Lien Bui Mean Reversion in International Stock Price Indices. An Error-Correction Approach. MSc Thesis

Transcription

1 Len Bu Mean Reverson n Inernaonal Sock Prce Indces An Error-Correcon Approach MSc Thess

2 Urech Unversy Urech School of Economcs MEAN REVERSION IN INTERNATIONAL STOCK PRICE INDICES AN ERROR-CORRECTION APPROACH Maser Thess Bu Huong Len Suden ID: MSc. Inernaonal Economcs and Busness Thess Supervsor: Prof. Dr. J.A. Bkker Second Supervsor: Prof. Dr. L. Sperdjk July, 2011

3 Absrac Ths research nvesgaes he mean reverng behavor of nernaonal sock prce ndces usng an errorcorrecon approach. Esmaons are conduced n boh me seres daases of ndvdual counres and panel daases of many developed counres. The ndvdual counres are: Denmark for he perod , Sweden for he perod and he Uned Saes for he larges perod from 1871 o Two panel daases are esablshed he pooled panel daase of he above menoned 3 counres for he perod , and he panel daase of 15 MSCI developed counres for he perod Consrucng many possble proxes for he fundamenal value of sock prces, we fnd srong evdence of mean reverson n sock prces for almos all proxes. The esmaed speed of mean reverson vares across daases, proxes and models bu n general s hgher han all prevous sudes on he same opc. Applyng he same approach of rollng-wndow esmaon of Sperdjk e al. (2010), hs hess also confrms her fndngs on he dynamcs of he mean reverson process. The hghes speed of mean reverson s found durng he me of World War I, he Grea Depresson and he sar of World War II. These resuls mply ha he speed of he mean reverng process of sock prces s dramacally hgher n he perods of an unsusanable economy han hose of normal economc condons.

4 Conens 1. Inroducon Theorecal framework on mean reverson Absolue mean reverson Relave mean reverson Mean reverson model Model specfcaon Smple mean reverson model Error-correcon model Choce of model Fundamenal value of socks Theores on fundamenal value of socks Proxes for fundamenal value process Tes for un roo and co-negraon Mehods of esmaon Tme seres daa Panel daa Daa Daa descrpon Tme seres daa Panel daa Dscoun rae calculaon Emprcal resuls Sock prce ndex Tes for un roos and co-negraon Tme seres daa Panel daa Esmaon resuls a

5 5.2.1 Indvdual counres Panel daa Robusness check for dscoun rae sensvy Tme-varyng mean reverson Panel daa 3 counres The Uned Saes Emprcal resuls Toal reurn ndex Tes for co-negraon Esmaon resuls Tme-varyng mean reverson Conclusons Ls of Fgures... Ls of Tables... References... b

6 1. Inroducon There s a long-sandng belef held n sock radng ha he sock prce ends o rse (fall) afer hng a mnmum (maxmum). Ths s referred among many analyss o mean reverson of sock prces where an exreme drop of sock prces s expeced o be followed by an ncrease, and vce versa. Many nvesors convnce hemselves o hs percepon n mean reverson of sock prces by lookng back a a hsorcally exreme sock marke hgh beng followed hereafer by a subsequen fall. For nsance, afer he 1974 marke low was reached, he sock marke was obvously overvalued and h s hgh n he summer 1987 and fnally endng s mean reverng process by he 1987 sock marke crash. Recenly fnancal analyss have debaed agan on he exsence of mean reverson n sock prces afer wnessng a severe drop n he sock marke n March 2009 durng he sub-prme crss ha followed he 2007 peak. The process connues wh he recovery of he marke a he end of 2009 whch srenghens he percepon of he mean reverng propery of sock prces among nvesors. Dscussons on he exsence of mean reverson of sock prces arac specal concerns of fnancal analyss and nvesors because of s mporance n nvesmen sraeges. Balvers e al. (2000) argue ha explong mean reverson of sock prces could help gan sgnfcan excess reurns n conraran radng sraegy 1. In addon, Campbell & Shller (2001) emphasze he crucal role of he mean reverng behavour of sock prces on he predcably of excess reurns. Furhermore, mean reverson n sock prces s economcally mporan o he aracveness of he sock marke for penson funds (Vlaar, 2005). If low reurns are followed by hgher expeced fuure reurns hanks o he presence of mean reverson, penson funds wll have more smulaon o nves n equy afer a downfall of he sock marke (Sperdjk e al., 2010), whch hen n urn helps accelerae s recovery. Is mporan economc mplcaons encourage a lo of auhors o fnd he answer on he exsence of he mean reverng process of sock prces. Ths process s esed by wo approaches n he leraure. One s he absolue mean reverson approach whch defnes a mean reverng process ndrecly by negave auocorrelaon n reurns. The oher s he relave mean reverson approach whch examnes mean reverson drecly by he relaonshp beween he sock prce and s fundamenals. Whle he emprcal evdence of absolue mean reverson are heerogeneous (mean reverson s frs found sascally sgnfcan by Fama & French (1988b) and Porerba & Summers (1988) bu laer rejeced by Joron (2003) and Malkel (2004)), emprcal resuls on relave mean reverson are conssen unl he momen. Balvers e al. (2000) and Sperdjk e al. (2010) adop a panel daa approach o esmae mean reverson hrough 1 Conraran sraegy s an nvesmen syle whereby an nvesor aemps o prof by gong agans he rend. 1

7 he saonary relaonshp beween he fundamenal processes of cross-counry sock prces and he sock prces by elmnang hose fundamenal processes, and hey boh esablsh sgnfcan mean reverson. Cochran & DeFna (1995), n conras, esmae mean reverson of sock marke prces drecly hrough he fundamenal process of sock prces by an error-correcon approach. Despe applyng a dfferen approach from Balvers e al. (2000) and Sperdjk e al. (2010), Cochran & DeFna (1995) sll confrm sascally sgnfcan evdence of mean reverson of sock prces. Mean reverson, f exss, s argued o be a me-varyng process. For nsance, Km e al. (1991) fnd ha mean reverson s a phenomenon ha exss only durng perods when he sock markes are hghly volale. Sperdjkk e al. (2010) reaffrm a non-consan speed of mean reverson hrough her fndngs ha he hghes speed of mean reverson s found n he perod of hgh economc uncerany ncludng he Grea Depresson and he sar of he World War II. Ths research s conduced wh wo man objecves. Frs, answers he queson of wheher he sock prce revers o s mean n he long run. Second, examnes he prevous fndng of Sperdjk e al. (2010) on a me-varyng mean reverson of sock prces. Followng Cochran & DeFna (1995), o examne he mean reverng behavor of sock prces we apply an error-correcon approach whch allows economc and fnancal fundamenals such as dvdends and earnngs o have sgnfcan ransory effecs on he sock prces. Moreover, as mean reverson s lkely o occur slowly and hus, can be deeced only n long me seres (Balvers e al., 2000), annual daa are used n hs research. Mean reverson s nvesgaed n boh me seres daases of 3 counres Denmark, Sweden and he Uned Saes and panel daases of hese 3 counres from 1922 o 2010 as well as of 15 developed counres from 1971 o Consrucng many possble proxes for he fundamenal value of he sock prce, we fnd srong evdence of mean reverson of sock prces for almos all proxes. Moreover, applyng he rollng wndow esmaon of Sperdjk e al. (2010) we confrm her fndngs on a non-consan speed of mean reverson over me. The remander of hs hess s organzed as follows. Secon 2 revews he leraure on mean reverson over he pas 20 years n whch wo dsnc approaches on mean reverson sudy are dscussed. The explanaon of he chosen approach for hs hess o es for mean reverson s presened n secon 3. Secon 4 provdes he descrpon of he daa used for emprcal analyss. Secon 5 and secon 6 dscuss he emprcal resuls for he wo represenaves of sock prces he sock prce ndex and he oal reurn ndex respecvely. Fnally, secon 7 concludes he hess wh he hghlghng of all he resuls from he prevous secons. Some mplcaons and lmaons wll be menoned along wh furher dscussons o exend he research. 2

8 2. Theorecal framework on mean reverson A grea number of sudes have been dedcaed o he research on mean reverson of sock prces over he pas 20 years. Two man economc heores domnang hose sudes n explanng he mean reverng behavor of he sock prces are he marke effcency, and he speculave process hypohess. Accordng o he effcen marke hypohess (EMH), all avalable nformaon s fully refleced n he sock prces (Fama, 1991) whch are hen deermned by he expeced reurns per share. Wh he fndngs ha he expeced reurns follow a saonary process (Conrad & Kaul, 1988), mean reverson of sock prces s possbly explaned under he hypohess of marke effcency (Fama & French, 1988b). In conras, Poerba & Summers (1988) argued ha he ransory componen n he sock prces s caused by nose radng under an neffcen marke. Ths argumen s confrmed by Culer e.al (1991) ha varaons n ex-ane reurns arse prmarly from nose raders who are no raonal n he convenonal sense of radng on he bass of all publcly avalable nformaon. Then mean reverson s he consequence of he speculave process self. Among he very frs leraure on mean reverson of sock prces, Summers (1986) proposed a basc model of he mean reverson process whch s hen furher developed by Fama & French (1988b): p = where * p + u, (2.1) p s he sock prce a me. Accordng o Summers (1986) and Fama & French (1988b), he sock prce p - a non-saonary process s he sum of a permanen prce componen or he fundamenal value of he sock prce a saonary process: * p whch follows a random walk, and a emporary componen u whch s modeled as u = u 1 + v, (2.2) where 0 1,.e. u follows a saonary frs-order auoregressve process, and v s a whe nose. Fama & French (1988b) argue ha each monh prce s shock s conrbued by a shock from * p whch s ncorporaed permanenly n he sock prce, and a emporary shock from u whch s elmnaed gradually. Accordngly, he slowly decayng saonary prce componen u mples a mean-reverng process of sock prces n he long-run. The saonary propery of hs process s esed by wo approaches n he leraure. One s he absolue mean reverson approach whch argues ha mean reverson of he 3

9 saonary prce componen u causes negave auocorrelaon n reurns. Therefore, n hs approach α s esmaed ndrecly from he regresson of he sock reurn on s lags o fnd he evdence of negave auocorrelaon n sock reurns whch n urn wll be he proof of mean reverson of sock prces (Fama & French, 1988b). In her paper, Fama & French (1988b) use connuously compounded real reurns for all New York Sock Exchanges socks and fnd srong evdence ha sock prces have a slowly decayng saonary componen. Alernavely, he relave mean reverson approach esmaes α drecly from equaon (2.2) afer he regresson of he sock prce on s fundamenal value n equaon (2.1). In hs case, he fundamenal value of he sock prce has o be specfed. Boh approaches es he null hypohess ha α s equal o one whch means ha he sock prce follows a random walk and hus does no rever o s mean, agans he alernave hypohess ha α s smaller han one whch mples mean reverson of sock prces. 2.1 Absolue mean reverson Inally when only shor me horzons 2 were used o esmae equaon (2.1), research on mean reverson of sock prce only comes o nconclusve resuls. Because mean reverson n equaon (2.1) s proved once negave auocorrelaon of sock reurns s found, or n oher words α s expeced o be close o one, long horzons are a prerequse condon for sascal nference. Fama & French (1988b) and Poerba & Summers (1988) were he frs o fnd sgnfcan resuls n esng mean reverson a long horzons. Wh several me horzons beween one year and en years, Fama & French (1988b) fnd sgnfcan evdence of mean reverson whch explans he 25-40% of he varaon of he reurns a 3- o 5-year horzon. Lke Fama & French (1988b), Poerba & Summers (1988) use monhly overlappng daa o ncrease he sample sze when examnng he saonary process of he emporary prce componen u. They use he varancerao es of Cochrane (1988) o fnd evdence of mean reverson. The varance rao s a measure of he randomness of a reurn seres. I s compued by dvdng he varance of reurns esmaed from longer nervals by he varance of reurns esmaed from shorer nervals, (for he same measuremen perod), and hen normalzng hs value o one by dvdng by he rao beween he duraon of he wo nervals. If he varance-rao s equal o one, he reurn varance s proporonal o he reurn horzon whch occurs when he logarhm of he sock prce follows a random walk. Wh sgnfcan evdence ha over long horzons reurn varances rse less han proporonal o he me nerval, Poerba & Summers (1988) conclude on mean reverson over long horzons. However, boh fndngs of mean reverson from Poerba 2 Tme horzon or laer menoned as horzon refers o he lengh of me over whch he sock reurns are calculaed. Normally shor horzons refer o he me nerval of less han 1 year whle long horzons menon he perods of more han 1 year. 4

10 & Summers (1988) and Fama & French (1988b) encouner wo man problems whch are crczed n he laer leraure. Frs, he relably of her fndngs s quesoned on he auocorrelaon of sock reurns. Boh ses of research are based on overlappng daa whch reduce he ndependence of he observaons. Ths n urn causes based and nconssen esmaes of he sandard errors, and reduces he accuracy of he auocorrelaon measuremen n he reurns. To accoun for he bas due o overlappng samples, he mehod of Hansen & Hodrck (1980) s appled where a movng average srucure of he sandard errors s aken no accoun. Ths mehod s based on he assumpon of he asympoc normal dsrbuon of he sock reurns whch s volaed due o he small sample propery of he long-run sock reurns. Addressng hs weak assumpon Km e al. (1991) fnd no sgnfcan evdence of mean reverson. The ssue of he small sample bas s also crczed by Rchardson & Smh (1991). They fnd ha he evdence of longerm mean reverson whch s proved by Fama & French (1988b) usng he generalzed mehod of momen (GMM) es procedures of Hansen & Hodrck (1980) dsappears f hey remove he small sample bas. Anoher ssue of usng monhly overlappng reurns s rased by Jegadeesh (1991). Hs regresson model usng 1-monh reurns as he dependen varable and he lagged mul-year reurns as he ndependen varables shows sgnfcan evdence of mean reverson n January only, whch suppors hs argumen of he seasonal paerns n sock prce mean reverson. Ths seasonaly s gnored by Fama & French (1988b) when all calendar monhs are consdered equally. Secondly, heeroskedascy s blamed for he overesmaon of mean reverson. Perods wh hgh volaly of sock prces show more rend of mean reverson whle n oher perods, evdence of mean reverson s nsgnfcan or even mean averson s found n he research of Joron (2003) on 31 counres where nerrupon-sufferng markes dsplay a larger han one varance-rao represen mean averson. Thus he ncluson of hose hghly-volale sock reurn perods would oversae he es resul. Fama & French (1988b) already realzed hs problem n her paper ha he srong negave auocorrelaon of perod maybe largely due o he frs 15 years. Bu due o he lmaon of he smple sascal echnque hey use, resoluon for heeroskedascy of sock reurns s only nroduced unl McQueen (1992) apples weghed leas squares (WLS) whch gves a lower wegh o observaons wh a hgher varably. 2.2 Relave mean reverson The absolue mean reverson leraure dscussed above examnes he mean reverng process of sock prces hrough fndng evdence for negave auocorrelaon n sock reurns. Many researches from many 5

11 auhors were devoed o fnd convncng evdence for he absolue mean reverson bu he fnal answer as o wheher he sock prce revers o s mean n he long run has no ye been arrved a. In he alernave approach of he relave mean reverson, Balvers e al. (2000) argue ha mean reverson may be deeced from sock prce ndces relave o a reference ndex under he assumpon ha dfference beween rend pah of one counry s sock prce ndex and ha of he reference ndex s saonary. Under hs approach, Balvers e al. (2000) ake no accoun he fundamenal process of he sock prce whch was gnored n he prevous research: p 1 - p = + ( p * - 1 p ) +. (2.3) 1 Equaon (2.3) s regarded as he smple mean reverson model or smple model laer n hs hess, where p represens he naural logarhm of he sock prce ndex for counry ha ncludes dvdends a he end of year so ha ( p 1 - p ) equals he connuously compounded reurn an nvesor realzes n perod (+1). p * 1 ndcaes he naural logarhm of he fundamenal value of he sock prce ndex n counry. s a counry-specfc consan and The parameer s a saonary shock erm wh an uncondonal mean of zero. 1 measures he speed of reverson. Mean reverson exss when > 0. Neverheless, argung ha he fundamenal process p * 1 s dffcul o specfy, Balvers e al. (2000) assume a saonary relaonshp beween he fundamenal process of one counry s sock prce and s reference s as follows: p * = r p * + z +, (2.4) where z s a counry-specfc consan, s a zero-mean saonary process ha can be serally correlaed. The superscrp r ndcaes a reference ndex. By usng hs assumpon and he oher ha he speed of mean reverson - across counres s equal o, he fundamenal value process s elmnaed and he followng relaon s esmaed nsead of equaon (2.3): r 1 - r r 1 = + ( p - r p ) +, (2.5) 1 where r 1 and r r 1 are connuously compounded reurns of counry and of he reference ndex a me (+1) respecvely. s a saonary process wh an uncondonal mean of zero. 1 6

12 Regresson of equaon (2.5) allows a drec esmaon of he mean reverson coeffcen λ whou he need o denfy he fundamenal process. By hs nnovave echnque n nvesgang he mean reverng process of sock prces, Balvers e al. (2000) fnd evdence a 5- o 1- percen sgnfcance level of mean reverson of he marke prce ndces of 18 counres from 1969 o Applyng he same approach, Sperdjk e al. (2010) use a longer annual daase of more han one cenury from 1900 o 2009 o analyze mean reverson n nernaonal sock markes. Ther fndngs of unbased esmaes of he speed of mean reverson dffer subsanally from he resuls obaned by Balvers e al. (2000) n ha hey observe a much lower speed of mean reverson. Boh Balvers e al. (2000) and Sperdjk e al. (2010) esmae mean reverson ndrecly hrough he saonary relaon beween he fundamenal processes of cross-counry sock prces and he sock prces by elmnang hose fundamenal processes. Cochran & DeFna (1995), n conras, esmae mean reverson of he sock marke prce drecly hrough he fundamenal process of sock prces by an errorcorrecon approach whch allows economc and fnancal varables o have sgnfcan ransory effecs on he sock prces. They use dvdends and ndusral producon growh raes as proxes for he fundamenals of he sock prces. The error-correcon erm s he long-run rend n he sock prce whch s descrbed by he regresson of he logarhm of he sock prce on dvdends and fuel prces. The mean reverson model s hen esmaed by he regresson of he connuously compounded reurn on he correcon erm, he dvdend growh and such oher economc conrol varables as lags and leads of he ndusral producon growh, erm spreads, defaul spreads, and so on. Wh he error-correcon approach, Cochran & DeFna (1995) conclude ha auocorrelaon n he proxes of he fundamenals - specfcally he dvdends n her research - could accoun for a poron of seral correlaon n sock prces. Or n oher words, mean reverson of sock prces s found n her sudy n he esmaon usng he daa on New York Exchange Secury socks for he perod from 1947 o The mos mporan facor n esmang he mean reverng process of sock prces by he error-correcon model s he deermnaon of he fundamenal values. Because hose values are unobservable, fndng suffcen proxes for hem s crucal o he accuracy of he esmaed model. Cochran & DeFna (1995) use only one proxy of dvdend for he fundamenal value of he sock prce. However, dependng on he curren dvdend -self could no yeld a relable resul for he es of mean reverson. The fundamenal value of a sock prce s expressed as he dscouned fuure cash flows earned by he sock whle curren dvdends only represen for curren cash flows. Defnng a correc specfcaon of he proxy s an essenal ask of he emprcal es on relave mean reverson. The beer proxy s found he more powerful he es wll be. Deermnaon on he proxy for he fundamenal of he sock prce s dscussed n deals n he nex par. 7

13 3. Mean reverson model In hs secon, varous models of he mean reverson process are dscussed frs, and based on gven argumens he choce for he models appled n hs hess s presened. Afer he decson on he model has been made, necessary procedures o esmae he model accuraely are consdered. 3.1 Model specfcaon Followng Cochran & DeFna (1995), Balvers e. al (2000) and Sperdjk e al. (2010) hs hess s neresed n he relave mean reverson process n whch he specfed mean of he sock prces are aken no accoun. However, here s of course no only one way o esmae relave mean reverson of sock prces Smple mean reverson model Balvers e al. (2000) s he frs o explcly consder he fundamenal process of sock prces whch s presened n equaon (2.3). Neverheless, he dea of hs equaon has acually been mpled n prevous leraure. In he sudy of reurn predcably, Fama (1991) forecass reurns from such valuaon raos as dvdend yelds (D/P), earnngs/prce raos (E/P). Ther esmaon of he predcve power of hese valuaon raos s n naure of he esmaon of equaon (2.3) (aforemenoned n he prevous secon) n whch he fundamenals of he sock prces are proxed by dvdends or earnngs: p 1 - p = + ( p * - 1 p ) +. (2.3) 1 If p * = 1 d * ( 1 d * 1 s he logarhm of dvdends), equaon (2.3) could be presened as he regresson of sock reurns on he logarhm of dvdend yelds: r 1 = + log ( D 1 P ) +, (3.1) 1 where r 1 = p 1 - p. r 1 s he connuously compounded capal gan f p s he sock prce ndex or he connuously compounded reurn f p s he oal reurn ndex. Equaon (3.1) s used by Fama & French (1988a) o nvesgae he power of he valuaon raos or parcularly dvdend yelds n her research n forecasng sock reurns. Ther fndngs show ha he slopes n he regresson of equaon (3.1) - ncrease wh he reurn horzons. The ncrease of he slope s approxmaely proporonal o he reurn horzon for he horzons o one year, bu less han proporonal 8

14 o he reurn horzons of wo years o four years. Ths behavor of he slopes suggess ha he expeced reurns are hghly auo-correlaed bu slowly mean-reverng, whch mples he mean reverson of he sock prce n long me horzons (Fama & French, 1988a). The regressor of equaon (3.1) could be he logarhm of he earnngs-prce rao (E/P) f earnngs are consdered as he proxy for he fundamenal value of he sock prces. A smlar o ha of he esmaon wh D/P ha here s a slow mean-reverng process n he sock prce s found when E/P s used o forecas sock reurns (Fama & French, 1988a). Campbell & Shller (1989) also agree on hs mean reverng process, wh he argumen ha f he socks are underprced relave o he fundamenal value, reurns end o be hgh subsequenly, he converse holds f socks are overprced 3. In comparson wh Fama & French (1988a), Campbell & Shller (1989) use boh dvdends and earnngs and en- and hry-year of earnngs as proxes for he fundamenals of sock prces. By he concluson on he sgnfcanly predcve power of hese valuaon raos of dvdend yelds, earnngs-prce raos and earnngs-prce raos based on a movng average of earnngs, Campbell & Shller (1989) confrm he evdence of mean reverson n sock prces. Boh he fndngs of Fama & French (1988a) and Campbell & Shller (1989) whch derve from he longhorzon reurns of one year o en years suppor he long-run mean reverng process. Noneheless, longer me horzon reduces he sample sze dramacally. As a resul, her esmaon s lkely o expose o a small sample bas. Culer e al. (1991), usng MSCI 4 daase of 13 equy markes, fnd much weaker resuls of mean reverson han expeced based on he earler fndngs of Fama & French (1988a) for he US marke afer small sample bas robusness. The small sample bas s generally he mos poenal problem n mos sudes on he long-erm mean reverng process of sock prces. Anoher problem n esmang mean reverson of sock prces by he smple mean reverson model employed by Fama & French (1988a) and Campbell & Shller (1989) s he sochasc properes of he explanaory varables or parcularly her orders of negraon. Accordng o Torous e al. (2004) f he regressor n he predcve equaon (3.1) conans a un roo, he gnorance of s non-saonary propery n he regresson would lkely lead o based esmaon. When hey accoun for he possbly of a un roo n he explanaory varables, conrary resuls o he earler fndngs of Fama & French (1988a) and Campbell & Shller (1989) are found ha he evdence of he predcably of he sock reurns by he varance rao or mean reverson of sock prces s relable a shorer raher han longer horzons. Ths lmaon of he smple mean reverson model could be mproved by usng he error-correcon model where he mean reverng process s esmaed n he presence of non-saonary varables. 3 Campbell & Shller (1988) page MSCI: Morgan Sanley Capal Inernaonal. 9

15 3.1.2 Error-correcon model The smple mean reverson model allows us o examne he mean reverng process hrough accounng for he valuaon raos whch have he sronges advanage of her avalably for esmaon. However some argumens agans he nsuffcency of he smple mean reverson model are found n he leraure. Accordng o Gropp (2004), usng valuaon raos could yeld downward bas of he mean reverson coeffcen. The expeced changes n he dvdends wll have an mmedae mpac on he fundamenal value of he sock prce whch s measured by he expeced fuure cash flows of he socks whle hose changes are only refleced n hose raos n he laer perods. Ths lag of essenal nformaon ncorporang n he varables shows he dsadvanage of he smple mean reverson model n esmang he mean reverng behavor of he sock prces. In addon, Marsh and Meron (1987) argue ha frm managers arge her dvdend-prce rao for a perod of me and make changes o dvdends n response o changes n he frm s sock prce n order o keep hs argeed rao unchanged. Thus, usng a relavely sable ndcaor of he valuaon rao o forecas a hghly flucuang sock prce could no yeld relable esmaes. One can sugges ha usng anoher valuaon rao - specfcally he earnngs-prce (E/P) rao - as a subsue would lkely be a good soluon o hs problem. Neverheless, Fama & French (1988a) ndcae ha E/P ends o have less explanaory power han D/P n esmang equaon (3.1). Ths argumen no only represens he dsadvanage of he smple mean reverson model bu also gves rse o a new approach o examne he mean reverson behavor of he sock prces an error-correcon approach whch could overcome he lmaon above menoned of he convenonal approach. Cochran & DeFna (1995) menon wo man advanages of usng he error-correcon model (ECM) for sock prce mean reverson esmaon. Frs, unlke prevous domnang approaches, allows he ncorporaon of a longerm consran on shor-erm sock prce movemens whch hen, n urn, helps specfy precsely he mean o whch sock prces rever. The approach of Balvers e al. (2000) s very powerful n examnng he mean reverng process of sock prces. Bu by ha way of esmaon where he fundamenal value s elmnaed n he esmaed equaon, he mean ha he sock prces rever o canno be denfed. The errorcorrecon approach, nsead, accouns for he presence of he mean. Second, because allows nonsaonary sochasc rends n he varables, as once dscussed above, s expeced o gve beer esmaon han he smple mean reverson model does. The error-correcon model and s closely relaed heory of a co-negraed process are nroduced by Engle & Granger (1987). Co-negraon s he process of wo or more varables whch boh have un roos and hus possess non-saonary sochasc rends. Whle hese varables are non-saonary, s possble ha a lnear combnaon of hem s saonary. If ha s he case, he varables are sad o be conegraed. Co-negraon mples a long-run relaonshp beween hese varables and consrans he 10

16 dynamcs of he sysem. Devaons (errors) from he long-run relaon can arse as ransory shocks mpac on each varable, bu snce he errors are saonary by defnon, hey should evenually be elmnaed. Engle & Granger (1987) show ha movemens n co-negraed varables can be represened as a model known as error-correcon model. If he wo varables y and x are co-negraed, hen y - β'x = u s a saonary process. An error-correcon model for y and x can be wren as follows: y = o y + 0 x + 1 * 1 + u, (3.2) where 1 s he correcon erm, 1 = y 1-2 x 1, (3.3) represens he long-run relaonshp beween 2 y and x, 1 < 0. Dependng on he avalably of he correcon erm or wheher he long-run relaonshp beween wo varables are denfed or no, here are wo ways o esmae equaon (3.2). If 1 s known, equaon (3.2) s used o esmae he co-negrang process of he wo varables. If 1 s unknown, he predced values of 1 s derved from esmang equaon (3.3) and hen (3.2) s esmaed wh hose predced values. Followng Cochran & DeFna (1995), he error-correcon model s employed n hs hess o esmae he mean reverng process of sock prces n he presence of he non-saonary sochasc rends of he varables: p = + 0 p * + * 1 + 1, (3.4) where, 1 s he correcon erm of counry a me (-1), = p - 2 p *, (3.5) 0 Shor erm effec of a change n p * on he change n p, 1 < 0 ~ mean reverson process. 1 measures he speed of mean reverson, 2 Long erm effec of a change n p * on he change n p. If 2 = 1 a 1% ncrease n p * wll lead o 1% ncrease n p n he long run whch mples a fully co-negrang relaon beween he sock prce and s fundamenal value. When 2 1 he conegrang relaon beween hem s no sable. 11

17 When 1 = 0 p 1 and p * 1 are n her equlbrum sae, 1 > 0 p 1 s above s equlbrum sae. Therefore p should decrease so ha p wll be brough back oward equlbrum. Ths requres 1 < 0. 1 < 0 p 1 s below s equlbrum sae. Therefore p should ncrease so ha p wll be brough back oward he equlbrum. Ths requres 1 < 0 as well. In hs very early leraure, Fama (1965) fnds he non-saonary propery of he sock prces. Also, dvdend yelds whch are he convenonal proxy for he fundamenal value of he sock prces are found o have un roos. Thus, he error-correcon model could be an approprae soluon o descrbe he sock prce movemens or he mean reverng process whch s of our neres. Followng Engle & Granger (1987), wo ways of esmaon dscussed above are aken no consderaon Engel & Granger Two-Sep Error-correcon model In her orgnal dea of he error-correcon model, Engle & Granger (1987) apply a wo-sep esmaor n her procedure. In he frs sep, he parameers of he co-negrang vecor are esmaed and n he second hese are used n he error correcon form (Engle & Granger, 1987). Applyng n he case of mean reverson of sock prces, s he error componen - he long-run relaonshp beween he sock prce and s fundamenals: = p - 2 p *. (3.5) In he second sep, predced value of he devaon of he sock prce from s fundamenal value - used o esmae mean reverson model (3.4): p = * p * ˆ s ˆ +. (3.6) Engle & Granger (1987) sae ha hs procedure s convenen because s no necessary o specfy he mean reverson process unl he error correcon srucure has been esmaed. Moreover, under her argumen he wo-sep esmaor of a sngle equaon of an error correcon sysem obaned by akng he predced value of he devaon of he sock prce from s fundamenal value ˆ as he rue value s proved o have he same lmng dsrbuon as he maxmum lkelhood esmaor usng he rue value of. Thus, leas squares sandard errors wll be conssen esmaes of he rue sandard errors. Agree wh Engle & Granger (1987), Bachmeer & Grffn (2006) reaffrm ha because 2 can be esmaed 12

18 super-conssenly by OLS regresson, he nference on he parameers of (3.6) can be proceeded as hough 2 s known as cerany Generalzed One-Sep Error-correcon model Accordng o Engle & Granger (1987) f s known, equaon (3.2) s used sngly o esmae he co- negrang process of he wo varables. Bu n almos cases, s unknown. However, we could sll esmae he mean reverson model by he error correcon approach n only one sep whch s referred o he generalzed one-sep error-correcon model. The generalzed one-sep error-correcon model (GECM) whch s frs nroduced by Banerjee e al. (1993, 1998) s a ransformaon of an auoregressve dsrbued lag (ADL) model: y = x ( y 1-2 x 1 ) + 2 x +. The error correcon erm n he GECM s gven by ( y 1-2 x 1 ). Unlke he wo-sep mehod, usng he sngle equaon GECM, he long-run relaonshp, he dsequlbrum and he shor-run dynamcs are esmaed smulaneously. Accordng o De Boef (2000) he sngle equaon GECM s boh heorecally appealng and also sascally superor o he wo-sep esmaor n many cases, because s asympocally equvalen o more complex, full-nformaon, maxmum-lkelhood and fully modfed esmaors. Wheher he sock prce movemen s a mean reverng process depends on he sgn of he coeffcen 1. Of mos neres s he esmaon of 1. Thus, applyng he sngle-equaon GECM we could plug (3.5) no (3.4) o esmae equaon (3.4) drecly as follows: p = + 0 p * + 1 ( Choce of model p 1-2 ) +. (3.7) p * 1 For s mprovemen agans prevous convenonal approaches n examnng he mean reverson behavor of he sock prces, he error-correcon model s our key mehod of esmaon. As aforemenoned he generalzed one-sep ECM s sascally superor o he Engle-Granger wo-sep esmaor, and herefore he generalzed one-sep s he preferred specfcaon. Noneheless, as he long-run relaonshp beween he sock prce and s fundamenal s unknown, he wo-sep ECM s also appled as he suggesed heory by Engle & Granger (1987). Fnally, he smple mean reverson model s also appled for comparson purpose. 13

19 3.2 Fundamenal value of socks Key varables n he Error-Correcon Model whch we wll use o examne wheher he sock prce movemen s a mean reverng process are he sock prce and s fundamenal value. Thus defnng he fundamenal value of he sock prce s a crucal par of our research Theores on fundamenal value of socks Fundamenal effcency heory The marke effcency hypohess s ofen referred o wheher a ceran se of nformaon s ncorporaed nsananeously no he curren marke prces. Noneheless, hs popular undersandng of hs hypohess only represens one aspec of he marke effcency - ha s nformaonal effcency. The mssng one of he wo dmensons of he sock marke effcency s called fundamenal effcency whch measures wheher sock prces exclusvely reflec he underlyng or fundamenal profs of a corporaon, condoned on he avalable nformaon (Ayres, 1991). Few auhors dsngush beween hese wo forms of marke effcency. Afer Ayres (1991), Elon and Gruber (1995) also make he dsncon beween he convenonal form of nformaonal effcency and he oher one whch s called marke raonaly n her paper. Accordng o hem, he fundamenal effcency or marke raonaly hypohess deals wh he queson wheher sock prces precsely reflec nvesors expecaons abou he presen value of fuure cash flows. In lne wh he defnon of Elon and Gruber (1995), Engelen (2005) argues ha fundamenal effcency requres ha secury prces are equal o he presen value of he fuure cash flows nvesors expec he company o generae, dscouned by he approprae rsk-adjused rae of reurn. He also lnks hese wo forms of effcency ha he nvesors base her expecaons abou fuure cash flows on avalable nformaon n he marke whch s based on he degree of nformaonal effcency of he marke. The applcaon of he fundamenal effcency heory s furher developed n he nex sub-secon on Asse Prcng Model where he relaon beween he sock prce and s fundamenals are presened physcally by he model Asse Prcng Model The mos basc Asse Prcng Model whch s called he marngale model or he random walk model of he sock prces s represened wh he assumpon ha an asse has a consan expeced reurn (Samuelson, 1965). The more general model developed laer ha allows a flucuan expeced reurn s: P = E ( P 1 D 1 r 1 1 ). (3.8) 14

20 E If he expeced dscouned fuure prce has a lm of zero or lm( P k k ) = 0, (3.8) becomes k (1 r ) he fundamenal value equaon of an asse prce (Campbell, 2000): k P = E D 1 (1 r 1), (3.9) where he rgh-hand sde of equaon (3.9) s called he fundamenal value of an asse prce (denoed as P ) (Campbell, 2000). Because an effcen marke s assumed, he sock prce s exacly equal o s * fundamenal value. More generally, f allows for he expeced reurn o change over me, he fundamenal value of he sock prce s descrbed as: * P = E 1 r ( ) 1 1 j 1 j D, (3.10) where D = D 1 (1+ g ) n whch g s he growh rae of he dvdends a me (+) In he specal case ha he dvdend growh rae and he expeced reurn are consan over me, he model of Gordon (1959) s obaned: * P = D 1 g. r g (3.11) Proxes for fundamenal value process As dscussed n he prevous sub-secon, he fundamenal value of he sock prce s he presen value of expeced fuure cash flows of he sock as descrbed n equaon (3.10). In order o denfy hs value precsely, we need o know all expeced fuure cash flows as well as he expeced reurn used o dscoun hese cash flows. However, s mpossble o accoun for nfne cash flows obaned n he fuure or n oher words he fundamenal value of sock prces s unobservable, and herefore we need o fnd a plausble proxy for he fundamenal. The followng sub-secons consder possble varables ha could be used as proxes for fundamenal values of sock prces and dscuss choces of he dscoun rae appled o dscoun he fuure cash flows. 15

21 Proxes for he fundamenal value of sock prces a. Curren dvdends Curren dvdend s a convenonal measure of he fundamenal value of sock prces. In her paper on mean reverson n sock prces usng an error-correcon approach, Cochran & DeFna (1995) employ dvdends along wh fuel prces as he proxes for he fundamenal value of he sock prce and fnd evdence of he long-run relaonshp beween he sock prces and he dvdends and he fuel prces a sgnfcance level of 5%. In spe of he fac ha curren dvdends conan lle nformaon o deermne sock prces, we sll follow Cochran & DeFna (1995) o examne he mean reverng process of he sock prces when dvdends are used as a measure of he fundamenal value and he resuls are compared o oher proxes. b. Smoohed earnngs The dea of usng smoohed earnngs nsead of yearly earnngs as he proxy for he fundamenal value of sock prces orgnaes from he suggeson of Graham & Dodd (1934). They recommend o shf he orgnal pon of deparure, or basc compuaon, from he curren earnngs o he average earnngs, whch should cover he perod no less han fve years, and preferably from seven o en years 5. In lne wh Graham & Dodd (1934), Campbell & Shller (1989) argue ha a movng average of earnngs s used nsead of yearly earnngs due o he fac ha yearly earnngs whch could have negave value are oo nosy o represen for he fundamenal value of he sock prces. Thus, hey employ en- and hry-year movng average of real earnngs n calculang he earnngs-prce rao whou accounng for a dscoun rae. Followng Campbell & Shller (1989), n hs hess, en- and hry-year movng average of real earnngs are used as poenal proxes for he fundamenal value of he sock prces. c. Perfec foresgh prce Sudyng volaly n sock prces, Shller (1981) nroduces a concep of a perfec foresgh sock prce whch s defned as he presen value of he acual subsequen dvdends over he fuure perods: * P = Dv 1 1 r 1 Dv ( 1 r 2 ) Dv Dv ( 1 r 3) ( 1 r 4 ) Dv ( 1 r 5) +. Ths measure s calculaed from he acual subsequen dvdends over me bu no from he expeced fuure dvdends based on avalable nformaon a he curren perod. Thus, employng hs measure as a proxy for he fundamenal value of he sock prces means ha we assume perfec nformaon abou fuure dvdends n he presen. Moreover, he furher he fuure cash flow, he smaller he dscoun facor appled 5 Graham & Dodd (1934) page

22 o or n oher words he less mporan s n deermnng he sock prce. Consderng he above menoned argumen of Graham & Dodd (1934) o use cash flows from no less han 5 years o calculae he prce, a 5-year horzon may be suffcen for he calculaon. Hence we use 5-year subsequen dvdends o calculae he perfec foresgh prce as he proxy for he fundamenal value of he sock prce. A smlar esmaon for he perfec foresgh prce suggesed by Shller (1981) s o use acual subsequen earnngs over he fuure perods nsead of subsequen dvdends. In ha case, he perfec foresgh prce s calculaed as follows: * P = Earnngs Earnngs r 1 ( 1 r 2 ) Earnngs ( 1 r 3) Earnngs ( 1 r 4 ) Earnngs ( 1 r 5) +. Compared o he calculaon of he smoohed earnngs, hs mehod s advanageous n akng no accoun he dscoun rae o dscoun all cash flows o he presen values. Therefore he proxy esmaed by hs approach s expeced o beer reflec he real fundamenal value of he sock prce han he proxes of dvdends and smoohed earnngs. d. Hndsgh prce Theorecally, only expeced fuure cash flows are aken no accoun n sock prcng. Neverheless, nvesors n realy normally base her expecaons on hsorcal nformaon of sock prces. In real sock radng, one can nss ha nvesors fnd no aracons from socks ha only show a decrease rend n he pas. Also n her arcle dsngushng he dfference beween he foresgh and he hndsgh prce, Saman & Sched (2002) emphasze he mporance of he hndsgh prce. Accordng o hem, forecased fuure cash flows as well as he expeced reurn ha s used o dscoun he expeced fuure cash flows o her presen values are based on he nformaon and he flucuaon of sock prces n he pas a a ceran level. Therefore, wh he assumpon ha nvesors base her expecaons on he hsorcal nformaon of he sock prces, dvdends and earnngs realzed n he neares years n he pas are employed o calculae anoher measure of he fundamenal value of he sock prce he hndsgh prce. Symmerc o he perfec foresgh prce, 5-year aneceden dvdends and earnngs are used n whch he mos recen one whch s consdered o have he mos predcve value for fuure expecaon s dscouned a lower rae. When he dvdends are used he esmaed hndsgh prce s calculaed as: * P = Dv 1 1 r 1 Dv ( 1 r 2 ) Dv Dv ( 1 r 3) ( 1 r 4 ) Dv ( 1 r 5) +. And here s he calculaon of he hndsgh prce when he earnngs are used: 17

23 * P = Earnngs Earnngs r 1 ( 1 r 2 ) Earnngs ( 1 r 3) Earnngs ( 1 r 4 ) Earnngs ( 1 r 5) +. e. Gordon model Gordon growh model s consdered he smples fundamenals-based approach o predc sock prces. Heaon & Lucas (1999) use he prce esmaed by hs model as a measure of he fundamenal of he sock prce when quanfyng he poenal mpac of fundamenal effecs on sock prce movemens. In lne wh Heaon & Lucas (1999), n examnng he mean reverson process of he sock prce, Manzan (2003) consder boh sac and dynamc Gordon growh model. In a sac model, boh he expeced reurn and he dvdend growh rae are assumed o be consan over me. In conras, he dynamc model allows for he dynamc adjusmen of he expeced reurn and he dvdend growh rae. However, Manzan (2003) fnd ha he resuls from hese wo approaches are no sgnfcanly dfferen from each oher. Consequenly, he sac model s used by Manzan (2003) o esmae mean reverson of sock prces usng he sock prce ndces from S&P 500 for he perod from 1871 o 2003, bu accounng for a srucural break of boh dscoun raes and dvdend growh raes n he 1950 s. Followng Manzan (2003) he sac Gordon model (equaon 3.11) s also employed n hs hess as a proxy of he fundamenal value of he sock prces. If a srucural break of he dscoun rae s no consdered n he whole perod he esmaon resul wh he naural logarhm of Gordon prce s exacly he same as he naural logarhm of dvdend. Noneheless, as laer found n sub-secon 4.2 a srucural break s found n he dscoun rae appled o calculae he Gordon prce. Hence, he resul from he regresson wh he Gordon prce s expeced o be dfferen from he dvdend. In addon o he orgnal sac Gordon model, an alernave approach s appled o calculae he Gordon prce n whch earnngs are used o esmae he Gordon prce nsead of dvdends. In ha case equaon (3.11) becomes: * P = Earnngs 1 g. r g Dscoun rae Good defned proxes need an appropraely esmaed dscoun rae whch s he requred rae of reurn n dscounng expeced fuure cash flows. Accordng o Fama & French (2002), average reurn on a broad porfolo of socks s ypcally used o esmae he expeced marke reurn. And n her paper, fundamenals of socks are used o esmae he expeced sock reurn. Fama & French (2002) consder wo alernaves o esmae expeced sock reurn. One s he average sock reurn whch s calculaed by addng he average dvdend yeld o he average rae of capal gan: 18

24 A ( R ) = A ( D P 1) + A ( GP ), (3.12) where A R ) s he average sock reurn, ( A ( D ) P 1 s he geomerc mean of dvdend yelds, A GP ) ( s he geomerc mean of he rae of capal gans. The oher s referred o growh model: A ( RD ) = A ( D P 1) + A GD ), (3.13) ( where, A R ) s he expeced sock reurn, ( A ( D ) P 1 s he geomerc mean of dvdend yelds, A( GD ) s he geomerc mean of he growh rae of dvdends. Fama & French (2002) argue ha he esmaon wh he fundamenals yelds a more precse expeced reurn. In order o examne he relably of he esmaes from wo above equaons, hey use S&P ndex from 1872 o 2000 and fnd ha he sandard error of he esmae from he dvdend growh model (3.13) s less han half from he average reurn (3.12). In he same aemp o use valuaon models o esmae expeced reurns, Claus & Thomas (2001) nsead use forecass by secury analyss o esmae he dscoun rae when calculang he expeced cash flows. However, Claus & Thomas (2001), n he end, fnd ha analyss forecass are based. In hs hess, he expeced reurn s calculaed n boh wo ways as n he research conduced by Fama & French (2002), and he comparson s made o choose he mos approprae one. The deals of dscoun rae esmaon wll be dscussed n Secon Tes for un roo and co-negraon In her n-deph research on he relaonshp beween he co-negraon propery and he error-correcon model, Engle & Granger (1987) sugges wo crucal condons for he error-correcon model o gve unbased and conssen esmaes. Frs, boh of he neresed varables dependen and ndependen are negraed of order one. Or n oher words he model requres he varables o conan a un roo. Anoher necessary requremen for a vald error-correcon mechansm s he co-negraon n he long-run relaonshp beween hese varables. Co-negraon mples ha devaons from he long-run equlbrum are saonary (Engle & Granger, 1987). Wh respec o he wo-sep ECM, De Boef (2000) emphaszes 19

25 ha esmaes usng Engel-Granger wo-sep esmaor perform well only under lmed condons: permanen memory, or un roos, and co-negrang regresson whou serally correlaed errors. Thus, es for un roos and co-negraon of all varables n he regresson s nevable. For he me seres daase, he augmened Dckey Fuller es (Dckey & Fuller, 1979) s appled. Neverheless, for he panel daases alernave procedures need o be mplemened. Accordng o Hoang & McNown (2006), hree mos common ess for un roos n panel daa n pracce are Levn-Ln (LL), Im-Pesaran-Shn (1997) (IPS) and Maddala-Wu (1999) (Fsher es) n whch LL es s hen mproved by he updaed verson of Levn-Ln-Chu (2002). Based on he characerscs of our balanced daase wh moderae panels 6, we use Levn e al. s (2002) approach (LLC) whch ess he null hypohess ha all he panels conan un roos agans he alernave hypohess ha each panel s saonary. However, he LLC es s proved o be oo resrcve n mos sudes whch compare he es power of dfferen ess for panel daa un roos (Hoang & McNown (2006), Maddala & Wu (1999)). The reason s ha n LLC es he null hypohess of un roos n all panels s esed wh an assumpon ha he auoregressve parameer s he same n all panels. Insead, he IPS es relaxes hs assumpon by allowng each panel o have s own auoregressve parameer. The Fsher es s drecly comparable o he IPS es (Maddala & Wu, 1999). The asympoc valdy of each es depends on dfferen condons. For he IPS es, wheher N (number of panels) s gong o nfny has a decsve effec on he asympoc resuls of he es whle for he Fsher es depends on wheher T (number of years) s gong o nfny. In hs research, he LLC es and he Fsher es s employed based on he characerscs of he daases. 3.4 Mehods of esmaon Tme seres daa A frs, he orgnal equaon s esmaed wh he Ordnary Leas Squares (OLS) esmaor. In order o asceran he esmaes from he regresson o be unbased and conssen, he emprcal procedure s followed by hree ess. The omed varables es (Ramsey, 1969) s mplemened o es wheher he orgnal regresson has omed varables. If here are omed varables, he esmaed coeffcens wll be based. In such cases, he lags of he dependen varables (capal gan/oal reurn) are added o he regresson and he Ramsey es s carred ou agan o make sure ha omed varables problem s mproved. In esng for auocorrelaon, hree dfferen ess are employed smulaneously: Durbn-Wason es (Durbn & Wason, 1950), Durbn's alernave es (Durbn, 1970), Breusch-Godfrey Lagrange Mulpler (LM) es (Breush & Godfrey, 1981). The purpose of mplemenng all hree ess s o ensure 6 A panel of moderae sze s defned by Levn e al. (2002) o have 10 o 250 ndvduals wh 25 o 250 me seres observaons per ndvdual. 20

26 ha he unbased esmaes are obaned relably. In all cases he resuls of he hree ess are conssen o each oher, whch mples ha we could solely rely on he resuls. The fnal es s he heeroskedascy es whch examnes wheher he varance of he error erms are consan across all observaons. If hs s no he case, he sandard errors repored from he regresson wll be based whch n urn affec he resuls of he es sascs for he sgnfcan evdence of he mean reverson process. In addon, hree ess for heeroskedascy are carred ou: LM es for auoregressve condonal heeroskedascy (ARCH), Whe's es for homoskedascy (Whe, 1980), Breusch-Pagan/Cook-Wesberg es (Breusch & Pagan, 1979) for consan varance. The mplemenaon of all hree ess no only affrms he rusworhy of he es resuls bu also helps fndng he heeroskedasc funcon of he error erms varance. To accoun for auocorrelaon and heeroskedascy, he soluon of addng lags of he capal gan s consdered frs. If hs soluon could correc for all problems he resul from he regresson wh he lags s repored as he fnal resul afer correcon. Oherwse, alernave procedures are followed. To correc for seral correlaon, Pras-Wnson/Cochrane-Orcu mehod (Pras & Wnsen (1954), Cochrane & Orcu (1949)), whch s very powerful n auocorrelaon correcon s appled. As for heeroskedascy problem, he Weghed Leas Squares (WLS) esmaor s mplemened. The WLS esmaon s eraed wh dfferen possble heeroskedasc funcons and he one ha could elmnae heeroskedascy and a he same me ensure no auocorrelaon s chosen as he bes esmaon Panel daa In each model esmaon, he orgnal regresson wh fxed effec or random effec (choce of regresson wh fxed effec or random effec based on Hausman es (Hausman, 1978)) s esmaed frs. In order o asceran he esmaes from he regresson o be unbased and conssen, he emprcal procedure s followed by he ess for auocorrelaon and heeroskedascy. As for he auocorrelaon es, he Wooldrdge es for auocorrelaon n panel daa s employed whle he Lkelhood-rao es s used o nvesgae he heeroskedasc propery of each regresson. If seral correlaon beween error erms s found afer he regresson wh fxed effecs or random effecs, he Pras-Wnsen esmaor (Pras & Wnsen, 1954) s appled o esmae he model wh panel-correced sandard errors. STATA allows esmang he Pras-Wnsen regresson wh and whou heeroskedascy. If he orgnal regresson appears o encouner boh seral correlaon and heeroskedascy wo soluons could be used o yeld unbased esmaes: Pras-Wnsen esmaon wh heeroskedascy, and feasble generalzed leas squares (FGLS) for panel daa whch allows esmaon n he presence of AR(1) auocorrelaon whn panels and cross-seconal correlaon and heeroskedascy across panels (Davdson & MacKnnon, 1993). In general he resuls from hese wo mehods are very smlar o each oher. 21

27 4. Daa 4.1 Daa descrpon Mean reverson, f exss, s lkely o occur slowly, and can herefore be deeced only n long me seres (Balvers e al., 2000). Thus, nsead of usng monhly daa as a radonal approach n mean reverson research, followng Balvers e al. (2000) and Sperdjk e al. (2010) yearly daa are used n hs hess o nvesgae he exsence of mean reverson n nernaonal sock prce ndces. Moreover, as n he fndngs of Fama & French (1988b) he mean reverng process of monhly sock prces s concenraed n January, usng annual daa s expeced o avod hs seasonal effec. Annual daa are colleced for wo knds of seres used n hs hess. The long daases whch cover he long me perod of 90 years and more are assembled for 3 counres: Denmark, Sweden and he Uned Saes. The shor panel daase covers a shorer me frame bu ncludes as large as 15 developed counres wh long-hsory sock markes. In all seres, fgures on he sock prce ndex, oal reurn ndex and dvdend yeld whch are our man varables of neres are compled. The U.S daase however, conans more nformaon. All ndces are year-end fgures and f no avalable as he real values n he orgnal daases hey wll be ranslaed o real fgures by daa on exchange raes and nflaon raes (or ndrecly from consumer prce ndces) o elmnae he dfference n nflaon s and exchange raes flucuaons beween counres. The real fgures, denomnaed n US Dollars, are obaned n he end for emprcal analyss. These daases are explaned n more deals n he followng sub-secons Tme seres daa The long daases for 3 counres are compled from dfferen sources. The Uned Saes The Uned Saes wh he mos developed and long-sandng sock marke has he longes daase from 1871 o 2010 whch s used by Shller (2000) 7. I ncludes daa on Sandard & Poor s sock prce ndex, dvdend, earnngs and also he U.S s consumer prce whch allows us o calculae real sock prce ndces, real dvdends and real earnngs. For he purpose of hs research on mean reverson, he dvdend yeld and he oal reurn ndex are hen calculaed from he avalable numbers. 7 Ths daase s downloaded from Shller s webse hp:// 22

28 Sweden Sarng n December 1918, he weekly Swedsh fnancal chroncle Affärsvärlden publshed a compose sock prce ndex ha would laer be called Affärsvärldens Generalndex (AFGX). AFGX s a capalweghed ndex and ncluded up o 1998 only frms on he Sockholm Sock Exchange A-ls. From 1998 onwards AFGX also ncludes frms on he so-called O-ls, conanng he (prevously) unlsed frms (Waldensröm, 2007). These daa on sock prce ndces, oal reurn ndces from 1919 o 2006 are provded by The Swedsh Rksbank - Sverges Rksbank. The orgnal daa obaned from he Swedsh Bank are nomnal fgures. Thus, exchange raes and nflaons raes from Dmson-Marsh-Saunon Global Reurns Daa (DMS Global) are employed o calculae he real values of hese ndces. Daa for he laer perod s compled from Morgan Sanley Capal Inernaonal (MSCI) and hen ncorporaed no he daase. Denmark Dansh Insue for Naonal Economy (Insu for Naonaløkonom) has publshed a long me seres of sock reurns for Denmark from 1922 o 1999 (Nelsen & Rsager, 2001). In her paper, he esmae of he oal annual sock reurn equals he sum of he dvdend yeld and he capal gan. The dvdend yeld s esmaed on he bass of a large sample of lsed socks whch accoun for 50 o 80 percen of he oal marke capalzaon on he Copenhagen Sock Exchange. The capal gan daa s consruced based on he value-weghed Dansh Share Prce Index publshed by Sascs Denmark. Ths daase afer beng ransformed no real US dollar value s also ncorporaed wh MSCI daa for he laer perod o esablsh he long daase from 1922 o Panel daa The frs panel daa s he daa base from MSCI. I ncludes he sock prce ndces, oal reurn ndces and dvdend yelds of he sock markes from 15 counres 8. The ndces are expressed boh n home currency as nomnal ndces and n real US dollar values (Real exchange raes whch accoun for he dfference n nflaon raes beween he US and oher counres are appled o ranslae nomnal ndces no real US dollar ndces). In he daase, he dvdend yeld s calculaed by he change n he oal reurn ndces (oal sock reurn) subraced by he change n he sock prce ndces (capal gan) whch s conssen wh he calculaon of he dvdend yeld n me seres daases used n hs hess. The second panel daase s he pool of hree dfferen me seres of Denmark, Sweden and he U.S. Among hem, he Dansh daabase has he shores me horzon from Therefore, we rese he base year of 1921 for he oher wo daases o oban a conssen panel daase of hree counres. 8 Ls of counres n MSCI daabase: he Neherlands, Germany, France, Belgum, Ausrala, Denmark, Ialy, Japan, Canada, Norway, Span, Sweden, Swzerland, Uned Kngdom, and he Uned Saes. 23

29 Table 1 represens some summary sascs of our daase. The geomerc mean of he connuously compounded capal gan (laer called as he capal gan) 9 and of he connuously compounded oal reurn (laer called as he oal reurn) 10 are compued for he panel daase of 15 counres from 1970 o Table 1 - Summary sascs of Panel daa 15 counres Counry Code Mean capal gan * Mean oal reurn * Mean oal reurn Balvers e al.(2000) Sandard Error of oal reurn* Mean Dvdend Yeld* Neherlands Germany Fance Belgum Ausrala Denmark Ialy Japan Canada Norway Span Sweden Swzerland UK US (*Source: Calculaed from MSCI daabase) The sascs of Balvers e al. (2000) on sock reurns from 1970 o 1996 from he same source of MSCI daa base are also presened n Table 1 for comparson purpose. In our daase, he oal reurn vares from he hghes of 19.8% for Ausrala o he lowes of 13.8% for he Uned Saes. Norway has he hghes sandard devaon of he oal reurn of 37.1% whle he Uned Saes has he smalles of 17.9%. 4.2 Dscoun rae calculaon As aforemenoned n sub-secon 3.2.2, he dscoun rae appled o calculae proxes for he fundamenal values of he sock prces s calculaed usng Fama & French s (2002) approach: 9 Connuously compounded capal gan n = ln(sock prce ndex) n ln(sock prce ndex) n Connuously compounded oal reurn n = ln(oal reurn ndex) n ln(oal reurn ndex) n-1. 24

30 A ( RD ) = A ( D P 1) + A GD ). (3.13) ( For he panel daa of 15 counres, he same dscoun rae s appled for he whole perod for each ndvdual counry. The summary sascs are shown n Table 2. Table 2 - Dscoun rae (n %) o calculae proxes Panel daa 15 counres Counry Code Dvdend growh rae (Geomerc mean) % Dvdend yeld (Geomerc mean) % Dscoun rae % (a) (b) (a)+(b) Neherlands Germany Fance Belgum Ausrala Denmark Ialy Japan Canada Norway Span Sweden Swzerland UK US Table 3 - Dscoun rae o calculae proxes - ndvdual counres Dvdend growh rae (%) Dvdend yeld (%) Dscoun rae (%) Gordon dscoun facor (a) (b) (a)+(b) Uned Saes Sweden Denmark

31 For he long me seres daase of 3 counres we fnd srucural breaks n he dvdend growh rae and he dvdend yeld boh for Denmark, Sweden and he US (Table 3). Sweden and Denmark has he same srucural break n he 1980s. The dfference beween he esmaed dscoun rae for he perods before and afer 1980 s very sgnfcan. The Uned Saes, noneheless, has dramacally dfferen dscoun raes for he perods before and afer 1950 whch s n lne wh he earler fndngs of Fama & French (2002). Ther esmae for he perod s 8.07% whle s 4.74% for he laer perod The dfference of he Gordon dscoun facor beween 2 perods before and afer he break s even larger. As a resul, we do no employ he same dscoun rae o esmae he proxes (he foresgh prce, hndsgh prce, Gordon prce) for he whole perod bu nsead calculae hem separaely o oban more precse esmaon. We confrm a srucural break n he dscoun rae of 3 counres whch resuls from he srucural breaks n he dvdend growh rae and he dvdend yeld by usng rollng wndow esmaes of he geomerc mean of he dvdend growh rae and he dvdend yeld wh he me nervals of overlappng 25 years. Fgure 1 dsplays he rollng wndow esmaes of he geomerc mean of hese seres over me, akng Denmark for llusraon. Fgure 1 - Rollng wndow esmaes - Geomerc mean of dvdend growh and dvdend yeld I s obvous from Fgure 1 ha he geomerc mean of he dvdend growh rae and he dvdend yeld of Denmark change dramacally afer The geomerc mean of he dvdend growh rae ncreases sharply afer he year 1980 whle he geomerc mean of he dvdend yeld decrease consderably afer hs year. However, because of much larger magnude of he dvdend growh rae compared o he dvdend yeld, he combnaon of hese oppose changes lead o a sgnfcan ncrease n he geomerc mean of he dscoun rae for he perod afer Thus, applyng dfferen dscoun raes for he perod before and afer 1980 s necessary o oban precse esmaes of mean reverson of he sock prce. 26

32 Dfferen dscoun raes appled o calculae he proxes for he fundamenal value of he sock prce s expeced o affec he emprcal resuls. Therefore n secon wll be found he robusness check for he dscoun rae sensvy n whch we use he alernave proxes calculaed wh a conssen dscoun rae for he whole sample perod for esmaons. 5. Emprcal resuls Sock prce ndex In hs hess, o nvesgae he mean reverng behavour of he sock prce wo varables are used as he marke sock prce he sock prce ndex and he oal reurn ndex. The sock prce ndex measures he prce performance of markes whou ncludng dvdends. The oal reurn ndex measures he prce performance of markes wh he ncome from consuen dvdend paymens 11. The emprcal resuls presened n hs secon are based on he regressons wh he sock prce ndex. The resuls wh he oal reurn ndex are found n Secon Tes for un roos and co-negraon Tme seres daa The Dckey Fuller (DF) es and he augmened Dckey Fuller (ADF) es are appled o es for un roos and co-negraon of he me seres daase of 3 counres. Two varables are used as he dependen varable: he naural logarhm of he sock prce ndex and he naural logarhm of he oal reurn ndex. Independen varables nclude all proxes of he fundamenal value of he sock prce (all are expressed n naural logarhm): dvdends, perfec foresgh prce, hndsgh prce and Gordon prce. In addon, because of he avalably of daa on real earnngs, 5 more proxes are used for he US daa base: foresgh earnngs, hndsgh earnngs, Gordon earnngs and smoohed earnngs of 30 years and of 10 years. The resuls of he ess for un roos and co-negraon are presened n Table 4 and Table 5. Table 4 - Un roo ess - ndvdual counres Varables (log) Denmark Sweden Uned Saes -sascs Un roo -sascs Un roo -sascs Un roo Sock prce ndex Y Y Y Toal reurn ndex Y Y Y Dvdends Y Y Y Foresgh prce Y Y Y 11 Index defnons from MSCI hp:// 27

33 Varables (log) Denmark Sweden Uned Saes -sascs Un roo -sascs Un roo -sascs Un roo Hndsgh prce Y Y Y Gordon prce Y Y Y Foresgh earnngs Y Hndsgh earnngs Y Gordon earnngs Y Smoohed earnngs Y Smoohed earnngs Y The ADF es s crcal value s (1% sgnfcance level), (5% sgnfcance level), (10% sgnfcance level). The ADF es ess he null hypohess ha he varable conans a un roo agans he alernave ha he varable s saonary. We could see from Table 4 ha we canno rejec he null hypohess even a 10% sgnfcance level for mos of he varables used. Thus, all hese varables are hen used o es wheher here s a co-negraon relaonshp beween he sock prce ndex and he fundamenal. The resuls for he es wh he oal reurn ndex are presened n Secon 6. Table 5 - Co-negraon es - ndvdual counres Varables (log) Denmark Sweden Uned Saes -sascs Conegraon -sascs Conegraon -sascs Conegraon Dvdends *** Y *** Y *** Y Foresgh prce ** Y *** Y ** Y Hndsgh prce * N ** Y ** Y Gordon prce N *** Y *** Y Foresgh earnngs *** Y Hndsgh earnngs ** Y Gordon earnngs -4.32*** Y Smoohed earnngs * N Smoohed earnngs *** Y *** Sgnfcan a 1% confdence level. ** Sgnfcan a 5% confdence level. * Sgnfcan a 10% confdence level. Across hree counres, 2 proxes show a conssen relaonshp of co-negraon wh sock prce ndex n all 3 counres: he dvdends and he foresgh prce. The es for he Gordon prce dsplays very sgnfcan resuls for Sweden and he Uned Saes. All addonal proxes for he Uned Saes show sgnfcan co-negraon relaonshps wh he sock prce ndex excep he smoohed earnngs of 30 years. All proxes of he fundamenals ha show a co-negraon relaonshp wh he sock prce ndex wll be used as he ndependen varables n he regressons usng he Engle-Granger wo-sep ECM whch requres a src condon of a co-negraon relaonshp beween varables n he model o yeld unbased 28

34 esmaes. The oher proxes whch are no co-negraed wh he sock prce ndex are sll used for esmaon usng he smple mean reverson model. The resuls of hree models esmaons are presened n he nex secon Panel daa As descrbed n Secon 3.3 he Levn-Ln-Chu es (LLC es) and he Fsher es are appled o es for un roos n panel daa. Resuls from he wo-ess are smlar. All varables are found o conan un roos n he shor panel daase of 15 counres as well as he long daase of 3 counres. Dealed resuls are found n Table 6. Table 6 - Un roo es - Panel daa Varables (log) Panel 15 counres Panel 3 counres LLC es Fsher es LLC es Fsher es Adjused * p-value Modfed ch-squared p-value Adjused * p-value Modfed ch-squared p-value Sock prce ndex Toal reurn ndex Dvdends Foresgh prce Hndsgh prce N/A N/A N/A N/A Gordon prce For he LLC es he adjused -sasc are he bass for he concluson on a un roo whle s he modfed ch-squared for he Fsher es. The resuls for un roo ess are conssen beween he LLC es and he Fsher es. Boh prove he exsence of a un roo n all varables. Thus, all proxes are employed n he es for co-negraon relaonshp wh he sock prce ndex. In hese wo panel daases, we use he Weserlund es for panel co-negraon (Weserlund, 2007) o es wheher he proxes of he fundamenals are co-negraed wh he sock prce ndex. The Weserlund (2007) es repors he resuls for he es of he null hypohess ha here s no co-negraon n he whole panel daa agans wo knds of alernave hypoheses ha here s co-negraon for a leas one cross-seconal un and ha here s conegraon for all cross-seconal uns. Therefore, rejecon of he null hypohess n he frs case would be aken as he evdence of co-negraon for he panel as a whole whle rejecon of he second null hypohess would be nerpreed as he evdence of co-negraon for a leas one cross-seconal un. The wo ess yeld he same resuls for all varables. Hence, only he resuls of he frs es ha demonsrae a 12 The Fsher es canno be appled o he hndsgh prce because he frs varable of he daase s mssng. 29

35 co-negraon relaonshp beween he proxes and he sock prce ndex for he whole panel are presened n Table 7. Table 7 - Co-negraon es - Panel daa Varables (log) Panel 15 counres Panel 3 counres Z-sascs p-value Z-sascs p-value Dvdends * Foresgh prce *** *** Hndsgh prce ** ** Gordon prce *** *** Among 4 proxes, he dvdends show he weakes evdence of he co-negraon relaonshp wh he sock prce ndex. The resuls are sgnfcan for he foresgh prce, he hndsgh prce and he Gordon prce n wo panels. 5.2 Esmaon resuls As menoned n Secon 3.1 o examne wheher he sock prce ndex revers o s mean n he long run hree models are esmaed and compared. The hree models are: Engle-Granger wo-sep error-correcon model: Frs sep: = Second sep: p - p = 2 + p *. 0 p * + 1 * ˆ +. In hs model, β 2 s esmaed hrough he frs sep and he predced values of he resduals are used o esmae mean reverson n he second sep. The resuls repored for hs model ncludes β o, β 1 and β 2. The sgn of β 1 concludes he exsence of mean reverson of he sock prce ndex. Generalzed one-sep error-correcon model: p = + 0 p * + 1 ( p 1-2 ) +. p * 1 Ths model s esmaed mmedaely wh 3 ndependen varables p *, coeffcens repored n hs model are,, (- from he regresson s esmaon s also repored. Smple mean reverson model: p = + ( p * - 1 p ) ). The coeffcen p 1 and p * 1. Therefore he 2 whch s calculaed ndrecly 30

36 In hs model, he esmaed value of s repored because deermnes wheher he sock prce revers o s mean n he long-run Indvdual counres Denmark The esmaon resuls for Denmark are presened n Table 8. Based on he co-negraon es resuls (Table 5), he hndsgh prce and he Gordon prce are no co-negraed wh he sock prce ndex. Therefore hey are no used for he Engle-Granger wo-sep ECM esmaon. Neverheless, hey are sll used for he generalzed one-sep ECM whch does no requre a src condon of co-negraon, and he smple model. The las column shows unbased esmaes of he coeffcens of neres. Table 8 - Esmaon resuls - Denmark GENERALIZED ONE-STEP ECM βo β 1 - β 1 β 2 β 2 Dvdends 0.486*** *** 0.177*** Foresgh prce ** 0.189*** Hndsgh prce Gordon prce ENGLE-GRANGER TWO-STEP ECM βo β 1 β 2 (+) 13 Dvdends 0.456*** * 1.441*** Foresgh prce * 1.479*** (+) shows sgnfcance level of β 2 s dfference from 1. SIMPLE MODEL Dvdends Foresgh prce Hndsgh prce Gordon prce 0.088** λ 13 The esmaes of 2 n he wo-sep ECM are derved from he frs-sep regresson of he sock prce ndex on he proxy of he fundamenal ( = p - 2 ransformaon of Equaon 3.5 s obaned and esmaed: p * - Equaon (3.5)). In order o es wheher 2 s dfferen from 1, he p - p * = + p * ( - 1) + (Equaon (3.5a)). Then we can use a normal -es o es 2 s sgnfcan dfference from 1 afer equaon (3.5a) esmaon. 2 31

37 In almos all regressons wh he Dansh daase no auocorrelaon s found whle mos of he me heeroskedascy s a problem. Thus, for Denmark he fnal resuls are manly obaned from addng lags of he capal gan and usng robus sandard errors or from he WLS esmaor. The esmaon wh he foresgh prce s he only one ha does no conan heeroskedascy or encouner omed varables bas (based on he resuls of Ramsey (1969) es for omed varables) n he orgnal regresson. Therefore s used as he fnal resul for hs proxy. In he orgnal regressons wh fxed effec or random effec, mean reverson s found for all proxes excep he hndsgh prce n he esmaon wh he generalzed one-sep ECM. However, afer correcng for omed varables, auocorrelaon and heeroskedascy, he resuls change n he oppose way for he dvdends and he Gordon prce. As for he dvdends, evdence of mean reverson s even more sgnfcan han before correcon. On he conrary, he resul wh he Gordon prce becomes nsgnfcan whch confrm he mporan propery of he co-negraon relaonshp of he wo varables n esmang he error-correcon model. The resuls from he Engle- Granger wo-sep ECM are smlar o he generalzed ECM alhough he evdence of mean reverson s less sgnfcan. In conras o he wo-sep ECM, he smple mean reverson model confrms he exsence of mean reverson only for he Gordon prce. Also shown n Table 8 he long-run relaonshp coeffcen β 2 esmaed on he Dansh daase s sgnfcanly dfferen from 1 whch mples ha he payou rao of dvdend (dvdend/sock prce) s decreasng over me. Fgure 2 shows an explcly decreasng rend n hs rao over he perod whch s n lne wh he fndngs of Bekaer e al. (2001). Holmen e al. (2007) also fnd smlar resuls on he decreasng payou rao of dvdend. Fgure 2- Dvdend payou rao across me Sweden The emprcal resuls for he Swedsh daase show more sgnfcan evdence of mean reverson of he sock prce han Denmark s. Afer correcng for omed varables, auocorrelaon and heeroskedascy, he exsence of mean reverson s found n he esmaons wh all proxes excep he hndsgh prce. 32

38 Especally he mean reverson coeffcen β 1 s sgnfcan a 1% confdence level for he dvdends, he foresgh prce and he Gordon prce. Table 9 presens dealed resuls for he esmaons wh hree models separaely. Table 9 - Esmaon resuls - Sweden GENERALIZED ONE-STEP ECM βo β 1 -β 1 β 2 β 2 Dvdends 0.708*** *** 0.349*** Foresgh prce ** *** 0.539*** Hndsgh prce * Gordon prce 0.631*** *** 0.297*** ENGLE-GRANGER TWO-STEP ECM βo β 1 β 2 (+) Dvdends 0.690*** *** 1.138*** Foresgh prce *** *** 1.161*** Hndsgh prce * 1.267*** Gordon prce 0.610*** *** (+) shows sgnfcance level of β 2 s dfference from 1. SIMPLE MODEL λ Dvdends 0.706*** Foresgh prce 0.179*** Hndsgh prce Gordon prce 0.689*** In comparson wh he resuls of Denmark, he Swedsh sock prce ndex shows much more sgnfcan evdence of he mean reverng process wh dramacally hgher values of he mean reverson coeffcen β 1. Noneheless, he long-run relaonshp co-effcen β 2 s no as far from 1 as n he case of Denmark. The exsence of mean reverson s found conssenly among he smple mean reverson model, he Engle- Granger wo-sep ECM and he generalzed one-sep ECM The Uned Saes Dfferen from he sock prce ndex daabase of Denmark and Sweden, he Uned Saes do no show auocorrelaon n all esmaons. Heeroskedascy (f any) s due o some oulers bu no he dependence of he varance of he error erms on he ndependen varables. As a resul, robus sandard 33

39 errors are used o correc for heeroskedascy. The esmaed coeffcens of he esmaons by hree models are found n Table 10. Table 10 - Esmaon resuls- The Uned Saes GENERALIZED ONE-STEP ECM βo β 1 - β 1 β 2 β 2 Dvdends Foresgh prce 0.908*** *** 0.298*** Foresgh earnngs *** 0.215*** Hndsgh prce Hndsgh earnngs Gordon prce ** 0.108** Gordon earnngs *** 0.088** Smoohed earnngs ** 0.103** Smoohed earnngs ENGLE-GRANGER TWO-STEP ECM βo β 1 β 2 (+) Dvdends *** Foresgh prce 0.885*** *** 1.443*** Foresgh earnngs *** Hndsgh prce *** Hndsgh earnngs Gordon prce ** 1.076** Gordon earnngs 0.252*** *** 0.818*** Smoohed earnngs *** Smoohed earnngs * *** (+) shows sgnfcance level of β 2 s dfference from 1. SIMPLE MEAN REVERSION λ Dvdends Foresgh prce 0.135*** Foresgh earnngs 0.183*** Hndsgh prce Hndsgh earnngs Gordon prce 0.108** Gordon earnngs 0.127*** Smoohed earnngs ** Smoohed earnngs * 34

40 Thanks o he avalably of real earnngs n he Uned Saes daabase, more proxes are used n all 3 models: foresgh earnngs, hndsgh earnngs, Gordon earnngs and wo smoohed earnngs of 30 and 10 years. Among hese addonal proxes, he esmaons wh he foresgh earnngs, he Gordon earnngs and he 30-year smoohed earnngs show sgnfcan evdence of mean reverson. The resuls wh he proxes esmaed from earnngs are conssen wh he resuls wh he proxes esmaed from dvdends. The values of β 1 for he US s are smlar o Denmark s esmaes. Moreover, for all 3 counres, he mean reverson coeffcen obaned from he esmaons wh he dvdends and he Gordon prce s dfferen because we apply a srucural break n he requred rae of reurn when calculang he Gordon prce. In addon, n almos all esmaons usng dvdends as he proxy, he Dansh and Swedsh daases show sgnfcan evdence of mean reverson whle no evdence s found for he US daase. The speed of mean reverson of he sock prces whch s measured by he coeffcen λ n he smple model s smlar o he fndngs of Balvers e al. (2000) whch s based on a shorer me perod As for he resuls from he esmaons by he wo-sep error-correcon model whch s n lne wh he approach of Cochran & DeFna (1995). Ther esmaed mean reverson coeffcen β 1 ranges n value from o sascally sgnfcan a 5% o 1% confdence level. Cochran & DeFna (1995) also nvesgae he mean reverson behavor of he sock prces for all NYSE socks bu hey use quarerly daa from a shorer perod compared o our US daase. Applyng he same approach of he wo-sep errorcorrecon model bu usng a shorer perod and dfferen frequency of daa, Cochran & DeFna (1995) fnd a smaller speed of he mean reverng process of he sock prce han our resuls whch range from o All he esmaed mean reverson coeffcens n her resuls are negave and sascally sgnfcan a 5% o 1% level for dfferen esmaons wh dfferen conrol varables Panel daa Denmark, Sweden and he Uned Saes To ake he advanage of he long-me perods of 3 counres Denmark, Sweden and he Uned Saes we pool hem ogeher o have a long panel daa from 1922 o All regressons wh he Panel 3 counres conan auocorrelaon bu no heeroskedascy. Thus, he Pras-Wnsen/ Cochrane-Orcu esmaor s employed o yeld unbased esmaes. The coeffcen from Pras-Wnsen regresson s found o be smaller han from he orgnal regresson wh fxed effec or random effec. The resuls wh all he generalzed one-sep ECM, he Engle-Granger wo-sep ECM and he mple model are presened n Table

41 Table 11 - Esmaon resuls - Panel 3 counres GENERALIZED ONE-STEP ECM βo β1 -β1β2 β2 Half-lfe 14 95% C.I Dvdends 0.492*** *** 0.12*** (4.5, 20.9) Foresgh prce *** 0.174*** (3.7, 11.2) Hndsgh prce Gordon prce 0.459*** *** 0.249*** (1.8, 3.9) ENGLE-GRANGER TWO-STEP ECM βo β1 Half-lfe 95% C.I Dvdends 0.471*** ** 19 (10.5, 91.6) Foresgh prce *** 7.5 (5.1, 13.6) Hndsgh prce Gordon prce 0.445*** *** 3.2 (2.3, 5) SIMPLE MODEL λ Half-lfe 95% C.I Dvdends 0.093*** 7.1 (4.3, 17.5) Foresgh prce 0.07*** 9.5 (5.3, 36.7) Hndsgh prce Gordon prce 0.344*** 1.6 (1.3, 2.2) The esmaed coeffcens are very conssen wh he resuls for ndvdual counres when hey are esmaed separaely as me seres daa. The esmaons wh he dvdends, he foresgh prce and he Gordon prce show very sgnfcan evdence of mean reverson a 1% confdence level. Agan when he hndsgh prce s ncluded n he regresson he exsence of mean reverson could no be proved for any of he hree models. Among he hree proxes ha show srong evdence of mean reverson he esmaed coeffcen β 1 from he regresson wh he Gordon prce s very sgnfcanly dfferen from zero. There s also large varaon n he esmaed speed of mean reverson when we use dfferen proxes for he fundamenal values of he sock prces. In hs panel daase of 3 counres he Gordon prce conssenly show he sronges evdence of mean reverson wh a very hgh value of λ of compared o he earler resuls of Balvers e al. (2000) and Sperdjk e al. (2010). Balvers e al. (2000) who also use he MSCI daase bu wh a shorer sample perod of for her esmaon repor he larges unbased esmae of λ equal o when he US ndex s used as he reference ndex. The resuls of Sperdjk e al. (2010) whch are based on Dmson e al. (2002) daase for 110 years from 1900 o All esmaes of he half-lfe and her correspondng 95% confdence nervals of he mean reverson process are presened n years. 15 Balvers e al. (2000) page

42 even show less sgnfcan speed of mean reverson wh he hghes repored speed equal o usng Germany ndex as he benchmark. In conras o he resul wh he Gordon prce, he esmaons wh he dvdends and he foresgh prce are no as hgh as he resuls from separae me seres regresson for each counry. The half-lfe of mean reverson s also presened for he regresson wh sgnfcan evdence of mean reverson. Ths half-lfe s calculaed by ln(0.5)/ln(1-λ) for he smple mean reverson model followng Balvers e al. (2000) and Sperdjk e al. (2010) whle s calculaed by ln(0.5)/ln(1+β 1 ) for he error-correcon model. The resuls are subsanally dssmlar beween dfferen proxes and dfferen models. Among he hree models, he generalzed one-sep ECM seems o yeld more relable resuls wh he esmaed half-lfe varyng from 2.5 years o 7.5 years. Ths means ha approxmaely 2.5 years o 7.5 years are requred o elmnae 50% of he prcng error. The correspondng 95% confdence nerval ranges from (1.8, 3.9) years o (4.5, 20.9) years. The esmaed half-lfe from he wo-sep model vares from 3.2 years o 9.1 years. However he 95% confdence nerval s much larger han ha from he onesep model wh he lowes value of 2.3 years and he hghes value of 91.6 years, whch shows he more uncerany of he esmaes from he wo-sep compared o he one-sep model. Our esmaes are larger han he average half-lfe of 1.75 years repored by Cochran & DeFna (1995). Compared o Balvers e al. (2000) and Sperdjk e al. (2010) our panel daa of 3 counres gve more dverse esmaes of mean reverson correspondng o dfferen proxes as well as dfferen models used o esmae. Ths sresses he effecs of choosng a suable proxy for he fundamenal of he sock prce on he emprcal evdence of sock prce mean reverson Panel 15 counres The second panel daase as descrbed n Secon conans he daa of 15 counres from 1970 o Despe of he dsadvanage of a shor me perod, he resuls from hs panel daa wh more counres could conrbue o he earler fndngs of Balvers e al. (2000) and Sperdjk e al. (2010) ha mean reverson of sock prce ndces f exs s no only he case for he US sock marke. In he frs regressons wh fxed effec or random effec, mean reverson s found n all esmaons even wh he hndsgh prce whch gves no evdence of mean reverson a all when beng ncluded n he prevous regressons wh he me seres and he panel 3-counry daases. However, n all orgnal regressons for he panel daase of 15 counres, very srong evdence of auocorrelaon and heeroskedascy s found. As aforemenoned, o accoun for hese problems, he FGLS esmaor and he Pras-Wnsen mehod s used smulaneously o oban unbased esmaes. The unbased esmaes are repored n Table Sperdjk e al. (2010) page

43 Table 12 - Esmaon resuls - Panel 15 counres GENERALIZED ONE-STEP ECM βo β 1 - β 1 β 2 β 2 Dvdends 0.746*** *** 0.107*** Foresgh prce -0.88*** *** 0.225*** Hndsgh prce Gordon prce 0.751*** *** 0.166*** ENGLE-GRANGER TWO-STEP ECM βo β 1 Dvdends 0.747*** ** Foresgh prce *** *** Hndsgh prce Gordon prce 0.751*** *** SIMPLE MODEL λ Dvdends 0.229*** Foresgh prce 0.113*** Hndsgh prce 0.06 Gordon prce 0.332*** Afer correcng for auocorrelaon and heeroskedascy, he resuls for he Panel daase of 15 counres from 1970 o 2010 are agan conssen wh our earler fndngs wh me seres daa of 3 ndvdual counres as well as he Panel daase poolng 3 counres from 1922 o Especally he speed of mean reverson esmaed wh he Gordon prce as he proxy for he fundamenal value of he sock prce s no dfferen from he resul of he frs panel. The repored value of λ s for he frs panel and for he second panel. And boh esmaes are sgnfcan a 1% confdence level. As for he esmaon wh he ECM, he resuls from he Engle-Granger wo-sep ECM are lower n comparson wh he generalzed one-sep ECM whch s also found n he esmaons whn dfferen used daases n general. In addon, excep for he esmaon wh he hndsgh prce, all regressons yeld unbased esmaes of all coeffcens wh expeced sgns. The long-run relaonshp coeffcen β 2 s posve whch mples he conegraon relaonshp beween he sock prce ndex and s fundamenal value across me. Neverheless, he value of β 2 s dfferen from one ha leads o he concluson ha 1% change n he fundamenals lead o more han 1% change n he sock prce ndex. Ths oucome could be explaned by he decreasng rend of dvdend payou rao n realy menoned n he prevous secon. 38

44 5.2.3 Robusness check for dscoun rae sensvy All he emprcal resuls presened above are based on he esmaons usng he proxes calculaed wh dfferen dscoun raes appled for dfferen perods. The dscoun rae used o esmae he proxes for he fundamenal value of he sock prce s expeced o affec he emprcal resuls. Thus, n hs secon, we wll use alernave proxes for esmaon o check wheher he dfferen dscoun rae appled has a sgnfcan effec on he evdence of mean reverson of sock prces. These alernave proxes are calculaed n he same way wh he curren used proxes usng he equaons dscussed n secon The only change n he calculaon s he dscoun rae. Ths robusness check of he dscoun rae sensvy s appled o he panel daa of 3 counres (Denmark, Sweden, he Uned Saes). So for nsance, nsead of applyng he wo separae dscoun raes of he wo perods and for Denmark we wll apply one conssen dscoun rae for he whole perod The dscoun rae appled for each counry s sll dfferen 17. The emprcal resuls for he alernave proxes are presened n Table 13. The resuls for he man proxes are re-presened for comparsons. Table 13 - Dscoun rae sensvy check - Panel 3 counres GENERALIZED ONE-STEP ECM βo β1 -β1β2 β2 Half-lfe 95% C.I Dvdends 0.492*** *** 0.12*** (4.5, 20.9) Foresgh prce *** 0.174*** (3.7, 11.2) Foresgh prce *** 0.224*** (2.7, 6.4) Hndsgh prce Hndsgh prce Gordon prce 0.459*** *** 0.249*** (1.8, 3.9) Gordon prce *** *** 0.128*** (4.3, 18.1) ENGLE-GRANGER TWO-STEP ECM βo β1 Half-lfe 95% C.I Dvdends 0.471*** ** 19 (10.5, 91.6) Foresgh prce *** 7.5 (5.1, 13.6) Foresgh prce *** 10.7 (7.3, 19.1) Hndsgh prce Hndsgh prce Gordon prce 0.445*** *** 3.2 (2.3, 5) Gordon prce *** *** 17.7 (10, 72.3) 17 Deals of he dscoun raes appled for he esmaons are presened n Table 3. 39

45 SIMPLE MODEL λ Half-lfe 95% C.I Dvdends 0.093*** 7.1 (4.3, 17.5) Foresgh prce 0.07*** 9.5 (5.3, 36.7) Foresgh prce *** 5.6 (3.6, 11.8) Hndsgh prce Hndsgh prce Gordon prce 0.344*** 1.6 (1.3, 2.2) Gordon prce *** 6.3 (4.0, 14.0) Foresgh prce 2, hndsgh prce 2 and Gordon prce 2 are he alernave proxes whch are appled one conssen dscoun rae n calculaon. The ohers are used for emprcal resuls presened n he prevous secon. I s apparenly seen n Table 13 ha smlar resuls are found when he alernave proxes are used. The esmaons wh he foresgh prce 2 show sgnfcan evdence of mean reverson n all hree models a 1% confdence level whch s exacly he same resul for he foresgh prce. Smlarly, he regressons wh he hndsgh prce 2 yeld he same resul wh he hndsgh prce. None of he esmaons wh hose wo proxes shows he exsence of mean reverson of he sock prce ndces. Lke he Gordon prce, he Gordon prce 2 also confrms mean reverson of he sock prces a 1% sgnfcance level. Therefore, usng dfferen dscoun raes does no affec he emprcal resuls on he evdence of he mean reverng process n he sock prce ndces. The resuls are conssen for he same knd of proxy. However, he magnude of he mean reverson coeffcens or he speed of mean reverson changes when we use he same dscoun rae for he whole sample perod, herefore, leads o changes n he half-lfe esmaon. Especally, he esmaed speed of mean reverson wh he Gordon prce 2 s dramacally smaller han he Gordon prce. The reason s he Gordon dscoun facor appled n he laer perod for each counry s much larger han ha n he frs perod and he average of he whole perod. As a resul, he Gordon prce 2 s larger n he frs perod and smaller n he laer perod compared o he Gordon prce. Fgure 3 llusraes hs comparson, akng Denmark as an example. Moreover, because of applyng he same dscoun rae for he enre sample perod, he resuls wh he Gordon prce 2 are smlar o he resuls wh he dvdends as expeced when we use all varables n he logarhm form. They are no exacly he same bu slghly dfferen because we use unequal dscoun raes for each counry. In shor, applyng dfferen dscoun raes does no affec he emprcal evdence on he exsence of mean reverson n he sock prce ndces. Noneheless, he speed of mean reverson vares when he dscoun rae changes. So he queson s whch dscoun rae gves more precse resul. The answer could be derved from Fgure 3. 40

46 Fgure 3 - Dscoun rae sensvy I s clearly seen ha he esmaed proxy followng he Gordon model usng he dscoun rae wh a srucural break s more n lne wh he sock prce ndex han usng one conssen dscoun rae for he enre sample perod. Consequenly, he sock prce ndex are observed o show clearer and faser mean reverng behavour o s fundamenal value when we use a flucuan dscoun rae raher han an unchanged one for he whole perod. In oher words, usng a srucural break n he dscoun rae beer reflecs he expeced fuure cash flows of he sock whch hen deermne he sock prce. Hence, we can conclude on he robusness of he emprcal resuls found wh he currenly used proxes for he fundamenal value of he sock prce. 5.3 Tme-varyng mean reverson Even hough he exsence of mean reverson of he sock prce ndces proves o be conssen a a very sgnfcan level among all above daases wh dfferen me spans, he speed of mean reverson s found o vary dramacally across hese daa. In addon, he fndngs of he flucuan dvdend growh raes and sock reurns n secon 4.2 also sugges ha here s a me-varyng speed of mean reverson across me. Furhermore, n her paper, Sperdjk e al. (2010) apply he same approach as Balvers e al. (2000) n examnng he mean-reverng propery of sock prces bu hey fnd large dfference n her resuls. The man reason s arbued o he dfference n he me nerval (Sperdjk e al. 2010). Whle based on MSCI daabase for he sock marke prce ndces whn he me nerval Balvers e al. (2000) fnd a sgnfcan speed of mean reverson wh he unbased esmae of λ equal o 0.182, he esmaed speed of mean reverson by Sperdjk e al. (2010) s only beween 1900 and 2008 (nearly four mes smaller han Balvers s e al. (2000)) for he case where he world ndex s he benchmark. Thus, s 41

47 plausble o argue ha he esmaes of he speed of mean reverson of sock marke ndces depend on he me nerval. Ths secon s devoed o research on he varyng speed of mean reverson across me by applyng a rollng wndow esmaon approach Panel daa 3 counres Many prevous sudes queson he assumpon of a consan speed of mean reverson over me. Km e al. (1991) argue ha mean reverson s a phenomenon of he perods when he sock markes are hghly volale, no a feaure of he pos-war era. In addon, accordng o Poerba and Summers (1988) he resuls of mean reverson change subsanally wh he ncluson and excluson of he Depresson years. Sperdjk e al. (2010) gve more nsgh on he flucuan speed of mean reverson of he sock marke ndex by applyng rollng wndow esmaons on he panel daa of 17 counres wh he overlappng me nervals of 27 years. They fnd apparen evdence of me-varyng mean reverson wh only 2 ou of 84 nervals show nsgnfcan evdence. The esmaes of he speed of mean reverson across dfferen nervals range from he larges value of n he nerval o he smalles of n he nerval whch s almos 10 mes smaller. Followng Sperdjk e al. (2010), we apply rollng wndow esmaons o our long panel daa of 3 counres wh he me nerval of overlappng 27 years long. The rollng wndow esmaon s only appled o he regressons ha yeld srong evdence of mean reverson. Fgure 4 dsplays he unbased esmaes of he mean reverson coeffcen for he error-correcon model and he smple model wh he Gordon prce, he foresgh prce and he dvdends and her correspondng 95% confdence nervals agans he md-year of he 27-year rollng-wndow perod 18. As for he smple model, he speed of mean reverson s he value of λ and s he absolue value of β 1 for he error-correcon model. All he esmaons show a clearly non-consan speed of mean reverson of he sock prce across me. Moreover he evdence of mean reverson s found for nearly 100% of all me nervals durng he whole sample perod. For he same proxy, he resuls from he ECM and he smple model are comparable o each oher. As for he esmaons wh he same model for dfferen proxes he resuls, on he one hand, show an analogous rend n he flucuaon of mean reverson across me, bu on he oher hand, dsplay dverse magnudes of he speed of mean reverson for he esmaons. For example, he smple model wh he Gordon prce yelds he hghes value of he speed of mean reverson equal o for he frs me nerval (md-year 1935) whle he same model wh he foresgh prce gves he hghes value of also for he me nerval The hghes value and he lowes value of he speed of mean reverson obaned from he esmaon by he ECM wh he Gordon prce are whch are 18 All rollng-wndow esmaes wh assocaed 95% confdence nervals ploed n laer fgures are also agans he md-year of he 27-year rollng-wndow perod. 42

48 found n he frs me nerval and n he me nerval of respecvely, hese values are lower compared o he regresson by he smple model. Fgure 4 - Rollng wndow esmaes - Panel 3 counres 43

49 Revewng he flucuaon of he speed of mean reverson dsplayed n he sx graphs, he common rend s apparenly recognzed wh he same peaks found n some of he same me nervals. Frs, he hghes values found among he oal of 63 me nervals are n he perods before 1945 whch s n lne wh he fndngs of Sperdjk e al. (2010) ( Fgure 5). In addon, hs me perod wh a very large speed of mean reverson concdes wh he mng of he Grea Depresson as well as he World War II whch made remendous mpac on he economes all over he world n general and on he nernaonal sock markes n parcular. The Grea Depresson sared n abou 1929 and lased unl he lae 1930s or early 1940s. The World War II whch nvolved mos of he world s naons lased from 1939 o Incorporang our fndngs wh he mng of hese noeworhy hsorcal and economc evens, we could srenghen he argumens of many prevous papers on hs opc ha n he perods of he nsable economy, he sock prce s lkely o rever o s fundamenal value faser han of normal condons. Sperdjk e al. (2010) also fnd he larges speed of reverson durng he md-year 1930 and conclude on a faser mean reverng process of he sock prce durng he perods of uncerany abou he susanably of he economy. Our fndngs also confrm par of he concluson of Km e al. (1991) who argue ha he mean reverson s prmarly a phenomenon of he perod whch ncludes he Grea Depresson and he World War II bu no lkely o occur n he pos-war era. Ther argumen ha he mean reverng process of sock prce manly occurs n he perods when he sock markes are hghly volale s srongly corroboraed by our resuls. Noneheless, n he normal marke condon, mean reverson of sock prce sll exss, hough wh a smaller magnude. In Fgure 5, he speed of mean reverson s esmaed usng he panel daa model n combnaon wh he boosrap approach. The sold lne represens he medan unbased esmae of he speed of mean reverson and he wo dashed lnes consue he correspondng 95% confdence nerval Sperdjk e al. (2010) page

50 Fgure 5 - Resuls of Sperdjk e al. (2010) Rollng-wndow esmaes of speed of mean reverson (benchmark: Wolrd Index) Fgure 6 - Rollng wndow esmaes - Panel 3 counres wh Gordon prce 45

51 As clearly seen n Fgure 6, he speed of he reverng process s hghes durng he Grea Depresson (he lae 1930s), reduces gradually aferwards, hen keeps s hgh value and slghly clmbs agan durng he World War II ( ). Afer ha decreases n he pos-war perods. The speed of reverson goes down afer he war bu sll keeps s value sable and sgnfcanly dfferen from zero. Moreover, beng conssen wh he fndng of Sperdjk e al. (2010), he mean reverng process of he sock prce reaches s oher peaks agan around he 1960s. These perods are n he mng of he wo phases of and of he Cold War as well as he sock marke crash due o he Venam War and he Waergae Scandal n he US. Afer hese hghly unsable perods, he mean reverng speed of he sock marke decreases agan n normal marke condons before movng o a slgh peak around he mdyear Some ensons could be arbued o hs small peak such as he 1987 Black Monday. However, hs marke crash lased for only a few days. So s effec and severy are sad o be no as large as he wors marke crash n he sock marke hsory or he crash whch all lased for more han 1 year. As a resul, he speed of mean reverson only slghly ncreases durng hese perods. In lne wh Sperdjk e al. (2010), he oucomes of our rollng regressons also show a decrease rend afer 1980 and hen he speed ncreases agan unl he md-year 1997 because of he mn-crash of he sock marke whch occurred due o he economc crss n Asa. Brefly speakng, based on he resuls from he rollng esmaons of he panel daa of 3 counres Denmark, Sweden and he Uned Saes durng he sample perod , we could conclude on he me-varyng propery of he sock marke ndces. The speed of he mean reverng process s dramacally hgher n he perods of unsusanable economy han n perods of normal economc condons The Uned Saes The me-varyng propery of he sock marke ndex s srongly confrmed wh he panel daase of 3 counres. Noneheless, he panel daase even hough akes he advanage of he large sample sze afer poolng he separae long me seres of 3 counres only covers he perod from 1922 o The sock daabase of he Uned Saes wh a long-sandng hsory of he sock marke whch covers a much longer me span from 1871 o 2010 s expeced o gve more nsgh on he flucuan speed of mean reverson of he sock prce. For ha reason, n hs secon we apply he rollng wndow esmaon o he Uned Saes daase o see f we fnd a conssen resul o he fndngs wh he panel daase of 3 counres. The me nerval of 27 years s also used for comparson purpose. The md-year of each me nerval ranges beween 1884 and The frs nerval covers he perod and he las nerval s of perod. The rollng wndow esmaes are dsplayed n Fgure 7 for regressons by he ECM and he smple model wh he foresgh prce and he foresgh earnngs whch show very srong evdence of mean reverson. 46

52 Fgure 7 - Rollng wndow esmaes - The Uned Saes The error-correcon model seems o gve more relable resuls when mean reverson s found n almos every me nerval for he whole sample perod. Only 11 ou of 114 me nervals show nsgnfcan evdence of mean reverson. Moreover, he esmaons by he ECM show more conssen resuls beween dfferen proxes han by he smple model. The rollng-wndow esmaes obaned from he smple model for me nervals for he md-year beween 1910 and 1920 flucuae n he oppose rend for he wo proxes he esmaes wh he foresgh prce only demonsrae a connuously ncreasng rend whle he esmaes wh he foresgh earnngs show a slgh decrease. Furhermore, a huge drop s observed for he esmaons wh he foresgh prce afer he md-year 1900 whle s a gradual decrease wh he foresgh earnngs. Esmaons wh he smple model acually gve larger dfference n he resuls wh dfferen proxes han he ECM. Therefore he error-correcon model s more relable n esmang he mean reverng process of he sock prce ndces. 47

53 In lne wh he fndngs of he panel 3 counres mean reverson s found o reach s hghes value over he whole sample perod durng he 1930s whch concdes wh he mng of he Grea Depresson. The speed of he mean reverng process s n a decreasng rend aferwards bu sll a a very hgh value durng he mng of he World War II. In addon, a small peak around he 1960s s also found for he US lke n he resuls of he Panel daa. Fgure 8 - Rollng wndow esmaes - The Uned Saes - Foresgh prce More neresng resul from he US s daa compared o ha from he panel daa s ha ncludes he me perod before 1922 or n oher words before he md-year 1935 whch could help compare our resuls wh he fndngs of Sperdjk e al. (2010) for larger me perods. The sar me nerval of Sperdjk e al. (2010) s he perod correspondng o he md-year Accordng o her resuls, he speed of mean reverson has a connuously ncreasng rend from he frs me nerval o he md-year Ths s found o be smlar o our resuls where he esmaed speed also follows a rsng rend from he md-year 1913 o 1929 whch conans he perod of he World War I ( ). In addon, our larger perod also shows he flucuaon of he reverng speed from he nerval o he nerval whch s no ncluded n he daa used by Sperdjk e al. (2010). The mean reverson of he sock prce ndex n he US sock marke s very sgnfcan around he 1900s. Ths could be plausbly explaned by he 1873 Depresson whch s a severe nernaonal economc depresson boh n Europe and he Uned Saes ha lased unl 1879, he sock marke crash because of he assassnaon of Presden Wllam McKnley and he sock marke crash due o a cred crunch n New York. In sum, he resuls from rollng wndow esmaons wh he US daabase agan confrm ha he mean 48

54 reverson of he sock marke ndex s a non-consan process wh he hghes speed found n perods of volale sock marke n he nsable economy. 6. Emprcal resuls Toal reurn ndex Boh Balvers e al. (2000) and Sperdjk e al. (2010) nvesgae he mean reverng behavor of he sock prce usng he marke oal reurn sock ndex whch ncludes he ncome from dvdend paymens. Our man emprcal resuls above presened n Secon 5 are based on he sock prce ndex whch measures he prce performance of markes whou ncludng dvdends. The queson s whch ndex provdes beer resuls for esmaon. For he research of Balvers e al. (2000) and Sperdjk e al. (2010) mean reverson s deeced from oal reurn ndces relave o a reference ndex. Therefore usng he oal reurn ndex for esmaon s plausble. However, n hs hess, we use a dfferen approach whch drecly uses he fundamenal value of he sock prce for esmaon o sudy he mean reverng behavor of he sock prce. The fundamenal of he sock prce s heorecally deermned whou he assumpon ha he dvdends are renvesed o purchase addonal uns of socks whch s used n he calculaon of he oal reurn ndex. As a resul, he esmaed fundamenal value of he sock prce wll beer reflec he sock prce ndex raher han he oal reurn ndex of he marke. The resuls wh he sock prce ndex n hs way of esmaon are more relable han hose wh he oal reurn ndex. Neverheless, for he robusness check and comparson purpose, hs secon shows he resuls of he regressons wh he oal reurn ndex. The emprcal procedures for he esmaons wh he oal reurn ndex are exacly he same as used wh he sock prce ndex n secon Tes for co-negraon In secon 5.1, he augmened Dckey Fuller es shows a un roo n he oal reurn ndex for all daases. Ths secon presens he es resuls for co-negraon relaonshp beween he oal reurn ndex and all proxes for he fundamenal value of he sock prce. 49

55 Table 14 - Co-negraon es wh oal reurn ndex - ndvdual counres Varables (log) Denmark Sweden Uned Saes -sascs Conegraon -sascs Conegraon -sascs Conegraon Dvdends * N * N *** Y Foresgh prce * N * N ** Y Hndsgh prce * N N *** Y Gordon prce ** Y ** Y * N Foresgh earnngs ** Y Hndsgh earnngs * N Gordon earnngs ** Y Smoohed earnngs N Smoohed earnngs N Table 14 shows ha he evdence of he co-negraon relaonshp beween he oal reurn ndex and he proxes for he fundamenal s less sgnfcan han he resuls wh he sock prce ndex. Smlar resuls are found wh he panel daa n Table 15. Table 15 - Co-negraon es wh oal reurn ndex - Panel daa Varables (log) Panel 15 counres Panel 3 counres Z-sascs p-value Z-sascs p-value Dvdends ** Foresgh prce *** *** Hndsgh prce * Gordon prce ** *** The less sgnfcan evdence of he co-negraon relaonshp wh he fundamenal found n he oal reurn ndex compared wh he sock prce ndex (all n naural logarhm) could also be llusraed by Fgure 9 of he sock prce ndex, he oal reurn ndex and one of he proxy he Gordon prce. 50

56 Fgure 9 - Co-negraon relaonshp beween he sock prce and he fundamenal Even hough he co-negraon relaonshp wh he Gordon prce s found boh n he sock prce ndex and he oal reurn ndex, can be apparenly seen from Fgure 9 ha he sock prce ndex s srongly conegraed wh he Gordon prce whle he oal reurn ndex show less clear evdence. The Z-sascs s ( ) for he sock prce ndex agans (-4.772) for he oal reurn ndex. The weaker he evdence of he co-negraon relaonshp, he less sgnfcan evdence of mean reverson of he sock prce s expeced. The followng secon wll gve emprcal suppors for hs argumen. 6.2 Esmaon resuls Based on he es resuls of he co-negraon relaonshp beween he oal reurn ndex and he proxes for he fundamenal value of he sock prce, only he proxes whch are co-negraed wh he oal reurn ndex are used for esmaons wh he Engle-Granger ECM. For he generalzed one-sep ECM all proxes are used. The resuls for he me seres daa of each counry as well as for he panel daa of 3 counres and 15 counres are presened below. 51

57 Table 16 - Esmaon resuls wh oal reurn ndex - Denmark GENERALIZED ONE-STEP ECM βo β 1 - β 1 β 2 β 2 Dvdends Foresgh prce *** Hndsgh prce Gordon prce 0.307** ** 0.100*** ENGLE-GRANGER TWO-STEP ECM βo β 1 β 2 (+) Dvdends 0.404*** *** Foresgh prce *** Hndsgh prce *** Gordon prce 0.278** ** 1.684*** (+) shows sgnfcance level of β 2 s dfference from 1. SIMPLE MODEL λ Dvdends Foresgh prce Hndsgh prce Gordon prce The resuls wh he Dansh daase only show evdence of mean reverson a 5% sgnfcance level for he regresson wh he Gordon prce usng he error-correcon model. The co-effcen s small (-0.054) and conssen beween he wo-sep and one-sep procedure. No mean reverson s found wh he smple model. Smlar resuls are found for he Swedsh daase. All esmaons wh he smple model show no evdence of mean reverson (Table 17). The ECM confrms he mean reverng process for only one proxy. However nsead of he Gordon prce, only he esmaons wh he foresgh prce show sgnfcan evdence of mean reverson. The value of he coeffcen or he speed of mean reverson s smaller han n he same regresson wh he sock prce ndex. The resuls are also conssen beween he one-sep and wo-sep procedure. 52

58 Table 17 - Esmaon resuls wh oal reurn ndex - Sweden GENERALIZED ONE-STEP ECM βo β 1 - β 1 β 2 β 2 Dvdends 0.641*** Foresgh prce *** *** 0.711*** Hndsgh prce Gordon prce 0.533*** ENGLE-GRANGER TWO-STEP ECM βo β 1 β 2 (+) Dvdends 0.644*** *** Foresgh prce *** *** 1.743*** Hndsgh prce *** Gordon prce 0.629*** *** (+) shows sgnfcance level of β 2 s dfference from 1. SIMPLE MODEL λ Dvdends Foresgh prce Hndsgh prce Gordon prce Lke he resuls wh he Swedsh daase, he Uned Saes only confrm he exsence of mean reverson of he marke sock prce when he foresgh prce and he foresgh earnngs are used as he proxes for he fundamenal value of he sock prce (Table 18). The regressons wh he dvdend, he hndsgh prce, ec. even show unexpeced sgn of he mean reverson co-effcen. Table 18 - Esmaon resuls wh oal reurn ndex - Uned Saes GENERALIZED ONE-STEP ECM βo β 1 - β 1 β 2 β 2 Dvdends ** ** Foresgh prce 1.097*** *** 0.31*** Foresgh earnngs ** 0.176** Hndsgh prce Hndsgh earnngs ** Gordon prce Gordon earnngs 0.204*** Smoohed earnngs Smoohed earnngs **

59 ENGLE-GRANGER TWO-STEP ECM βo β 1 β 2 (+) Dvdends 0.235* 0.072*** 5.183*** Foresgh prce 1.025*** *** Foresgh earnngs *** Hndsgh prce *** 4.837*** Hndsgh earnngs *** 3.461*** Gordon prce *** 2.689*** Gordon earnngs 0.235*** *** (+) shows sgnfcance level of β 2 s dfference from 1. SIMPLE MEAN REVERSION λ Dvdends Foresgh prce Foresgh earnngs Hndsgh prce Hndsgh earnngs Gordon prce Gordon earnngs Smoohed earnngs Smoohed earnngs In conras o he resuls of Denmark and Sweden wh he smple model, he US daase shows a rgh sgn of he mean reverson coeffcen. Noneheless, none of he regresson shows sgnfcan evdence of he mean reverng process of he sock prce. Analogous o he resuls wh me seres daase of ndvdual counres, he panel daase of 3 counres and he panel daase of 15 counres do no presen as srong evdence of mean reverson wh he oal reurn ndex as ha wh he sock prce ndex. Table 19 and Table 20 show he dealed resuls wh he panel daase of 3 counres and he panel daase of 15 counres respecvely. Table 19 - Esmaon resuls wh oal reurn ndex - Panel 3 counres GENERALIZED ONE-STEP ECM βo β1 -β1β2 β2 Half-lfe 95% C.I Dvdends 0.471*** Foresgh prce * 0.065** Hndsgh prce Gordon prce 0.441*** ** 0.056** (11.2, 112.5) 54

60 ENGLE-GRANGER TWO-STEP ECM βo β1 Half-lfe 95% C.I Foresgh prce ** 51.3 (28.4, 250.9) Gordon prce 0.436*** * SIMPLE MODEL λ Dvdends Foresgh prce Hndsgh prce Gordon prce The esmaons wh he oal reurn ndex for he panel daa of 3 counres yeld very dfferen resuls from he esmaons wh he sock prce ndex. Due o nsgnfcan evdence of mean reverson he esmaed half-lfe s very large rangng from 20.5 years o 63.8 years whle s from 2.5 years o 19 years for he sock prce ndex. Moreover, based on he correspondng 95% confdence nervals of he esmaed half-lfe of hese esmaons, he esmaes from he regressons wh he oal reurn ndex are more unceran compared o hose wh he sock prce ndex. Table 20 - Esmaon resuls wh oal reurn ndex - Panel 15 counres βo β 1 - β 1 β 2 β 2 Foresgh prce *** *** 0.252*** Gordon prce 0.746*** *** 0.095*** ENGLE-GRANGER TWO-STEP ECM βo β 1 Foresgh prce *** *** Gordon prce 0.745*** * SIMPLE MODEL λ Dvdends 0.119*** Foresgh prce 0.069* Hndsgh prce 0.06 Gordon prce 0.116*** The resuls wh he panel of 15 counres (Table 20) are approxmae o he same esmaons wh he sock prce ndex (Table 12). For nsance, he esmaed speed of mean reverson wh he foresgh prce 55

61 s equal o for he oal reurn ndex and for he sock prce ndex. Neverheless, lke he esmaons wh he oher daases, he absolue value of β 1 from he regressons wh he oal reurn ndex s smaller han ha wh he sock prce ndex. In sum, he esmaons wh he oal reurn ndex show less sgnfcan evdence of mean reverson n sock prces compared o hose wh he sock prce ndex. The reason s he sronger co-negraon relaonshp beween he esmaed fundamenal value of he sock prce and he sock prce ndex han he oal reurn ndex whch s clearly seen n Fgure 9. More neresngly, he longer perod he daase covers, he larger he dfference s observed beween he esmaons wh he sock prce ndex and he oal reurn ndex. For nsance, he esmaes obaned from he regressons wh he US daase he longes daase among ohers show he larges dfference beween one wh he sock prce ndex and one wh he oal reurn ndex whle he esmaons from he panel 15 counres daase whch covers he shores perod from 1970 o 2010 yeld smlar resuls for he wo ndces. Ths could be plausbly explaned by he dvdend payous whch s accumulaed n he oal reurn ndex over me bu s no accouned for n he sock prce ndex. Hence, s he accumulaed dvdend payou ha makes he gap beween he wo ndces become larger and larger and whereby leads o more dssmlar esmaon resuls derved from hese wo seres over me. 6.3 Tme-varyng mean reverson The me-varyng mean reverson of he oal reurn ndex s examned wh he panel daa of 3 counres and he resuls are compared wh he sock prce ndex. Because only he esmaons wh he foresgh prce and he Gordon prce applyng he ECM show sgnfcan evdence of mean reverson of he sock prce, hese wo regressons are used for rollng wndow esmaons wh he nervals of overlappng 27 years. Fgure 10 dsplays he rollng wndow esmaes of he speed of mean reverson over me obaned from he wo regressons. The wo graphs on he lef dsplay he rollng wndow esmaes wh he oal reurn ndex. And on he rgh are found he rollng wndow esmaes wh he sock prce ndex whch are also presened n secon 5.3. In general, he rsng or decreasng rend across me of he speed of he mean reverng process s smlar beween he sock prce ndex and he oal reurn ndex. However, he me of he hghes peak s oally dfferen. Whle he regressons wh he sock prce ndex show he larges value of he mean reverson speed a he frs me nerval of , he regressons wh he oal reurn ndex dsplay he hghes peak around he md-year Ths me perod ncludes some ensons of he Cold War (such as he Berln Crss of 1961 and he Cuba Crss of 1962) as well as he Ol Crss of 1973 whch 56

62 hen lead o he sock marke crash. Neverheless, hose hsorcal and economc evens are sad no o leave such severe affec o he sock markes as he Grea Depresson n he lae 1930s and he World War II ( ). Consequenly, he resuls from he esmaons wh he sock prce ndex are more relable and could be more plausbly explaned han he oal reurn ndex. Fgure 10 - Rollng wndow esmaes - Sock prce ndex and Toal reurn ndex comparson Furhermore, he me-varyng propery of he sock prce ndex s more conssen among dfferen esmaons wh dfferen proxes han he oal reurn ndex. I s obvously seen from Fgure 10 ha he hghes peak of he rollng esmaes wh he foresgh prce s found before he md-year 1970 whle afer he md-year 1970 for he rollng esmaes wh he Gordon prce. Ths nconssence agan confrms ha he sock prce ndex gves more relable evdence of he mean reverng behavour of he marke sock prce han he oal reurn ndex. 57