Estimating the Dynamics of Weak Efficiency on the Prague Stock Exchange Using the Kalman Filter *

Transcription

1 JEL Classificaion: C, D53, G4 Keywords: GARCH, Kalman filer, maringale, weak-efficiency Esimaing he Dynamics of Weak Efficiency on he Prague Sock Exchange Using he Kalman Filer * Ví POŠTA Universiy of Economics, Prague (posav@vse.cz) Absrac The paper builds on he maringale represenaion of he marke efficiency hypohesis and, wih he use of an E-GARCH model of he volailiy of he PX and PX-GLOBAL daily reurns, a sae-space model is formulaed. Using he Kalman filer, he ime-varying dependency of he daily reurns on heir lagged values is esimaed. The esimaion of his parameer shows how quickly he Prague Sock Exchange, represened by is PX index and PX-GLOBAL index, has gradually moved oward he condiion of weak efficiency.. Inroducion The usual approach aken when assessing marke efficiency is o examine wheher or no a marke is efficien wih respec o a paricular and fixed period of ime. In his paper I ake a differen view of he problem and I aemp o esimae he evoluion of marke efficiency in he environmen of he Prague Sock Exchange. The heoreical background I make use of is he maringale represenaion of he weak-efficiency hypohesis. According o his view, a marke is weak-efficien when oday s reurns are no dependen on lagged pas reurns. Insead of running a ime series regression o check he value of he paricular coefficiens, I will esimae how his parameer of he dependency of presen reurns on lagged pas reurns has been changing over ime. Thus, one can obain a good picure of he evoluion of he naure of he Czech capial marke (he Prague Sock Exchange). The main idea of he esimaion is o formulae a sae-space model where he sae variable is he esimaed ime-varying dependency of presen reurns on lagged pas reurns. Such a ime-varying regression funcion also has ime-varying residuals, whose variance will be modeled by a paricular GARCH model. The sae variable is hen esimaed using he Kalman filer. Such an esimaion is based on an observable variable he reurns. Before he sae-space model is formulaed, I es several versions of he GARCH model o find ou which models will be mos suiable for use in he sae-space model. I will esimae he dynamics for he wo Prague Sock Exchange indices: he PX and he PX-GLOBAL. This approach using Kalman filering is based on he analyses by Hall (00), Li (00, 003), and Rockinger (000). Rockinger made a comparison among ransiion economies including he Czech economy and repored slow convergence of he Czech capial marke oward he condiion of weak efficiency. Wih he developmen of he Czech capial marke since ha ime, especially wih respec o in- * This paper is a par of a research projec financed by IGA Universiy of Economics, Prague 48 Finance a úvr - Czech Journal of Economics and Finance, 58, 008, no. 5-6

2 creasing liquidiy, i is reasonable o expec ha he model esimaed in his paper should repor an increasing level of convergence oward weak efficiency. Three papers analyzing he weak-efficiency condiion of he Czech capial marke have been published recenly. In (Poša, Hackl, 007) he weak-efficiency condiion is esed by comparing Mone Carlo simulaions of sock prices wih he real behavior of sock prices. The inerpreaion of such ess is no sraighforward, bu he resuls of he ess indicae ha he marke migh be considered o be close o weak efficiency. Tran (007) and Hájek (007) do no es he hypohesis wihin he maringale represenaion, bu raher use random walk as a baseline model. This approach is no usually preferred nowadays; for furher discussion see (LeRoy, 989). Tran finds ha he marke does no mee he condiion of weak efficiency represened by he random walk, especially when non-linear mehods are used. Hájek focuses on inernaional comparison and also does no consider he Czech marke o be weak- -efficien, in conras o he Hungarian marke. The srucure of he paper is as follows: in he nex secion mehodology I sar wih he maringale formulaion of he weak-efficiency hypohesis. Then, I proceed wih a presenaion of he GARCH models I will laer use o es and form he sae-space models. A he end of secion I describe he daa used for he analysis. In he resuls secion, I firs presen he empirical resuls of he GARCH esing and hen give a general formulaion of he sae-space model. The empirical esimaions of he models are given and he final oupu in he form of he esimaed ime-varying dependency coefficiens is presened. Finally, in he conclusion I make a few remarks on he evoluion of he esimaed parameers and hus on he weak-efficiency hypohesis in he Czech environmen.. Mehodology To es he marke efficiency hypohesis, i is necessary o clearly sae he model wihin which he concep is considered. I will assume he maringale exposiion of he marke efficiency hypohesis. I sress ha he random walk represenaion of weak- -efficiency is sricer a deailed exposiion of he problem is given in (LeRoy, 989). Le s assume a price process {P } and an informaion se I which consiss of all he pas realizaions of he price process {P }. The price process {P } is a maringale if he following condiion holds: EP / I P () Expression () saes ha he condiional expecaion of he fuure price based on he given informaion se is equal o he curren price. This can be equivalenly expressed by he concep of he fair game: Er / 0 I () which saes ha he condiional expecaion of he fuure reurn r + based on he given informaion se is equal o zero. Boh expressions () and () imply ha i is impossible o use he pas evoluion of he price process o make such predicions of he fuure prices or reurns so as o earn sysemaic exra yields (yields above he marke reurn). Finance a úvr - Czech Journal of Economics and Finance, 58, 008, no

3 The fuure reurn may be expressed as is curren condiional expecaion based on he informaion se of he whole evoluion of he pas reurns plus he realized exra yield: r / E r Y (3) where is he informaion se of he pas reurns and Y is he realized exra yield. Equaion (3) saes ha he ex-pos fuure reurn r + consiss of is ex-ane expecaion, which, of course, usually differs from he real fuure price. The difference is represened by he realized exra yield, which may be posiive or negaive. Le s express he condiional expecaion in (3) wih respec o he beginning of he process. Le s assume he process sared n periods ago (which of course may be an arbirarily chosen poin in ime), hen he condiional expecaion in (3) may be expressed as: E r / E E r / /... E... E r /... / (4) n n while he following holds: nn... and n rn. In oher words, due o he ieraed condiioning propery of he condiional expecaions he curren expecaion of he fuure reurn may be expressed as he saring value of he reurn, as i is he only member of he firs informaion se n : E r / (5) Keeping he maringale concep in mind, he realized exra yield in (3) mus be purely sochasic from he poin of view of period : Y (6) The maringale model does no require o be whie noise, because he maringale model, as opposed o he random walk model, does no exclude he possibiliy of predicing he variabiliy of he price process based on is pas evoluion. Subsiuing (5) and (6) in (3) he fuure reurn may be expressed as: r (7) where r + is he reurn on an index. According o such a represenaion of he weak- -efficiency hypohesis, he marke is weak-efficien if he bea coefficiens in he following regression funcion are zero or saisically insignifican: p p jri j0 i r (8) As already indicaed, my aemp is no o run a simple regression such as ha of (8) and esimae he beas, bu raher o build on equaion (8) and, wih he help of he sae-space represenaion, o esimae he ime-varying beas. The version of equaion (8) I will use in he modeling is as follows: r r (9) I is based on a simple regression which proved ha only he one-lagged value of he pas reurns is significan. The oher lags up o a lag of 0 were no signi- 50 Finance a úvr - Czech Journal of Economics and Finance, 58, 008, no. 5-6

4 fican. This applies for boh he PX index and he PX-GLOBAL index. The residuals in (9) are assumed o behave as follows: ~ N 0, h (0) Tha is o imply ha he variance of he residuals will no be aken as a given number bu will be modeled using a GARCH specificaion. I will es five GARCH represenaions on he series of pas reurns on he sock indices. The firs wo will be general GARCH represenaions GARCH (,) and GARCH (,). Then GARCH-M, E-GARCH, and TARCH will be esed. The general represenaion of he models consiss of equaion (9) excep he GARCH-M specificaion, as will be made clear below and he paricular variance equaion according o he GARCH version. In he case of he general GARCH, he variance equaions for GARCH (,) and GARCH (,) are, respecively: h h () h h h () 3 4 where are residuals from he mean equaion. The GARCH-M model adds a special parameer ino he mean equaion so ha he mean equaion (9) becomes: r r h (3) where measures he sensiiviy o he expeced risk of he sock index (in his analysis). As he variance equaion I use equaion (). Furher, I es he E-GARCH specificaion, which uses he variance equaion in he following form: ln h 3ln h (4) h h The E-GARCH specificaion enables us o es he so-called leverage effec, which saes he hypohesis ha negaive shocks in he form of negaive residuals have a larger impac (and also persisence) on he variance (volailiy) han posiive ones. This effec is embodied in coefficien. When i is negaive and saisically significan, he leverage effec is proved. A big advanage of his model is ha, due o is exponenial naure, i does no give negaive values of volailiy. The las model of volailiy esed in his paper is he TARCH (or Threshold GARCH) model. The variance equaion is defined as: h h (5) 3 where is one when he shock (residuals) is negaive and zero oherwise. Therefore, as in he case of he E-GARCH model, he leverage effec is presen. If coefficien is posiive, hen negaive shocks have bigger impac on volailiy han posiive ones. Finance a úvr - Czech Journal of Economics and Finance, 58, 008, no

5 TABLE Descripive Saisics for he Reurns on Indices Index Mean S. dev. Skewness Kurosis The GARCH models will be assessed according o he saisical significance of he esimaed parameers and also according o he presence of serial correlaion in he residuals. The Durbin-Wason saisic and he Ljung-Box Q-saisic will be used: Q T T k j (6) T J where j is he j-h auocorrelaion, T is he number of observaions, and k is he chosen lag. When he auocorrelaion beween he residuals is measured, his saisic can be used o es he specificaion of he mean equaion. Of course, no saisically significan serial correlaion should be presen. When he squared residuals are used in he compuaion, his saisic can be used o es he specificaion of he variance equaion. Again, no serial correlaion should be presen. In addiion, I use he ARCH LM es o check for he remaining condiional heeroskedasiciy in he residuals. No saisically significan heeroskedasiciy should be presen if he variance equaion is correcly specified. As already saed, he sae-space model will be presened in he nex secion afer he GARCH models have been esed. The las par of his secion will be dedicaed o a descripion of he daa used. The esimaion was carried ou wih he help of he Prague Sock Exchange indices he PX and he PX-GLOBAL. The PX sared on April 5, 994 and is a direc coninuaion of he former main index he PX-50. This index consiss of blue-chip socks, so is base is narrow. On he oher hand, he PX-GLOBAL has a broader base and also includes less liquid sock issues which do no have o mee such sric requiremens as he ones included in he PX. Boh indices are compued as price indices and do no ake accoun of dividend yields. For he analysis I use he hisory of he indices from January 5, 995 o July 4, 007. As is clear from he heoreical presenaion of he analysis, daily reurns, which are measured as differences in logarihms, are used. Table repors he basic saisics for he wo series of daily reurns. In Table, *,**, and *** show he rejecion of he null hypohesis of a normal disribuion (Jarque-Bera) or he exisence of a uni roo (ADF) a significance levels of 0 %, 5 %, and %, respecively. The Jarque-Bera saisics show ha he reurns do no follow a normal disribuion. This is caused especially by he high values of kurosis of he disribuion. In addiion o descripive saisics, he ADF (augmened Dickey-Fuller) -saisic is repored o show he saionariy of he series. The null hypohesis of a uni roo is rejeced a he % significance level. Thus, he original series of he sock indices are I() inegraed of order. j Jarque- -Bera ADF -saisics PX *** *** PX-G *** *** 5 Finance a úvr - Czech Journal of Economics and Finance, 58, 008, no. 5-6

6 TABLE GARCH(,) TABLE 3 GARCH(,) Index PX PX-G Index PX PX-G Coefficien Value Value Coefficien Value Value *** *** *** ** 0.58 *** *** *** *** 3.0E-06 ***.97E-06 *** 5.3E-06 *** 3.44E-06 *** *** *** *** *** *** *** ** ** ** *** Saisic Value Value Saisic Value Value DW DW Resuls Firs. he resuls for he GARCH models for boh he PX and he PX-G (PX- -GLOBAL) will be repored. The firs wo models esed were he sandard (.) and (.) GARCH models. Tables and 3 repor he esimaions of he paricular coefficiens. In all he ables below. *.**. and *** show he rejecion of he null hypohesis of insignificance of he paricular parameer a significance levels of 0 %. 5 %. and %. respecively. To make he inerpreaion of he resuls more convenien I again sae he mean and variance equaions of he models. GARCH (.): r r h h and GARCH (.): r r h h h 3 4 In he case of GARCH (.) all he coefficiens are significan a he % level of significance. In he case of GARCH (.) one coefficien is no saisically significan. Table 4 gives he empirical esimaion of he GARCH-M model saed as: r r h h h The only difference beween GARCH-M and GARCH (.) as I used i is he coefficien, which measures he sensiiviy of he reurns o he expeced risk. The esimaion of his parameer is saisically insignifican. The las wo esed models were E-GARCH and TARCH. Tables 5 and 6 repor he resuls for E-GARCH: r r ln h ln h 3 h h Finance a úvr - Czech Journal of Economics and Finance, 58, 008, no

7 TABLE 4 GARCH-M Index PX PX-G Coefficien Value Value *** *** E-06 ***.97E-06 *** *** *** *** *** Saisic Value Value DW TABLE 5 E-GARCH TABLE 6 TARCH Index PX PX-G Index PX PX-G Coefficien Value Value Coefficien Value Value ** *** ** ** *** *** 0.55 *** *** *** *** 3.57E-06 ***.5E-06 *** *** *** *** *** *** *** *** *** *** *** *** *** Saisic Value Value Saisic Value Value DW DW and TARCH: r r h h 3 As is clear from he resuls, all he esimaed parameers are saisically significan. Also, one can see ha he leverage effec was proved. In he case of E-GARCH, i is coefficien. As i was esimaed o be negaive, he impac of negaive shocks on volailiy is higher han ha of posiive shocks. In he case of he TARCH model i is again coefficien, which here mus be posiive if he leverage effec is presen. All ables on he GARCH models repor he Durbin-Wason saisic o check he serial correlaion of he residuals. I is approximaely. in all cases, which is accepable. Based on he significance of he esimaed coefficiens, only GARCH (,), E-GARCH, and TARCH are suiable for he sae-space models. All hree models also show no serial correlaion in he squared residuals as measured by he Ljung- -Box Q-saisic, and he ARCH LM es shows no remaining heeroskedasiciy (I do no presen he resuls of he ess here). 54 Finance a úvr - Czech Journal of Economics and Finance, 58, 008, no. 5-6

8 TABLE 7 Model PX TABLE 8 Model PX-Glob Coefficien Value Coefficien Value ** ** *** *** *** *** *** *** ** *** Saisic Value Saisic Value Log Likelihood Log Likelihood AIC AIC Based on hese resuls, he sae-space models are formulaed. I presen here only he sae-space model wih he E-GARCH specificaion, firs as an example on which i is easy o visualize he oher represenaions and, second as he model which gave he bes resuls in boh cases and which will be discussed furher below. The sae-space formulaion is as follows: r r, ~ N 0, h ln h ln h 3 h h, ~ N 0, e The firs equaion is he mean equaion (and also he regression funcion (9)) described above. The ime subscrip,, behind indicaes ha his coefficien is no o be esimaed as a single value bu as a ime-varying parameer, forming a series. The second equaion is he variance equaion of he E-GARCH model, which describes he behavior of he variance of he residuals in he firs equaion. The hird equaion describes he behavior of he bea coefficien. I is supposed o follow a random walk whose variance is defined by he exponenial funcion. Parameer is also o be esimaed. The covariance of he residuals is se o zero. The Kalman filer is a good way of esimaing an unobserved (sae) variable from he observed variable(s). The idea behind he Kalman filer is presened in Appendix. As I have already saed, he sae-space models wih oher variance specificaions gave poor resuls. In he Table 7 and 8 I presen he resuls for he model defined by (7) for he PX and PX-GLOBAL indices. All he parameers excep are saisically significan. I jus recall ha his parameer poins o wheher or no negaive shocks have a larger impac on he volailiy of he reurns han posiive shocks. The parameer was esimaed as negaive, which would indicae he leverage effec. However, i was no significan, even a he 0% level. (7) Finance a úvr - Czech Journal of Economics and Finance, 58, 008, no

9 FIGURE Bea Coefficien Series for PX Value Trading Day FIGURE Bea Coefficien Series for PX-Glob Value Trading Day Higher values of he log likelihood funcion are preferred. The value of he log likelihood funcion eners he Akaike informaion crierion, of which lower values are preferred. I is defined as: l k AIC T T (3) where l is he value of he log likelihood funcion, T is he number of observaions, and k is he number of parameers esimaed. Of course, here are no precise values of hese parameers which should be reached. As compared wih he oher esimaed models, he value of he log likelihood funcion is he highes and he AIC is sufficienly low, alhough he main deficiency of he oher models consised in he low saisical significance of more han one of he esimaed parameers. No saring values for he parameers or for he covariance marix were used. The esimaed sae variable in he form of he ime-varying parameer for he cases of he PX index and PX-GLOBAL index is depiced in Figures and. The iniial esimaed value of he bea coefficien was 0.45 in he case of he PX index and 0.63 in he case of he PX-GLOBAL, which indicaes clear weak- 56 Finance a úvr - Czech Journal of Economics and Finance, 58, 008, no. 5-6

10 TABLE 9 (Year over Year) Indices of Traded Volumes Year/Year Main Marke Secondary Marke Free Marke 00/ / / / / / inefficiency. The value of bea hen kep falling almos coninuously o he level of 0.04 as of April 6, 00. The evoluion of he bea for he PX-GLOBAL is almos idenical, excep i reached a slighly lower value a ha ime. The bea for he PX sared rising again in 00, reaching 0. in March 004. Approximaely in he hird quarer of 006 i again sared falling significanly, reaching 0.0 on June 9, 007. The bea for he PX-GLOBAL followed a similar paern and sared rising sharply in July 00, reaching 0.4 in February 004. In May 006 i sared falling significanly and reached 0.0 on June 9, 007. The evoluion of he dependency of he reurns on one-lagged pas reurns clearly shows how he Czech capial marke, as approximaed by he Prague Sock Exchange, has become weak-efficien. Even hough he beas are no sricly zero, he dependency of curren reurns on pas reurns is insignifican. The analysis also shows ha he evoluion is no smooh. Indeed, here is an apparen reversal approximaely from he las quarer of 00 o he firs half of 006. The possible explanaion of his shif lies in he economic developmen of he Czech economy ogeher wih he liquidiy of he marke. Of course, hese facors are parly inerlinked. The marke probably reaced o he economic slowdown ha began a he end of 000 and coninued ill he end of 00. We migh have expeced he marke o reac much sooner, bu his laer reacion jus poins o is semi-srong inefficiency. This apparen economic slowdown was parly accompanied by a decline in he liquidiy of he marke, which is repored in Table 9. In Table 9 here is an apparen decrease in he volume raded on he marke beween 003 and 00. The lower liquidiy suppors he possibiliy of he exploiaion of pas prices o predic fuure prices, because he prices are more sable. From Figures and one can see ha mos of he increase in he beas occurred in his period. In Figures and we can also see ha here was no such an abrup reacion of he beas o he economic recession in 997 and 998, which migh cas a lile doub on he presened explanaion. However, one mus ake accoun of he absolue values of he coefficiens, which were much higher a ha ime, poining o apparen inefficiency of he marke. To assess he possible impac of economic developmen and liquidiy on he beas more rigorously, I formulae wo auoregressive models. The daa used for The issues raded in he main, secondary, and free marke differ in erms of he requiremens he issue mus mee and he disclosure duies. On July, he main marke and secondary marke merged and were named he main marke. The indices are calculaed as year-over-year indices of he volume raded on he paricular marke. Finance a úvr - Czech Journal of Economics and Finance, 58, 008, no

11 TABLE 0 MODEL - Bea-PX MODEL - Bea-PX-GLOB Coefficien Value Coefficien Value Bea-PX (-).6794 *** Bea-PX-GLOB (-3) * Bea-PX (-) *** HDP (-) * HDP (-) * Saisic Value Saisic Value LM es saisic.575 LM es saisic.845 he models include he esimaed beas, GDP in levels, and he raded volume of socks and unis. While he beas and raded volumes are on a monhly basis, GDP is on a quarerly basis. To exploi he longer monhly series, I disaggregae he GDP series o a monhly basis (using he second-order polynomial so ha he sum of hree monhs amouns o he value of he respecive quarer). All series were seasonally adjused and enered in logs and firs-differenced. Table 0 presens he resuls for he auoregressive models which include GDP as a regressor (he Breusch-Godfrey LM saisic is presened as an indicaor for serial correlaion; he null hypohesis of no serial correlaion was no rejeced). The GDP parameer, which accouns for economic developmen, eners wih lag (-) and is coefficien is negaive and significan a he 0% level. This means ha a decline in economic aciviy leads o a rise in he beas, which suppors he hypohesis formulaed above. I is noeworhy ha he GDP parameer eners wih lag (-) o (-4) wih saisical significance up o 0 %, wih he bes resul presened in he able. I repor his fac because of he frequency ransformaion. Regression wih he liquidiy parameer did no prove he parameer of ineres o be saisically significan a reasonable lags, so I do no presen he resuls. However, i is imporan o make wo remarks. Firs, he coefficien signs were negaive a lags (-) o (-3), which is in line wih he above reasoning. And second, i is imporan o bear in mind ha i was no possible o exrac he raded volume of unis from he liquidiy parameer (he volume raded). 4. Conclusion In he paper I presened a sae-space model and he Kalman filering echnique o esimae he evoluion of he dependency of he curren reurns on he PX and PX-GLOBAL indices on heir one-lagged pas values. The sample sars in 995 and he resuls of he analysis show ha he marke was clearly weak-inefficien a ha ime. The resuls also show how he marke has neared weak-efficiency since ha period. The analysis also shows he speed a which his naure of he capial marke has changed. As a preliminary analysis, several models of volailiy were esed. The resuls suppor he leverage effec hypohesis, indicaing ha negaive shocks may have a larger impac on he volailiy of he reurns han posiive shocks. However, his resul was no suppored wihin he sae-space models. I was impossible o exclude unis from he figures on he raded volume. 58 Finance a úvr - Czech Journal of Economics and Finance, 58, 008, no. 5-6

12 The developmen of he sensiiviy of he curren reurns on he pas values was discussed in he broader conex of he economy. This discussion poined ou he relaionship beween he developmen of weak-efficiency and economic developmen ogeher wih he liquidiy of he capial marke. This hypohesis was esed wihin an auoregressive model. The liquidiy parameer was no found o be saisically significan, which may be due o mehodological problems. The role of economic developmen as measured by GDP proved o be a significan facor of he evoluion of marke efficiency. Based on he resuls of his analysis and my oher recen paper, I conclude ha he Czech capial marke approximaed by he Prague Sock Exchange may be considered weak-efficien. Anoher imporan noe is ha he condiion of weak-efficiency should no be considered o be saic. Even a once weak-efficien marke may go hrough periods characerized by weaker fulfillmen of he condiion. This may be caused by he macroeconomic developmen of he economy, which resuls in emporary microeconomic changes in he framework wihin which he marke operaes. APPENDIX Kalman Filer Appendix provides a basic represenaion of he Kalman Filer mehod. The Kalman Filer consiss of wo sages: filering and smoohing. The firs equaion below describes he observed variable and he second describes he unobserved (sae) variable: Y ZX RD X TX where Z, R, and T are he coefficien marices, Y and X are vecors of he observed and unobserved variables, respecively, D is a vecor of exogenous variables, and and are sochasic variables wih variance/covariance marices H and Q, respecively: H Q r r is called he signal-o-noise raio. As new informaion on he observed variables is released, he filering procedure creaes esimaes of he unobserved variables. Le s assume A is an opimal esimae of he vecor X and P is he corresponding variance/covariance marix. Then, knowing A - and P -, he opimal esimae may be expressed as: A T K Z A K Y D / / K TP/ ZF a F ZP/ Z H P / T P/ P/ ZF ZP/ TQ kde: Based on hese esimaes he esimaion error is compued: Y ZA RD / Finance a úvr - Czech Journal of Economics and Finance, 58, 008, no

13 The esimaion error eners he log likelihood funcion, which is o be maximized. This is he crierion for he opimal esimae: l log log F F The second sage uses all he available informaion. I is a backward recursive compuaion which sars wih he las esimae of he filering procedure and goes back o he beginning of he sample: A/ T AP A/ TTA P/ T P P P/ TP/ P P PT P / REFERENCES Bollerslev T (986): Generalized Auoregressive Condiional Heeroskedasiciy. Journal of Economerics, 3: Engle RF (98): Auoregressive Condiional Heeroskedasiciy wih Esimaes of he Variance of U.K. Inflaion. Economerica, 50: Engle RF, Lilien DM, Robins RP (987): Esimaing Time Varying Risk Premia in he Term Srucure: The ARCH-M Model. Economerica, 55: Fama EF (970): Efficien Capial Markes: A Review of Theory and Empirical Work. The Journal of Finance, : Fama EF (99): Efficien Capial Markes: II. The Journal of Finance, 5: Glosen LR, Jaganahan R, Runkle D (993): On he Relaion beween he Expeced Value and he Volailiy of he Normal Excess Reurn on Socks. Journal of Finance, 48: Hájek J (007): Tes slabé formy efekivnosi sedoevropských akciových rh. Poliická ekonomie, 6: Hall S, Urga G (00): Tesing for Ongoing Efficiency in he Russian Sock Marke. hp:// LeRoy SF (989): Efficien Capial Markes and Maringales. Journal of Economic Lieraure, 4: Li X (00): China s Evolving Sock Marke Efficiency Reconsidered: The Kalman Filer Analysis. Massey Universiy Commerce Working Paper, no Li X (003): China: Furher Evidence on he Evoluion of Sock Markes in Transiion Economies. Scoish Journal of Poliical Economics, 3: Nelson DB (99): Condiional Heeroskedasiciy in Asse Reurns: A New Approach. Economerica, 59: Poša V, Hackl Z (007): Informaion Efficiency of he Capial Marke: a Sochasic Calculus Approach: Evidence from he Czech Republic. Finance a úvr-czech Journal of Economics and Finance, 57(5-6): Rockinger M, Urga G (000): The Evoluion of Sock Markes in Transiion Economies. Journal of Comparaive Economics, 3: Shreve SE (005): Sochasic Calculus for Finance I: The Binominal Asse Pricing Model. Springer. ISBN Tran Q (007): Tesování slabé formy efekivnosi na eském akciovém rhu. Poliická ekonomie, 6: Finance a úvr - Czech Journal of Economics and Finance, 58, 008, no. 5-6