Have bull and bear markets changed over time? Empirical evidence from the US-stock market

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1 Journal of Finance and Invesmen Analysis, vol.1, no.1, 2012, ISSN: (prin version), (online) Inernaional Scienific Press, 2012 Have bull and bear markes changed over ime? Empirical evidence from he US-sock marke Klaus Grobys 1 Absrac This conribuion analyzes bull and bear markes from 1954:1-2011:2 in he US-sock index S&P 500. Thereby, a 2-Sae-Markov-Swiching model is applied o figure ou bull and bear marke regimes wihin he laer period, whereby he esimaed sae probabiliies are used o esimae a dummy variable model by employing operaional crieria. A sample-spli analysis, where he daa se is divided ino wo samples of equal lengh, gives evidence for a srucural break in he expecaion of reurns being associaed wih bull marke regimes whereas no srucural break can be ascerained concerning bear marke regimes. This oucome has srong implicaions for modern asse allocaion heory which akes he presence of regime swiching ino accoun as invesors who expec a significan increase in sock reurns would allocae a higher weigh o socks even hough hey would face bull marke regimes a he ime poin when deciding on asse allocaions. 1 Swedish Research Associaion of Financial Economics, Klaus.grobys@srafe.se Aricle Info: Received : January 25, Revised : February 12, 2012 Published online : February 28, 2012

2 152 Have bull and bear markes changed over ime? JEL classificaion numbers: G12, G14 Keywords: Sock marke, Bull- and bear-markes, Markov-Swiching model, Srucural break, Sample-spli analysis, Dummy-variable model 1 Inroducion In recen years, sudying financial cycles has become an imporan issue in he academic financial lieraure and aracs more and more aenion. According o Gonzalez e al [6] academics agree on ha bull markes are associaed wih persisenly rising share prices, srong invesor ineres, and raised financial well-being. Aroa and Buza [2] conclude ha bull markes are usually associaed wih a period of prosperiy, when he fuure seems brigh and invesors have easy access o money. They argue furhermore ha he same mass psychology evoking he expecaion ha every do-com company will be profiable and, hence, creaed he boom in he sock marke during , was accounable for he crash in he US-sock marke in January-March The U.S economy began o slow down during he end of he year 2000, and he res of he world followed, resuling in a worldwide recession as poined ou by Aroa and Buza [2]. While bull markes involve an enhancemen of he invesors financial well being, he opposie akes place when he direcion of he equiy price changes alers and he invesors face bear markes. Moivaed by he global scale of he Ocober 1987 sock marke crash and he subsequen Asian and Russian crises of , he undersanding of how rends in sock marke reurns change over ime has led o new implicaions concerning he asse allocaion problem. Claessens, Koese and Torrones [5] highligh ha asse prices influence consumpion hrough he heir effec on household wealh, and can especially effec invesmen by changing companies ne worh as well as he marke value of he capial sock relaive o is replacemen value. They conclude ha in paricular

3 Klaus Grobys 153 he financial crisis ha sared in 2007 in he Unied Saes have been accompanied by an inensive debae over is impac on he broader economy. In heir sudies hey analyze linkages beween key macroeconomic and financial variables around business cycles where 21 OECD counries over he period are aken ino accoun. Their findings give evidence ha he ineracions beween macroeconomic and financial variables can play a role in deermining he magniude and lengh of a recession. Guidolin and Timmermann [8] explore he asse allocaion and uiliy cos implicaions of reurn predicabiliy from a small, expeced uiliy maximizing invesor wih a muliple-period horizon. Thereby, he invesor opimizes he asse allocaion problem while facing hree asses - namely socks, bonds and cash. In conras o he radiional invesmen advice recommending an increased exposure of socks he longer he invesmen horizon, Guidolin and Timmermann [8] conclude ha he opimal allocaion o socks declines as a funcion of he invesmen horizons given he invesors face bull markes. The opimal asse allocaions vary considerably over ime being dependen on wheher he invesors face bull- or bear markes. However, Guidolin and Timmermann s [8] findings do no accoun for srucural changes in he underlying disribuions of bull- and, respecively, bear markes. Have he properies of bull- and bear markes changed over ime? If he properies of financial cycles significanly change over ime, he laer has o be imbedded in he asse allocaion problem and, hus, he exising asse allocaion lieraure ough o be revised. Even hough Gonzalez e al [6] conclude ha he disribuions of bull- and bear markes have changed beween he las wo cenuries, heir resuls can be challenged due o he daa se being employed, for insance. The sock index being employed for he period ress upon Schwer s [15] index of sock prices and is mainly composed of railroad, insurance and banking socks. Their sudies sugges ha capial marke reurns increased by almos 7.20% p.a. in he 20h cenury, while bear mean reurns

4 154 Have bull and bear markes changed over ime? increased in erms of absolue amoun by13.32% p.a. I is worh noing ha hey repor ha heir resuls are saisically no significan (i.e. significan a he 10% level only) when he earlies par of he daa se prior o 1835 is excluded. Moreover, poenial differences which may be assessed beween differen cenuries may imply neglecing impac on he invesmen decision concerning he ime horizon which an invesor usually faces. There are no sudies available ha analyze srucural changes in bull- and bear markes over ime while figuring ou poenial implicaions for invesors who maximize heir uiliies. The following conribuion fills his gap in he lieraure and is organized as follows. Firs, a lieraure review is presened where differen approaches will be considered being employed o esimae low frequency rends in he daa such as bull and bear markes. Secondly, a convenien model is suggesed which capures bull and bear marke reurns and allows for esing poenial srucural breaks while accouning for an unknown form of heeroskedasiciy. Aferwards he resuls will be presened and discussed. The las secion concludes. 2 Lieraure Review In order o esimae bull and bear markes, Gonzalez e al [6] and Claessens, Koese and Torrones [5] employ a model resing upon he daing algorihm as inroduced by Bry and Broschan [4]. Gonzalez e al [6] menion ha Bry and Broschan s [4] algorihm basically replicaes he NBER business cycle urning poins and hence generaes formal daing rules in order o deermine local peaks and roughs in sochasic ime series. Thereby, he minimum lengh of a cycle is defined o las 15 monhs from he peak o he rough and back o he peak, whereas one phase (expansion or conracion) is supposed o las a leas 5 monhs. Pagan and Sossounov [14] exend he required minimum duraion of a financial cycle o be 16 monhs raher han 15 monhs, whereby each phase is defined o las

5 Klaus Grobys 155 a leas 4 monhs. Furhermore, sharp sock price movemens are accouned for by disregarding he minimum phase lengh if he sock index falls by more han 20% in a single monh as menioned by Gonzalez e al [6]. Furhermore, Claessens, Koese and Torrones [5] employ quarerly daa in order o analyze recessions, credi conracions, house price declines and equiy price declines. Thereby, he daing algorihm inroduced by Harding and Pagan [10] is employed exending he Bry and Broschan s [4] algorihm o idenify also urning poins in financial ime series. The algorihm suggesed by Harding and Pagan [10] deermines local maxima and minima ha mee defined censoring rules, requiring a cerain minimum lengh, respecively, duraion for financial cycles. Claessens, Koese and Torrones [5] resric financial cycles o las a leas 20 monhs while a sole expansion, respecively, conracion is supposed o las a leas 8 monhs. Their sudies sugges an equiy price declines (i.e. bear markes) average duraion of monhs, whereby he average duraion of equiy price buss is esimaed a monhs. The laer is defined as equiy price decline which is in duraion ha is from peak-o-rough wihin he op quarile of all price declines. Lunde and Timmermann [12] who invesigae he duraion dependence in bull and bear markes esimae he average duraion of bull and bear markes o be and 9 monhs respecively. However, hey repor ha he shores bull and bear markes lased one week only, whereas he longes duraions are esimaed o be and 34 monhs respecively. Their saisical model reas he variable driving he sock marke as observable while he probabiliy ha a bull or bear marke lass for a cerain period of ime is derived from hazard raes which is esimaed via discree survivor funcions. Lunde and Timmermann [12] employ daily sock price daa in he US from 2/17/ /31/1997. Hence, Lunde and Timmermann [12] sudies involve he same drawback as he sudies of Gonzalez e al [6] as hey also employ Schwer s [15] index. In conras o Lunde and Timmermann [12], Guidolin and Timmermann [8]

6 156 Have bull and bear markes changed over ime? invesigae he asse allocaion problem in he presence of regime swiching while disinguishing beween four differen sock marke regimes where he firs regime is a low reurn, high volailiy crash/bear sae, regimes 2 and 3 are bullish saes wih low-volailiy and regime 4 is a recovery sae wih high volailiy which ends o follow crash regimes. Guidolin and Timmermann [8] employ a 4-Sae-Markov-Swiching model in order o accoun for he laen naure of sock marke regimes while employing monhly sock marke daa. While exhibiing persisency he sock markes regimes are seen o change frequenly over ime. If he invesor faces a slow growh sae (i.e. regime 2) which is seen o be he mos persisen regime, Guidolin and Timmermann [8] conclude ha invesors who are assumed o inves on longer horizons will hold he less socks he longer he horizons. In conras o formal daing algorihms, Markov-Swiching models accoun for he laen naure of low frequency rends as he variable driving he reurns is assumed o unobservable, whereby ransiions beween saes are governed by a Markov chain. The Markov-Swiching model ress upon he pioneering work of Hamilon [9] who has according o Alexander and Dimiriu [1] provided he firs formal saisical represenaion of he idea ha economic recessions and expansions influence he behaviour of economic variables. In he financial lieraure, Markov-Swiching models are widely employed. Furhermore, employing formal daing algorihms while accouning for resricions such as a minimum lengh will ignore evens such as he sock markes crash of Ocober 1987 where according o Gonzalez e al [6] only hree monhs separaed he peak and he rough. In he following bull and bear marke regimes are esimaed by employing a 2-Sae-Markov-Swiching model which will no be resriced. Aferwards an operaional crieria is incorporaed o ranslae he probabiliy esimaes of he Markov-Swiching model ino a dummy-variable model being used o es he hypohesis wheher bull and bear markes show a srucural break in heir disribuions over ime.

7 Klaus Grobys Economeric Mehodology In he following he saisical mehodology is described. Firs, an ordinary 2-Sae-Markov-Swiching model is employed accouning for a consan ransiion probabiliy marix. Aferwards, he esimaed smoohed sae probabiliies are employed o consruc a dummy-variable model. Then, differen hypoheses are esed o assess wheher saisical properies of he sock marke under consideraion (i.e. S&P 500 sock index) have changed over ime. Thereby, an ordinary sample-spli analysis is performed. 3.1 Applying he 2-Sae-Markov-Swiching Model o empirical sock marke daa In he following sock marke cycles are assumed o be he oucome of differen regimes. Following Ang and Baeker [3] and Alexander and Dimiriu [1] sock marke reurns are modeled while accouning for wo regimes, bull- and bear marke regimes. During bull marke regimes he invesor expecs posiive reurns on average while he bear marke regime is expeced o be associaed wih negaive reurns. Then he model can be wrien as y (1) S, S, where y = vecor of log-reurns S, = vecor of expeced reurns depending on he regime S S, = vecor of disurbances, assumed normal wih ime-varying sae dependen variance 2 S, Since sock markes reurns of developed counries do basically no exhibi serial correlaion equaion (1) does no involve any lagged variables of y. Furhermore,

8 158 Have bull and bear markes changed over ime? he ransiion probabiliies for he wo saes are in line wih Hamilon [9] assumed o follow a firs order Markov chain which is consan over ime and given by 1, 2 1,... 1 P S j S i S i P S j S i p (2) ij The marix of ransiion probabiliies S is in line wih Alexander and Dimiriu [1] given by S p p p 1 p p12 p22 1 p11 p22 Le a Markov chain be given by p ij. wih 1, 0 ' when S 1 and 0,1 ' when S 2 hen he condiional expecaion of 1 given for S i may be given by pi 1 E 1 S i S. pi 2 The condiional probabiliy densiy funcion of colleced in a 2x1 vecor, 1 2 y are assumed o be normal and where 1 f y S i, is he normal densiy funcion where he parameer vecor is condiional on he sae 1/ i 2 i exp i /2 i so ha y. The condiional sae probabiliies are in line wih Alexander and Dimiriu [1] obained recursively by ˆ where ˆ 1' ˆ 1 1 wih ˆ ˆ 1 S (3) = vecor of condiional probabiliies for each sae esimaed a ime ˆ =predicion of he same condiional probabiliies for ime 1 ˆ 1 1 = vecor of ones In equaion (3) he symbol denoes he elemen by elemen muliplicaion. Furhermore, he i h elemen of he produc ˆ 1 can be considered as he

9 Klaus Grobys 159 condiional join disribuion of he vecor y and S in equaion (3) denoes he densiy of he observed vecor i whereas he numeraor y condiional on he curren informaion se. The condiional densiy of he error vecor is hen given by T T, log, log ' ˆ 1 1. L S f y S Using he esimaed smoohed sae probabiliies o se up a dummy-variable model The esimaed sae probabiliies of he Markov-Swiching model as given by equaion (1)-(3) can be used o define operaional crieria o ransfer he model ino one which accouns for dummy variables as following. Le he dummy variable 1 d be equal o one if a probabiliy hreshold indicaes ha wih 0,1 and analogously he variable 2 PT S d =1 indicaing sae S 2 if PT S1 PT S1 (1 ) and zero oherwise. Then he following model can be esimaed: y d d u (4) 1 1, s 2 2, s where u is assumed o be a whie noise error erm. The parameer 1 and 2 denoe he expecaion of he log-reurns y for bull- and, respecively, bear markes concerning he overall sample. As bull- and bear markes are assumed o be persisen, he model will be resriced as following:! 1, s 1 d if PT S1, PT 1 S1 and 1 1 and analogously! 2, s 1 PT S d if PT S, PT S and PT S, 1 1 1

10 160 Have bull and bear markes changed over ime? which means ha ouliers (i.e. if he regime swich has a duraion of one monh only) will no be considered as a swich in he regime. As a consequence, a regime is assumed o have a minimum duraion of a leas hree monhs. Then he sample is divided ino wo subsamples of equal lengh where T1 1,..., T* and T 2 T* 1,..., T. In equaion (5) he dummy variables d 3 and d 4 accoun for he same operaional crieria as in equaion (4) bu ake he value zero if T1 : y d d d d u wih (5) d 1 1, s 2 2, s 3 3, s 4 4, s d d 3, s, 4, s 3, s 4, 0 s d T Hypoheses esing: Have firs or second order momens of bull- and bear markes changed over ime? If bull- and bear markes have no changed over ime, he parameers d 3, s and d 4, s should no be significan. Consequenly, hree pairs of hypohesis will be esed: H1: The parameers in bull and bear markes have no changed over ime H2: The parameer in bull markes has no changed over ime H3: The parameer in bear markes has no changed over ime The corresponding pairs of hypoheses are given by H1: H : 0 d3, d4, 0 versus H : 1 d3, d4, 0 (6) s s H2: H 0 : d3, 0 versus H 1 : d3, 0 (7) s H3: H : 0 d4, 0 versus H 1 : d4, 0 (8) s s where he es saisic 1 concerning equaion (6) is under he null hypohesis asympoically chi-square disribued wih wo degrees of freedom whereas he s s s

11 Klaus Grobys 161 es saisics 2 and 3 concerning equaions (7) and (8) are under he null hypohesis chi-square disribued wih one degree of freedom. Since he error erm of sock marke reurns is due o empirical sudies no normally disribued he es saisics are boosrapped while accouning for q boosrapping samples. Thereby he wild boosrapping mehodology is applied o he daa as described in deail in Godfrey [7]. Under he assumpion of he errors independency, he wild boosrapping mehodology exhibis in accordance o Liu [11] and Mammen [13] and asympoicially valid p-values and accouns moreover for he presence of an unknown form of heeroskedasiciy. For he resampling procedure he picking disribuion as suggesed by Liu [11] is employed where i and g N e g g E g E g g N 1/2 17/6 1/6,1/2 2 1/2 17/6 1/6,1/2 as well as 1. According o Godfrey [7] a ypical observaion for a wild boosrapping scheme may be given by y * d d d d ue. (9) 1 1, s 2 2, s 3 3, s 4 4, s i where y * denoes he sock marke reurns under he null hypohesis. Godfrey [7] argues ha he randomness of y * abou is condiional mean depends only on he specificaion of he picking disribuion. 4 Empirical Resuls of he US-sock marke In he Analysis he US-sock index S&P 500 is aken ino accoun. Following Guidolin and Timmermann [8] monhly sock marke daa is employed accouning for daa from 1954:1-2011:2 corresponding o 686 observaions. The Markov-Swiching model which employs he log-reurns of he ime series gives he following esimaes (sandard errors are given in parenhesis):

12 162 Have bull and bear markes changed over ime? S S e 04 2 S e 04 S (9) (10) S (11) Bull- and bear markes are highly persisen. The average duraion of bull- and bear-markes are esimaed a monhs and 7.64 monhs respecively. Tha is he expeced duraion of bear markes is only 15% of he bull marke s duraion. The probabiliy hreshold. PT which is employed o assess he exac ime poin of a change in he regime is used wih Consequenly all observaions where he ime varying sae probabiliy of bull-sae exceeds 0.50 are classified as bull-regimes whereas all oher observaions are couned among bear marke regimes. This approach shows ha he longes bull marke is beween 1975:3-1986:6 corresponding o 137 monhs whereas he longes bear marke is esimaed from 1973: :2 and corresponds o 15 monhs. Table 2 shows ha he overall ime window (i.e. 1954:1-2011:2) accouns for eleven bear- and bull markes where he median lengh of bear- and bull markes is esimaed a seven and 30 monhs, respecively. 2 Figure 1 shows he smoohed sae probabiliies from 1954:1-2011:2. Since he esimaed sae probabiliies for bull saes are 0.58 and 0.66 for 1980:3 and 1980:5, respecively, and 0.27 for 1980:4, he observaion a 1980:4 is reaed as oulier and hence assumed o be a par of he bull marke from 1973: :2. The dummy variable model (see equaion 2 The 25%- and 75%-quaniles for bear- and bull-markes are esimaed a four and en monh for bear markes and seven and 64 monhs for bull markes.

13 Klaus Grobys 163 4) ha is employed o esimae low frequency rend being involved in he daa esimaes he parameer of he log-reurns as following (-saisics are given in parenhesis). y d d u (12) 1, s 2, s The parameer esimaes show ha he low frequency rends being esimaed by he Markov-Swiching model are saisically significan. The amoun of he parameer 2 ha indicaes he bear marke regime is 2.76 imes larger in comparison o he bull marke parameer 1. In order o figure ou a change in he disribuions expecaions he overall sample is divided ino wo subsamples where he firs sample corresponds o daa from 1954:2-1982:7 (i.e. 342 observaions) and he second sample corresponds o monhly sock marke daa from 1982:8-2011:2 (i.e. 343 observaions). The dummy variables d 3, s and d 4, s indicae if he expecaions of he log-reurns have changed during bull- and bear-marke regimes in he second sample. y d d d d u (13) 1, s 2, s 3, s 4, s Table 1 shows he resuls from esing he pairs of hypohesis (i.e. H1-H3). Only H2 can be rejeced on a common significance level of =5%. Even hough esing he bear-marke parameer (H3) suggess ha he srucural change in he second sample is no significan (boosrapped p-value is ) a simulaneous es of boh, bull- and bear parameer rejecs he null hypohesis on a 10% significance level. In order o suppor H2 equaion (5) is re-esimaed under he resricion ha! 4, s 0 d. Then he resriced model which is given by equaion (14) y d d d u (14) 1, s 2, s 3, s is esed wheher he change in he bull marke parameer is significan concerning he second sample (H4). Applying he F-es presens a es saisic of (p-value is ). Wild boosrapping of he es saisic resuls in

14 164 Have bull and bear markes changed over ime? a p-value of whereby he number of boosrapped samples is q Since H2 and H4 clearly rejec he null hypohesis on a common significance level of 5% here is srong evidence for a srucural change concerning he expeced reurns during bull markes. Equaion (14) suggess ha he mean during bear markes has increased by % wihin he second sample (i.e. 1982:8-2011:2). Furhermore, he absolue amoun of he negaive expeced reurns in bear markes d 2, s is clearly larger in comparison o bull markes expecaions concerning all esimaed models. In erms of ordinary reurns he bull- marke reurns in he firs sample are esimaed a 9.26% p.a. wih an annual volailiy of 12.44% whereas he corresponding figures for he second sample are esimaed a 18.24% p.a. wih volailiy of 11.74% p.a. In bear markes beween 1954:2-1982:7 invesors faced a mean reurn of % p.a. wih volailiy 23.18% whereas he corresponding figures for he period 1982:8-2011:2 represen % and 24.65%. Ineresingly applying he same probabiliy hreshold o he Markov-Swiching model sugges a oal bear marke ime of 27 monhs from 1954:2-1982:7 whereas from 1982:8-2011:2 he invesors sayed 57 monhs in bear marke regimes during he second sample which is an increase of 211%. However, bull markes summed up duraion are esimaed a 315 monh in he firs sample and 286 monhs in he second sample corresponding o a sligh decrease of 9.02%. Tha means he invesor faced 7.89% of ime bear markes concerning sample 1, whereas he corresponding figure for sample 2 is 16.62%. Table 2 shows ha he ordinary reurns of boh regimes are normally disribued as he Jarque-Bera es saisics are close o hree and consequenly he null hypohesis of normaliy canno be rejeced. The esimaed skewness in bull markes is slighly posiive in he second sample and slighly negaive wihin he firs sample. The volailiy in bear markes is abou wice as much as he volailiy in bull markes, whereas he laer has only marginally decreased wihin he second sample concerning bull markes.

15 Klaus Grobys 165 Figure 1: Smoohed sae-probabiliies of he Markov-Swiching Model Table 1: Hypoheses esing Hypohesis Tes saisic i p-value Boosrapped p-value* H H H *The number of boosrapped samples is q Discussion Bull marke regimes have saisically changed in a significan way over ime

16 166 Have bull and bear markes changed over ime? and he sudy here gives srong evidence for a srucural break in bull markes as he expeced reurn given bull markes has srongly increased during he second sample. Table 2: Summary saisics Period 1954:2-1982:7 1982:8-2011:2 1954:2-2011:2 Type Bull Bull Bear Mean* Observaions Median* Minimum* Maximum* Sd. Dev.* Skewness Kurosis Jarque-Bera p-value *Noe: In monhly erms The expeced reurn given a bull marke regime is 94.56% higher during 1982:8-2011:2 in comparison o he previous sample (i.e. 1954:2-1982:7), whereas

17 Klaus Grobys 167 he bull marke volailiy has no changed over ime. 3 The expeced reurn in bear markes hough is clearly negaive and from a larger magniude in erms of amoun. The volailiy in bear markes is wih 24.05% p.a. abou wice as high as he bull marke volailiy which is esimaed a 12.45% p.a. (i.e. based on sample 1). In conras o bull markes expeced reurns, expeced reurn in bear markes have no changed over ime. The previous oucome may have srong implicaions concerning he asse allocaion problem as discussed by Guidolin and Timmermann [8] who conclude ha invesors who are assumed o inves over longer periods of ime will hold fewer socks he longer he invesmen period. Since he invesors expec according o he resuls of his sudy higher reurns in he fuure, hey would allocae a higher weigh o socks even hough hey may face a curren bull marke since bull markes in he fuure exhibi higher reurns han curren bull markes. In conras o Lunde and Timmermann [12] and Claessens, Koese and Torrones [5] who employ daily and quarerly daa in order o esimae financial cycles, he sudy suggesed here employs monhly daa which is also in line wih Gonzalez e al [6] and Guidolin and Timmermann [8]. This daa frequency is adequae as Gonzalez e al [6] highligh ha bull and bear markes are broad marke movemens and would be bes capured by employing low-frequency daa. As he Markov-Swiching model provide esimaes of he probabiliies of being in a bull or bear markes raher han providing peak o rough daes, he regimes being esimaed by he 2-Markov-Swiching model do no exacly mach he peak o rough daes as provided by Gonzalez e al [6]. For insance, he well known sock marke crash in Ocober 1987 and he associaed bear marke is according o 3 Esimaing a 95%-confidence inerval for he bull marke variance concerning sample 1 resuls in VAR( y ) [ ; ] while aking ino accoun he chi-square disribuion wih (315-1) degrees of freedom. As he bull marke variance concerning sample 2 is (i.e. on a monhly base), he null hypohesis of volailiy equaliy canno be rejeced on a significance level of 5%.

18 168 Have bull and bear markes changed over ime? Gonzalez e al [6] from 1987:8-1987:11 corresponds o four monhs. The corresponding bear marke probabiliies for he laer period are , , and Employing operaional crieria as suggesed in he sudy here, resuls in an esimaed corresponding bear marke period from 1987:9-1987:12 as he esimaed probabiliies of being in a bear marke for hese monhs are higher han 0.50 (i.e , , , ). Furhermore, Gonzalez e al [6] repor bull and bear markes kurosis a and respecively. As he esimaed kurosis of he bull marke reurn disribuion is wo o hree imes higher han he bear marke kurosis, Gonzalez e al [6] argue ha bull markes are more likely o show larger movemens as hey are seen o be prone o ouliers. However, he 2-Sae-Markov-Swiching model being employed in his sudy esimaes boh disribuions, he disribuion of he ordinary bull and bear marke reurns o be close o normaliy (see able 2). Since he absolue amoun of he mean and he volailiy are 72.96% and % higher in bear markes compared o bull markes (i.e. based on he sample 2), bear markes are more likely o involve ouliers raher han bull markes. Moreover, Gonzalez e al [6] repor ha bull marke capial reurns increased by 7.20% p.a. when comparing he las wo cenuries. However, heir resuls become insignifican on a 5% significance level when he earlies par of he daa se prior o 1835 is excluded. This sudy suggess an increase of 8.87% wihin he las 60 years, whereas here is no evidence given for a change in he bear marke disribuion. The model being employed here is a 2-Sae-Markov Swiching model, whereas Guidolin and Timmermann [8] esimae a 4-Sae-Markov Swiching model where he bull marke regime is pared in hree sub-regimes exhibiing differen properies. However, Guidolin and Timmermann s [8] approach involves wo drawbacks. Firs, one runs he risk of overfiing which means he model esimaes saes he daa acually does no conain. The second poin is ha he saes become he less persisen he more saes will be included. Therefore, he

19 Klaus Grobys 169 choice of a 2-Sae-Markov-Swichung model is preferred in his sudy even hough sub-regimes which are ofen referred o as marke-correcions (i.e. due o heir shor-run characer) may exis. 6 Concluding Remarks Bull markes have significanly changed wihin he las hree decades. This is he oucome of a saisical analysis. However, here sill remains a need for reasonable economic explanaions. Fuure research may consider poenial linkages beween changes in macroeconmic variables and changes in he financial markes. If changes in he sock markes canno be explained economically oher sciences may be considered ha explain empirical facs from a behavioral or psychological poin of view, for insance. The oucome of his sudy has srong implicaions concerning he asse allocaion problem because invesmens in socks become he more aracive he higher he expeced reurns in he fuure. A significan increase in he reurn disribuion corresponds o a drif in he inegraed ime series from a saisical poin of view. There may be also need for fuure research o figure ou he underlying sochasic processes of sock markes agains he background of srucural breaks. Furhermore, his mehodology can be also applied o oher sock markes in order o invesigae wheher he oucome of his sudy is suppored in an inernaional conex, oo. This sudy here gives evidence for a drif erm being incorporaed in he inegraed ime series. Therefore anoher avenue of fuure research may aim a figuring ou he ime poin when he break occurred or wheher he break is raher a poenial proxy for a linear or non-linear rend in he daa.

20 170 Have bull and bear markes changed over ime? References [1] C. Alexander and A. Dimiriu, Indexing, Coinegraion And Equiy Marke Regimes, Inernaional Journal of Finance and Economics, 10, (2005), [2] H.K. Arora and M.P. Buza, Unied Saes Economy & The Sock Marke, Journal Of Business And Economics Research, 1(1), (2003), [3] A. Ang and G. Bekaer, Regime Swiches in Ineres Raes, Journal of Business and Economic Saisics, 20, (2002), [4] G. Bry and C. Boschan, Cyclical Analysis of Time Series: Seleced Procedures and Compuer Programs, New York: NBER, [5] S. Claessens, M.A. Kose and M.E. Terrones, Wha happens during recessions, crunches and buss?, Economic Policy, 24, (2009), [6] L. Gonzalez, J.G. Powell, J. Shi and A. Wilson, Two cenuries of bull and bear marke cycles, Inernaional Review of Economics & Finance, 14, (2005), [7] L. Godfrey, Boosrap Tess for Regression Models, Palgrave Macmillan, New York, [8] M. Guidolin and A. Timmermann, Asse Allocaion under Mulivariae Regime Swiching, Journal of Economic Dynamics and Conrol, 31, (2008), [9] J. Hamilon, A New Approach o he Economic Analysis of Nonsaionary Time Series and he Business Cycle, Economerica, 57, (1989), [10] D. Harding and A. Pagan, A comparison of wo business cycle daing mehods, Journal of Economics Dynamics and Conrol, 49, (2002), [11] R.Y. Liu, Boosrap procedures under some non i.i.d. models, Annals of Saisics, 16, (1988),

21 Klaus Grobys 171 [12] A. Lunde and A. Timmermann, Duraion dependence in sock prices: An analysis of bull and bear markes, CAF Universiy of Aarhus, Working Paper Series, No. 70, (2000). [13] E. Mammen, Boosrap and wild boosrap for high dimensional linear models, Annals of Saisics, 21, (1993), [14] A.R. Pagan and K.A. Sossounov, A simple framework for analyzing bull and bear markes, Journal of Applied Economerics, 18, (2003), [15] W.G. Schwer, Indexes of U.S. sock prices from 1802 o 1987, Journal of Business, 63, (1990),