A three regime model of speculative behaviour: modelling the evolution of the S&P 500 composite index

Transcription

1 A hree regime model of speculaive behaviour: modelling he evoluion of he S&P 500 composie index Aricle Acceped Version Brooks, C. and Kasaris, A. (2005) A hree regime model of speculaive behaviour: modelling he evoluion of he S&P 500 composie index. The Economic Journal, 115 (505). pp ISSN doi: hps://doi.org/ /j x Available a hp://cenaur.reading.ac.uk/20554/ I is advisable o refer o he publisher s version if you inend o cie from he work. Published version a: hp://dx.doi.org/ /j x To link o his aricle DOI: hp://dx.doi.org/ /j x Publisher: Wiley All oupus in CenAUR are proeced by Inellecual Propery Righs law, including copyrigh law. Copyrigh and IPR is reained by he creaors or oher copyrigh holders. Terms and condiions for use of his maerial are defined in he End User Agreemen.

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4 A Three-Regime Model of Speculaive Behaviour: Modelling he Evoluion of he S&P 500 Composie Index May 2004 Chris Brooks and Aposolos Kasaris Absrac In his paper we examine wheher a hree-regime model ha allows for dorman, explosive and collapsing speculaive behaviour can explain he dynamics of he S&P 500 Composie Index for he period We exend exising wo-regime models of speculaive behaviour by including a hird regime ha allows for a bubble o grow a a seady growh rae, and examine wheher oher variables, beyond he deviaion of acual prices from fundamenal values, can help predic he level and he generaing sae of reurns. We propose abnormal volume as an indicaor of he probable ime of he bubble collapse and hus include abnormal volume in he sae and he classifying equaions of he surviving regime in he explosive sae. We show ha abnormal volume is a significan predicor and classifier of reurns. Furhermore, we find ha he spread of he 6-monh average of acual reurns above he 6-monh average of fundamenal reurns can help predic when a bubble will ener he explosive sae. Finally, we examine he financial usefulness of he hree-regime model by sudying he Sharpe raios of profis of a rading rule formed using inferences from i. Use of he hree-regime model rading rule leads o higher Sharpe raios and end of period wealh han hose obained from employing exising models or a buy and hold sraegy. Keywords: speculaive behaviour models, fundamenal values, dividends, regime swiching, speculaive bubble ess, rading rules. JEL Classificaions: C51, C53, G12 The auhors are boh of he ISMA Cenre, Universiy of Reading. Conac: Chris Brooks, ISMA Cenre, Universiy of Reading, PO Box 242, Whieknighs, Reading, RG6 6BA, UK. Tel: +44 (0) ; Fax: +44 (0) ; E- mail: C.Brooks@rdg.ac.uk We are graeful o hree anonymous referees for exensive commens on a previous version of his paper. We are also graeful o Phillip Xu, Ryan Davies, David Oakes, Menelaos Karanasos, Karim Abadir, Aposolos Filippopoulos, Saranis Kalyviis and seminar paricipans a he Deparmen of Economics and Relaed Sudies a he Universiy of York, and a he Deparmen of Inernaional and European Economic Sudies a he Ahens Universiy of Economics and Business for useful commens and suggesions. We are responsible for any remaining errors. 2

5 1. Inroducion The evoluion of prices in equiy markes during he 1920 s, he 1980 s and he lae 1990 s, has puzzled economiss and marke paricipans. During hese periods, marke prices displayed significan growh followed by abrup marke collapses. I is hard o reconcile his behaviour wih he variaion of fundamenals during hese periods and hus wo alernaive heories have been developed ha ry o explain why sock markes appeared o be moving wihou fundamenal jusificaion. The firs approach aribued hese abrup changes in he marke rend o a non-linear relaionship beween acual prices and fundamenal values. The second approach suppors he view ha self-fulfilling expecaions and speculaive bubbles caused he significan and increasing divergence of acual prices from fundamenal values. Iniially, sudies on he relaionship beween acual prices and fundamenal values focused on he indirec idenificaion of speculaive bubbles in financial daa (see Shiller (1981), Blanchard and Wason (1982), Wes (1987), Diba and Grossman (1988)). However, indirec ess of bubble presence suffered from poenial problems of inerpreaion since bubble effecs in sock prices could no be disinguished from he effecs of unobservable marke fundamenals. For his reason, direc bubble ess, which es direcly for he presence of a paricular bubble specificaion on sock marke reurns, were developed (see Flood and Garber (1980), Flood, Garber and Sco (1984), Summers (1986), Culer, Poerba and Summers (1991), McQueen and Thorley (1994), Salge (1997), van Norden and Schaller (1999)). Under hese ess, researchers selec he ype of bubble ha hey suspec migh be presen in he daa and hen examine wheher his form of speculaive bubble has any explanaory power for sock marke reurns. Alhough several differen ypes of bubbles have been examined in he lieraure, periodically collapsing speculaive bubbles, firs proposed by Blanchard and Wason (1982), have araced increasing aenion, especially in he lae 1990 s. Models of periodically collapsing speculaive bubbles are able o capure several sylised characerisics of hisorical accouns of manias and panics (Kindleberger (1989)). The common characerisic of many such periods is ha prices iniially diverge from fundamenal values in a sysemaic and increasing fashion. As ime passes, he rae of divergence acceleraes and hus prices increase wihou bound and his exponenial rend is followed by a sharp reversal of marke prices o fundamenal values when marke paricipans realise ha he curren price level is unsusainable. In order o empirically examine he presence of periodically collapsing bubbles, researchers in recen years have focused on he consrucion of direc bubble ess ha can idenify such sochasic bubble processes in financial and macroeconomic daa. More specifically, Evans (1991) and van Norden and Schaller (1993) show how periodically collapsing speculaive bubbles can induce regime swiching behaviour in asse reurns. Regime swiching in asse reurns and speculaive behaviour have been linked 1

6 in several sudies. Van Norden and Schaller (1993) and van Norden (1996) show ha a wo-regime speculaive behaviour model has significan explanaory power for sock marke and foreign exchange reurns during several periods of observed marke over- and under-valuaions. Hall, Psaradakis and Sola (1998) es for he presence of collapsing speculaive bubbles in Argeninean moneary daa using a univariae Markov-swiching ADF es and find evidence of a speculaive bubble in consumer prices in he period June 1986 o Augus van Norden and Vigfusson (1998) compare he van Norden and Schaller (1997) approach wih he Hall, Psaradakis and Sola (1998) swiching saionariy es for he presence of speculaive bubbles and find ha boh models have significan power in deecing periodically collapsing speculaive bubbles. However, boh approaches concenrae on he explosive phase of he speculaive bubble since i is during his phase ha asse prices display significan paerns of bubble behaviour. For his reason, boh models are consruced o idenify periods of explosive sock marke growh followed by a sharp reversal. This specificaion of he bubble ess implicily assumes ha a speculaive bubble will always display explosive growh, alhough his will be limied for small bubble sizes. This is an unrealisic assumpion since here are periods during which asse prices display consan growh or mimic he behaviour of fundamenals. During such periods he speculaive bubble can be assumed o be dorman since i grows a a seady rae. A speculaive bubble process ha can replicae his swich from a dorman o an explosive sae is described in Evans (1991). However, Evans uses his bubble process only in a simulaion sudy and does no provide empirical evidence concerning he presence of such speculaive bubbles in asse prices. In wha follows, we ry o fill his gap by showing how he van Norden and Schaller (1999) model can be exended o allow for asse price behaviour of he form described in Evans. We do his by incorporaing a hird regime in which he bubble grows a he fundamenal rae of reurn. We hen examine wheher he hree-regime speculaive behaviour model has explanaory power for U.S. sock marke reurns. Addiionally, we show ha oher variables, such as abnormally high volume, can be used in he van Norden and Schaller framework in order o model he probabiliy of a bubble collapse more effecively 1. Furhermore, exan research has focused only on he issue of idenifying he presence of speculaive bubbles. Alhough he idenificaion of a speculaive bubble is useful in deermining wheher marke paricipans are raional and wheher markes are efficien, i is also ineresing o examine he financial usefulness of speculaive bubble models by esing wheher such models can be used o generae abnormal rading profis or reurn profiles ha ypical invesors would consider more desirable han hose of a buyand-hold sraegy. We do his by formulaing a rading rule ha explois informaion abou he implied probabiliy of a sock marke crash or rally derived from swiching regime speculaive behaviour models. 1 Brooks and Kasaris (2002) have examined he usefulness of volume as a classifier and predicor of excess reurns in he conex of he van Norden and Schaller (1999) 2-regime model. 2

7 This allows us o evaluae he predicive abiliy of our hree-regime model agains he van Norden and Schaller model in a financially inuiive manner: by deermining which model can lead o higher Sharpe raios. Very lile exising research has been able o pinpoin major marke downurns, alhough one excepion is Maheu and McCurdy (2000), who use a Markov swiching model o idenify bull and bear runs in sock markes, alhough hey do no examine he model s financial usefulness. We also examine he marke iming abiliy of speculaive behaviour models by comparing he resuls of he rading rule o hose of a buy and hold sraegy and hose of randomly generaed rading rules. Finally, we employ a longer sample han in he original van Norden and Schaller sudy, examining he reurns on he S&P 500 for he period January 1888 o January The res of his paper is organised as follows. In secion 2, we derive he hree-regime speculaive behaviour model. Secion 3 presens he daa and he mehodology used o consruc fundamenal values. Secion 4 presens he resuls of he speculaive behaviour models while in secion 5 we examine he ou of sample forecasing abiliy and profiabiliy of he speculaive behaviour rading rules. Secion 6 concludes. 2. A Three-Regime Speculaive Behaviour Model An exensive heoreical lieraure exiss concerning he developmen of raional speculaive bubbles (see, for example, Blanchard (1979), Blanchard and Wason (1982), Wes (1988), Diba and Grossman (1988), and Kindelberger (1989)). In he Blanchard and Wason (1982) model, he speculaive bubble would burs o zero in he collapsing sae, and herefore i could no regenerae. The possibiliy of negaive bubbles was also explicily ruled ou. An imporan recen innovaion was he model proposed by van Norden and Schaller (1999), where he size and probabiliy of bubble collapse are dependen on bubble size and where boh posiive and negaive bubbles are permied, and where parial bubble collapses are permissible (he laer are also considered by Evans (1991) and Hall and Sola (1993)). van Norden and Schaller esimae he model using daa on he value-weighed index of all socks from he Cenre for Research on Securiy Prices (CRSP) daabase for he period January 1926 o December Their resuls show ha here is non-linear predicabiliy in sock marke reurns and ha he deviaions of acual prices from fundamenal values are a significan facor in predicing boh he level and he generaing sae of reurns. They find ha he model has explanaory power for several periods of apparenly speculaive behaviour of he daa, since he probabiliy of a crash increases significanly prior o large sock marke declines while he probabiliy of a rally is high prior o large sock marke advances. Alhough he van Norden and Schaller approach can be used direcly o es for he presence of periodically parially collapsing speculaive bubbles, i has several limiaions. Firsly, van Norden and Schaller do no provide any informaion on he ou of sample forecasing abiliy of heir model. Furhermore, heir model provides limied informaion as o he likely ime of a bubble collapse, since he probabiliy of a collapse is solely dependen on he bubble size. This leads o long periods of high 3

8 probabiliies of collapse, when he bubble deviaion is persisenly high. Moreover, alhough he power of he speculaive behaviour model o deec bubbles of he form described by Evans (1991) has been examined by van Norden and Vigfusson (1998), he financial usefulness of his model has no, o our knowledge, been examined. I would hus be ineresing o deermine wheher his model can be used in order o ime large marke falls. Finally, he van Norden and Schaller model examines only he explosive sae of a speculaive bubble since i is during his sae ha asse prices display apparen bubble behaviour. This implies ha heir model assumes ha he bubble will always induce explosive behaviour in asse prices; he asse price will eiher grow wih explosive expecaions or will reverse o fundamenal values. However, i is possible (and casual observaion of bubbly series provides suppor for he idea) ha speculaive bubbles in sock markes may alernae beween dorman and explosive saes 2. During he dorman sae, he bubble grows a he required rae of reurn wihou explosive expecaions since he probabiliy of a crash is zero or negligible, as in he Evans (1991) model. In such periods, he acual reurns on he asse are equal o he reurns on he fundamenals, plus a random disurbance and hus acual prices mimic he behaviour of fundamenal values. This bubble behaviour is more consisen wih a hreeregime speculaive model. This swiching beween a dorman bubble sae and an explosive bubble sae can be observed in he early 1920 s, he early 1950 s, he 1960 s and he early 1990 s amongs oher periods, when bubbles alernae beween growing a a small seady growh rae and an increasingly explosive growh rae. However, as seen from he resuls of van Norden and Schaller, heir model will always assign a small bu posiive probabiliy of he bubble collapsing, and his probabiliy will affec he expeced reurns on he asse. This will cause a posiive bias in he esimaes of he probabiliy of collapse a every poin in ime and especially during periods when he bubble displays consan growh. I is herefore preferable o consruc a speculaive behaviour model ha explicily allows for dorman periods as well as for explosive bubble growh. We will now show how he van Norden and Schaller model can be exended o a hree-regime speculaive behaviour model. The laer describes a process in which he expeced size of he bubble in he nex ime period can be generaed from one of hree regimes: a deerminisic (or dorman ) regime (D), a survivingexplosive regime (S), and a collapsing-explosive regime (C). I is worh saing a he ouse ha our model, whils i is a regime-swiching model, i is no a Hamilon-syle Markov-swiching model, since he probabiliy of being in a paricular regime a ime +1 does no depend direcly on he sae a ime. The bubble of he nex ime period can be generaed from any of he hree regimes. In order o classify he behaviour ino one of he hree regimes, several variables may be significan, alhough he relaive size of he bubble is expeced o play a predominan role. The model will now be defined. 2 Bohl (2000) suggess he use of a hreshold auoregressive model in a coinegraion framework o separae periods of deerminisic from explosive bubble growh. 4

9 Following Evans (1991), in regime D, he bubble size is small and hus invesors believe ha he bubble will coninue o grow a a consan mean rae (1+i): E ) ( b 1W 1 D) (1 i b (1) where b is he size of he bubble (he difference beween he acual and fundamenal price) a ime, i is a consan discoun rae, and W is an unobserved indicaor ha deermines he sae in which he process is a ime. This regime implies ha invesors believe ha he bubble has a negligible probabiliy of collapse and hus hey do no expec o be rewarded for his probabiliy wih an excess reurn. Evans (1991) assumes ha once a bubble crosses a cerain hreshold, i erups ino an explosive regime in which he bubble coninues o grow or collapses o a smaller value. Evans (1991) arbirarily assumes a hreshold value, whereas we model he probabiliy of being in regime D. The probabiliy of being in regime D in ime +1 is denoed n, and is dependen on he bubble s relaive size and on oher variables observed a ime 3 ; his probabiliy will be defined precisely laer. Even when he bubble is in he dorman regime, invesors know ha here is a probabiliy ha he bubble migh ener an explosive sae in which he bubble coninues o grow wih explosive expecaions or collapses o a smaller value. The probabiliy of being in his explosive sae is sae, here are wo underlying regimes: he surviving regime ha occurs wih probabiliy collapsing regime ha occurs wih probabiliy is given by: E ) 1 n. In his explosive q and he 1 q. In he collapsing regime (C), he size of he bubble a ( b 1W 1 C) g( B p (2) ( B where g ) is a coninuous and everywhere differeniable funcion such ha, g ( 0) 0 and 0 g( B ) B 1 4, B is he relaive size of he bubble in period ( B b p a ), and a p is he acual asse price a ime. This funcion is only for heoreical use and will no be imposed on he daa. From (1) and (2) and since: E ( b W D) (1 n ) q E ( b W S) (1 q ) E ( b W ) E ( b ) n (3) we can show ha he expeced size of he bubble in he surviving regime will be: 1 C (1 i) (1 q ) E ) a ( b 1W 1 S) b g( B p (4) q q 3 Noe ha he n is used o denoe he probabiliy of being in regime D raher han d, since he laer conflics wih he sandard noaion ha we adop laer for dividend paymens. 4 a In he original van Norden and Schaller model (1993), he bubble size in he collapsing regime is u ( B ) p where 0 u ( B ) B 1. This implies ha in he collapsing regime, he original van Norden and Schaller model saes ha he bubble in period +1 is expeced o be equal o or smaller han he bubble in period. We use a slighly differen noaion and a assume ha he bubble size is g ( B ) p where 0 g( B ) B (1 i). This implies ha he bubble in he collapsing regime is smaller han he bubble in he dorman sae and herefore, when he bubble does no yield he required rae of reurn, i is considered o be in he collapsing sae. 5

10 A any poin in ime, he condiional probabiliy of observing he surviving regime in he nex ime period is given by (1- n ) q and he probabiliy of observing he collapsing regime is (1- n )(1- q ). Grouping ogeher (1), (2) and (4), he bubble of he nex ime period will be generaed by he following sochasic process: b 1 Mb M b q ( 1 q ) wih probabiliy n a g( B ) p q wih probabiliy (1- a ( B p wih probabiliy (1- g ) where M = (1+i) is he gross fundamenal reurn on he securiy. n ) q (5) n )(1- q ) A his poin, i is useful o noe hree imporan differences beween his model and he Evans (1991) daa generaing process. Firs, we allow for he exisence of negaive (price decreasing) bubbles, and second, in Evans, a sricly posiive bubble mus cross he arbirary hreshold in order o ener he explosive regime. Finally, he assumes ha he probabiliy of collapse in sae D is zero, whereas we do no acually impose his on he daa. In (5), we assume ha when he bubble is in he dorman regime, he bubble follows a deerminisic process. This is because he probabiliy of observing a collapse when he bubble is in regime D is so small ha invesors decide o ignore i Noe ha our model produces a very small probabiliy of being in regime C in he nex ime period ((1- n )(1- q )) if he probabiliy of being in regime D in he nex ime period is very high. We can show ha he expeced gross oal reurn of he nex period, r +1, if he bubble is generaed by he dorman regime (D), is 5 : E( r 1 W 1 D) M (6) Equaion (6) saes ha he expeced gross reurn on he asse, if he bubble is generaed by he dorman regime in he nex ime period, is equal o he required rae of reurn on he bubble-free asse. Gross reurns are calculaed as r +1 = (P a +1 + d +1 ) / P a and hey herefore include dividend paymens. However, as he bubble grows, he probabiliy of being in he seady growh regime diminishes and hus he probabiliy of being in he explosive sae (surviving or collapsing regime) increases. In his explosive sae, as he bubble increases, he probabiliy of being in he surviving sae diminishes, causing he probabiliy of collapse o increase geomerically. In he explosive sae, invesors perceive ha he bubble can now collapse and hus ake ino accoun he probabiliy of a crash. The expeced reurn in he surviving regime is: 5 Proofs of he equaions are no presened here in he ineres of breviy bu are available in an appendix from he auhors upon reques. 6

11 E (1 q ) r 1 W S) M MB g( B ) (7) q ( 1 In equaion (7), invesors adjus heir expecaions for he nex period reurn o ake ino accoun he probabiliy of collapse. If he bubble collapses, he gross expeced reurn is given by: E ( r 1 W 1 C) M g( B ) MB (8) In equaion (8), he gross reurn in he collapsing regime is a funcion of he required reurn on he bubbly asse and he relaive size of he bubble in he collapsing regime. The magniude of he reurn depends on he funcion g B ), a funcion ha does no require specificaion since he model will subsequenly be linearised. ( I is sraighforward o see ha he above bubble model collapses o he original van Norden and Schaller model if we se he probabiliy of being in he dorman regime equal o zero. Furhermore, if we fix he probabiliy of survival o a consan value and se he bubble size in he collapsing regime equal o zero hen he model collapses o he original Blanchard and Wason (1982) model. Finally, he probabiliyweighed expeced reurn on he asse a any poin in ime is M, and his is also he ex ane bubble growh rae. This comprises he expeced reurn in sae D, which is M, and a reurn higher han M in sae S and lower han M in sae C. Nohing has ye been said abou he probabiliies n and q apar from ha hey are negaive funcions of he size of he bubble. Le us examine hese probabiliies in more deail. From he original Evans (1991) model, as he bubble grows, he probabiliy of being in he dorman regime (D) decreases. Here we consider boh posiive and negaive bubbles and herefore we formulae he probabiliy n as a funcion of he absolue size of he bubble. However, he size of he bubble may no be he only variable ha invesors examine in order o decide wheher he bubble is in he dorman or he explosive sae. In order o model he probabiliy of being in he dorman regime in he nex ime period, we base our inuiion on he resuls of Harvey and Siddique (2000) and Chen, Hong and Sein (2001), who find ha when reurns have been high in he recen pas, he skewness in fuure reurns is more negaive. This implies ha high average reurns will end o be followed by large negaive reurns. From equaion (6), he expeced bubble reurns in he dorman regime are indisinguishable from he expeced fundamenal reurns. However, when he bubble eners he explosive sae and survives, we know from (7) ha i yields increasingly larger reurns han he bubble-free asse. For his reason, i is plausible o assume ha when invesors observe larger average acual reurns han average fundamenal reurns in he near pas hey will conclude ha he bubble mus have enered he explosive sae. The probabiliy of being in he dorman regime falls as he bubble grows, so ha ime spen in he surviving regime will reduce he probabiliy of being in regime D, ceeris paribus, since i ends o increase he size of he bubble. This means ha large 7

12 posiive reurns imply a smaller probabiliy of being in he dorman regime in he near fuure. Chen, Hong and Sein (2001) find ha he predicive power of pas reurns for skewness is larger if one considers he las six monhs reurns. They consider acual reurns, however, and we wan o parially separae bubble reurns from fundamenal reurns. For his reason, we include he spread of he average 6-monh acual gross reurns over he average 6-monh fundamenal gross reurns as a variable in he probabiliy of being in he dorman regime. I should be expeced ha he larger he value of he spread, he lower he probabiliy of he bubble coninuing o be in he dorman regime. Since we examine boh posiive and negaive bubbles, we ake he absolue values of he averages before we calculae he spread. This ensures ha when a negaive bubble is enering he explosive sae, we correcly idenify he explosive growh of such a bubble. Furhermore, his variable will help o idenify periods of seady bubble growh when he esimaed bubble deviaion is arbirarily large. This poin will be elucidaed shorly. In order o ensure ha esimaes of he probabiliy of he dorman sae are bounded beween zero and one, we follow van Norden and Schaller in using a probi specificaion. Under his seing, he probabiliy of being in he dorman regime in he nex ime period is given by: f, a Pr( W 1 D) n ( n,0 n, B B n, SS ) (9) where is he sandard normal cumulaive densiy funcion and f a S, is he absolue value of he average 6-monh acual reurns minus he absolue value of he average 6-monh reurns of he esimaed fundamenal values. Turning now o he probabiliy of he bubble surviving in he explosive sae ( q ), we saed in he previous secion ha in he van Norden and Schaller model his probabiliy is a funcion of only he absolue size of he bubble. As in mos of he speculaive bubble models, heir approach assumes ha a bubble collapse is a random even ha may or may no be fuelled by he arrival of news. In effec, mos raional speculaive bubble models implicily assume ha invesors randomly organise and decide o sell a he same ime hus causing he bubble o collapse. However, alhough invesors can esimae he probabiliy of a bubble collapse from he size of he bubble, hey sill face uncerainy abou he ime of he collapse. We herefore conjecure ha invesors observe oher, non-price, variables in an effor o draw inferences abou he probable ime of collapse, and hus o idenify he opimal ime o exi from he marke. Our suggesion is ha bubble collapses occur because invesors observe a signal ha leads hem o he conclusion ha he marke is no longer expecing he bubble o coninue o exis. Once hey observe his signal, hey sar o liquidae heir holdings and hus cause he bubble o collapse. We consider abnormal rading volume as a possible signal and hus as a predicor of he ime of he bubble collapse. The relaionship beween rading volume and sock reurns has been exensively researched in he lieraure (see Karpoff (1987) for a meiculous survey of he lieraure unil 1987). Ying (1966) and Morgan (1976) find ha large increases in volume are usually followed by increased variance of reurns, a 8

13 resul ha leads Morgan (1976) o conclude ha volume is associaed wih sysemaic risk. One possible explanaion for his finding is ha rading volume is a proxy for he degree of disagreemen in he sock marke 6. Alhough Karpoff (1986) claims ha abnormal volume is no proof of invesor disagreemen (eiher ex ane or ex pos) abou informaion ha is available, Shalen (1993) argues ha volume and reurn volailiy have a posiive relaionship wih he ex ane dispersion of expecaions abou fuure prices. Furhermore, He and Wang (1995) claim a high degree of uncerainy abou fundamenal values leads o an increase in observed rading volume. Moreover, Marsh and Wagner (2000) sae ha, especially for he U.S. sock marke, abnormal volume can help explain increases in he size of boh he negaive and he posiive ails of he reurn disribuion. Alhough hey find ha his effec is symmeric, Hong and Sein (1999) and Chen, Hong and Sein (2001) claim ha divergence of informaion, approximaed by abnormal rading volume, only causes an increase in he negaive ail of he fuure reurn disribuion. Based on he above, we propose ha abnormally high rading volume is a signal of changing marke expecaions abou he fuure of a speculaive bubble. This implies ha abnormal volume signals an increase in he probabiliy of he bubble collapsing, hus causing an increase in he probabiliy of observing a large negaive (posiive) reurn if a posiive (negaive) speculaive bubble is presen. The main difference beween our model and he models of price volume relaionship described above is ha he laer sae ha volume increases boh ails of he disribuion of expeced reurns or i signifies a decrease in he skewness of fuure reurns. We consider abnormal volume as a sign ha oher invesors are selling he bubbly asse 7. For abnormal volume o signal an imminen bubble collapse would require he assumpion ha invesors have differen endowmens implying ha here are agens in he economy ha do no hold equiy and may decide o do so a a fuure dae. The effec would be he same if he number of invesors changes over ime. Furhermore, we assume ha some invesors face shor selling consrains and ha in he shor run, he supply of equiy from firms, hrough IPOs and SEOs is limied. The laer assumpion combined wih he unwillingness of invesors o sell he bubbly asse because of he expecaion of high reurns in he surviving regime, cause he supply curve o be relaively inelasic. Under his seing, speculaive bubbles are a form of demand side inflaion in sock prices. As he bubble coninues o grow, he probabiliy of a crash increases and hus some invesors will decide o liquidae heir holdings for profi aking or because hey perceive a crash o be imminen. If a sufficienly large number of invesors decide o sell he bubbly asse, supply will increase significanly and hus volume will be abnormally high while he rae of increase in prices will slow. This abnormally high volume will signal ha a bubble collapse is imminen. Under his seing, we model he probabiliy of he 6 The reader is referred o Harris and Raviv (1993), Kandel and Pearson (1995) and Odean (1998) for oher models wih he same feaure. 7 Abnormally high volume could also be considered a sign ha invesors are buying he bubbly asse. 9

14 bubble coninuing o be in he surviving regime (S) as a negaive funcion of boh he absolue relaive size of he bubble deviaion, and a measure of abnormal volume: x Pr( W 1 S) q ( q,0 q, B B q, BV ) (10) where x V is a measure of unusual volume in period. Grouping all he equaions ogeher yields he following non-linear swiching model of gross sock marke reurns 8 : E( r 1 W 1 D) M (11.1) (1 q ) E ( r 1 W 1 S) M MB g( B ) q (11.2) ( r 1 W 1 C) M g( B ) MB (11.3) W D) n W S) (1 n ) q (11.5) E Pr( 1 (11.4) Pr( 1 Pr( W 1 C) (1 n )(1 q ) (11.6) The model described by equaions (11.1) hrough (11.6) is highly non-linear, so in order o esimae i, we linearise i by aking he firs order Taylor series expansion of equaions (11.1), (11.2) and (11.3) around x an arbirary B 0 and V 0. The resuling linear swiching regression model of gross reurns is: D D r 1 D,0 u 1 (12.1) S x S r 1 S,0 S, BB S, VV u 1 (12.2) C C r 1 C,0 C, BB u 1 (12.3) f, a n,0 n, B B n, SS f, a x n,0 n, B B n, SS q,0 q, B B q, VV f, a x 1 B S B V Pr( W 1 D) (12.4) Pr( W 1 S) 1 (12.5) Pr( W 1 C) n,0 n, B n, S 1 q,0 q, B q, V (12.6) We esimae he augmened model under he assumpion of disurbance normaliy using maximum likelihood. The likelihood funcion of each observaion is: S C D S u C D u 1 u n q n q n (13) 1 ( r 1 ) 1 s c D 8 A derivaion of hese equaions is available from he auhors upon reques. The reader may also find i useful o consul o he working paper by van Norden and Schaller (1997) for a derivaion of heir model, which is available a he Bank of Canada web sie: hp:// 10

15 where r 1 S,0 S, B B S, V ( S S S u ) 1, C C C u ) 1 S C V X r 1 C,0 C, BB (, and r 1 D,0 D D D ( u ) 1 denoe he probabiliy densiy funcions of an observaion condiional on i D being generaed by a given regime. is he sandard normal probabiliy densiy funcion (pdf), D, S, C are he sandard deviaions of he disurbances in he dorman, surviving and collapsing regimes respecively. In order o esimae he model, we direcly maximise he above log-likelihood funcion using a consrained opimisaion algorihm. The only consrain imposed is ha he sandard deviaions of he residuals are posiive, hus bounding he log likelihood funcion away from infiniy 9. From he Taylor series expansion, several esable implicaions can be derived concerning he sign and / or size of he coefficiens of he sae equaions. These condiions should be saisfied if he hree-regime model of speculaive bubbles has explanaory power for gross reurns, and are: C, B 0 (i) (ii) S, B C, B S, V 0 (iii) q, B 0 (iv) q, V 0 (v) n, B 0 (vi) n, S 0 (vii) Resricion (i) saes ha as he bubble increases in size, he expeced reurns in he collapsing regime should decrease (increase) if a posiive (negaive) bubble is presen, since he bubble mus collapse in regime (C). Furhermore, he speculaive behaviour model implies ha resricion (ii) mus hold since as he bubble increases in size we expec he difference of expeced reurns across he surviving and he collapsing regimes o increase as well. Resricion (iii) saes ha he expeced reurns in he surviving regime mus increase if abnormal volume is observed since i signals an increase in he probabiliy of a bubble collapse. Resricions (iv) and (v) should hold if our speculaive behaviour model is valid since he probabiliy of he bubble surviving mus decrease when he absolue size of he bubble or abnormal volume in he marke increase. The same holds for resricions (vi) and (vii) because he probabiliy of being in he dorman regime (D) mus decrease as he bubble grows larger in absolue size or he reurns of he marke in he las 6 monhs are larger han he reurns on he fundamenal values. 9 Noe ha he log-likelihood funcion is unbounded if he sandard deviaions of he error erms in he hree regimes become 11

16 In order o es he power of he model o capure bubble effecs in he reurns of he S&P 500, we follow van Norden and Schaller (1999) and es he hree-regime speculaive bubble model agains wo alernaive models ha are nesed wihin he model of speculaive behaviour. These alernaive models capure sylised facs of sock marke reurns and herefore we examine wheher our model has any explanaory power beyond hese simpler specificaions. Firsly, we examine wheher he effecs capured by he swiching model can be explained by a more parsimonious model of volailiy regimes. In order o es his, we examine he following condiions: (14.1) D, 0 S,0 C,0 0 (14.2) S, B C, B S, V q, B q, V n, B n, S (14.3) D S C Resricion (14.1) implies ha he inerceps across he hree sae equaions are he same while resricion (14.2) saes ha he bubble deviaions, he measure of abnormal volume and he measure of he spread of acual reurns above he fundamenal reurns have no explanaory power for he reurns of period +1 or for he probabiliy of swiching regimes. The laer poin suggess ha here is a consan probabiliy of swiching beween a low, medium and high variance regime as his is saed in resricion (14.3). The volailiy regimes model examines he join hypohesis ha he inerceps are he same across he hree regimes and ha reurns and he generaing sae of reurns are unpredicable if we use he variables under consideraion. I is ineresing o separae he wo hypoheses and o examine wheher reurns can be characerised by a simple mixure of normal disribuions model ha allows boh reurns and variances o differ across he hree regimes. This mixure of normals model implies he following resricions: 0 (15) S, B C, B S, V q, B q, V n, B n, S Finally, we augmen he original vns model ino a hree-regime model of speculaive behaviour ha does no include he measure of abnormal volume and he spread of reurns in he sae or ransiion equaions (hereafer referred o as he AvNS model). If hese volume and spread variables have explanaory power for he nex period s reurns, he es should rejec his simpler specificaion. The resricions of his las es are: 0 (16) S, V q, V n, S 3. DATA AND FUNDAMENTAL VALUES The daa we use o es for he presence of speculaive bubbles are 1381 monhly observaions on he S&P 500 for he period January 1888 January In order o calculae fundamenal values and real gross reurns, we use daa aken from Shiller (2000) 10. The measure of monhly abnormal volume is calculaed zero. 10 Daa available a: hp:// For a descripion of he daa used see also Shiller (2000) and he descripion online. Shiller s sample ends in January 2000, bu we updae his sample unil January 2003 using daa obained 12

17 as he monhly average of daily share volume, repored by he NYSE 11, and hen he percenage deviaion of las monh s volume from he 6 monh moving average is aken 12. This moving average is consruced using only lagged volume figures ha would have been included in agens informaion ses. The monhly dividend and price series are ransformed ino real variables using he monhly U.S. Consumer Price - All Iems Seasonally Adjused Index repored by Shiller (2000). Finally, as noed earlier, he spread beween acual and fundamenal reurns ( f a S, ) is calculaed as S r r f, a a,6 f,6, where a,6 r is he average 6- monh acual reurns and f,6 r is he average 6-monh reurns of he esimaed fundamenal values. In order o consruc fundamenal values, we follow van Norden and Schaller (1999) and use a mahemaical manipulaion of Campbell and Shiller s (1987) VAR model of he dividend componen of prices. The Campbell and Shiller model allows for predicable variaion in he dividend growh rae, alhough i assumes consan discoun raes. 13 The Campbell and Shiller model saes ha he spread beween sock prices and a consan muliple of curren dividends is he opimal forecas of a muliple of he discouned value of all fuure dividend changes: f 1 i 1 i 1 s ( (1 ) ) p d E g d g (17) i i g 1 i Using he VAR mehodology creaed by Campbell and Shiller, we examine wheher changes in dividends can be forecased by he spread beween prices and he muliple of curren dividends. If he changes in dividends canno be forecased by he spread, his would imply ha invesors use only pas dividends o form expecaions abou fuure dividends. If, on he oher hand, invesors include oher variables in heir informaion se hen his informaion will be refleced in pas prices and hus pas realisaions of he spread. This would imply ha he spread has power o forecas fuure dividend changes. Under his approach, he measure of relaive bubble size is hus given by: 1 i s d 1 i B 1 (18) p Figure 1 presens he bubble deviaions calculaed from equaion (18) for he enire sample (afer adjusing for he number of lags). Noe ha he bubble deviaions are increasingly large in 1929, 1987 and he lae 1990 s. The bubble deviaions are significanly negaive in 1917, 1932, 1938, 1942 and Overall, he Campbell and Shiller measure of bubble deviaions displays significan shor-erm variabiliy bu also large and persisen broad swings. from Daasream. In order o verify ha he wo daases are consisen, we compare Shiller s daa from January 1965 o January 2000 wih he values from Daasream and find no differences. 11 Daa available a: hp:// 12 We also examined unusual rading volume measures using 3, 12 and 18 monh moving averages bu found ha he deviaion from he 6-monh moving average has he highes explanaory power in predicing boh he level and he generaing sae of reurns. The resuls for he oher measures of abnormal volume are no presened for breviy and are available upon reques from he auhors. 13 We also examined a simple dividend muliple measure of fundamenals and found ha our resuls are qualiaively unchanged. The resuls are no presened here for breviy bu can be provided in he form of an appendix from he auhors upon reques. 13

18 4. RESULTS The resuls of he hree-regime model of speculaive behaviour are presened in he firs par of Table 1 alongside he resuls of he vns model, and he AvNS model for comparison. The AvNS model is a 3- regime exension of he van Norden and Schaller model ha is equivalen o our hree-regime model bu does no conain any addiional variables oher han he bubble deviaions in he sae and ransiion equaions. The second panel of he able conains he resuls of he likelihood raio (LR) ess of he resricions on he coefficiens implied by he speculaive behaviour model while he hird panel of Table 1 presens he resuls of he likelihood raio ess of he volailiy regimes, mixure of normals, and he exended van Norden and Schaller alernaive models. From he firs par of Table 1, all he coefficiens of he hree-regime model have he expeced sign and a financially meaningful magniude. More specifically, he esimae of he inercep in he dorman regime ( D, 0 ) is , and i is highly significan as seen from he sandard error. This implies ha he expeced reurn in he dorman regime, which is equal o he required fundamenal reurn, is 0.31% per monh (3.78% on an annual basis). We consider his figure reasonable in erms of real fundamenal reurns, and he corresponding value of his coefficien for he AvNS model is quie similar (1.0033), implying a mean reurn in he surviving regime of 4.03% on an annual basis. However, when he price series eners he explosive regime, is behaviour becomes more exreme. The equilibrium gross reurn in he surviving regime ( S, 0 ), when he bubble size and he measure of abnormal volume are equal o zero, is significanly higher han in he dorman regime, 1.18% per monh (15.12% on an annualised basis. The equilibrium reurn in he surviving sae in he vns model is beween he equilibrium reurns of he dorman and he surviving regimes in he hree-regime models. This could be evidence ha he equilibrium reurn in he wo-regime model is he resul of he mixure of he dorman and explosive growh regimes. Turning now o he equilibrium reurn in he collapsing sae, we can see ha he expeced equilibrium reurn in sae C for he hree-regime model is -2.16% per monh (-23.09% on an annualised basis). This is consisen wih he heory of speculaive bubbles since he expeced reurn in he collapsing sae should be negaive. Examining he slope coefficiens in he sae equaions, we noe ha he coefficiens on he relaive bubble size in he surviving and he collapsing regimes ( and C, B ) have he expeced signs 14 and are saisically significan a he 1% level. The speculaive bubble model requires he reurn in he collapsing S, B 14 I is no possible o derive an expeced sign for he bubble coefficien in he surviving sae since he speculaive bubble model only implies ha i should be greaer in value han he bubble coefficien in he collapsing regime. Neverheless, we should expec ha as he bubble increases in size, invesors demand a higher reurn o compensae hem for he increased risk of bubble collapse. 14

19 regime o be a negaive funcion of he size of he bubble ( 0 ) while he coefficien of he bubble C, B size in he surviving regime mus be greaer han he corresponding coefficien in he collapsing regime ( S, B C, B ). From he second panel of Table 1, we can see ha boh of hese condiions for model plausibiliy are suppored by he daa. The null hypohesis ha he bubble coefficien in he collapsing regime is equal o zero is rejeced a he 5% level (p-value of LR es 0.02), implying ha as he bubble size increases, he reurns in he collapsing regime decrease 15. Furhermore, we can see ha resricion (ii) is saisfied since a he 5% level (in he second panel of Table 1 he p-value of he LR es is 0.019). S, b C, b The hree-regime model also incorporaes abnormal volume in he surviving sae equaion. The poin esimae of he abnormal volume coefficien in he surviving sae ( S, V ) is saisically significan (pvalue ), and has he expeced sign according o condiion (iii) ( 0 a he 10% significance S, V level). This implies ha as volume increases, expeced reurns for he nex period increase, consisen wih our conjecure ha increased abnormal volume signals increased risk. The coefficien esimaes of he equaion for he probabiliy of being in he seady growh regime, equaion (23.4), are in favour of he hree-regime speculaive behaviour model. For he hree-regime model, he inercep coefficien ( n, 0 ) implies ha here is a 15.43% probabiliy of swiching o he explosive sae when he size of he bubble and he spread of acual reurns are boh equal o zero. This probabiliy is calculaed as 1 ( n, 0 ) using he poin esimaes shown in Table 1. The corresponding probabiliy for he AvNS model is 13.83%. However, as he bubble grows, he probabiliy of being in he dorman sae in period +1 decreases since he coefficiens on he absolue bubble size and he spread of reurns are negaive. More specifically, he poin esimae of he coefficien on he absolue bubble size in he equaion for he probabiliy of he dorman regime ( n, B ) is and is significan a he 5% level. This implies ha as he bubble grows in size, he probabiliy of being in he explosive regime increases 16. Our model also incorporaes he spread of he average six-monh acual reurns over he six-monh average of fundamenal reurns in he ransiion equaion. This variable helps o disinguish periods of explosive growh, especially when he esimaed bubble deviaions are large. The coefficien esimae on he measure of he spread ) is negaive, as expeced, and significan a he 5% level. Furhermore, he ( n,s 15 The opposie holds for negaive bubbles since hey collapse by yielding posiive abnormal reurns, alhough i would require a negaive bubble size in excess of -15% in order for he reurns of he collapsing regime o become posiive. 16 Alhough he uncondiional probabiliy of enerning he exposive sae is quie low, i is of ineres o consider he cumulaive effec ha his could have over a period of ime. For example, he probabiliy of remaining in he dorman sae for he nex six 6 monhs would be equal o 36.7%, and would be given by ( 1 (1 ( n, 0))) 15

20 LR es resuls show ha he condiions on he signs of he coefficiens in he ransiion equaion (vi) and (vii), are saisfied a he 1% and 5% level respecively as seen from par wo of Table 1. The LR es rejecs he hypohesis ha he spread of acual reurns does no affec he probabiliy of being in he dorman regime and confirms ha he probabiliy of consan growh decreases as he bubble growh acceleraes. Turning now o he probabiliy of being in he surviving regime, he coefficien esimaes of he classifying equaion (12.5) for he hree-regime model and for he AvNS model are in favour of he presence of periodically collapsing speculaive bubbles. As he bubble grows, he probabiliy of being in he surviving regime in period +1 decreases since he coefficien on he absolue bubble size is negaive ( ) and saisically significan a he 1% level. Furhermore, he LR es shows ha condiion (iv), which saes ha his coefficien should be negaive ( 0 ), is saisfied a he 10% level. This is consisen wih Evans (1991) and he noion ha when he bubble is relaively small, he probabiliy of he bubble collapsing is small and hus invesors do no ake i ino accoun in he pricing of he asse. As he bubble grows and / or abnormal volume increases, he probabiliy of he bubble collapsing increases geomerically. The poin esimae of he abnormal volume coefficien in he equaion for he probabiliy of survival ( q, V q, B ) is negaive, as expeced, and saisically significan ( wih p-value ). The LR es confirms ha he probabiliy of he bubble coninuing o grow wih explosive expecaions is a negaive funcion of abnormal volume. From he above, he probabiliy of he bubble collapsing should increase significanly prior o a bubble collapse if our hree-regime model is superior a forecasing regime changes. Indeed, in Augus 1982 when a large negaive bubble is presen, he probabiliy of he bubble collapsing, esimaed from he hree-regime model, increases by 530% o a value of 2.08%. The AvNS model, predics, for he same monh, a probabiliy of collapse of 2.22%, which is only 1.3% higher han for he previous monh. The probabiliies of being in a given regime calculaed from he poin esimaes of he hree-regime model are presened in Figure The corresponding probabiliies from he AvNS model are presened in Figure 3. I is apparen ha he hree-regime model incorporaing he volume and spread variables yields a more variable probabiliy of being in he seady growh regime, which decreases considerably during periods of significan marke advances or declines. Furhermore, he probabiliy of being in he collapsing regime increases significanly before several bubble collapses, namely Augus 1929, June 1932, Augus 1982 and Ocober These probabiliies are calculaed as Pr( W 1 D) n, Pr( W 1 S) (1 n ) q, and Pr( W 1 C) (1 n )(1 q ) using he poin esimaes of Table Noe ha some of hese periods were followed by marke rallies. This is because we are also examining price decreasing bubbles, which collapse yielding posiive reurns. The probabiliies of collapse for boh models are also high in oher periods ha were no followed by bubble collapses. This could be evidence agains he speculaive bubble model. However, we will show in he nex secion ha he hree-regime bubble model has significan predicive abiliy and can be used o ime marke reversals. 16

21 The sandard deviaions of he residual erms from he hree-regime model are consisen wih he heoreical predicions since he sandard deviaion of he error erm in he collapsing regime is greaer han in he surviving regime. This is because bubbles ofen collapse by yielding exreme negaive reurns (or posiive reurns in he case of price decreasing bubbles). Furhermore, according o Evans (1991) and van Norden and Vigfusson (1998), he sandard deviaion of he errors in he dorman regime should be small, since i is periodically collapsing speculaive bubbles ha cause an increase in he variance of reurns. The sandard deviaion of he residuals in he seady growh regime is 2.95%, in he surviving regime i is 4.71%, while in he collapsing regime i is 10.91% on a monhly basis. The hird panel of Table 1 presens he resuls of he LR ess of he augmened model agains simpler models ha capure well-documened properies of sock marke reurns. The LR es resul rejecs he volailiy regimes alernaive specificaion a he 1% level implying eiher ha he mean reurns are differen across he hree regimes, or ha speculaive bubbles, abnormal volume and he spread of acual reurns have predicive power for he reurns of period +1 or for he probabiliy of swiching regimes. In order o separae he wo resricions, we es he speculaive behaviour model agains a mixure of normal disribuions model and he resul of he LR es shows ha he daa rejec he mixure of hree normal disribuions alernaive in favour of he hree-regime periodically collapsing speculaive bubble model. This suggess ha he measure of bubble deviaions, he measure of abnormal volume and he spread of acual reurns have significan forecasing abiliy for nex period reurns and for he probabiliy of swiching regimes. The LR es saisic signifies a rejecion of he null of he mixure of normal disribuions a he 1% significance level. We also examine he hree-regime model agains he AvNS model in order o see wheher abnormal volume and he spread of acual reurns should be used in order o forecas he level and he generaing sae of reurns. Again, our model rejecs his simpler specificaion. 5. Predicive and Profiabiliy Analysis Alhough in he previous secion we showed ha he hree-regime speculaive behaviour model has significan explanaory power for he S&P 500 reurns, he abiliy of his model o forecas hisorical bubble collapses has no ye been examined. In a previous sudy, van Norden and Vigfusson (1998) examined he size and he power of bubble ess based on regime swiching models, and found ha he ess are conservaive, bu have significan power in deecing periodically collapsing speculaive bubbles of he form described in Evans (1991). However, heir echnique only examines he economeric reliabiliy of he swiching speculaive bubble model developed by van Norden and Schaller. In his secion, we will examine he ou of sample forecasing abiliy of he hree-regime model and compare i wih he forecasing abiliy of he vns and he AvNS models. We will hen invesigae wheher regime swiching speculaive behaviour models can be used o deermine opimal marke enry and exi 17