TEST OF THE WEAK FORM EFFICIENT MARKET HYPOTHESIS FOR THE ISTANBUL STOCK EXCHANGE BY MARKOV CHAINS METHODOLOGY

Transcription

1 TET OF THE WEAK FOM EFFICIENT MAKET HYPOTHEI FO THE ITANBUL TOCK EXCHANGE BY MAKOV CHAIN METHODOLOGY Öğr.Gör.Dr. üleyma Bilgi KILIÇ Çukurova Üiversitesi İktisadi ve İdari Bilimler Fakültesi Ekoometri Bölümü ÖZET Bu çalışma İstabul Mekul Kıymetler Borsası 00 edeksie ait gülük getirilerii rassal yürüyüş gösterip göstermediği Markov zicirleri yötemi ile test edilmektedir. Eğer bir piyasada zayıf formda etkilik hipotezi geçerli ise hisse seedi getirileri rassal yürüyüş özelliği gösterecektir. assal yürüyüş teorisi hisse seedi getirilerii tarihsel fiyat verileriyle tahmi edilemeyeceğii ögörür. Böylece rassal yürüyüş özelliği göstere bir piyasada tarihsel fiyat verilerie dayaılarak gerçekleştirile tekik aaliz yötemleri geçersiz olacaktır. ABTACT I this study, Markov chai methodology is used to test whether or ot the daily returs of the Istabul tock Exchage (IE) 00 idex follows a martigale (radom walk) process. If the Weak Form Efficiet Market Hypothesis (EMH) holds i ay stock market, stocks prices or returs follow a radom walk process. The radom walk theory asserts that price movemets will ot follow ay patters or treds ad that past price movemets caot be used to predict future price movemets hece, techical aalysis is o use.. Itroductio Efficiet Market Hypothesis (EMH) is a issue of itese debate amog academics ad fiacial professioals. Much of the theory o these subects ca be traced to Frech mathematicia Louis Bachelier whose Ph.D. dissertatio titled "The Theory of peculatio" (900). EMH evolved by Fama (965) who proposed three forms of the efficiet market hypothesis: () The "Weak" form asserts that all past market prices ad data are fully reflected i securities prices. I other words, techical aalysis is of o use. () The "emistrog" form asserts that all publicly available iformatio is fully reflected i securities prices. I other words, fudametal aalysis is of o use. (3) The Bachelier came to the coclusio that "The mathematical expectatio of the speculator is zero" ad he described this coditio as a "fair game." Ufortuately, his isights were so far ahead of the times that they wet largely uoticed for over 50 years util his paper was rediscovered ad evetually traslated ito Eglish ad published i 964 ( 333

2 "trog" form asserts that all iformatio is fully reflected i securities prices. I other words, eve isider iformatio is of o use. The debate about EMH has resulted i hudreds of empirical studies attemptig to determie whether specific markets are i fact "efficiet" ad if so to what degree. We summarize below oly those studies which utilizes Markov Chai methodology i aalyzig the stock prices ad testig radom walk hypothesis. I a early study, ya (973) explaied the relevace of the theory of Markov processes to the aalysis of stock price movemets ad stated that Markov theory is see to be relevat to the aalysis of stock prices i two ways:. As a useful tool for makig probabilistic statemets about future stock price levels. I this role it costitutes a alterative to the more traditioal regressio forecastig techiques to which it is, i may ways, superior.. As a extesio of the radom walk hypothesis. McQuee ad Thorley (99) used Markov chai model to test the radom walk hypothesis stock prices ad showed that aual real returs exhibit sigificat o radom walk behaviors i the post war period i the New York tock Exchage (NE). Los (998) ivestigated the oparametric efficiecy testig of Asia stock markets ad illustrate that all six Asia stock markets have strog price tred behavior ad ca be profitably exploited by techical aalysis with first-order Markov filters. Mills ad Jordaov (003) examied the predictability of size portfolio returs supplemetig covetioal autocorrelatio aalysis by Markov chai processes ad foud that predictabilities appear for the largest size portfolios rather tha the smallest. Most of the other studies aimed to explore the uderlyig patters of ecoomic mechaisms that geerate the time series of stock returs by usig Markov chais regime-switchig methodology; Hamilto (989) first ivestigate to capture discrete chages i the uderlyig (uobservable) ecoomic mechaism that geerates the fiacial time series data by Markov regime switchig model. Driffill ad ola (998) show that a Markov-switchig model is a more appropriate represetatio of stock divideds ad that regime-switchig provides a better explaatio for stock prices tha the bubble. Kaas (003) examied the forecast performace of stock retur of the Markov regime switchig model for U stock market usig aual observatios ad cocluded that the Markov regime switchig model is the most preferable o-liear empirical extesio of the preset-value model for the stock retur forecastig. Takaki (004) proposed a two-step procedure for predictig itraday returs cosistig of the method of pricipal compoets ad the EM algorithm to estimate the model parameters as well as the uobservable states. First, a rate of retur of a `stock' i a sigle day is assumed to be geerated by several commo factors plus some additive error (`itraday equatio'). ecodly, the oit distributio of those commo factors is assumed to deped o the hidde state of the day, which fluctuates accordig to a Markov chai. I the ext sectios of this study, Markov chai methodology is used to test whether or ot the daily returs of the IE 00 idex follows a radom walk process. Daily returs of IE 00 idex is assumed to be a stochastic process with four discrete state space with Markov chai structure that, the coditioal probability of ay future retur give ay past retur ad the preset retur, is idepedet of the past retur ad depeds oly o the preset retur of the process. After determiig the steady state 334

3 probabilities, give i ay state, probabilities of goig i either directios that are below ad above expected retur are tested.. The sample ad ormality test of the returs Daily values of Istabul tock Exchage (IE) 00 idex were obtaied from the electroic data delivery system of Cetral Bak of Turkey ( The IE 00 idex ca be cosidered as a large diversified portfolio that covers the stocks of 00 leadig firms, which are beig treaded i the IE. Hece, the idex sufficietly represets the IE. The idex values cover 434 workdays of 7 years for the period Daily returs ( ) are computed as a percetage chage of the IE 00 idex: P P () P where, P is the IE idex value i day, (.,,,.434) We iitially estimated the daily expected retur ( µ ) ad stadard deviatio ( σ ) of the IE 00 idex. We also ivestigated the distributioal property ad volatility of the daily returs ad tested that the stock returs are ormally distributed. Table gives the descriptive statistics for the daily stock returs. The expected value of the daily retur is % ad stadard deviatio is 3.46%, that is relatively high whe compared to the mea. This meas that the daily returs exhibit high volatility. Miimum ad maximum daily returs are -0% ad 30% respectively for the period cosidered. Table :Descriptive statistics for the ormality test ample size Mi.Val. Max.Val. Exp. etur ( µ ) td. Dev. ( σ ) etur ( ) We applied the Oe-ample Kolmogorov-mirov ormality test to determie whether or ot the daily returs are ormally distributed. This test compares the observed cumulative distributio fuctio for the stock retur with the cumulative ormal distributio; the Kolmogorov-mirov Z statistic is computed from the largest differece (i absolute value) betwee the observed ad theoretical cumulative ormal distributio fuctios. This goodess-of-fit test tests whether the observatios could reasoably have come from the ormal distributio. 335

4 Figure : Histogram of the daily returs Table : Oe-ample Kolmogorov-mirov Test results Kolmogorov-mirov Z statistic Asymp. ig. (-tailed).000 Figure shows the histogram of the daily returs, ad the Table gives the ormality test results. I this test the ull hypothesis is that the distributio is ot ormally distributed. Calculated Z statistics i Table is ad correspodig twotailed sigificat level is almost zero that we ca strogly reect the ull hypothesis. The daily returs of the IE 00 idex are ormally distributed. The results of ormality test above support the well-kow empirical evidece for stock markets that the distributios of loger-horizo returs are closer to the ormal, (Takaki, 004). 3. Markov chai aalysis for testig of radom walk hypothesis After the calculatio of mea ( µ ) ad stadard deviatio ( σ ) of the daily returs of IE 00 idex ( ), we trasformed the returs ito four discrete state space ad aalyzed these states as Markov chais. Table 3, presets the retur states ( ), retur itervals ad descriptios of the states. We assumed that oe stadard deviatio above the mea retur as high the retur. Table 3: etur iterval ad descriptios of the states etur states ( ) etur iterval Descriptio < 0 Negative retur 0 < µ Positive low retur 3 µ µ + σ Betwee mea ad high retur 4 > µ + σ Above high retur 336

5 We trasformed the daily returs to the four states accordig to the fuctio below ad computed the frequecies of the occurrece of the trasitios from states i to i Table 4., if < 0, if 0 < µ () 3, if µ µ + σ 4, if > µ + σ Table 4: Frequecies of the occurrece of the trasitios from states i to 3 4 ow totals From the frequecies table (Table 4) we ca compute the four state (4x4), oe step (oe day) trasitio probability matrix ( P ) from state i to by dividig the row elemets to row totals: P i (3) I the above oe step trasitio probability matrix, the daily retur states of IE 00 idex ( ) is assumed to be a stochastic process with four discrete state space {,, 3 ad 4 } with Markov chai structure that, the coditioal probability of ay future retur state ( + ), give ay past retur state ( 0 i0,..., i ) ad the preset retur state ( i), is idepedet of the past retur ad depeds oly o the preset retur of the state: P{ i i,..., i i i + 0 0,, { i} P + (4) for all i,, ad for 0,, } 337

6 Markov chai methodology do ot require that the daily stock returs to be ormally distributed but require the Markov chai to be statioary which is defied as costat trasitio probabilities i the log ru. After the modelig the system as Markov chai we ca aalyze the log ru behavior of the retur states to determie the steady state probabilities. ice the above oe step probability matrix (3) have all positive probability values it has the property of regular ergodic chai. A ergodic Markov chai ca have oly oe ivariat distributio, which is also referred to as its equilibrium distributio (see Neal (993) for the properties of ergodic Markov chais). This meas that after eough umber of steps ( days) a give retur state will ted to occur a fixed percet of time. where π π The steady state probabilities ca be stated as follows: For a regular ergodic Markov chai: lim P π ( ) i s are the steady state probabilities ad this limit is idepedet of i. The s satisfy the followig steady state equatios: π > 0, M π, M π π P, For.,,,. M. i i i π We ca fid the value of s that is idepedet of the iitial probability distributio after a eough umber of trasitios. As becomes larger, the values of the P moves to fixed limit ad each probability vector of ted to become equal for all i values of i. Thus, each of the four rows of P i has idetical probabilities: 338

7 P (5) (6) P P (7) P (8) P (9) We ca see from the above probability matrixes that after the five trasitios (5), the values of the P moves to fixed limit ad each of the four rows of P has i idetical probabilities. These results idicate that there is a limitig probability that the retur states will be i steady state coditio after the 5 days ad this probability is idepedet of the iitial state of i. From 5 P i, the steady state probabilities are: 5 i 3 4 π (0) [ ] 339

8 From the steady state probabilities we ca compute the expected recurrece time for each of the retur states T ). Expected recurrece times are equal to the ( reciprocal of the expected steady state probabilities, T, for,,3,4. π or, T π ( ) Hece, expected recurrece times for the states,, 3 ad 4 are., 7.5,.7 ad 8 days respectively. This result idicates that the state of egative retur ( ) occurs most frequetly that stock returs experieces egative retur for each of the days; whe the IE is i egative retur state, it ca be expected that it will be i egative state after two days later. imilarly, whe the IE is i 3, it ca be expected that it will be i state 3 after.74 days later. Whe IE is i high retur state ( 4 ) is expected that it will be i that state after approximately 8 days later. The state have the least frequecy ad expected to be occurrig for each of the 7.49 days. However, give i ay state, probabilities of goig i either directios that are below ad above expected retur ( µ ) are appears to be same. From the steady state probabilities ( π ) we ca see that the sum of the probability of the states that are below the expected retur {P ( )+P( )} ad above the expected retur {P( 3 )+P( 4 )} are ad 0.49 respectively. These probabilities are very close to each other ad suggest that give i ay state; probabilities of goig i either directios below ad above expected retur are same. This situatio ca be treated as a oe-dimesioal radom walk or a martigale with steps equally likely i either directio: Let N represets the umber of days. Let p be the probability of takig a step to the below expected retur states, q the probability of takig a step to the above expected retur states, the umber of steps take to the below, ad the umber of steps take to the above expected retur states. The expected quatities p, q,,, ad N are related by p+q, pq/0.50, + N ad /N. By applyig the maximum likelihood goodess of fitess test, we ca rigorously determie whether or ot the probabilities of retur states goig below ad above the expected retur are equal to From Table 4, we ca calculate observed ad expected umber of frequecies of retur states: N434 (total umber observatios) ad ) E N 434 E 7 (Expected umber of frequecies of 340

9 O 57 (Observed umber of frequecies of ) O 077 (Observed umber of frequecies of ) The ull hypothesis is 7 ad that we ca calculate chi-square ( statistics to test the ull hypothesis: ( O E ) ( O E ) x ) x + E E (57 7) (077 7) Calculated x statistic is almost zero ad statistically isigificat that we ca ot reect the ull hypothesis of equal probabilities. The daily retur states of IE 00 idex follow a radom walk (martigale) process with steps equally likely i either directio of below ad above expected retur. This results are cosistet with the our previous study (Kılıç, 997) that we applied Augmeted Dickey-Fuller test to the series of IE idex, ad foud that the series have a uit root ad follow a radom walk process. Existece of radom walk i IE supports weak form efficiecy hypothesis. The results of this paper do ot idicate that the existece of semi strog form or strog form efficiecy i IE; Muradoğlu ad Meti (996) applied coitegratio test i IE ad foud that the stock prices ad moetary variables coitegrate; IE assimilates publicly available iformatio o moetary variables with a lag. Hece, stock returs could be predicted by moetary variables. This suggest that IE is iefficiet i the semi strog form util 993 because the data of the study covers the period of I aother our previous study, Cabaş et al. (00) we ivestigated the relatioship betwee fiacial characteristics of the idustrial firms ad their aual stock returs i IE, ad foud that three fiacial characteristics (liquidity, profitability to shareholders ad growth) are useful for predictig stock returs; publicly available fiacial data did ot reflected i the stock prices ad it is possible to outperform i IE by fudametal stock aalysis. 4. Coclusio esult of this study hold the weak-form EMH that at ay give time, stock prices fully reflect all the available historical iformatio. Uder a radom walk, historical data o prices ad volume have o value i predictig future stock retur. I other words, statistical aalysis ad "techical aalysis" is useless. Buyig ad sellig stocks by ust depedig oly o historical stock prices i a attempt to outperform above the market retur will effectively be a game of chace rather tha skill. Further research could be coducted toward the high frequecy returs, such as five miute itraday returs. I this way we ca see if there is a opportuity to outperform by itraday buyig ad sellig strategy. 34

10 efereces Cabaş,., Düzakı, H. ad Kılıç.B., 00. Fudametal ad macroecoomic iformatio for commo stock valuatio: The Turkish case. Yapı Kredi Ecoomic eview, Vol. 3, No.. Driffill, J., ad ola, M., 998. Itrisic bubbles ad regime-switchig. Joural of Moetary Ecoomics, Vol. 4 Issue, p357. Eugee F.F., 965. adom Walks i tock Market Prices, Fiacial Aalysts Joural, eptember/october. Hamilto, J.D., 989. A ew approach to the ecoomic aalysis of ostatioary time series ad the busiess cycle, Ecoometrica 57, Kaas, A., 003, No-liear Forecasts of tock eturs. Joural of Forecastig, Vol. Issue 4, p 99. Kılıç,.B., 997. İMKB de Zayıf Etkilik ve assal Yürüyüş, III. Ulusal Ekoometri ve İstatistik empozyumu Dergisi, 9-30 Mayıs, 997, Uludağ Üiversitesi, Bursa. Los, C.A.., 998. Noparametric Efficiecy Testig of Asia tock Markets Usig Weekly Data. Yale chool of Maagemet's Ecoomics esearch Network, p, 30p. McQuee, G., ad Thorley,., 99. Are tock eturs Predictable? A Test Usig Markov Chais. Joural of Fiace, Vol. 46 Issue, p39. Mills, T.C. ad Jordaov, J.V., 003. The size effect ad the radom walk hypothesis: evidece from the Lodo tock Exchage usig Markov Chais. Applied Fiacial Ecoomics, Vol. 3. Muradoğlu, G. ad K. Meti., 997. "Efficiecy of the Turkish tock Exchage with respect to Moetary Variables : A Coitegratio Aalysis", Europea Joural of Operatioal esearch, Vol. 90, No. 3, p Neal,.M., 993. Probabilistic iferece usig Markov chai Mote Carlo methods. Techical eport CG-T-93-, Departmet of Computer ciece, Uiversity of Toroto. adford M.N., 993. Probabilistic Iferece Usig Markov Chai Mote Carlo Methods, Techical eport CG-T-93-, Departmet of Computer ciece, Uiversity of Toroto, 5 eptember 993. ya, T.M., 973. ecurity Prices As Markov Processes. Joural of Fiacial & Quatitative Aalysis, Vol. 8 Issue, p7, 0p. Takaki H., 004. A discrete-time model of high-frequecy stock returs. Quatitative Fiace, Vol. 4 Issue, p40. 34