Time Series Econometrics. Heteroskedasticity in Stock Return Data: Volume and Number of Trades versus GARCH Effects

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1 DEPARTMENT OF ECONOMICS Uppsala Universiy Maser Thesis Auhor: Chriser Rosén Supervisor: Lennar Berg December 007 Time Series Economerics Heeroskedasiciy in Sock Reurn Daa: Volume and Number of Trades versus GARCH Effecs

2 Absrac The resul of Lamoureux and Lasrapes and Omran and McKenzie are exended o he Swedish sock marke, and his paper examines heir findings ha GARCH modelling capures he serial dependence in informaion flow ino he marke. Moreover, his paper also examines if (as a proxy for informaion flow) he number of rades can challenge he volume of rade in order o explain GARCH effecs in financial ime series. Using daa on 5 large socks ha are raded on The Nordic Sock Exchange, his paper finds ha even hough he parameer esimaes of he GARCH model becomes significanly lower for abou half of he companies in his sudy when volume of rade or he number of rades is used in he condiional variance of reurn equaion, he auocorrelaion of he sandardized residuals sill exhibi a highly significan GARCH effec in more han 1/3 of he companies when hese wo addiional explanaory variables are included in he condiional variance equaion. The serial dependence in volume of rade and number of rades does no eliminae he need for GARCH modelling of volailiy.

3 Conens Absrac... Conens Inroducion Background The informaion flow hypohesis...6. Previous sudies This sudy Marke efficiency Theory of ARCH and GARCH models Daa and mehodology Empirical resuls and analysis Conclusions References...17 Appendix A.1: Correlaion beween volume of rade/ number of rades and sock reurn daa Appendix A.: Correlaion beween volume of rade and number of rades....0 Appendix B.1: Esimaes of GARCH (1,1) model wihou volume of rade or number of rades...1 Appendix B.: Esimaes of GARCH (1,1) model wih volume of rade..... Appendix B.3: Esimaes of GARCH (1,1) model wih number of rades

4 1. Inroducion Knowledge abou volailiy forecasing is very imporan in financial markes, and i has been under consideraion by academics and praciioners for he las wo decades (Poon and Granger, 003). Much has been wrien abou forecasing performance of various volailiy models. Good volailiy models have applicaion in areas such as invesmen, securiy valuaion, risk managemen and moneary policy making. A good forecas of he volailiy in he asse under consideraion over he invesmen holding period is a good saring poin when evaluaing invesmen risk. Volailiy is one of he mos imporan facors in he pricing of derivaive securiies. To price an opion accurae we need o know he volailiy of he underlying asse from now ill he opion expires. Volailiy forecasing has also aken a cenral roll in financial risk managemen; his has made correc volailiy forecasing a compulsory exercise for many financial insiuions around he world (Poon and Granger, 003). Financial marke volailiy can also have a wide repercussion on he economy as a whole, for his reason many policy makers rely on marke esimaes of volailiy as a baromeer for he vulnerabiliy of he financial markes and he economy. The Chicago Board Opions Exchange Volailiy Index (VIX- index) measure he implied volailiy of S&P 500 index opions. This VIX- index aims o measure he markes volailiy over he nex 30 days and is naurally valuable informaion o invesors. In he Unied Saes, he Federal Reserve explicily akes ino accoun similar volailiy forecass of socks, bonds, currencies and commodes when seing is moneary policy ( Nassar, 199). Financial ime series such as sock prises can ofen appear o have periods wih large swings followed by periods wih relaively calmer swings. This is someime refered o as volailiy clusering in economeric lieraure. One hypohesis which ries o explain hese auo correlaion in swings, is he informaion flow hypohesis. In shor i saes ha when new informaion arrives o he marke, asse prices evolve. So, if informaion o he marke varies he variance of he asse prices will vary. Therefore, informaion flow can help explain volailiy clusering. 4

5 Two sudies; Lamoureux and Lasrapes (1990) and Omran and McKenzie (000) uses his informaion flow hypohesis in a formal way in order o examine if he degree of informaion o he marke can explain he degree of volailiy swings in asse prices. The aim for his paper is o analyse if such volailiy clusering described above measured by he General auoregressive condiional heeroscedasiciy (GARCH (1,1)) model can be explained by he informaion flow ino he Swedish sock marke (volume of rade and number of rades will be used as a proxy for informaion flow) for hese socks. The focus will be on answering he quesion if he volume of rade and/or he number of rades is accounable for he GARCH (volailiy clusering) effecs. This paper will limi iself o he Swedish sock marke and will use a daa se of 5 differen large socks raded on The Nordic Sock Exchange. The daa se include; daily reurns, volume of rade and number of rades during he period from o Volume of rade is he number of shares raded for a paricular sock on a paricular day, and he number of rades is he number of realized buying and selling orders for a paricular sock on a paricular day. Volume is chosen since i is he same variable used by Lamoureux and Lasrapes (1990) and Omran and McKenzie (000), and herefore i is a scope for comparison beween hese sudies. The variable number of rades is a conribuion made by his paper in order o challenge he volume of rade variable in explaining he GARCH effecs in financial ime series. This paper is organized as follow: Firsly, in secion wo, a review of he informaion flow hypohesis is presened. In addiion, earlier sudies on he subjec are briefly discussed ogeher wih how his sudy differeniaes o hem. Secondly, in secion hree, some economeric and financial conceps are examined. Thirdly, in secion four, a specificaion regarding he model used in order o es he hypohesis under consideraion is presened ogeher wih daa and mehodology. Secion five provides analysis of he empirical resuls. Finally, a conclusion is presened. 5

6 . Background This secion includes a presenaion of he informaion flow hypohesis, earlier sudies made in his area and also how his paper will differ from hem. A good undersanding of his par will also jusify he model specificaion used in he empirical secion..1 The informaion flow hypohesis The posiive correlaion beween volume of rade and asse reurns in equiy markes has been documened in lieraure (Karpoff, 1987). This saemen migh no longer be valid due o changes in he financial marke. Appendix A.1 indicaes his and shows he correlaion beween volume and reurns and for number of rades and reurn daa for he samples used in his ex. The informaion flow hypohesis discussed here is neverheless one possible explanaion regarding he variance relaionship beween informaion and he financial marke. Because daily reurns are generaed by he sum of wihin day equilibrium reurns, and because he number of wihin day reurns, n, is random, daily reurns are condiional o n (Omran and McKenzie, 000). Furher i is believed ha prices evolve when new informaion arrives ino he marke and n is se o represen he number of informaion arrivals in he marke on a cerain day. A possible explanaion for he success of GARCH models in modelling sock reurns is he informaion flow hypohesis. If i is assumed ha he number of informaion arrival and herefore he wihin day equilibrium reurns variable, n, forms a serially dependen sequence, hen i is possible ha GARCH is capuring he emporal dependence in his variable. To explain how GARCH migh capure he effec of ime dependency in informaion arrivals o he marke, he following heoreical discussion is presened. In he GARCH model he condiional variance of a ime series depend upon pas squared residuals of he process. 6

7 A possible model for daily sock reurns is: r = µ 1 + 1, ε (1) ε ( ε ε,...) ~ N(0,h ) () h = α 0 + α 1 (L) ε 1+ α (L)h -1 (3) Where r represens he rae of reurn, µ 1 is he mean r condiional on pas informaion, L is he lag operaor, and α 0 is a consan. If he parameers of he lag polynominals α 1 (L) and α (L) are posiive, hen shocks o volailiy persis over ime. The degree of persisence is deermined by he magniude of hese parameers. To moivae he empirical ess of his paper, le ψ i denoe he ih inraday equlibrium price change in day, which implies n ε = i= 1 ψ i (4) The n is he he random variable, represening he sochasic rae a which informaion flows ino he marke, so, equaion (4) implies ha daily reurns are generaed by a subordinaed sochasic process, in which ε is subordinaed o ψ i and n is he direcing process. (see Harris (1987).) Furher, if ψ i is i.i.d. wih mean zero and variance σ, and he informaion flow ino he marke is sufficienly large, hen ε n ~ N(0, σ n ). GARCH may be explained as an expression of ime dependence in he rae of evoluion of inraday equilibrium reurns. In order o make his poin very clear, assume ha he daily number of informaion arrivals is serially correlaed, which can be expressed as follows: n = k + b(l)n -1 + φ (5) Where k is a consan, b(l) is a lag polynomial of order q, and φ is whie noise. Shocks in he informaion flow o he marke persis according o he auoregressive srucure of b(l). 7

8 Define Ω = E( ε n ). If he informaion flow model is valid, hen represenaion of (5) ino his expression for variance yields Ω = σ n. Subsiuing he Ω = σ k + b(l) 1 Ω + σ φ (6) Equaion (6) capures he ype of persisence in condiional variance ha can be picked up by esimaing a GARCH model. To be precise, shocks o he informaion process lead o momenum in he squared residuals of daily reurns.. Previous sudies The ARCH process discovered by Engle in 198 has been shown o provide a good fi for many financial ime series Bollerslev (1987), Lamoure and Lasrapes (1988), Baillie and Bollerslev (1989) and Lasrapes (1989). ARCH modelling pus an auoregressive srucure on condiional variance, allowing volailiy shocks o persis over ime. This persisence capures he cluser behaviour of reurns over ime and can explain he well-documened non-normaliy and nonsabiliy of empirical asse reurn disribuions (Fama, 1965). As suggesed by Diebold (1986), Gallan, Hsieh, and Tauchen (1988), and Sock (1987, 1988), GARCH migh capure he ime series properies (e.g. serial correlaion) of he wihin day reurns variable. One previous sudy ha ried o examine he validiy of his explanaion for daily sock reurns is ha of Lamoureux and Lasrapes (1990). The sudy of Lamoureux and Lasrapes (1990) used an empirical sraegy o exploi ha GARCH effec in daily sock reurn daa reflecs ime dependence in he informaion flow o he marke. The sudy used daily rading volume as a proxy for he informaion flow, and used a sample of 0 common socks. I was found ha he GARCH effecs vanished when volume was included as an explanaory variable in he condiional variance equaion. In conclusion he Lamoureux and Lasrapes paper provides empirical suppor for he hypohesis ha GARCH is an expression for he daily ime dependence in he rae of informaion arrival o he marke for individual socks. Thus, he resul found in he Lamoureux and Lasrapes (1990) paper properly moivaes he use of GARCH models o sudy he behaviour of asse prices. 8

9 In a sudy made by Omran and McKenzie (000), he resul of Lamoureux and Lasrapes 1990 are exended o he UK sock marke, and ha sudy also finds evidence ha GARCH modelling capures he serial dependence in informaion flow o he marke. Omran and McKenzie uses daa on 50 UK companies and found ha alhough he parameer esimaes of he GARCH (1,1) model become insignifican when volume of rade is used in he condiional variance of reurn equaion, he auocorrelaion of he squared residual sill exhibi a highly significan GARCH effec, somehing ha was no examined by Lamoureux and Lasrapes. 1 In conclusion he sudy by Omran and McKenzie find consisen resul wih Lamoureux and Lasrapes 1990, ha he volailiy persisence, as measured by he GARCH model, become negligible when volume of rade is inroduced in he variance equaion of reurns. However, he hypohesis of uncorrelaed squared residuals (no GARCH effec) is rejeced. There is sill a highly significan GARCH paern in he squared sandardized residuals of he model for all bu four ou of 50 companies. Therefore, hey conclude ha GARCH effecs canno be explained only by he serial dependence in volume of rade..3 This sudy As already saed briefly in he inroducion, his paper conains a daa se of 5 frequenly raded socks on he Nordic Sock Exchange. The crieria ha he socks mos be frequenly raded is aken from Lamoureux and Lasrapes (1990). Moreover, socks wih splis during he period of sudy are excluded o eliminae possible problems from spli effecs on volume and number of rades. The daa se includes 1543 observaions. The variable number of rades is used in order o es if his conains a differen kind of informaion han does he volume of rade variable. If he number of rades are few bu he volume is high means ha every selling or buying order is relaively big. Individuals ha rade in his way migh have access o differen ypes of informaion. One difference beween his paper and he papers by Lamoureux and Lasrapes (1990) and Omran and McKenzie (000) is ha in hese wo papers he parameer esimae of he variance 1 Evidence is also found ha here is a srong associaion in he iming of innovaion ouliners in reurns and volume. The resul suggess ha a hreshold model for volume and reurn could prove a useful roue o pursue in fuure research (Omran and McKenzie, 000). 9

10 equaion is consruced o be nonnegaive. This paper does no have his resricion. The reason for his is ha he resricion is no availible in he Eviews saisical sofware, which is used by his paper for esimaion. 3. Marke efficiency In finance, volailiy is ofen referred o sandard deviaion or variance compued from a se of observaions. In financial applicaions he condiional variance is more relevan. Because his paper is concerned wih ime series economerics he condiional variance is naurally used. Marke efficiency is a heory abou wih which precision he marke prices incorporaes new informaion. If prices respond o all relevan new informaion in a rapid fashion, we say ha he marke is relaively efficien. Under he weak form of he efficien marke hypohesis (EMH), sock prices are assumed o reflec any informaion ha may be conained in he pas hisory of he sock price iself. Under he weak form of EMH he yield follows a random walk see equaion (7) below. R = C + ε where ε ~ N ( 0, σ ) (7) Where R is he sock price a ime, C is a consan and ε is a normal disribued error erm wih expeced value zero and a consan variance. I has been found empirically ha sock reurn disribuion has hicker ales (lepokurosis) han a normal disribuion. A hicker ale means ha exreme movemens are more common han a normal disribuion can explain. I has also been found ha volailiy in financial asses end o appear in cluser. Periods in which heir prices show wide variaions for an exended ime period followed by periods in which here is relaive calm. This means ha he variance is auocorrelaed in ime. For equiies, i is ofen observed ha downward movemens in he marke are followed by higher voliliies han upward movemens of he same magniude. The variance in a financial asse oday is dependen on yeserday s variance in he financial asse. When asse 10

11 prices behave in his way i is reasonable o assume ha he ime series variance follows an GARCH process (Alexander, 005). One poin o make clear is ha he EMH ses no resricions regarding volailiy movemens; i can be auocorrelaed wihou he EMH is rejeced. 3.1 Theory of ARCH and GARCH models ARCH and GARCH models are used o measure volailiy in financial ime series. As already been poined ou financial ime series, such as sock prices, exchange raes, inflaion raes, ec. ofen exhibi he phenomenon of volailiy clusering. Tha is, periods in which heir prices show wide swings for an exended ime period followed by periods in which here is relaive calm (Gujarai, 003). Knowledge of volailiy is of grea imporance when analysing he risk of holding an asse or when pricing an opion. In order o model financial ime series ha experience volailiy clusering one usually has o ake he firs difference of he logarihm of he financial ime series under analysis o make hem saionary and possible o exend in a meaningful way. Mos financial ime series are random walks in heir log level form, Tha is, hey are nonsaionary and is behaviour can only be sudied for he ime period of he acual series. As a consequence, i is no possible o generalize i o oher ime periods. The series used in his paper are saionary in heir firs difference log level form and a formal es for his has been made bu is no presened in he appendix. In order o model varying variance he GARCH (1,1) can be used (Gujarai, 003). In developing a GARCH model wo specificaions mus be provided, one for he condiional mean and one for he condiional variance. 11

12 A general GARCH(q, p) model can be wrien as; r = α + 1 β + ε, (8) r ε ( ε ε,...) ~ N(0,h ) (9) 1, p q = + α + 0 α iε i i= 1 j= 1 σ γ σ (10) j j where (8) is he mean equaion and (10) is he condiional variance equaion. The mean equaion given in (8) is wrien as a funcion of an exogenous variable and an error erm. Since σ is he one-period ahead forecas variable based on pas informaion, i is called he condiional variance. The condiional variance equaion specified in equaion (10) is a funcion of hree erms: The mean: (α 0 ). News abou volailiy from he previous period, measured as he lag of he squared residual from he mean equaion:ε -1 (he ARCH erm). Las period s forecas variance: σ -1 (he GARCH erm). The (q,p) in GARCH (q,p) refers o he presence of he order GARCH erm and he order ARCH erm. An ordinary ARCH model is a special case of a GARCH specificaion in which here is no lagged forecas variance in he condiional variance equaion. If he sum of ARCH and GARCH coefficiens (α+γ ) is close o one, volailiy shocks are quie persisen over ime. Furher, if α+γ 1 he variance is saionary, if α+γ >1 he variance is explosive, and if he α 0 and γ 0 he condiional variance is non-negaive. Because his resricion of non-negaiviy is no available in Eviews a formal es o examine if he condiional variance is saionary has been made bu no presened in he Appendix. The condiional variance series obained afer he GARCH(1,1) modell is runned was esed by he usual ADF es. The series showed ha he series was saionary for all series and he variance is herefore no explosive. 1

13 4. Daa and mehodology The daa se comprises daily reurns, volume of rade and number of rades for 5 Swedish companies during he period from o These companies were among he bigges in Sweden during he period of he sudy. The daa was obained from OMX. Volume of rade is he number of shares raded for a paricular sock on a paricular day. Volume of rade is chosen since i is he same variable used by Lamoureux and Lasrapes (1990) and Omran and McKenzie (000), and herefore here is a scope for comparison beween he sudies. Moreover, his paper adds he variable number of rades, which is he number of rades ha occurred for a paricular sock on a paricular day. In he firs sage of he analysis, he following model is esimaed for each sock in he sample: Mean equaion: r = µ 1 + ε (11) Employing hree differen specificaions of equaion (3) Variance equaions: h = α 0 + α 1 (L) ε 1+ α (L)h -1 (1) h = α 0 + α 1 (L) ε 1+ α (L)h -1 +ω 1 V (13) h = α 0 + α 1 (L) ε 1+ α (L)h -1 +ω 1 T (14) Where r is 100*log e (P /P -1 ), and P is he sock price a ime. Equaion (11) allows for an auoregression of order 1 in he mean of reurns since mos of he reurns daa exhibi a small bu significan firs order auocorrelaion (Omran and McKenzie (000)). Equaions (1), (13), and (14) models he condiional variance of he unexpeced reurns,ε, as a GARCH(1,1) process, wih he volume, V and number of rades, T, included in equaion (13) and (14). In equaion (1) hese wo variables are se o zero. 13

14 Following he same mehodology as Lamoureux and Lasrapes (1990) and Omran and McKenzie (000). Firs, he resriced model of Equaion (1) is esimaed by seing he coefficien of volume of rade and number of rades o zero, hereafer fiing a GARCH (1,1) model o he ε. of he mean equaion. In he second sage, he unresriced models of Equaion (13) and (14) are esimaed. If volume of rade or number of rades is serially correlaed, and works as a proxy for informaion arrivals o he marke, hen i can be anicipaed ha ω 1 > 0 in hose wo models, and he persisence in volailiy as measured by he sum of α1 and α becomes negligible. The ARCH LM es is used o es he hypohesis of no GARCH effecs in he residuals from he hree condiional variance models and is presened in he ables of appendix B.1, B. and B Empirical resuls and analysis Appendix B.1 shows he resul of he GARCH (1,1) model (resriced) of equaion (1). This able shows he resul of esimaing he GARCH (1,1) model o he daa se. The GARCH model suggess ha here is volailiy persisence as measured by he sum of α 1 and α because mos of he sums is close o 1. The able also shows he ARCH LM es a lag 10 o se wheher he sandardized squared residuals (SSR) exhibi addiional serial correlaion. If he variance equaion is correcly specified, here should be no effec of SSR. When he variance equaion is specified as a GARCH (1,1) model he SSR do no show any significan effecs for any of he 5 companies. Appendix B. shows ha he coefficien of volume of rade is highly significan for all companies bu hree. Furher, volailiy persisence becomes less for only slighly more han half of he socks, when compared wih he resuls repored in Appendix B.1. Moreover, when checking he ARCH LM es in order o deec serial correlaion in he SSR afer fiing he variance equaion including volume of rade, here is sill a highly significan serial correlaion in he SSR of he model for 11 ou of he 5 companies. These resuls show ha volailiy persisence decrease for abou half of he companies when volume of rade is included in he The daa was also esed agains he EGARCH and he resul was unaffeced. 14

15 variance equaion, bu ha he SSR shows serial correlaion in 11 ou of 5 companies. In summery GARCH paerns canno fully be explained by volume of rade. Appendix B.3 shows ha he coefficien of number of rades is significan for 18 ou of 5 companies and volailiy persisence becomes less for abou half of all companies. Moreover, he ARCH LM es ells ha 10 ou of he 5 companies experience serial correlaion in he SSR afer fiing he variance equaion including he number of rades as an explanaory variable. The resul from his model specificaion indicaes ha volailiy persisence decrease for mos companies versus all companies when he GARCH (1,1) model was used. Furher, serial correlaion in he SSR becomes presen. Similar o he inference drawn from he esimaes in Appendix B., he GARCH srucure is no fully explained by he addiional variable in he condiional variance equaion. One possible explanaion of hese resuls lies in he complex srucure of equaion (13) and (14). These include pas values of boh condiional volailiy h -1 and volume of rade V or number of rades T as explanaory variables. The complicaion arises because h -1 is iself a funcion of boh V -1, and T -1. Moreover, V and T are highly correlaed wih is own pas values, which can lead o a mulicollineariy problem beween he explanaory variables used h -1 and V or h -1 and T (Omran and McKenzie (000)). 15

16 6. Conclusions The papers empirical resuls, based on daa drawn from he Swedish sock marke, are o some degree differen from Lamoureux and Lasrapes (1990) and Omran and McKenzie (000). I is possible ha he difference arises because Omran and McKenzie (000) use a resriced parameer space, whereas no resricion was assumed for he esimaions in his paper. The resuls are no consisen wih heirs in ha he volailiy persisence, as measured by he GARCH componens, become negligible for all companies under sudy when volume of rade is inroduced in he condiional variance equaion. The resul from his paper find ha volailiy persisence decrease for abou 50% of he companies regardless if volume of rade or number of rades is used in he condiional variance equaion. A second difference beween his paper and he Omran and McKenzie (000) paper is ha hey found ha serial correlaion in he SSR was presen in 46 ou of he 50 companies under sudy. The numbers for his paper are 11 ou of 5 and 10 ou of 5 for volume of rade and nr of rades respecively. Because of hese resuls, his paper concludes ha GARCH effecs canno consisenly be fully explained by he serial dependence in eiher volume of rade nor he number of rades. 16

17 References Akgiray, V. (1989) Condiional heeroscedasiciy in ime series of sock reurns, evidence and forecass, Journal of Business, 6, Alexander, Carol, (005), Marke Models, John Wiley & Sons Ld. Baillie, Richard T. and Tim Bollerslev, 1989, The message in daily exchange raes: A condiional variance ale, Journal of Business and Economic Saisics, Forhcoming. Bollerslev, Tim, 1987, A condiionally heeroskedasic ime series model for speculaive prices and raes of reurn, Review of Economics and Saisics 69, Damodar N Gujarai. (003). Basic Economerics, 7 h -ediion. New York, McGraw-Hill. Diebold, Francis X., 1986, Commen on modelling he persisence of condiional variance, Economeric Reviews 5, Engle, Rober E., 198, Auoregressive condiional heeroskedasiciy wih esimaes of he variance of Unied Kingdom inflaion, Economerica 50, Eviews, Eviews 5 Help. Fama, Eugene F., 1965, The behavior of sock marke prices, Journal of Business 38, Franses, P. & van Dijk, D., (000), Non-Linear Time Series Models in Empirical Finance, Cambridge Universiy Press. Gallan, A. Ronald, David Hsieh, and George E. Tauchen, 1988, On fiing a reclaciran series: The pound/dollar exchange rae, , Mimeo, Duke Universiy. Harris, Lawrence E., 1987, Transacion daa ess of he mixure of disribuions hypohesis, Journal of Financial and Quaniaive Analysis, Karpoff, Jonahon M., 1987, The relaion beween price changes and rading volume: A survey, Journal of Financial and Quaniaive Analysis, Lamoureux, Chrisopher G. and William D Lasrapes, 1988, Persisence-in-variance, srucural change and he GARCH model, Journal of Business and Economic Saisics, Forhcoming. Lasrapes, William D., 1989, Exchange rae volailiy and U.S. moneary policy: An ARCH applicaion, Journal of Money, Credi and Banking 1, Lamoureux, C. G. and Lasrapes, W. D. (1990) Heeroscedasiciy in sock reurns daa: volume versus GARCH effecs, Journals of Finance, 45,

18 M. F. Omran and E. McKenzie. (000) Heeroscedasiciy in sock reurns daa revisied: volume versus GARCH effecs, Applied Financial Economics, 10, Markowiz, Harry M. (1991). Porfolio Selecion. Blackwell Publishers. Nasar, Sylvia. (199), For Fed, a New Se of Tea Leaves, New York Times. OMX, Rober A. Haugen. (001), Modern Invesmen Theory. 5 h -ediion. New Jersey, Prenice- Hall, Inc. Ser-Huang Poon. and Clive W. J. Granger, 003, Forecasing Volailiy in Financial Markes: A Review, Journal of Economic Lieraure, Vol. 41. No.. (Jun., 003), pp Sock, J. H. (1987) Measuring business cycle ime, Journal of Poliical Economy, 95(6), Sock, J. H. (1988) Esimaing coninuous ime processes subjec o ime deformaion, Journal of The American Saisical Associaion, 83(401),

19 Appendix A.1 Appendix Correlaion beween beween reurns and Volume of rade Company Correlaion Company Correlaion ASSA B -0,18 SWMA 0,4 HM B -0,01 VOLV B 0,6 NCC B 0,41 HOLM B 0,0 NDA SEK -0,06 SCA B 0,10 STE R 0,17 SAAB B 0,0 TREL B 0,13 PEAB B -0,04 VOST SDB 0,60 HOGA B 0,01 AZN -0,14 MTG B -0,09 ALIV SDB -0,14 AXFO 0,7 ERIC B -0,38 SHB B -0,0 INVE B -0,10 TIEN 0,08 NOKI SDB 0,37 OMX -0,0 SCV B 0,1 Mean correlaion in absolu figures: 0,17 Minus signs: 11 Correlaion beween reurns and Number of rades Company Correlaion Company Correlaion ASSA B -0,19 SWMA 0,68 HM B 0,18 VOLV B 0,67 NCC B 0,80 HOLM B 0,48 NDA SEK 0,33 SCA B 0,36 STE R 0,03 SAAB B 0,61 TREL B 0,55 PEAB B 0,40 VOST SDB 0,73 HOGA B 0,4 AZN -0,08 MTG B 0,45 ALIV SDB 0,5 AXFO 0,47 ERIC B 0,16 SHB B 0,09 INVE B 0,31 TIEN 0,17 NOKI SDB 0,6 OMX 0,10 SCV B 0,67 Mean correlaion in absolu figures: 0,38 Minus signs: 19

20 Appendix A. Correlaion beween Number of rades and Volume of rade Company Correlaion Company Correlaion ASSA B 0,80 SWMA 0,64 HM B 0,77 VOLV B 0,74 NCC B 0,57 HOLM B 0,60 NDA SEK 0,47 SCA B 0,75 STE R 0,67 SAAB B 0,5 TREL B 0,69 PEAB B 0,47 VOST SDB 0,88 HOGA B 0,4 AZN 0,87 MTG B 0,55 ALIV SDB 0,71 AXFO 0,81 ERIC B 0,58 SHB B 0,3 INVE B 0,44 TIEN 0,75 NOKI SDB 0,87 OMX 0,75 SCV B 0,47 Mean correlaion: 0,63 0

21 Appendix B.1 GARCH (1,1) Model Nr Company ARCH (α 1) GARCH(α ) α 1+α ARCH LM Tes 1. ASSA B 0,07 9,000. HM B 0,011 6, NCC B 0,081 5, NDA SEK 0,11 9, STE R 0,047 6, TREL B 0,070 5, VOST SDB 0,13 9, AZN 0,035 7, ALIV SDB 0,150 13, ERIC B 0,078 1, INVE B 0,118 7, NOKI SDB 0,014 8, SCV B 0,107 8, SWMA 0,0 6, VOLV B 0,066 5, HOLM B 0,056 5, SCA B 0,153 7, SAAB B 0,078 8, PEAB B 0,198 7, HOGA B 0,056 9, MTG B 0,095 7,400. AXFO 0,103 8,60 3. SHB B 0,094 8,10 4. TIEN 0,055 8, OMX 0,099 10,910 0,971 41,900 0, ,80 0,84 33,180 0,880 79,370 0, ,850 0,85 38,880 0,807 48,500 0,951 15,750 0,833 55,180 0,94 147,030 0,859 5,050 0,98 669,0 0,837 47,950 0,975 64,930 0,897 53,950 0,86 3,770 0,743 6,880 0,910 88,390 0,576 13,480 0, ,980 0,894 69,960 0,87 43,560 0,893 77,110 0,96 110,870 0, , No ARCH No ARCH 0.93 No ARCH 0.99 No ARCH No ARCH 0,9 No ARCH No ARCH No ARCH No ARCH 1,00 No ARCH No ARCH 0,996 No ARCH No ARCH No ARCH 0,963 No ARCH 0.88 No ARCH No ARCH No ARCH No ARCH No ARCH No ARCH No ARCH No ARCH No ARCH 0,999 No ARCH

22 Appendix B. GARCH (1,1) Model wih Volume of rade Nr Company ARCH (α 1) GARCH(α ) α 1+α ARCH LM Tes Volym of Trade 1. ASSA B 0,19 9,467. HM B 0,14 7, NCC B 0,130 5, NDA SEK 0,146 9,4 5. STE R 0,099 6, TREL B 0,08 1, VOST SDB 0,197 6, AZN 0,303 7, ALIV SDB 0,085 4, ERIC B 0,115 14, INVE B 0,117 7, NOKI SDB 0,015 0, SCV B 0,150 7, SWMA 0,39 7, VOLV B 0,0 1, HOLM B 0,11 3, SCA B 0,59 8, SAAB B 0,076 8, PEAB B 0,0 7,06 0. HOGA B 0,36 7, MTG B 0,167 9,04. AXFO 0,45 8, SHB B 0,089 8,01 4. TIEN 0,00 16, OMX 0,103 10,859 0,767 39,414 0,076 1,916-0,19-3,347 0,841 58,545 0,801 33,00-0,150-4,76 0,078 1,399 0,349 9,5-0,061 -,05 0, ,758 0,861 51,894-0,064 -,446 0,750 30,11 0,341 6,777-0,18-4,697 0,06 1,349 0,579 19,498 0,913 89,567 0,333 8,053 0,33 6,861 0,793 44,111 0,615,10 0,898 80,148 0,746 46,008 0,897 10, No ARCH 0,164 13, ARCH 1,089 0, ARCH 6,53 10, No ARCH 0,014 4, No ARCH 0,300 6, ARCH 5,038 13, ARCH 1,617 9, ARCH 0,903 13, ARCH 6,19 0, No ARCH 0,00 5, No ARCH -0,00-0, ARCH 1,971 5, No ARCH 0,093 4, ARCH 0,300 6, ARCH 1,670 1, No ARCH 10,079 19, No ARCH 0,374 8, No ARCH -0,09-1, ARCH 13,40 10, ARCH 11,59 11, No ARCH,900 8, No ARCH 3,1 15, No ARCH -0,10-6, No ARCH 3,399 7, No ARCH 0,071 1,194

23 Appendix B.3 GARCH (1,1) Model wih Number of rades Nr Company ARCH (α 1) GARCH(α ) α 1+α ARCH LM Tes Number of Trades 1. ASSA B 0,13 9,659. HM B 0,174 6, NCC B 0,049 4, NDA SEK 0,11 9, STE R 0,197 6,53 6. TREL B 0,044 7, VOST SDB 0,14 8, AZN 0,76 6, ALIV SDB 0,00 1, ERIC B 0,38 7, INVE B 0,118 7, NOKI SDB 0,006 0, SCV B 0,11 7, SWMA 0,03 5, VOLV B 0,073 5, HOLM B 0,07, SCA B 0,56 8, SAAB B 0,078 8, PEAB B 0,195 6, HOGA B 0,06 8, MTG B 0,09 7,09. AXFO 0,306 8, SHB B 0,095 8, TIEN 0,194 15, OMX 0,103 10,916 0,77 40,41-0,01-0,419-0,41-8,757 0,881 76,91 0,55 6,067-0,359-7,859 0,799 4,39 0,1 5,661-0,91-14,35 0,563 0,404 0,859 5,060-0,109-4,136 0,86 4,137 0,975 19,804 0,888 47,051-0,08-0,556 0,588 19,156 0,91 91,515 0,183 5,374 0,97 134,303 0,900 66,061 0,440 11,684 0,891 75,549 0,763 46,87 0,896 10, No ARCH 0,065 11, ARCH 0,98, ARCH 1,08 11, No ARCH -0,000-0, ARCH 0,80 14, ARCH 1,01 13, No ARCH 0,016, ARCH 0,19 15, ARCH 0,911 3, ARCH 0,061 11, No ARCH -0,000-0, ARCH 0,486 5, No ARCH 0,005, No ARCH -0,000-0, No ARCH 0,00 1, No ARCH 0,80 14, No ARCH 0,049 9, No ARCH 0,05, ARCH,313 14, No ARCH 0,059 6, No ARCH -0,005-0, ARCH 0,337 15, No ARCH 0,033 0, No ARCH 0,56 7, No ARCH 0,011 0,983 3