Ieraoal Coferece O Appled Ecoomcs ICOAE 0 463 INFOMATION ENTOPY AND EFFICIENT MAKET HYPOTHESIS DANIEL TAIAN PELE,ANA-MAIA ȚEPUȘ Absrac Ths sudy ams o demosrae ha he exreme values of reurs dsrbuo are mosly assocaed wh pacular perods of sock markes effcecy, whe her level of uceay reaches a local mmum We propose a esmaor of uceay, hrough he eropy of probably desy fuco of reurs The relaoshp bewee he level of uceay of a sock marke ad exreme values of reurs dsrbuo s llusraed rough a bary logsc regresso model esmaed for ma dexes of four sock markes from Ceral ad Easer Europe JEL codes: G0 - Facal Crses, G4 - Iformao ad Marke Effcecy; Eve Sudes, G5 - Ieraoal Facal Markes Keywords: eropy, eropy of a fuco, effce marke hypohess Iroduco Modelg he capal markes s closely lked o he effce marke hypohess, a cocep foud cojuco wh he raoaly of vesor behavor Sce Bacheler's groudbreakg sudy(900), mos of he weeh ceury academc research face has bee formed aroud he paradgm of effce markes From he classcal defo of Fama(970), ul more rece developmes of Tmmerma ad Grager(004), Effce Marke Hypohess(EMH) s separable from he cocep of formao ad he mechasm of corporao of a cea se of formao he radg prce of a facal assenumerous sudes have vesgaed how he effce marke hypohess, as a heorecal model, s observed a he level of emprcal realy Sglz(98) shows ha he hypohess of a effce marke, where prces fully reflec avalable formao (Fama, 970), s o cosse wh he oo of Pareo opmum Moreover, he formao refleced sock prces are jus formao ha does o requre a cos o oba hem Summers (986) argue ha sascal ess commoly used for esg he Effce Marke Hypohess have a relavely low power Thus, a basc sascal law says ha f we cao rejec he ull hypohess, we cao auomacally accep as a vald hypohess From hs po of vew, are forms of effcecy ha ca o be dscrmaed by he usual sascal ess, ad we cao deduce from hese ess a cocluso abou he valdy of Effce Marke Hypohess Malkel (003) dscusses he effcecy of capal markes hrough he ccsms ha have bee made over me He clearly descrbes he radom walk hypohess: he logc of hs hypohess s ha f he formao s mmedaely refleced sock prces, he omorrow's prce chage wll reflec oly he omorrow s formao ad wll be depede of oday s prce chage As he formao s upredcable, he prce chages mus be upredcable ad radom The lk bewee a cea measure of complexy ad he Effce Marke Hypohess s que clear: f we assume a effce marke ( weak form), he he sock prce follows a radom walk model, e he me seres of reurs s a whe ose process I erms of quaave measures, such a whe ose process has he hghes level of complexy; o he corary, f he Effce Marke Hypohess s o me, he he prce s o loger a radom walk process ad cosequely, he level of complexy of he marke wll be lower For sace, f he prce s a purely deermsc process, compleely predcable, he a mmum level of complexy s acheved; f he prce s a purely radom process, compleely upredcable, he we are dealg wh he maxmum level of complexy sso(008) uses eropy as a measure of complexy o vesgae he hypohess ha sock marke crashes are mos closely assocaed wh perods of low eropy The relaoshp bewee sock marke crashes ad formaoal effcecy s as follows: f he marke s effce, so he formao s o saaeously refleced prces, he local reds appears he evoluo of he prce Bu oce he formao s corporaed o sock prce, he vesor s reaco may lead o a sgfca collapse of sock prce Our sudy ams o demosrae ha boh he decle, as well as he sgfca growh of he radg prces ca be explaed by lower levels of sock marke complexy, relao o he Effce Marke Hypohess The paper s orgazed as follows: Seco we descrbe he cocep of eropy as a measure of complexy ad propose a mehod of esmag eropy usg probably desy fuco of reurs; Seco we descrbe he heorecal model used o vesgae he relaoshp bewee he level of complexy of sock marke ad he occurrece of exreme values of reurs; Seco 3 are preseed he resuls of he esmaed model for sock markes of omaa, Bulgara, Hugary ad he Czech epublc ad he las seco s for coclusos Iformao Eropy as a measure of complexy Eropy s boh a measure of uceay ad complexy of a sysem, wh umerous applcaos physcs ( he secod prcple of hermodyamcs), formao heory, bology (DNA sequece complexy), medce, ecoomcs (complexy of a sysem) PhD, Lecurer, Depame of Sascs ad Ecoomercs, Uversy of Ecoomcs, Buchares, Emal: dapele@asero PhD Caddae, Depame of Moey, Uversy of Ecoomcs, Buchares, Emal: aamaraepus@yahoocom
464 Ieraoal Coferece O Appled Ecoomcs ICOAE 0 [] If X s a dscree radom varable, wh probably dsrbuo p x x X : p p, where, he Shao Iformao Eropy s defed as follows: p P X x ) 0 p (, ad H( X ) p log p H( X ) (/ )log (/ ) log For uform dsrbuo he Shao Eropy reaches hs maxmum:, whle he mmum value s aaed for a dsrbuo lke he followg: x x X : 0, H( X ) 0 I oher words, hgh levels of eropy are obaed for suaos wh hgh uceay ad low levels of eropy are assocaed wh suaos wh lower uceay The relao bewee eropy ad capal markes s sraghforward(sso, 008) r p p log P log P Le he logreur of a asse ad le a radom varable assocaed o bullbear saes The for a cea perod of me, oe ca defe he formao eropy of he 0 ad sequece: H plog p ( p)log ( p ) p P( s ) P( s 0) s, 0 0, 0, where I oher words, we ca defe he eropy as a measure of he complexy of he sock marke, by rasformg he me seres of logreurs o a sequece of 0 ad From our po of vew, he mehodology should be exeded by akg o accou he couous aure of reurs dsrbuo ad he followg we propose such a exeso of hs mehodology Eropy of a fuco Ulke he case of a dscree radom varable, he eropy of a couous radom varable s dffcul o defe If X s a couous radom varable wh probably desy fuco f (x), he we ca defe, by aalogy wh he Shao formao eropy, dffereal eropy: H f ) log dx ( A, [] where A s he suppo se of X A ave esmaor of dffereal eropy could be used o quafy he complexy level of a sock marke, ad he resuls for BET Idex of Buchares Sock Exchage(Pele, 0) shows a sgfca correlao bewee he values of hs esmaor ad probably of exreme egave values of daly logreurs dsrbuo Ufouaely, dffereal eropy does o have all he propees of Shao eropy: ca ake egave values ad addo s o vara o lear rasformaos of varables However, we ca defe he eropy of a fuco ha sasfes cea propees, hrough a rasformao called quazao We prese he esseal elemes of hs rasformao, as hey appear Lorez (009) Defo Le f : I [ a, b] * a couous real-valued fuco, le N x a h ad defe ( / ), for 0,,, S( f )( ) f ( x ) where h ( b a) / The sampled fuco for f s for 0,, Quazao process refers o creag a smple fuco ha approxmaes he orgal fuco Le q 0 a quaum The he Q f x q followg fuco defes a quazao of f: q ( )( ) ( / ), f [ q,( ) q) Defo Le f measurable ad esseally bouded o he erval [ a, b] ad le q 0 I [ q,( ) q) Le B ( ) ad f I The H ( f ) ( B )log ( ( B )) q he eropy of f a quaum q s, where s he Lebesgue measure Followg hese defos, oe ca compue he eropy of ay couous fucos defed o a compac erval If x o [0,], he for a fxed quaum q / H f, q ( ) log, he maxmum value of he eropy The followg heorem (Lorez, 009) provdes a cocepual framework for defg a esmaor of eropy of a fuco Theorem
Ieraoal Coferece O Appled Ecoomcs ICOAE 0 465 S ( f ) Le f couous o [a,b] ad le / samplg space Le Q S he sampled fuco ad q S a quazao of wh quaum q 0 c () Deoe {( / ) qq S } card q (umber of values ( / )q q Q S p () ) ad le probably of value : c( ) c( ) p( ) c( j) j p )log p ( ) H ( f ) lm ( q The h 0 [3] The above heorem assures us ha regardless of quazao ad samplg, we oba a cosse esmaor of he eropy of a fuco Eropy of probably desy fuco of reurs Ths cocepual framework ca be used order o defe he eropy of probably desy fuco(pdf) of reurs dx Le f a couous real-valued fuco such as 0 ad The he hypohess of Lorez heorem are H ( f ) fulflled ad we ca compue q I realy, he aalycal expresso of probably desy fuco s ukow, so we ca esmae he desy usg a oparamerc approach, such as Kerel Desy Esmao (KDE) ^ x x K Thus, KDE esmaor of pdf s h h [4] K s a kerel fuco, wh he followg propees: xk ( x) dx 0 K ( x) 0, x, K ( x) K( x), x, K( x) dx The parameer h s a scale parameer, whose choce deermes he qualy of he esmae (s also kow as smoohg parameer or badwdh) Bascally, our mehodology volves he followg seps o esmae he eropy of he probably desy fuco of reurs: r Le he me seres of logreurs for a me perod T, ad le f (x) he probably desy fuco k If f ( x ), we esmae pdf wh Kerel Desy Esmao, obag values for 0,, S Les sampled fuco as S( f )( ) f ( x ) for 0,, k q Q S f j q Le a quaum; he defe q ( )( ) ( / ) f ( x j ) [ q,( ) q), f c card f x q q ( ) c( ) { ( j ) [,( ) )} p( ) c ( j) j Compue probables H ˆ q ( f ) p( )log p( ) Oe ca esmae he eropy of probably desy fuco as [5] Ths esmaor of eropy of he probably desy fuco of reurs wll be called, wha follows, PDF Eropy Noes xmax xm h Acually, we have chose x x h ad m ( / ), for 0,, For compuaoal reasos, we have used k 7 8, ad KDE was doe usg a Gaussa kerel: K( x) exp( x / )/ There are may dsrbuos for whch he probably desy fuco s o ecessarly bouded (Ch-square dsrbuo s oe such example); moreover, eve pdf s bouded, hs rage s dffere from dsrbuo o dsrbuo To esure comparably bewee he resuls of he esmao varous markes, we proceeded o sadardze he values esmaed by KDE: f ( x ) m f f ( x ) max f m f, where max f ad m f are he exreme esmaed values of probably desy fuco ad
466 Ieraoal Coferece O Appled Ecoomcs ICOAE 0 Oe ca defe a ormalzed esmaor of eropy of probably desy fuco of reurs (Normalzed PDF Eropy): Hˆ N ( f ) q p ( )log log p ( ) [6] Ths wll esure comparably amog dffere markes; fac, hs esmaor of complexy wll be used below o llusrae he relaoshp bewee he degree of complexy of he marke ad he lkelhood of exreme eves 3 The heorecal model of eropy as a predcor of exreme values of reurs dsrbuo Eropy ca be regarded as a measure of formaoal effcecy of a sock marke; f he marke s weak form effce, he he prce follows a radom walk process, herefore he bull ad bear marke suaos are lkely probable I erms of formao eropy, marke effcecy s equvale o he suao of maxmum eropy, maxmum complexy or maxmum uceay Coversely, whe he prce exhbs a predoma red(upwards or dowwards), he level of ceay s hgh, ad such perods are descrbed by lower values of eropy Our workg hypohess s ha he lkelhood of boh als of reurs dsrbuo could be explaed by lower values of eropy To verfy hs hypohess we esmae he followg logsc regresso model: * exp( 0 H ) P( Y ) exp( 0 H ) [7] I he above equao, we have: Y * * - Y {, ( ) 00} {, ( ) 00}, where P P (upper ad lower al) H - s marke formao eropy a me, quafed by Normalzed Shao Eropy ad by Normalzed PDF Eropy Normalzed Shao Eropy was esmaed usg he mehodology from sso(008) s, 0 0, 0 Thus, for a cea me perod T, le Usg a rollg wdow T p P( s ), oe ca compue he probably The he Normalzed Shao Eropy for he ere erval of legh T s H [ p log p ( p )log ( p )]/log ( ) [8] Also, we have esmaed Normalzed PDF Eropy for several me ervals T; moreover, sce he me seres of daly reurs s very osy, he model was esmaed usg reurs calculaed o a local me wdow : As we eed o dscrmae amog several models, we should use a performace dcaor of he logsc regresso model I geeral, such a dcaor s defed by comparg he lkelhood fuco of esmaed model wh he lkelhood fuco of he model whe he exogeous varable s removed Oe ca defe pseudo-, as a measure of model s performace(nagelkerke,99): exp{[log L( M) log L(0)]/ }, where L(M ) ad L(0) are lkelhood fuco of he model, wh ad whou he exogeous varable ewg he expresso as formao due o explaaory varable Ufouaely, adjusme s made(nagelkerke,99): adj /[ exp(log L(0)/ )] log( ) [log L( M) log L(0)] / r p p, hs could be erpreed as he surplus of wll ever reach, o eve for a perfec model, so he followg We choose he model ha bes descrbes he correlao bewee realy ad heorecal hypohess usg as cero he adj maxmzao of 4 esuls We esmaed he Normalzed Shao Eropy ad he Normalzed PDF Eropy usg varous rollg-wdows ad varous me perods, for ma dexes of sock markes from four coures of Ceral ad Easer Europe (omaa, Bulgara, Hugary, ad Czech epublc) There are several sudes he leraure dealg wh he marke effcecy of hose coures, ad he resuls are que coradcory Emerso e al(997), foud varyg levels of effcecy ad varyg speeds of moveme owards effcecy wh a sample of four shares, seleced from Sofa Sock Exchage
/5/997 /5/998 /5/999 /5/000 /5/00 /5/00 /5/003 /5/004 /5/005 /5/006 /5/007 /5/008 /5/009 /5/00 /5/997 /5/998 /5/999 /5/000 /5/00 /5/00 /5/003 /5/004 /5/005 /5/006 /5/007 /5/008 /5/009 /5/00 Ieraoal Coferece O Appled Ecoomcs ICOAE 0 467 ockger ad Urga(000), a sudy coverg he perod 993-999, argue ha Hugara marke always sasfes weak effcecy, whle for he Czech marke, hey docume covergece oward effcecy Hájek(007), o a sudy for he perod 995-005, cocludes ha he weak form of he EMH cao be valdaed o he Czech sock marke, sce daly prce chages of boh dvdual socks ad dces are sysemacally learly depede Also, he level of effcecy creases for weekly or mohly reurs Dragoă ad Mrcă(004) ad Dragoă e al(009) foud dffere coclusos regardg he effcecy of omaa sock marke; whle he frs paper, he Effce Marke Hypohess s rejeced, he secod paper reveals a sgfca moveme oward effcecy Our aalyss shows ha are dffere degrees of effcecy amog he four sock markes vesgaed ad hs resul could be explaed usg formao eropy as a measure of sock marke uceay 4 Esmao resuls for BET Idex of Buchares Sock Exchage To esmae he model, we have used daly logreurs of BET Idex, for me perod 9 Sepember 997-5 March 0(3364 daly observaos) Table : Adjused Pseudo - for he wo esmaors of complexy for BET Idex Normalzed Shao Eropy Normalzed PDF Eropy 3 4 5 3 4 5 T=60 00 0008 005 006 0003 0056 0044 006 00 0008 T=00 007 004 004 003 000 0043 003 000 0009 0009 T=50 00 0006 000 005 000 003 004 000 0006 006 T=00 00 0007 000 005 0004 0009 000 0000 0000 0006 T=40 0005 0004 0007 000 0004 0003 0000 0000 0000 000 adj values shows, he bes resuls are obaed usg Normalzed PDF Eropy as explaaory varable, wh 60 As he T ad The fac ha he bes resuls for BET Idex were obaed for T 60 suggess ha he omaa sock marke has o log emporal memory, he local emporal coex beg mos releva I addo, marke complexy esmao usg eropy of probably desy fuco of reurs provdes beer resuls ha he classcal Shao eropy 05 0 005 0-005 -0-05 095 09 085 08 075 07 Graph : Daly logreurs of BET Idex The resuls of he opmum model, for Normalzed PDF Eropy, 60 Table : Esmao resuls of logsc regresso for BET Idex Graph : Normalzed PDF Eropy( T 60, T ad ), are preseed below Parameer Esmae Sadard Error Wald Ch-Square Pr > ChSq 0 09884 09 470 <0000-57776 4034 430948 <0000 adj Observaos 3304 Pseudo- 00558 Ch-Square(Hosmer-Lemeshow - Log L 7038 Goodess of F Tes) 3 Pseudo- 00 Pr > ChSq 0048
7/5/00 /5/003 7/5/003 /5/004 7/5/004 /5/005 7/5/005 /5/006 7/5/006 /5/007 7/5/007 /5/008 7/5/008 /5/009 7/5/009 /5/00 7/5/00 /7/00 8/7/00 /7/003 8/7/003 /7/004 8/7/004 /7/005 8/7/005 /7/006 8/7/006 /7/007 8/7/007 /7/008 8/7/008 /7/009 8/7/009 /7/00 8/7/00 468 Ieraoal Coferece O Appled Ecoomcs ICOAE 0 Aalyzg he resuls of he esmao, we ca see ha he eropy adversely affec he lkelhood of exreme values of daly reurs Thus, f eropy creases by 0 us (eg from 08 o 09), he he odds of occurrece of exreme values of BET reurs drops by aroud 80% Table 3: Normalzed PDF Eropy of BET Idex(als ad body of reurs dsrbuo) Descrpve Sascs N Mea Meda Mode Sd Devao Mmum Y * 78 0967 0934 0747 00486 0747 78 Maxmum Y * 0 36 09449 0958 09054 00355 0764 36 Moreover, he average eropy s sgfcaly lower he days correspodg o he exreme values of reurs dsrbuo ha he oher days 4 Esmao resuls for SOFIX Idex of Sofa Sock Exchage To esmae he model, we have used daly logreurs of SOFIX Idex, for me erval 6 ovember 00 4 Jauary 0 (39 daly observaos) Table 4: Adjused Pseudo - for he wo esmaors of complexy for SOFIX Idex Normalzed Shao Eropy Normalzed PDF Eropy 3 4 5 3 4 5 T=60 008 009 0004 000 000 0077 008 07 0078 0089 T=00 00 0040 003 005 0004 05 0087 045 0069 0040 T=50 008 0039 0047 0056 0045 0089 0064 078 0079 0058 T=00 0070 009 06 044 00 04 004 06 0090 006 T=40 009 000 00 0087 0088 0098 000 03 007 0093 As ca be see from he values of Esmaor for 50 adj for esmaed logsc regresso models, he bes performace offers Normalzed Eropy T ad 3 as well as Normalzed PDF Eropy Esmaor for T 60 ad 3 T ad 3 The resuls are subsaally smlar, bu based o he resuls of Hosmer-Lemeshow es we chose he model wh 60, sce he oher model preses a lack of f For SOFIX Idex also, marke complexy esmao usg eropy of probably desy fuco of reurs provdes beer resuls ha he classcal Shao eropy 0 005 0-005 -0-05 098 096 094 09 09 088 086 084 08 08 Graph 3: Daly logreurs of SOFIX Idex Graph 4: Normalzed PDF Eropy( T 60, 3 ) The resuls of he opmum model, wh Normalzed PDF Eropy as exogeous, T 60 ad 3 Table 5: Esmao resuls of logsc regresso for SOFIX Idex Parameer 0 Esm ae 758 33-337489 Sadard Wald Ch- Pr > Ch Error Square Sq 3750 5594 <0000 40649 689309 <0000, are preseed below
/5/997 /5/998 /5/999 /5/000 /5/00 /5/00 /5/003 /5/004 /5/005 /5/006 /5/007 /5/008 /5/009 /5/00 /5/997 /5/998 /5/999 /5/000 /5/00 /5/00 /5/003 /5/004 /5/005 /5/006 /5/007 /5/008 /5/009 /5/00 Ieraoal Coferece O Appled Ecoomcs ICOAE 0 469 Observa os - Log L Pseudo- 7 0 adj 9 Pseudo- 70 349 Ch-Square(Hosmer- 85 34 Lemeshow Goodess of F Tes) 538 00 03 96 Pr > ChSq 83 Aalyzg he resuls of he esmao, we ca see ha he eropy egavely affec he lkelhood of exreme values of daly reurs Thus, f eropy creases by 0 us (eg from 08 o 09), he he odds of occurrece of exreme values of SOFIX reurs drops by aroud 97% Table 6: Normalzed PDF Eropy of SOFIX Idex(als ad body of reurs dsrbuo) Descrp ve Sascs N Mea a Med e Mod Sd Devao Mm um Maxm um Y * 4 090 09 084 0034 0843 09759 95 3 3 Y * 37 095 095 089 008 08404 09954 0 7 87 48 4 Moreover, he average eropy s sgfcaly lower for he days wh exreme values of reurs (090) ha he oher days (095) 43 Esmao resuls for BUX Idex of Budapes Sock Exchage To esmae he logsc regresso model, we have used daly logreurs of BUX Idex, for me horzo Aprl 997 4 Jauary 0 (344 daly observaos) Table 7: Adjused Pseudo - for he wo esmaors of complexy for BUX Idex Normalzed Shao Eropy Normalzed PDF Eropy 3 4 5 3 4 5 T=60 000 000 000 000 009 069 067 07 087 093 T=00 0000 0000 000 0005 004 083 085 056 005 090 T=50 0000 000 0000 0003 008 083 089 090 037 07 T=00 0005 0004 0006 0000 0004 038 04 056 098 084 T=40 006 006 008 0003 0000 067 054 037 063 07 adj As ca be see from he values of for esmaed logsc regresso models, he bes performace offers Normalzed Eropy Esmaor for T 50 ad 4 For BUX Idex also, marke complexy esmao usg eropy of probably desy fuco of reurs provdes beer resuls ha he classcal Shao eropy 0 05 0 005 0-005 -0-05 -0 098 096 094 09 09 088 086 084 08 08 Graph 5: Daly logreurs of BUX Idex Graph 6: Normalzed PDF Eropy( T 50, 4 ) The resuls of he opmum model, wh Normalzed PDF Eropy as exogeous, T 50 ad 4 Table 8: Esmao resuls of logsc regresso for BUX Idex Parameer 0 Esm ae 89 7-358376 Sad ard Error 909 5 37 5 Wald Ch- Pr > Ch Square Sq 987797 <0000 00030 <0000, are preseed below
30/0/995 6/0/995 0/06/996 9/0/997 4/0/997 7/06/998 //999 7/0/999 7/6/000 5//00 3/0/00 4/6/00 3//003 6/09/003 8/05/004 6/0/005 0/09/005 9/05/006 7/0/007 3/09/007 5/05/008 3/0/009 8/9/009 0/5/00 30/0/995 6/0/995 0/06/996 9/0/997 4/0/997 7/06/998 //999 7/0/999 7/6/000 5//00 3/0/00 4/6/00 3//003 6/09/003 8/05/004 6/0/005 0/09/005 9/05/006 7/0/007 3/09/007 5/05/008 3/0/009 8/9/009 0/5/00 470 Ieraoal Coferece O Appled Ecoomcs ICOAE 0 Observa os 39 07 adj Pseudo- 0 49 Ch-Square(Hosmer-Lemeshow 4 Goodess of F Tes) 5 - Log L Pseudo- 004 3 Pr > ChSq 09 47 Aalyzg he resuls of he esmao, we ca see ha he eropy egavely affec he lkelhood of exreme values of daly reurs of BUX Idex Thus, f eropy creases by 0 us (eg from 08 o 09), he he odds of occurrece of exreme values of BUX Idex reurs drops by aroud 98% Table 9: Normalzed PDF Eropy of BUX Idex(als ad body of reurs dsrbuo) Descrp ve Sascs Y * Y * 0 N Mea 64 088 46 38 094 46 a Med 088 095 05 e Mod 08 4 086 3 Sd Devao 5 4 0040 0034 M mum 4 08 087 Max mum 09668 09906 For BUX Idex, average eropy s sgfca hgher he body of reurs dsrbuo (094), ha he upper ad lower al(088) 44 Esmao resuls for PX Idex of Prague Sock Exchage To esmae he bary logsc regresso model, we have used daly logreurs of PX Idex, for me erval 7 Sepember 993 5 Jauary 0 (484 daly observaos) Table 0: Adjused Pseudo - for he wo esmaors of complexy for PX Idex Normalzed Shao Eropy Normalzed PDF Eropy 3 4 5 3 4 5 T=60 000 0000 000 000 0000 0033 008 0067 0039 0099 T=00 0000 0004 000 000 0000 007 04 05 0050 0 T=50 00 006 00 005 006 0063 073 03 0077 05 T=00 0009 0005 0000 0003 000 0086 04 06 0075 047 T=40 0007 0003 0000 0000 0000 06 057 03 0075 08 As ca be see from he values of Esmaor for T 50 ad adj for esmaed logsc regresso models, he bes performace offers Normalzed Eropy as well as Normalzed PDF Eropy Esmaor for T 00 ad 3 T ad 3 The resuls are subsaally smlar, bu based o he resuls of Hosmer-Lemeshow es we chose he model wh 00, sce he oher model preses a lack of f For PX Idex also, marke complexy esmao usg eropy of probably desy fuco of reurs provdes beer resuls ha he classcal Shao eropy 05 0 005 0-005 -0-05 -0 095 09 085 08 075
Ieraoal Coferece O Appled Ecoomcs ICOAE 0 47 Graph 7: Daly logreurs of PX Idex Graph 8: Normalzed PDF Eropy( T 00, 3 ) The resuls of he opmum model, wh Normalzed PDF Eropy as explaaory varable, 00 below Table : Esmao resuls of logsc regresso for PX Idex T ad 3 Sadard Parameer Esmae Error Wald Ch-Square Pr > ChSq 838 8960 93993 <0000 0-468 505 30977 <0000 adj, are preseed Observaos 3983 Pseudo- 06 Ch-Square(Hosmer-Lemeshow Goodess - Log L 653879 of F Tes) 3840 Pseudo- 0083 Pr > ChSq 0809 Aalyzg he resuls of he esmao, we ca see a egave correlao bewee eropy ad he lkelhood of exreme values of daly reurs of PX Idex Thus, f eropy creases by 0 us (eg from 08 o 09), he he odds of occurrece of exreme values for PX Idex reurs drops by aroud 9% Table : Normalzed PDF Eropy of PX Idex(als ad body of reurs dsrbuo) Descrpve Sascs N Mea Meda Mode Sd Devao Mmum Maxmum Y * 78 09060 09037 086 00455 086 09833 Y * 0 3905 0935 0940 08937 0030 08048 09899 For PX Idex, average eropy s sgfca hgher he body of reurs dsrbuo (093), ha he upper ad lower al(090) 5Coclusos Iformaoal effcecy of he sock markes s a esely debaed opc rece years, especally he coex of curre ecoomc ad facal crss From a formao heory po of vew, capal marke effcecy may be assocaed wh a hgh degree of uceay Ideed, f a sock marke s effce, meag ha formao s rasmed saly ad s compleely corporaed radg prces, s vually mpossble o acpae fuure evoluos prces or reurs Ths raslaes o a hgh degree of uceay, specfc o a radom walk behavor of sock prces As radom walk process s he mos complex erms of predcably, s aural o use eropy as a measure of uceay, relaoshp o marke effcecy The ma cocluso of he sudy s ha perods characerzed by a sharp drop eropy, whe he marke complexy level reaches a local mmum, ad uceay s low, are assocaed wh he occurrece of exreme values of reurs I hs sudy we aalyzed he relaoshp bewee complexy ad predcably of sock marke crashes sock usg eropy of probably desy fuco of reurs as a measure of complexy esuls of he esmaed models show ha hs complexy esmaor produces beer resuls ha classcal Shao eropy The eropy of probably desy fuco of reurs ca be used wo ways: as a ool o measure he degree of marke effcecy, as well as o compare, erms of effcecy, wo or more sock markes As a ool for measurg he degree of sock marke effcecy, we proposed a dcaor who ca deec, a a me, he level of effcecy, akg o accou he local emporal coex Ufouaely, hs dcaor has a umber of dsadvaages, amog whch he dffculy o deduce some heorecal propees, sce he probably desy fuco does o ake values a sadardzed erval, for ay dsrbuo Aoher major dsadvaage s ha depeds very much o he local emporal coex, whch ca be mproved by esmag he dcaor usg raday daao he oher had, as a ool for comparso, PDF Normalzed Eropy may be useful evaluag he relave effcecy of wo markes by comparg he sesvy of he chace of occurrece of exreme values of he reurs dsrbuo o a chage eropy Accordg o Effce Marke Hypohess, a growg level of marke uceay should be refleced a adverse chage he chaces of occurrece of exreme values of reurs Lower he magude of expeced odds chage as a resul of oe u crease of eropy, lower he degree of marke effcecy From hs po of vew, he omaa sock marke s he leas effce relao o capal markes aalyzed hs sudy Accordg o he esmaed model, f eropy creases by 0 us (eg from 08 o 09), he he chace of occurrece of exreme values of BET reur drops o aroud 80%
47 Ieraoal Coferece O Appled Ecoomcs ICOAE 0 For he Bulgara marke hs expeced chage s approxmaely 97%, for he Hugara marke s abou 98%, ad for he Prague Sock Exchage Idex he expeced chage s abou 9% The explaaos for hese dffereces bewee he coures surveyed ca be foud he exsg equales erms of ecoomc ad capal marke developmes Ackowledgmes Ths paper was cofaced from he Europea Socal Fud, hrough he Huma esources Developme Operaoal Secoral Program 007-03, projec umber POSDU/89/5/S/5984, Performace ad excelece he pos-docoral ecoomc research omaa 6efereces Bacheler, L(900), Théore de la Spéculao (frs wo pages of Eglsh raslao), Aales Scefques de l'ecole Normale Supereure, I I I -7, -86 (Eglsh Traslao;- Cooer (ed), (964) adom Characer of Sock Marke Prces, Massachuses Isue of Techology pp7-78) Bouzebda, S, Elhaab, I(009), Uform badwdh cossecy of he kerel-ype esmaor of he Shao's eropy, Compes edus Mahemaque, Volume 348, Issues 5-6, March 00, Pages 37-3, ISSN 63-073X Dragoă, V, Soa, A, Pele, D T, Mrcă, E, Besafa, M(009), The Developme of he omaa Capal Marke: Evdeces o Iformao Effcecy, omaa Joural of Ecoomc Forecasg, vol 0 (), 009, pp 47-60 Dragoă, V, Mrcă, E(004), Emerge capal marke s effcecy: The case of omaa, Europea Joural of Operaoal esearch, 55, 004, pp 353-360 Emerso,, Sephe, H, Zalewska-Mura, A(997), Evolvg Marke Effcecy wh a Applcao o Some Bulgara Shares, Ecoomcs of Plag, Volume 30, Issue, Pages 75-90 Fama, E(970), Effce Capal Markes: A evew of Theory ad Emprcal Work, Joural of Face, 5(), 383 47 Hájek, J(007),"Czech Capal Marke Weak-Form Effcecy, Seleced Issues", Prague Ecoomc Papers, Uversy of Ecoomcs, Prague, vol 007(4), pages 303-38 Lorez, (009), O he eropy of a fuco J Approx Theory 58, (Jue 009), 45-50 Malkel, B(003), The Effce Marke Hypohess ad Is Ccs, Joural of Ecoomc Perspecves, Volume 7, Number, Wer 003, pp 59-8 Maeev, M(004), CAPM Aomales ad he Effcecy of Sock Markes Traso: Evdece from Bulgara, Souh- Easer Europe Joural of Ecoomcs, (004) 35-58 Nagelkerke, N J D(99), A oe o a geeral defo of he coeffce of deermao, Bomerka (99) 78(3): 69-69 Parze, E(96), O The Esmao Of Probably Desy Fuco Ad Mode, The Aals Of Mahemacal Sascs, 33, 065-076 Pele, DT(0), Iformao eropy ad occurrece of exreme egave reurs, Joural of Appled Quaave Mehods, Fohcomg, 0 sso, A (008), The Iformaoal Effcecy ad he Facal Crashes, esearch Ieraoal Busess ad Face, Vol, pp 396-408 ockger, M, Urga, G(000), The Evoluo of Sock Markes Traso Ecoomes, Joural of Comparave Ecoomcs, Volume 8, Issue 3, Sepember 000, Pages 456-47 Sa, S(994), Adapve Kerel Desy Esmao, PhD Thess, Houso, Texas Sapoa, G(990), Probablé, Aalyse des Doées e Sasque, EdTechp Sco, D W(99), Mulvarae Desy Esmao: Theory, Pracce Ad Vsualsao, New York, Joh Wley Slverma, B W(986), Desy Esmao For Sascs Ad Daa Aalyss, Lodo, Chapma ad Hall Sglz, J(98), The Allocao ole of he Sock Marke: Pareo Opmaly ad Compeo, The Joural of Face, Vol 36, No, Papers ad Proceedgs of he Thy Nh Aual Meeg Amerca Face Assocao, Dever, Sepember 5-7, 980 (May, 98), pp 35-5 Summers, L(986), Does he Sock Marke aoally eflec Fudameal Values?, The Joural of Face, 4(3), 59 60 Tmmerma, A, Grager, C(004), Effce marke hypohess ad forecasg, Ieraoal Joural of Forecasg, 0(), 5 7 *** Webse of Sofa Sock Exchage: wwwbse-sofabg *** Webse of Buchares Sock Exchage: wwwbvbro *** Webse of Budapes Sock Exchage: wwwbsehu *** Webse of Prague Sock Exchage: wwwpsecz