Weekly Predictability of Daily Returns: A Periodic Autoregressive Model Approach. Louhelainen, Mika. ISBN ISSN X no 23

Transcription

1 Weekly Predicabiliy of Daily Reurns: A Periodic Auoregressive Model Approach Louhelainen, Mika ISBN ISSN X no 3

2 Weekly Predicabiliy of Daily Reurns: A Periodic Auoregressive Model Approach Mika Louhelainen Universiy of Joensuu Deparmen of Business and Economics Yliopisokau 7, Box, FIN mika.louhelainen@joensuu.fi Absrac A radiional assumpion regarding he raes of he reurns of sock marke specifies ha he expeced raes of reurns are idenical for all days of he week. In his aricle, ha presumpion is quesioned and predicabiliy of daily reurns from he previous weekday s reurns is esed wih he Periodic Auoregressive (PAR) model. The daa used consis of daily indices from nine inernaional sock markes from period 990 o 003. Empirical evidence implies ha disribuion of daily raes of reurn is non-normal. To ensure reliabiliy of he empirical resuls of hypohesis ess, coefficien confidence inervals are boosrapped. Resuls illusrae ha a leas some weekday reurns are periodically prediced. Acknowledgemens: The auhor would like o hank Mika Linden and Markku Lanne for helpful commens and suggesions. Financial suppor from he Jenny and Ani Wihuri -foundaion is graefully acknowledged.

3 I. Inroducion Numerous sudies have documened differen kind of unexplained ime dependen and seasonal behavior in he sock prices over pas decades. From he view of financial heory seasonals denoe ha sock markes are inefficien. On he oher hand, regulariies in sock prices imply profi opporuniy from invesors view. According efficien marke hypohesis, sock prices fully reflec all available informaion and follows he maringale behavior, which infer ha sock prices are no predicable. Afer several anomalies were revealed in sock markes, Fama (970) allocaed efficiency in sock markes in ree subses: weak form (can pas reurns predic fuure reurns), semi form (how quickly do sock prices reflec o new informaion) and srong form (are invesors privae informaion fully refleced in o sock prices). Well known examples from seasonal anomalies are Day of he Week and January effecs. Researchers have deeced in many sudies ha daily reurns are no equal, and in January reurns are larger han in oher monhs. The lieraure before 970 s exhibied suggesive evidence ha daily, weekly and monhly reurns are predicable from pas reurns. For example, Fama (965) finds posiive firs-order auocorrelaion of daily reurns for 3 of he 30 socks in Dow Jones. Fisher (966) found in his research ha auocorrelaion of monhly reurns on porfolios are posiive and larger han hose for individual socks. From he view of oday s saisical knowledge he evidence for predicabiliy in he early research ofen lacks of saisical power. Lo & MacKinlay (988) and LeBaron (99) among ohers coninued o examine sock index auocorrelaions. They found posiive auocorrelaed a high frequencies. Several reasons are proposed for anomaly concurrency. Typically, in anomaly sudies ransacion coss are assumed o be zero. Grossman and Sigliz (980) supposed ha a precondiion for he efficien marke hypohesis is ha informaion and rading cos in sock marke boh are always zero. However, ransacion coss are and will be a par of sock markes in real life. Jensen (978) proposed economically more sensible version of he efficien marke hypohesis saying ha prices reflec informaion o he poin where he marginal benefis of acing on informaion do no exceed he marginal cos. See M. Gibbons and P. Hess (98), A. Agrawal and K. Tandon (994), R.Bhardwaj and L.D. Brooks (99), for Day of he week and January anomaly.

4 Neverheless, he sock markes do no auomaically come efficien. Invesors have o deec he inefficiency on he sock marke and rade off i away. This paper models weekday anomaly wih hree differen ways. The firs model, simple PAR() model, ess if previous week daily reurns have predicion power over his weekday daily reurns a he same weekdays. The second model is VAR() model where all weekday raes of reurns are explained by all weekday reurns from he previous week. Third model includes in PAR() model AR() parameers. The paper is organised as follows: Secion describes he models used, Secion 3 presens algorihm of boosrap esimaion, and how confidence inervals for coefficien esimaes are defined wih boosrap mehod, Secion 4 presens he empirical resuls and Secion 5 concludes. II. Models Usually in weekday anomaly sudies daily raes of reurn are esimaed from he equaion where only he dummy variables are predicaors for reurns. Such model is useful when we wan o es if he raes of reurn for every weekday are equal. The model produces merely average raes of reurns for weekdays, bu no much predicion informaion. Sock reurn auocorrelaions give predicion informaion. A periodic auoregression 3, PAR model, provides esimaes ha can be used o predicae weekday s nex reurn from he weekday s previous observaion. The PAR family of models was originally inroduced by Thomas and Fiering (96) for monhly river flow modelling and simulaion. In economics ime series PAR models are used by among ohers Franses (996) and Osborn and Smih (989) in modelling macroeconomic daa. While PAR-models have some advanages agains nonperiodic models hey are no wide used. Osborn (99) derived he heoreical resuls on he effecs of misspecificaion of periodiciy and showed ha orders of he ime-invarian models can be higher han ha of he PAR-model. Also he neglecing periodiciy could lead o overac of lags in a nonperiodic model and misspecificaion may resul in biased A.Agrawall & K.Tandon (996), R.Gibbons & P.Hess (98) and A.Kamara (997) among ohers. 3 Early references o PAR model are Gladyshev (96), Cleveland and Tiao (979). 3

5 forecass. The ignoring of periodic srucure may cause ha esimaed error process is periodical heeroskedasic. PAR model exends a nonperiodic AR model by allowing he auoregressive parameers o vary wih he season. In oher words, PAR model assumes ha he observaions in each of he season can be described using a differen model. Such propery is useful in his case, because Linden and Louhelainen (004) showed ha inernaional sock raes of reurns vary in differen weekdays. Consider a univariae rae of reurn x, which is observed daily i.e. =,,, n. A periodic auoregressive model of order PAR() for five weekday can be wrien as x =α +β x +ε x =α +β x +ε x =α +β x +ε x =α +β x +ε x =α +β x +ε () where α s is a weekday varying inercep erm, and where β s s are auoregressive parameers up o order which may vary wih he weekday s,s =,, 3, 4, 5. The ε s, are he errors for each five model and hey are assumed o be NID i.e. ( ) ε N 0, σ for s and =,,,n. However, in periodic auoregressive models, errors may have seasonal variances σ s and errors in differen models may correlae wih each oher. Using more han one lag in PAR model changes he model ino mulivariae form. A he same ime, numbers of parameers increase s imes, bu number of equaions is equal. Mulivariae PAR() presenaion for lags and for five weekdays can be wrien as follows, x. = α+ β. x + β. x + ε. x = α + β x + β x + ε x = α + β x + β x + ε x = α + β x + β x + ε x = α + β x + β x + ε () 4

6 Neverheless, he periodic ime series models () and () are no saionary models since he variance and auocovariances can ake differen values in differen season. To invesigae saionary properies of x i is more convenien o rewrie equaion () wih lag operaor L defined by Lx k = x for k =,,..., p. Le k process X ' ( x,x,x,x,x ) X be he ( 5 ) vecor = =,,,n, where x is he observaion in weekday s. s in week, s =,, 3, 4, 5. The residuals ε ' ( ε., ε., ε3., ε4., ε5.) i.e. ε N ( 0, σ s, s ) and α ' = ( α, α, α, α, α 3 4 5) are ( 5 ) = are assumed NID vecor of consan erms. where Φ ( L) X = α + ε (3) β L β L Φ ( L) =Φ Φ L= β L β L β L 5. (4) Same way he PAR() process in equaion () can be rewrien wih lag operaor i.e. Φ ( L) X = α + ε (5) where ( L) L L Φ =Φ Φ Φ β L β L = β L β L β L 5. β L β L β L β L β L 5. (6) 5

7 There are few approaches o invesigaing he saionary properies of a PAR process parameerisaion, i.e. invesigaing wheher i conains uni roos. For a PAR (p) process, presence of uni roos in x amouns o invesigaing he soluions o characerisic equaion of (4) Φ Φ Φ = (7) k L L 0 0 k The PAR(p) process is saionary if sabiliy condiion (7) doesn hold. In he oher k words, Φ Φ L Φ L has no roos in and on he complex uni circle. 0 k Characerisic equaion for PAR() model (3) is 5 Φ ( L) = ( L β ) = 0. (8).s s= Hence if he parameers of PAR() model equals o uniy, he PAR() model conains a single uni roo. Respecively for PAR() characerisic equaion is 5 ( L) ( Lβ Lβ.s.s ) Φ = = 0 s= PAR() model conains a single uni roo (i.e. L=), when he parameer resricions (9) are valid. { } β. s + β. s = for all s =,,,5 (0) There are a leas wo approaches o es periodic properies of ime series. The firs is o invesigae periodic properies of residuals from non-periodic models. The oher approach is simply o esimae a PAR(p) model, where p is seleced using convenional model selecion crieria, and o es wheher here is periodic variaion in he auoregressive parameers. The residual periodiciy can be invesigae by graphical and es mehods. A graphical approach is proposed in Hurd and Gerr (99) and Vecchia and Ballerini (99). While some plos show periodic behaviour, i migh be difficul o deermine he naure of he 6

8 seasonaliy from he graphs. Insead, one can use some modified version of he Box- Pierce Q-es, which is proposed by Lobao e al. (00). Anoher conceivable es is LM es suggesed by Franses (993). Residuals ˆε from a nonperiodic AR(k) model o x are modelled as k 5 x ( D... D ) () ˆ ε = η + ψ ˆ ε + + ψ ˆ ε + µ i i s s s i= s= where η and i ψ are parameers and i D is dummy variable for every five weekdays ha s. examine presence of periodiciy. The F-es procedure can be formalized as ψ = 0,..., ψ = 0 and null hypohesis is ha no periodic auocorrelaion of order m..s m.s Under H 0, F-saisic follows a sandard F(5m,n-k-5m)-disribuion. The residuals, which nonperiodic AR(k) model yields can also be used o check for seasonal heeroskedasiciy. The auxiliary regression ges following form ˆ ε = ω + ω D + ω D + ω D + ω D + ε () for five weekdays. The F-es for ω = 0,..., ω = 0can be used o es he null hypohesis 4 of no seasonal heeroskedasiciy. Now under ha hypohesis, F-saisic follows a sandard F- disribuion wih (4,n-k) degree of freedom. While he periodiciy is revealed from he ime series he simple PAR(p) model is no necessarily he bes model. An alernaive model is one where weekdays previous week s reurns have periodiciy power on his week s weekday reurns. Now he model ges a following form x = α + β x + β x + β x + β x + β x + ε x = α + β x + β x + β x + β x + β x + ε x = α + β x + β x + β x + β x + β x + ε x x = α + β x + β x + β x + β x + β x + ε = α β x β x β x β x β x ε (3) or in marix noaion model is 7

9 x α β β β β β x, ε....., α β β β β β x..... x, ε, x = α + β β β β β x + ε , 3, x α β β β β β x4, ε4, α β β β β β x x ε 5, 5, (4) Model is more general han simple PAR() model and is looks like VAR() model, while every equaion in model conains PAR componen. The null hypohesis has a form H : β = =,, β. Under null hypohesis χ -saisic follows a sandard χ s.s disribuion wih 4 degree of freedom. However he generalized PAR() model above does no include nonperiodic AR componens. Therefore a PAR() model wih AR() componens is suggesed also: x = α + β x + β x + β x + ε x = α + β x + β x + β x + ε x = α + β x + β x + β x + ε x = α + β x + β x + β x + ε x = α + β x + β x + β x 3. + ε (5) In his model, weekday reurns are prediced by previous wo weekdays reurns and weekdays own previous reurns. III. Boosrapping he coefficiens confidence inervals To find confidence ha coefficien is significan i has o be esed. Usually hypohesis such as H 0 : β = 0 and H : β = β are esed wih - or F ess. The disribuion of he F saisic relies on he assumpion of he normally disribued regression errors. Wihou his assumpion, he exac disribuion of his saisic depends on he daa and esimaed parameers. If he residuals are no normally disribued he correc rejecion size of H 0 : β = 0 and H : β = β ess are no warraned. Wih boosrap mehod i is possible accurae confidence inervals wihou making normaliy assumpions. Consider he AR () model 8

10 x = α + βx + ε (6) Fiing he daa o he model (8) yields esimaors ˆα and ˆβ for α and β. The error erms ˆε in model are assumed o be IID from an unknown disribuion. The boosrap algorihm is following. A firs a random sample error erms ˆε * wih replacemen is drawn so ha each ˆε belongs o random sample wih probabiliy. A second, he new daase * * x is ge by fiing he random error erms ˆε ino he model = ˆ + ˆ + ˆ (7) * * α βx ε x where he values of regression coefficiens ˆβ and ˆα and variable x is se o be fixed. Fiing he new daase * x in o he model * * x x = α + β + ε (8) yields new regression coefficiens * * ˆβ and ˆα. Replicaion of his algorihm B imes enables o ge disribuion of boosrapped esimaes. As B increases, also he accuracy of disribuion of esimaors increases. There are few mehods 4 such as boosrap-, percenile, and ABC, which ry improving he accuracy of boosrap coefficiens confidence inervals. Efron & Tibshirani (993) showed ha BC a mehod is more accurae han percenile mehod. The BC a mehod correcs he percenile inerval for median bias and skewness. This mehod requires an esimae of he acceleraion a, which is relaed o he skewness of he sampling disribuion and bias-correcion parameer z 0. The inerval of inended coverage µ for esimaor ˆβ, is given by 4 The mehods for confidence inervals are inroduced in books: B. Efron & R. Tibshirani, An Inroducion o he Boosrap, Chapman and Hall, New York (993) and A.C.Davison & D.V.Hinkley Boosrap Mehods and heir Applicaions, Cambridge Universiy Press, (997) 9

11 ˆ ˆ ˆ ( µ ) ˆ ( µ ) β β = β β (9) (, ) (, ), low up where and ( µ ) z0 z µ ψ + = z 0 + ( µ ) a(z 0 + z ) (0) µ ψ z + z. () a(z z ) ( µ ) 0 = z 0 + ( µ ) 0 + In hese formulas, z ( µ ) h is he 00µ percenile poin of a sandard normal disribuion and ψ is cumulaive normal disribuion funcion, e.g. ψ (.645) = 0.95 and 0.95 z =.645. The confidence inerval is given by aking he appropriae percenile of he boosrap disribuion * ˆβ, e.g. if values for µ and µ are 0.05 and 0.95 respecively, hen he confidence poins when using 999 boosrap replicaions will be he 50h and 950h ordered values of ˆβ. The acceleraion and bias correcion are approximaion by, where [ ] b= z 0 =Φ () B + B ( ˆ * ) # β b β * Φ i is he sandard normal quanile funcion, and #( β β) /( B ) ˆ b + is he proporion of boosrap replicaes a or below he original sample esimae β and # denoes he number of imes he even occurs. The skewness correcion, in oher words he acceleraion can be wrien as ( ˆ βi β) n 3 i a = = (3) n 3 6 i= ( ˆ βi β) More generally a is one-sixh he sandardized skewness of he disribuion of ˆβ. A same way confidence inervals can be calculaed o oher esimaors. 0

12 IV. Empirical resuls The daa used hroughou his paper consis of daily indices from nine 5 sock markes. The ime periods are from beginning of 990 o end of February 003 conaining over 3000 observaions per sock marke. Daily raes of reurn are calculaed as ln x = ln x ln x (4) Mulivariae OLS and Seemingly Unrelaed Regression (SUR) are he alernaive mehods for esimaion of PAR model. As SUR mehod allows for conemporaneous cross equaion correlaion and equaion specific error variances, SUR is preferred o mulivariae OLS. Table. Tesing periodic paerns in he esimaed residuals from nonperiodic AR() models Sock marke Diagnosic es saisic χ PH Canada.08. * Finland Holland * Ialy * Japan. * 6.80 * Singapore * Dow Jones NASDAQ SP& The diagnosic es saisic concerns residual auocorrelaion of order in model x = α + x + ε and ( χ ) is periodic heeroskedasiciy, ( χ ) PH periodic residual auocorrelaion of order. *) Significan a he 5 % level (based on boosrapped es disribuion). χ The empirical resuls of he wo es mehods, () and (), for periodic properies of series are presened in Table. The resuls show ha periodic heeroskedasiciy can be found only in Japan. Residuals are periodically auocorrelaed in mos of sock markes. Only in SP&500 he boh χ -es values are no rejeced. Sill under es resuls in Table is clear ha sock marke daa have periodic properies. 5 Canada (TSE), Finland (HEX), Ialy, (MIBTel), Japan (Nikkei), Holland (AEX), Singapore (STI), Unied Saes (Dow Jones, NASDAQ and S&P 500)

13 SUR disurbances (Figures -9 in Appendix II) do no follow normal disribuion excep for Wednesday in Ialy. The conclusion is found also wih he kurosis values, which are beween 3.50 and.57, and wih skewness values, which are beween.45 and.0 (Table in Appendix I). For he normal disribuion, skewness value is zero and kurosis value equals o 3. The resuls of Bera-Jarque ess also confidence SUR disurbances are no normally disribued. Therefore, boosrap mehod is validaed o ensure righ decisions of hypohesis ess. All regressions used 000 boosrap replicaions. Table 3 (Appendix I) presens SUR esimaion resuls o equaion () for all nine sock marke and for all weekdays. Rejecion of H 0 : βs =0 bases on boosrapped confidence inervals of coefficiens β s a level 95 %. Boosrapped resuls are also used o es H : β=,...,= β5. A leas some weekdays reurns can be predic from same weekdays reurns from previous week. In Ialy, hose days are Tuesday and Thursday. Previous Tuesday s coefficien has a predicabiliy power on Tuesday s reurns in Holland and in NASDAQ. The oher significan coefficiens are Thursday in Japan, Monday in S&P 500 and Friday in Dow Jones. Hypohesis is also rejeced in all hose sock markes. However in Finland and Singapore here is no sign of predicabiliy. I is hard o find similariies in resuls beween differen sock markes. However, here are in hree sock markes wih significan coefficiens for Tuesday. They all are negaive. This implies negaive reurns for nex week s Tuesday if his week s reurns were posiive and vice versa. There are also wo sock markes wih significan coefficiens for Thursday. They are posiive. I means ha negaive reurns on Thursday are also negaive in nex Thursday and vice versa, respecively. Hypohesis es for coefficiens equaliy is acceped only in Finland. Elsewhere i is rejeced. Idea ha is more general is o examine if he oher previous weeks reurns give significan predics for his weeks weekdays reurns, i.e. model (4). SUR esimaion resuls are presened in Table 4 (Appendix I). I is a noable ha usually Monday and Tuesday reurns can be prediced from previous weeks raes of reurns in every sock markes. Oher weekdays have only a few significan coefficiens. In mos of cases

14 previous week Friday predics posiively Monday reurns. Thus his is a sign of auocorrelaion in reurns. The resuls in Table 3 and Table 4 are very similar in hose pars where weekday s previous reurns predic same weekday s reurns on his week. Only difference in Table 4 is ha Ialy s previous week Monday is significan predicor for his week s Monday. For Tuesday, significan coefficiens are previous Thursday in Japan, previous Monday in Singapore and previous Wednesday in Dow Jones. There are some resuls for oher weekdays bu hese are irregular. Survey of he resuls in Table 4 reveals ha mos of predicaive raes of reurns are found in Ialy and in NASDAQ. Correspondingly, leas predicaive raes of reurns are in Finland and Holland. According o Table 4, reurns of Wednesday and Friday are unpredicable excep for some occasional cases. Table 5 (in Appendix I) repors he resuls of regression model where o PAR() model is added AR() componens, i.e. model 5. For Singapore i is found significan AR() parameers for four weekdays and in Canada for ree weekdays. In oher sock markes here are only one or wo significan AR() parameers. A significan AR() parameer can be found only in Wednesday for Ialy. Weekends seem o have some influence o sock reurns. In five sock markes, Friday raes of reurns have predicaive power on Monday reurns. In addiion, Friday reurns can be explained by Thursday reurns also in five sock markes. In he middle of he week significan AR parameers are found for Canada, Singapore and Japan. SP&500 is he only sock marke wih no significan AR parameers. Thus in general he simple PAR() model is adequae for mos sock markes. In modified PAR() models (4) and (5) coefficien of deerminaion is greaer han in simple PAR() model. Ialy seems o have mos significan PAR and AR parameers. Insead, for Finland i is found only one significan parameer. There Monday raes of reurns can be explained by raes of reurns of previous Friday. For Singapore we found merely significan AR parameers bu for SP&500 here were no a single. 3

15 Wihou boosrapping confidence inervals he number of significan coefficiens would be much more larger. The 95% criical level for coefficien would be around +/ insead of +/- 0.. Thus by using he normal error assumpion for example in Table 3 previous week s Tuesday, Wednesday and Thursday would be significan in Canada. In he same way in Table 4 Holland would have en significan coefficiens insead of one. V. Conclusions I is well-known ha daily sock marke reurns in several sock markes are auocorrelaed and daily reurns are no equal. This paper esed he predicabiliy of he daily sock reurns over he previous week s daily reurns. A he same ime weak form of he efficien marke hypohesis is quesioned. All sock markes excep SP&500 had weekday periodic properies. This was revealed by ess for nonperiodic AR() residuals. The paper conained hree es models: a simple PAR() where weekday rae of reurns were modeled wih same weekday s previous week s reurns, a VAR() model ha included PAR() componens wih all weekday reurns, and a PAR() model ha has added wih nonperiodic AR() componens. However, he disurbances of he models were no normally disribued. Therefore he boosrap mehod was used o ensure he reliable conclusions of hypohesis esing. The bes esimae for omorrow s sock marke reurn is no oday s reurn as he random walk heory ofen assumes. Coefficiens of some weekday s previous week s observaions are significan in mos analyzed sock markes. Raes of reurn are predicable over he week a leas for one weekday in seven of nine sock markes. The resuls of VAR() model revealed ha he simple PAR() model was no adequae. From he previous week significan effecs were found on several weekdays. Also he hird model alernaive fied beer han simple PAR() model. Usually Monday s and Friday s reurns have significan one day AR componen. In he middle of week significan AR parameers can be found only in some sock markes. 4

16 The weak form of he efficien marke hypohesis can be rejeced in all analyzed nine sock markes. Previous day s reurns have predicaive power on curren reurns. However he reurn predicabiliy is no only limied o firs wo AR componens. I can be also found periodically, i.e. over one week. References Abraham A. and Ikenberry D. L. 994: The Indusrial Invesor and he Weekend Effec Journal of Financial and Quaniaive Analysis vol Agrawal A. and K. Tandon K. 994: Anomalies or Illusions? Evidence from sock markes in eigheen counries. Journal of Inernaional Money and Finance vol Ariel R.A. 987: Monhly effec in sock reurns. Journal of Financial Economics vol : High Sock Reurns before Holidays: Exisence and Evidence on Possible Causes The Journal of Finance vol Bhardwaj R.K and Brooks L.D 99: The January Anomaly: Effecs of Low Share Price. Transacion Coss. and Bid-Ask Bias The Journal of Finance Vol Cleveland W.P. and Tiao G.C. 979: Modelling seasonal ime series Revue Economie Appliqu'ee vol Cross F. 973: The Behaviour of Sock Prices on Fridays and Mondays Financial Analysis Journal vol Davison A.C. and Hinkley D.V. 997: Boosrap Mehods and heir Applicaions Cambridge Universiy Press. Cambridge Efron B. and Tibshirani R.J. 993: An Inroducion o he Boosrap Chapman & Hall. New York Fama E. 965: The behaviour of sock markes prices Journal of Business. Vol : Efficien capial markes: a review of heory and empirical work. Journal of Finance vol : Efficien capial markes: II. The Journal of Finance vol Fisher L. 966: Some new sock-marke indexes Journal of Business vol Franses P.H. 995: "Quarerly U.S. Unemploymen: Cycles. Seasons and Asymmeries" Empirical Economics vol : Periodiciy and sochasic rends in economic ime series Oxford U.P. Oxford.. 998: Time series models for business and economic forecasing Cambridge Universiy Press. Cambridge French D. W. 980: Sock Reurns and he Weekend Effec. Journal of Financial Economics vol Gibbons M. R. and Hess P.J. 98: Day of he Week Effecs and Asse Reurns. Journal of Business vol Gladyshev E.G. 96: Periodically Correlaed Random Sequences Sovie Mahemaics vol

17 Grossman S.J and Sigliz J.E 980: On he Impossibiliy of Informaionally Efficien Markes American Economic Review vol Hurd H.L. and Gerr N.L 99: Graphical Mehods for Deermining he Presence of Periodic Correlaion. Journal of Time Series Analysis. vol Jaffe J. and Weserfield R. 985: The Week-end Effec in Common Sock Reurns: he Inernaional Evidence Journal of Finance. vol Jensen M.C. 978: Some Anomalous Evidence Regarding Marke Efficiency Journal of Financial Economics vol Kamara A. 997: New Evidence on he Monday Seasonal in Sock Reurns The Journal of Business vol Linden M. and Louhelainen M. 004: Tesing for weekday anomaly in inernaional sock index reurns wih non-normal errors Deparmen of Business and Economics, WP-4, Universiy of Joensuu. Lo A.W. and MacKinlay A.C. 988: Sock Marke prices do no follow random walks: Evidence from a simple specificaion es Review of he Financial Sudies vol Lobao I, Nankervis J.C. and Savin N.E. 00: Tesing for Auocorrelaion Using a Modified Box-Pierce Q-es Inernaional Economic Review. Vol LeBaron B. 99: Some Relaions beween Volailiy and Serial Correlaions in Sock Marke Reurns. Journal of Business. vol Osborn D.R. 99: The implicaions of Periodically Varying Coefficiens for Seasonal Time Series Processes, Journal of Economerics. vol and Smih J.P 989: The Performance of Periodic Auoregressive Models in Forecasing Seasonal UK Consumpion, Journal of Business and Economic Saisics. vol Pagano M. 978: Periodic and Muliple Auoregressions Annals of Saisics. vol Trouman B.M 979: Some Resuls in Periodic Auoregression Biomerika vol Vecchia A.V. and Ballerini R. 99: Tesing for Periodic Auocorrelaions in Seasonal Time Series Daa Biomerika vol

18 APPENDIX I Table. The resuls of Bera-Jarque es, skewness, and kurosis values for residuals of PAR() model in 9 sock markes. Sock marke Bera-Jarque Skewness Kurosis Canada Mon 653* Tue 75* Wed 3636* Thu * Fri 7* Finland Mon 5* Tue 38* Wed 93* Thu 84* Fri 3* Holland Mon 433* Tue 97* Wed 0* Thu 57* Fri 5* Ialy Mon 54* Tue 6* Wed Thu 57* Fri * Japan Mon 07* Tue 470* Wed 59* Thu 67* Fri 68* Singapore Mon 8903* Tue 478* Wed 473* Thu 549* Fri 47* Dow Jones Mon 88* Tue 0* Wed 75* Thu 60* Fri 745* NASDAQ Mon 69* Tue 465* Wed 498* Thu 06* Fri 97* SP&500 Mon 7* Tue 5* Wed 37* Thu 35* Fri 7* *)Null hypohesis is rejeced a 95% confidence level 7

19 Table 3. Predicabiliy of weekday reurns on weekday s previous reurns in nine sock marke Canada Finland Holland Ialy Japan Singapore NASDAQ S&P 500 Dow Jones Monday Mon- -0.8* * Cons Tuesday Tue * -0.4* * Cons Wednesday Wed Cons Thursday Thu * 0.8* Cons Friday Fri * Cons *) Coefficien differs from zero a 95% confidence level. The resuls based on boosrapped confidence inervals. 8

20 Table 4. Predicabiliy of daily reurns on previous week s all daily reurns in nine sock marke Variable Canada Finland Holland Ialy Japan Singapore Dow Jones NASDAQ S&P500 Monday Mon * * * Tue Wed * Thu Fri * 0.63 * * * * 0.05 * Cons χ -es a 3.0 * * * * 3.4 * Tuesday Mon * Tue * * Wed * Thu * Fri Cons χ -es * 6.56 * * Wednesday Mon Tue Wed Thu Fri * Cons χ -es Thursday Mon Tue Wed * * -0.3 Thu * 0. * Fri Cons χ F-es.39 * Friday Mon Tue Wed Thu * Fri * Cons χ -es *) Coefficien differs from zero a 95% confidence level. The resuls base on boosrapped confidence inervals. a) Chi-squared -es is for H 0 :β Mon- =,,= β Fri-. 9

21 Table 5. Resuls of PAR() and AR() combined model in nine sock marke Canada Finland Holland Ialy Japan Singapore Dow Jones NASDAQ SP&500 Monday Mon * * Fri * 0.6 * * * * 0.0 Thu Cons χ -es.89 * 9.0 * * 9.98 * 7.55 * *.95 * Tuesday Tue * Mon * Fri Cons χ -es * Wednesday Wed Tue 0.48 * * Mon Cons χ -es 7.5 * * Thursday Thu * Wed * Tue Cons χ -es Friday Fri * Thu 0.83 * * 0.77 * * * Wed Cons χ -es.73 * *.07 * * 9.46 * The combined model is represened a (5).The resuls base on boosrapped confidence inervals *) Coefficien differs from zero a 95% confidence level and null hypohesis for χ -es ha coefficien esimaes for each weekday are equal. 0

22 Table 6. The coefficien of deerminaion and Durbin Wason saisic for hree models Models PAR() VAR() PAR()- AR() DW R DW R DW R Canada Monday Tuesday Wednesday Thursday Friday Finland Monday Tuesday Wednesday Thursday Friday Holland Monday Tuesday Wednesday Thursday Friday Ialy Monday Tuesday Wednesday Thursday Friday Japan Monday Tuesday Wednesday Thursday Friday Singapore Monday Tuesday Wednesday Thursday Friday Dow Jones Monday Tuesday Wednesday Thursday Friday NASDAQ Monday Tuesday Wednesday Thursday Friday SP&500 Monday Tuesday Wednesday Thursday Friday

23 APPENDIX II Figure. Regression residuals densiy and normal (hin) in PAR() model for Canada daa Figure. Regression residuals densiy and normal (hin) in PAR() model for Finland daa Figure 3. Regression residuals densiy and normal (hin) in PAR() model for Holland daa

24 Figure 4. Regression residuals densiy and normal (hin) in PAR() model for Ialy daa Figure 5. Regression residuals densiy and normal (hin) in PAR() model for Japan daa Figure 6. Regression residuals densiy and normal (hin) in PAR() model for Singapore daa 3

25 Figure 7. Regression residuals densiy and normal (hin) in PAR() model for Dow Jones daa Figure 8. Regression residuals densiy and normal (hin) in PAR() model for NASDAQ daa Figure 9. Regression residuals densiy and normal (hin) in PAR() model for SP&500 daa 4