INTERNATIONAL JOURNAL OF BUSINESS, 11(4), 2006 ISSN: 1083 4346 Calendar Corrected Chaotc Forecast of Fnancal Tme Seres Alexandros Leonttss a and Costas Sropoulos b a Center for Research and Applcatons of Nonlnear Systems Unversty of Patras, 26500 - Ro, Patras, Greece & Unversty of Ioannna Department of Educaton 45110 - Dourout, Ioannna, Greece me00743@cc.uo.gr b Department of Busness Admnstraton, Unversty of Patras, 26 500 Ro Patras, Greece csro@otenet.gr ABSTRACT Usng daly returns from the NASDAQ Composte and TSE 300 Composte ndces from 1984 to 2003, we specfy a method that corrects the chaotc forecastng of fnancal tme seres takng nto account the day-of-the-week, the turn-of-the-month and the holday effects. When calendar effects are present n the seres, the forecastng ablty of the model leads to proftable opportuntes compared to a buy-and-hold strategy. JEL Classfcaton: C22, C53, G14 Keywords: Calendar effects; Forecastng; Least medan of squares; Tradng rules; Chaos * The authors are grateful to Dr. M. J. P. Cooper from the Insttute of Mathematcs and Statstcs of the Unversty of Kent at Canterbury (UK) for hs comments on the manuscrpt, and an anonymous referee of the Journal.
368 Leonttss and Sropoulos I. INTRODUCTION Emprcal studes on fnancal tme seres have revealed the presence of calendar effects n the behavor of stock returns. Calendar studes questoned whether rregulartes exst n the rates of return durng the calendar year. Knowng ths, would better allow nvestors to predct returns on stocks. Accordng to the Effcent Market Hypothess (EMH) such seasonal patterns should not persst snce ther exstence mples the possblty of obtanng abnormal returns applyng market-tmng strateges. The day-of-the week effect, frst documented by Osborne (1962); the weekend effect (sgnfcantly lower returns over the perod between Frday s close and Monday s close), frst documented by French (1980); the January effect (relatvely hgher returns n January), frst reported by Wachtel (1942); the tradng month effect studed by Arel (1987); and the holday effect documented by Lakonshok and Smdt (1988), are among the most mportant calendar effects. These calendar effects have been studed extensvely n nternatonal level (e.g. Dubos and Louvet (1995), Hrak and Maberly (1995), Aggarwal and Schatzberg (1997), Mookerjee and Yu (1999), Mlls et al. (2000)) and the general concluson s that there are partcular perods of tme where the nvestors' behavor changes sgnfcantly, affectng the dstrbuton of returns. Gven the exstence of the aforementoned market anomales, that provde evdence of market neffcences, the fundamental queston s how ths nformaton can be utlzed by forecastng models, leadng to better portfolo performance. Jensen (1978) hghlghts the mportance of tradng proftablty when assessng market effcency: f a tradng rule s not strong enough to outperform a buy and hold strategy on a rsk-adjusted bass then t s not economcally sgnfcant, whle Roll (2000) argues that f calendar tme anomales represent evdence of market neffcences, then they ought to represent an explotable opportunty. Extendng prevous work of Ls and Medo (1997), Cao and Soof (1999), among others, who appled non-lnear technques n fnancal applcatons provdng better results compared to the random walk forecasts, ths study utlzes a nonlnear chaotc forecastng method on the NASDAQ and Toronto Stock Exchange Composte ndces, takng nto account specfc stylzed rregulartes of stock returns, reported n emprcal fnance lterature, such as the calendar effects. The methodology appled n the present study overcomes the lmtatons of prevous emprcal work, n whch ether the calendar effects were not taken nto account or the predctve ablty of the forecastng models was not tested extensvely. The rest of the study s organzed as follows: Secton 2 descrbes the data set. Secton 3 ntroduces the algorthm that takes nto account the day-of-the-week, the turn of the month and the holday effect, whle Secton 4 presents the results of the proposed method. Fnally, Secton 5 concludes proposng drectons for future research. II. THE DATA SETS AND PRELIMINARY DIAGNOSTICS The dataset used s comprsed of the NASDAQ Composte and the Toronto Stock Exchange 300 Composte (TSE 300), both ndces belongng to mature markets havng a hgh tradng volume. It s noted that because the frst one s a U.S. ndex whle the
INTERNATIONAL JOURNAL OF BUSINESS, 11(4), 2006 369 second one a Canadan one they do not share the same holdays. Addtonally, they belong to two extremes: NASDAQ s a technology-based ndex, therefore a speculatve behavor s expected, whle TSE 300 s an ndustry-based ndex, therefore a more stable behavor s expected. Indcatvely, the standard devaton of NASDAQ s about 165% the standard devaton of TSE 300, whch means that NASDAQ s behavor s more speculatve. Also, a hgh kurtoss value for each ndex s observed. Table 1 Descrptve statstcs NASDAQ TSE 300 Observatons 4,796 4,775 Max 0.0576 0.0376 Mn -0.0523-0.0521 Mean 0.0002 0.0001 Std. dev. 0.0063 0.0038 Skewness -0.2499-1.1800 Kurtoss 8.3209 17.7803 The perod under study for both ndces covers the perod from February 1984 tll December 2003. We transformed the daly closes to frst logarthmc dfferences, so each tme seres has nearly 4,800 observatons. On Table 1 the descrptve statstcs for each ndex are reported. III. CHAOTIC FORECAST Gven the exstence of calendar anomales, the fundamental queston s how ths nformaton can enhance portfolo performance and fnancal forecastng. Ths secton descrbes the smple chaotc forecastng method, and then ntroduces an algorthm that embodes the calendar effects. A. Smple Chaotc Forecast The dynamcs of a tme seres { x t} N t= 1 are reconstructed usng the vectors x t =(x t,x t+1,...,x t+m-1 ), where m s the embeddng dmenson and the total number of the vectors s T=N-(m-1). The components of these vectors should be lnearly uncorrelated otherwse the calculated dmenson wll approxmate 1 regardless of ts true value. If m s more than 2 tmes the dmenson of the orgnal (and unknown) dynamc system, then the reconstructed dynamc system preserves all the characterstcs of the orgnal one. In expermental stuatons such as the reconstructon of the phase space of the fnancal dynamcs we have to relax the prevous condton and try dfferent
370 Leonttss and Sropoulos reconstructons (.e. m may vary). For the emprcal results we set m=3 snce for that value of m the best results for each ndex are reported (see also Sropoulos and Leonttss 2002). The predcton s obtaned as follows: For the last pont of the phase space (x T ) we consder the K nearest neghbors of x T, denoted as x T(k), k=1,2, K. Ths s a hypersphere of radus r. The forecast of the next value of the tme seres (x N+1 ) s the mean of x T(k)+(m-1)+1. Ths model can be expanded to take nto account lnear relaton present n the data of each neghborhood, but snce the nose level s large (Sropoulos and Leonttss 2002) the estmated lnear relatons can be spurous. B. Introducng the Calendar Effects The calendar effects dstort not only the dstrbuton of a fnancal tme seres but also ts dynamcs. Therefore an accurate model should be able to restore the tme seres at least n mean and varance. We consder 3 knds of calendar effects: the day-of-theweek, the frst and the last tradng month days, and the holday effects. In a case that a day falls nto 2 or more knds, the latter holds. The above algorthm s tested for dfferent values of K. For each K, a mnmzaton algorthm helps to restore the tme seres. Snce we have 8 locaton parameters and 8 scale parameters we work on a mnmzaton space of 16 dmensons. In ths space an optmum combnaton of these parameters s able to enhance the performance of the algorthm, gven that the calendar effects exst on the tme seres. Snce the calendar effects do not follow the normal dstrbuton (see Table 1), we standardze each calendar effect n a robust way estmatng the robust locaton and the robust scale parameters usng the Least Medan of Squares (LMS) method proposed by Rousseeuw and Leroy (1987). The LMS locaton parameter mnmzes the medan of the squared errors, and the scale parameter s a functon of the medan of the squared resduals. Thus, we obtan the calendar corrected tme seres after subtractng each effect s bas on locaton and dvdng by each effect s bas on scale, as n eq. (1). ~ x loc x sca = (1) where x s an element of the vector of the fst logarthmc dfferences that belong to a partcular calendar event, loc and sca are the locaton and the scale parameters of the calendar effect, wth = Monday, Tuesday,..., Last day of tradng month, Holday, and x~ s the calendar corrected observaton. Then we rerun the algorthm and measure ts performance. Fnally we do the nverse transformaton to the predcted value that s multpled by sca and add loc : pˆ ~ + = p sca loc (2)
INTERNATIONAL JOURNAL OF BUSINESS, 11(4), 2006 371 where ~ p s a predcted value for a calendar event based on the data of the rght hand sde of eq. (1), and pˆ s the calendar-corrected predcted value. IV. EMPIRICAL RESULTS AND COMPARISON OF METHODS Ths secton apples the method analyzed above to the NASDAQ Composte and TSE 300 Composte ndces, and dscusses the results. Intally, out-of-sample forecasts on the two ndces for the year 2003 and for the 4-year perod 2000-2003 are appled. Every forecasted value s the result of the nearest neghbor methodology appled n all the prevous values of the forecasted observaton. The most common way to evaluate the forecastng accuracy of a model, how close are the predcted values to the actual ones, s by usng the normalzed mean squared error. We also focus on the correct forecast of next day s sgn (.e. to forecast f the tme seres wll go up or down). We adopt the followng tradng rule for evaluatng the performance of our forecasts: Start wth nvestng 1$ on the frst tradng day of the perod under study. Do the next day's predcton. If t s postve buy the next day's return, otherwse keep captal unchanged. Tables 2 and 3 summarze the results. Table 2 Performance of the smple chaotc forecast algorthm wth and wthout calendar correcton of the tme seres of NASDAQ Composte versus the buy-and-hold strategy NASDAQ Composte Test perod 2000-2003 Test perod 2003 Performance Buy-and hold 0.49 Performance Buy-and hold 1.50 Wthout 1.29 (K=3) Wthout 1.52 (K=3) Wth 1.89 (K=3) Wth 1.81 (K=49) Effect Locaton Scale Effect Locaton Scale Mon 0 1 Mon 0.002198 1.004 Tue 0 1 Tue -0.0022 1 Wed 0 0.996 Wed 0 1 Thu 0 1 Thu -0.0022 1 Fr 0 1 Fr 0 1 Frst 0.002198 0.996 Frst 0 1 Last 0 0.996 Last 0 1.004 Hol 0 1.012 Hol 0 1
372 Leonttss and Sropoulos Table 3 Performance of smple chaotc forecast algorthm wth and wthout calendar correcton on the tme seres of TSE 300 Composte versus the buy-and-hold strategy TSE 300 Cmp. Test perod 2000-2003 Test perod 2003 Performance Buy-and hold 1.00 Performance Buy-and hold 1.22 Wthout 1.35 (K=4) Wthout 1.18 (K=2) Wth 1.97 (K=4) Wth 1.32 (K=6) Effect Locaton scale Effect locaton scale Mon 0 0.996 Mon 0 1 Tue 0 1.008 Tue -0.00179 1 Wed 0 1 Wed 0 1 Thu 0 1 Thu 0 1 Fr 0 0.988 Fr 0 1 Frst 0 1.004 Frst 0 1 Last 0 1 Last 0 1 Hol 0.001794 0.996 Hol 0.001794 1.004 The buy-and-hold strategy serves as a benchmark to the performance of the chaotc forecast. The number of neghbors s optmzed to result the maxmum possble proft on the tranng perod accordng to the above-mentoned tradng rule. One would expect NASDAQ Composte to be a more calendar senstve ndex than TSE 300 Composte, due to the senstve nature of the stocks t represents. Nevertheless, the results ndcate that both ndces are subject to calendar effects, although these are less evdent n the TSE 300 ndex. Ths means that the dynamcal estmaton and correcton of the calendar effects may add a new nsght to ths ssue. V. DISCUSSION AND CONCLUSIONS Over the past twenty years fnancal economsts have documented numerous stock return patterns related to calendar tme. The lst ncludes patterns related to the monthof-the-year, day of-the-week, day-of-the-month, and market closures due to exchange holdays to name a few. Calendar anomales are not n accordance wth the concept of the Effcent Market Hypothess. However, the smple observaton of these anomales are far from convncng, and do not contradct the EMH unless they can be used to provde explotable proft opportuntes. Ths paper presents a nonlnear forecastng method for fnancal tme seres takng nto account calendar effects, mprovng the qualty of the forecasts and leadng to the development of proftable tradng strateges (excludng taxes, transacton and other costs). The study shows that there s an mprovement on out-of-sample forecastng results, for calendar-corrected tme seres. More precsely, both ndces were found to
INTERNATIONAL JOURNAL OF BUSINESS, 11(4), 2006 373 be calendar-senstve. Ths means that many systematc bases affect ther structure. If one subtracts all these peces of bas from the tme seres, makes forecasts, and adds them back, the results are mproved compared to the buy-and hold strategy or to noncorrected results. It turns out that the calendar effects dstort the determnstc structure of a tme seres. However, for the TSE 300 Composte ndex, where the presence of calendar effects s weaker compared to the NASDAQ ndex, the forecastng ablty of the model s lmted. Future research may focus on the fact the nonlnear forecastng model s a parameter senstve technque whch results are hghly dependent on the selecton of the parameters, n contrast to other stochastc technques such as ARIMA and ARFIMA. Also, focus may be pad to a possble mprovement of the proposed neghbor selecton method. Fnally, another route for future research concerns the applcaton of the proposed methodology to other mature and emergng markets, and the ncluson of more calendar anomales. REFERENCES Aggarwal R., and J.D. Schatzberg, 1997, Day of the Week Effects, Informaton Seasonalty, and Hgher Moments of Securty Returns, Journal of Economcs and Busness 49: 1 20. Arel, R., 1987, A Monthly Effect n Stock Returns, Journal of Fnancal Economcs 18: 161-174. Cao, L., and A. Soof, 1999, Nonlnear Determnstc Forecastng of Daly Dollar Exchange Rates, Internatonal Journal of Forecastng 15: 421 430. Dubos, M., and P. Louvet, 1995, The Day-of-the-week-effect: The Internatonal Evdence, Journal of Bankng and Fnance 20: 1463 1484. French, K., 1980, Stock Returns and the Weekend Effect, Journal of Fnancal Economcs 8: 55 70. Hrak, T., and E.D. Maberly, 1995, Are Preholday Returns n Tokyo Really Anomalous? If So, Why?, Pacfc-Basn Fnance Journal 3: 93 111. Jensen, M. C., 1978, Some Anomalous Evdence Regardng Market Effcency, Journal of Fnancal Economcs 6(2/3): 95 101. Lakonshok, J., and S. Smdt, 1988, Are Seasonal Anomales Real? A Nnety-year Perspectve, Revew of Fnancal Studes 1(4): 403 425. Ls, F., and A. Medo, 1997, Is A Random Walk the Best Exchange Rate Predctor?, Internatonal Journal of Forecastng 13:255-267. Mlls, T. C., C. Sropoulos, R.N. Markellos, and D. Harzans, 2000, Seasonalty n the Athens Stock Exchange, Appled Fnancal Economcs 10: 137 142. Mookerjee, R., and Q. Yu, 1999, An Emprcal Analyss of the Equty Markets n Chna, Revew of Fnancal Economcs 8: 41 60. Osborne, M. F., 1962, Perodc Structure n the Brownan Moton of Stock Returns, Operatons Research 10: 345 379. Roll, R., 2000, Are Markets Effcent? The Wall Street Journal, Dec. 28. Rousseeuw, P.J., and A.M. Leroy, 1987, Robust Regresson and Outler Detecton. Wley and Son.
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