.68) (1.560) (0.530) Momens of e-ceneed olume Momens of e-ceneed ades 30 Mnue 30 Mnue Baae momens wh olume (0.084) (0.311) (0.975) (0.090) (0.466) (0.10) Baae momens wh ades (0.084) (0.339) (0.989) (0.14) (0.54) (0.318) Momens of e-ceneed olume Momens of e-ceneed ades 60 Mnue 60 Mnue Baae momens wh olume (0.10) (0.35) (0.894) (0.069) (0.386) (0.157) Baae momens wh ades (0.081) (0.63) (0.070) (0.79) (0.196) Momens of e-ceneed olume Momens of e-ceneed ades Noes: 1. GMM esmaes ae based on he followng 9 condons - he fs hee momens of euns ( ), he fs hee momens of "acy" a and he coaance s of ( a, ), (, a), (, a). and m ae he second and hd momens of he nfomaon flow. The alues n m 3 backes below and m ae sandad eos. m 3 3. J denoes he es of oe-denfyng escons and s dsbued as a χ. A 3 (3) degees of feedom he ccal alue a he 5 % sgnfcance leel s The alues n backes below he J-es ae p-alues

21 Table 3. GARCH, GARCH + lagged olume and GARCH + lagged numbe of ades Csco Inel Models GARCH α ( 0.004) α 0.74 ( 0.004) GARCH + (-1) GARCH + n (-1) GARCH GARCH + (-1) 10 Mnue 10 Mnue (0.00) (0.000) (0.005) (0.006) (0.003) α (0.000) (0.003) α (0.000) (0.003) (0.007) (0.000) GARCH n + (-1) (0.006) (0.010) (0.000) α + α AIC Mnue 30 Mnue α ( 0.008) (0.015) (0.015) (0.017) (0.01) (0.00) α ( 0.009) (0.011) α (0.006) (0.011) α (0.005) (0.0) 0. (0.0) 0.01 (0.000) 0. (0.01) (0.000) α + α AIC Mnue 60 Mnue α ( 0.011) (0.009) (0.010) (0.014) (0.014) (0.01) α ( 0.013) 0.83 (0.018) α (0.000) (0.013) α (0.001) (0.07) (0.06) (0.000) 0.85 (0.05) (0.000) α + α AIC Noes: 1. The GARCH mean equaon s gen by = c+ u. denoes olume and n denoes he numbe of ades.. Thee specfcaons fo he GARCH aance equaon ae consdeed: a) α α α b) = α + α + α + α = + +, ,c) = α + α + α + α n

22 Table 4. Regesson esmaes fo olume and he numbe of ades 1 = α +, ˆ ˆ δ + ε s = ε j= 1 j j 1 j= 1 π s = a+ β ˆ + ρ j s j +e 1 s = b+ γ n + ρ s + u ˆ j j j= 1 s = c+ β + γ n + ρ s +η ˆ j j j= 1 Csco 1 n and n β R γ R β γ R 10 Mnue Esmaes Sandad (0.006) (0.006) (0.0038) (0.0039) eos 30 Mnue Esmaes 0.38e e e e Sandad (0.33e-03) (0.579e-04) (0.491e-03) (0.5943e-04) eos 60 Mnue Esmaes e Sandad eos (0.1743e-0) (0.1999e-0) (0.5546e-0) (0.6383e-0) Inel 10 Mnue Esmaes Sandad (0.0015) (0.0011) (0.0018) (0.0016) eos 30 Mnue Esmaes 0.181e e e e Sandad (0.181e-03) (0.1141e-04) (0.810e-03) (0.486e-04) eos 60 Mnue Esmaes e Sandad eos (0.9506e-03) (0.899e-03) (0.971e-0) (0.809e-0) Noes: 1. Numbes shown n alcs ae no sgnfcan a he 5% leel.. denoes euns, olume, and n s denoes he Schwe (1990) daly olaly measue, ˆ denoes he numbe of ades. denoes

23 Table 5. Recoeng nomaly usng e-ceneed (olume and he numbe of ades) Csco Inel 10 Mnue 10 Mnue Skewness Kuoss JB 4.8e e e p-alues (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) 30 Mnue 30 Mnue Skewness Kuoss JB p-alues (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) 60 Mnue 60 Mnue Skewness Kuoss JB p-alues (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) Noes: 1. JB denoes Jaque-Bea es sasc.. and n denoes olume and ades and whch hae been e-ceneed po o he sandadzaon so ha o hae a mean of uny. n n

24 Table 6. Coelaon coeffcens fo olume and numbe of ades wh GARCH models Csco (60 Mnue) Leel - Coelaon gach egach gach pgach fgach fegach gach n gach egach gach pgach Log - Coelaon fgach fegach gach n gach egach gach pgach Inel (60 Mnue) Leel - Coelaon fgach fegach gach n gach egach gach pgach Log - Coelaon fgach fegach gach n Noes: 1. denoes olume and n denoes he numbe of ades.. gach denoes olaly fom GARCH, denoes olaly fom exponenal GARCH, denoes olaly fom heshold GARCH, denoes olaly gach fom GARCH n-he-mean, fgach egach pgach n n n n denoes olaly fom faconal GARCH, denoes olaly fom faconal exponenal GARCH, and fegach gach olaly fom wo componen GARCH. denoes

25 Table 7. Coelaon coeffcens fo olume and numbe of ades wh ealzed olaly Csco (Daly) Leel - Coelaon n Log - Coelaon n Leel - Coelaon Inel (Daly) Log - Coelaon n Noes: 1. JB denoes Jaque-Bea es sasc, denoes ealzed olaly, denoes olume, and n denoes he numbe of ades. n