An Econometric Model of Nonlinear Dynamics in the Joint Distribution of Stock and Bond Returns

Transcription

1 WORKING PAPER SERIES An Economeric Model of Nonlinear Dynamics in he Join Disribuion of Sock and Bond Reurns Massimo Guidolin and Allan Timmermann Working Paper A hp://research.slouisfed.org/wp/2005/ pdf January 2005 FEDERAL RESERVE BANK OF ST. LOUIS Research Division 411 Locus Sree S. Louis, MO The views expressed are hose of he individual auhors and do no necessarily reflec official posiions of he Federal Reserve Bank of S. Louis, he Federal Reserve Sysem, or he Board of Governors. Federal Reserve Bank of S. Louis Working Papers are preliminary maerials circulaed o simulae discussion and criical commen. References in publicaions o Federal Reserve Bank of S. Louis Working Papers (oher han an acknowledgmen ha he wrier has had access o unpublished maerial) should be cleared wih he auhor or auhors. Phoo couresy of The Gaeway Arch, S. Louis, MO.

2 An Economeric Model of Nonlinear Dynamics in he Join Disribuion of Sock and Bond Reurns Massimo Guidolin Universiy of Virginia Allan Timmermann Universiy of California San Diego July 2004 Absrac This paper considers a variey of economeric models for he join disribuion of US sock and bond reurns in he presence of regime swiching dynamics. While simple wo- or hree-sae models capure he univariae dynamics in bond and sock reurns, a more complicaed four sae model wih regimes characerized as crash, slow growh, bull and recovery saes is required o capure heir join disribuion. The ransiion probabiliy marix of his model has a very paricular form. Exis from he crash sae are almos always o he recovery sae and occur wih close o 50 percen chance suggesing a bounce-back effec from he crash o he recovery sae. 1 Inroducion This paper sudies a variey of economeric models for he join disribuion of US sock and bond reurns. We show ha alhough here are well-defined regimes in he marginal We hank wo anonymous referees and he edior, Dick van Dijk, for many helpful suggesions. We are also graeful o seminar paricipans a CERP Universiy of Turin, Universiy of Houson, Universiy of Rocheser, Federal Reserve Bank of S. Louis and a he Tinbergen Cenenary conference for commens on an earlier version of he paper.

3 disribuions of boh sock and bond reurns, here is very lile coherence beween hese regimes. This complicaes models for he join dynamics of sock and bond reurns and suggess ha a richer model wih several saes is required. We sudy in deail a richly specified model wih four regimes broadly corresponding o crash, slow growh, bull and recovery saes. Unforunaely he vas majoriy of work on regime swiching considers univariae models. Examples include sudies of economic variables such as exchange raes (Engel and Hamilon (1990)), oupu growh (Hamilon (1989)), ineres raes (Gray (1996), Ang and Bekaer (2002b)), commodiy indices (Fong and See (2001)), and sock reurns (Rydén, Teräsvira, and Asbrink (1998), Turner, Sarz and Nelson (1989), and Whielaw (2001)). Excepions o he focus on univariae models include Ang and Bekaer (2002a) and Perez-Quiros and Timmermann (2000) who consider bivariae regime swiching models fied o sock marke porfolios racking eiher counry indices or porfolios based on marke capializaion. Hamilon and Lin (1996) also consider a bivariae model for sock reurns and growh in indusrial producion. There appears o be no clear guidelines for how o generalize univariae nonlinear models o he general mulivariae case, however. Simple generalizaions easily yield overwhelmingly large models. To see his, suppose ha sock reurns are divided ino wo saes based on periods of high and low volailiy, while bond reurns are divided ino recession, low growh and high growh saes. Also suppose ha he pair of sae variables are only weakly correlaed. In his case a six-sae model comprising low and high volailiy recessions, low and high volailiy saes wih low growh and low and high volailiy saes wih high growh is required o capure he join disribuion of sock and bond reurns. In general such models are no feasible o esimae or will be poorly idenified since mos saes are likely only o be visied very few imes during he sample. 1 The plan of he paper is as follows. Secion 2 sudies regimes in he individual asse reurns. Secion 3 considers heir join disribuion and discusses a some lengh a four-sae specificaion. Secion 4 exends ou seup o include addiional predicor variables such as 1 For a furher discussion of mulivariae regime swiching models see Franses and van Dijk (2000), pp

4 he dividend yield. Secion 5 concludes. 2 Sock and bond reurns under regime swiching: Univariae models In his secion we consider he dynamics in he univariae or separae disribuions of sock and bond reurns. An undersanding of he univariae dynamics of he reurns for he individual asse classes is an imporan saring poin for an analysis of heir join disribuion. We sudy hree major US asse classes, namely socks, bonds and T-bills alhough we simplify he analysis o jus socks and bonds by analyzing heir excess reurns over and above he T-bill rae. We furher divide he sock porfolio ino large and small socks in ligh of he empirical evidence suggesing ha hese socks have very differen risk and reurn characerisics across differen regimes, c.f. Perez-Quiros and Timmermann (2000). 2.1 Daa All daa is obained from he Cener for Research in Securiy Prices. Our analysis uses monhly reurns on all common socks lised on he NYSE. The firs and second size-sored CRSP decile porfolios are used o form a porfolio of small firm socks, while deciles 9 and 10 are used o form a porfolio of large firm socks. We also consider he reurn on a porfolio of 10-year T-bonds. Reurns are calculaed applying he sandard coninuous compounding formula, y +1 =lns +1 ln S, where S is he asse price, inclusive of any cash disribuions (dividends or coupons) beween ime and +1. To obain excess reurns, we subrac he 30-day T-bill rae from hese reurns. Dividend yields are also used in he analysis and are compued as dividends on a value-weighed porfolio of socks over he previous welve monh period divided by he curren sock price. Our sample is January December 1999, a oal of 552 observaions. 3

5 2.2 Regimes in he individual series Before proceeding o he join model for sock and bond reurns we consider he presence of regimes in he individual asse reurn series. The objecive is o assess he degree of coherence across he sae variables characerizing he regimes (if any) in he reurns on small and large firms and on long-erm bonds. A high degree of coherence would naurally sugges a subsanial reducion in he overall number of regimes, k, requiredinajoinmodel for sock and bond reurns. Each of he univariae reurn series (indexed by i =1,..,n, where n is he number of asses), y i, is modeled as a simple Markov swiching process whose parameers are driven by an asse-specific sae variable, S i, aking values s i =1,.., k i, where k i is he number of saes for he ih series: px y i = µ isi + a ij,si y i j + σ isi u i, i =1,..., n, u i IIN(0, 1), (1) j=1 where sae ransiions are governed by a consan ransiion probabiliy marix P (S i = s i S i 1 = s i 1 )=p si s i 1, s i,s i 1 =1,...,k i. (2) Thus each regime is assumed o be he realizaion of a firs-order, homogeneous, irreducible and ergodic Markov chain. For each series, y i, he number of saes, k i, is a key parameer in he proposed model. If k i =1, we are back o he sandard linear model used in much of he lieraure. As k i rises, i becomes increasingly easy o fi complicaed dynamics and deviaions from he normal disribuion in asse reurns. However, his comes a he cos of having o esimae more parameers which can lead o deerioraing ou-of-sample forecasing performance. Economic heory offers lile guidance o he mos plausible non-linear model capable of adequaely fiing he daa. If recurren shifs only affec he diversifiable componen of porfolio reurns (idiosyncraic risk), regime swiching in well-diversified porfolios such as hose we sudy here should only show up in he form of regime-dependen heeroskedasiciy giving rise o a model of he ype y i = µ i + σ si u i. (3) 4

6 On he oher hand, when shifs occur in he sysemaic risk componen, hen mos economic models would sugges regime dependence boh in he risk premium (µ) and in he variance: px y i = µ is + a ij,si y i j + σ si u i, (4) j=1 The presence of auoregressive lags may proxy for omied sae variables racking imevarying risk premia. This ambiguiy abou he correc heoreical model suggess we should consider a wide range of models. To deermine k i, we underake an exensive specificaion search, considering values of k i =1, 2, 3 and differen values of he auoregressive order, p. We consider up o hree saes because of he exising evidence in he lieraure of eiher wo (Schwer (1989) and Turner, Sarz, and Nelson (1989)) or hree (Kim, Nelson and Sarz (1998)) regimes in he mean and volailiy of U.S. asse reurns (see also Rydén e al. (1998)). I is of course imporan o deermine wheher muliple saes are needed in he firs place, i.e. wheher k i > 1. Tesing amodelwihk i saes agains a model wih k i 1 saes is complicaed because some of he parameers of he model wih k i saes are unidenified under he null of k i 1 saes and es saisics follow non-sandard disribuions. 2 To check if he linear model (k i =1) is misspecified, we compued he es proposed by Davies (1977) which accouns for he unidenified nuisance parameer problem. To deermine he number of saes, we adoped he Hannan-Quinn informaion crierion for model selecion (c.f. Rydén e al. (1998)). This rades off he improved fi resuling from adding more parameers as k i grows agains he decreasing parsimony. Table 1 repors he parameer esimaes of wo- and hree-sae models fied o he reurns on our hree porfolios along wih lineariy ess and values of he Hannan-Quinn informaion crierion. The lef panels (A and C) se p =0(no auoregressive erms), while he righ panels (B and D) assume ha p =1. For all hree asses he single-sae model is srongly rejeced in favour of a mulisae model. 3 The Hannan-Quinn crierion poins o 2 See, e.g., Davies (1977), Garcia (1998) and Hansen (1992). 3 In addiion o he Hannan-Quinn informaion crierion we also considered he Akaike and Schwarz informaion crieria. Two of hree informaion crieria applied o he univariae series suggesed a wo-sae model for sock reurns while all crieria seleced a hree-sae model for bond reurns. 5

7 a wo-sae specificaion for boh sock marke porfolios and a hree-sae specificaion for bonds. Furhermore, here is evidence of firs-order auoregressive erms in he small sock andbondreurnseries. Each of he wo regimes idenified in he wo sock reurn series has a clear economic inerpreaion. The firs regime capures a bear sae wih high volailiy and low expeced reurns: large socks are characerized by negaive mean excess reurns and an annual volailiy of 22.2%, small socks by relaively low mean excess reurns of 5.4% per annum and volailiy of 29.5%. Conversely, he second - more persisen - regime is associaed wih high mean reurns (large socks earn an annualized premium of 11.7%, small socks a premium of 13%) and low volailiy. The esimaes of he ransiion probabiliy marices for small and large socks are also quie similar alhough small socks end o say longer in bear saes. The saes idenified in he bond reurns have a similar inerpreaion: Regime 1 capures economic recessions during which ineres raes end o fall or say roughly consan so ha long-erm bonds earn low bu posiive average excess reurns (1.8% per annum), while heir volailiy is above-average (8.5%). Regime 2 capures economic booms wih rising ineres raes and negaive excess reurns on bonds. To furher assis wih he economic inerpreaion of hese saes, Figure 1 shows smoohed sae probabiliy plos for he wo-sae models fied o he individual reurn series. Alhough he maching beween he high volailiy saes idenified for he wo sock porfolios is by no means perfec, here are clearly srong similariies beween he wo and many wellknown hisorical episodes rigger similar regime swiches in boh porfolios, e.g., he Vienam War in he 1960s, he oil shocks of he 1970s, he volailiy surge of , he early 1990s recession, and he Asian flu of As a resul, he correlaion beween he smoohed probabiliy of sae 1 across he wo sock reurn series is In conras, here is no much similariy beween he regimes idenified in he sock and bond reurn series. Indeed he correlaions beween he smoohed sae probabiliies inferred from bond reurns and he probabiliies implied by boh small and large sock reurns are close o zero ( ). This impression is furher enhanced by he scaer plos of smoohed sae probabiliies shown in Figure 2, indicaing no srong correlaions beween he saes 6

8 idenified in sock and bond reurns. Furhermore, many episodes associaed wih regime swiches in he sock marke porfolios (e.g., he early 1980s recession and he 1987 crash) are no reflecedinsimilarswichesinbondreurns. Of course, his analysis may no fully reveal possible similariies beween he nonlinear componens in sock and bond reurns since we idenified hree saes in bond reurns. We herefore nex consider hree-sae models for sock and bond reurns. Panels C and D in Table 1 repor parameer esimaes for hese models while Figure 3 plos he smoohed sae probabiliies for he univariae hree-sae models fied o he wo sock reurn series and bond reurns. Inerpreaion of he hree saes in sock reurns is difficul. As we move from regime 1 o 3 he risk premium on large socks changes from -20.3% o 44.5% per annum and he volailiy declines from 25% o 6.3%. For small socks here is no grea difference in he volailiy esimaes for saes 1 and 3 while heir mean reurns (-29.3% and 104% per annum, respecively) are very differen. In conras, he hree-sae model marks a clear improvemen over he wo-sae model fied o bond reurns. In his case he hree saes are easier o inerpre. Regime 1 has relaively high volailiy (11.8%) and high mean excess reurns (3.6%), and herefore represens periods of declining shor-erm ineres raes and srong growh following a recession. Regime 2 corresponds o periods of rising shor-erm ineres raes (leading o negaive mean excess reurns on long-erm bonds) and downward sloping, sable yield curves. The hird sae is he mos frequenly visied regime in our sample, characerized by moderaely posiive mean excess reurns (0.7%) and moderae volailiy (6.2% per annum). The seady growh of he 1990s wih sable ineres raes and moneary policy falls almos enirely in his regime. This classificaion of he sample period ino regimes is more sensible han ha provided by he wo-sae model for bond reurns. 4 4 There is in fac an ineresing associaion beween some of he regime shifs appearing in Figure 3 for bonds and changes in moneary policy. For insance, ou of roughly 15 major swiches, as many as four can be linked o he classical Romer and Romer (1989) (conracionary) moneary policy shock daes, in he sense ha hese swiches occur wihin six monhs of Romer and Romer s daes. In paricular, he 1968:12 and 1979:10 episodes are associaed wih almos conemporaneous shifs o regime 2, consisen wih igh moneary condiions and increasing ineres raes; similarly, he 1955:09 and 1974:04 daes precede swiches 7

9 Thelackofcoherencebeweenregimesinsockandbondreurnsencouneredinhe wo-sae models is even clearer in he hree-sae models. The correlaion beween he smoohed sae probabiliies for sock and bond reurns shown in Figure 3 is sysemaically negaive or close o zero, irrespecive of how he saes are ordered. Ineresingly, all of he resuls on he presence of regimes in sock and bond reurns, heir inerpreaion, and he coherence beween regimes in sock and bond reurns are insensiive o he inclusion of auoregressive erms. For insance, he coefficiens of correlaions across sock and bond porfolios are similar o hose repored above when p =1and he sae probabiliies resuling from his model are pracically indisinguishable from hose in Figure 2. In conclusion, while here is a srong correlaion beween he process driving regimes in large and small firms sock reurns, bond reurns appear o be governed by a very differen process. This is already suggesed by he fac ha a wo-sae model is seleced for sock reurns while a hree-sae model is chosen for bond reurns and is furher sressed by he difference in he sae ransiion probabiliy esimaes of he wo-sae models. 5 The fac ha a hree-sae specificaion fis excess bond reurns much beer han a simpler, woregime model and ha hese saes are weakly correlaed wih hose idenified in he sock porfolios indicaes ha muliple regimes are needed o capure he join disribuion of sock and bond reurns. 3 A join model for sock and bond reurns Earlier sudies of regime swiching in sock and bond reurns focused on separaely modeling sock reurns or he evoluion in ineres raes, bu do no consider heir join disribuion. When considering he join sochasic process of reurns on socks and bonds, we have o o sae 3, in which bond reurns are moderae. As explained by Romer and Romer (1989), heir daes are supposed o deec only pure, conracionary moneary shocks. This explains why we find more shifs han heir daes. We hank an anonymous referee for leading us o explore hese issues. 5 While bond reurns imply ha he average duraion of a bear marke is almos 13 monhs, he sock reurns sugges an esimae beween four (large socks) and nine (small socks) monhs. 8

10 carefully deermine he number of saes in heir join disribuion and need o pay aenion o differences in heir individual sae characerisics. To capure he possibiliy of regimes in he join disribuion of asse reurns, consider an n 1 vecor of reurns in excess of he T-bill rae, y =(y 1,y 2,..., y n ) 0. Suppose ha he mean, covariance and possibly also serial correlaion in reurns are driven by a common sae variable, S, ha akes ineger values beween 1 and k: px y = µ s + A j,s y j + ε. (5) j=1 Here µ s =(µ 1s,..., µ ns ) 0 is an n 1 vecor of mean reurns in sae s, A j,s is he n n marix of auoregressive coefficiens associaed wih lag j 1 in sae s, and ε =(ε 1,...,ε n ) 0 N(0, Ω s ) follows a mulivariae normal disribuion wih zero mean and sae-dependen covariance marix, Ω s,givenby "Ã! 0 # px px E y µ s A j,s y j!ãy µ s A j,s y j s = Ω s (6) j=1 j=1 Regime swiches in he sae variable, S, are assumed o be governed by he ransiion probabiliy marix, P, wih elemens Pr(S = s,s 1 = s 1 )=p ss 1, s,s 1 =1,.., k. (7) Each regime is hus he realizaion of a firs-order Markov chain wih consan ransiion probabiliies. While simple, his model allows asse reurns o have differen means, variances and correlaions in differen saes. This means ha he risk-reurn rade-off can vary over saes in a way ha can have srong implicaions for invesors asse allocaion. For example, knowing ha he curren sae is a persisen bull marke will make mos risky asses more aracive han in a bear sae. Likewise, if sock marke volailiy is higher in recessions han in expansions, equiy invesmens are less aracive in recessions unless heir mean reurn rises commensurably. Esimaion of he parameers of he join model is relaively sraighforward and proceeds by opimizing he likelihood funcion associaed wih (5) - (7). Since he underlying sae 9

11 variable, S, is unobserved we rea i as a laen variable and use he EM algorihm o updae our parameer esimaes, c.f. Hamilon (1989). 3.1 Deerminaion of he number of saes Before urning o he selecion of he number of saes for he join model, we firs consider he implicaions of he analysis of he univariae series in Secion 2. Suppose ha each of he n univariae reurn series is governed by a Markov swiching process of he form (1) - (2). Also assume ha he innovaion erms are simulaneously correlaed, "Ã! # px px E y i a ij,si y i j µ isi!ãy m a mj,sm y m j µ msm s i,s m = σ imsi s m, j=1 j=1 alhough for all i 6=m and all q 6= 0, E[(y i q P p j=1 a ij,s i y i q j µ isi q )(y m P p j=1 a mj,s m r m j µ msm )] = 0 (no serial correlaion or cross-correlaion). Under no furher resricions on he relaionship beween he individual sae variables {s 1,...,s n } he saes (S ) for he join process {y 1,..., y n } can be obained from he produc of he individual saes: (8) S = ny S i = S 1 S 2... S n. (9) i=1 This gives a oal of k = Q n i=1 k i possible saes and k(k 1) sae ransiion probabiliies. Under independence beween he individual saes, he ransiion probabiliy marix defined on he join oucome space is simply he Kronecker produc of he individual ransiion marices and he number of ransiion probabiliy parameers o be esimaed reduces o P n i=1 k i(k i 1) which can be considerably smaller han k(k 1) when n is large. For example, in he bivariae case (n =2)wehave Pr(s 1 = a, s 2 = a s 1 1 = b, s 2 1 = b )=P ab [1] P a b [2]. (10) Obviously, he original n-variable Markov swiching process wih Q n i=1 k i saes is perfecly equivalen o a modified univariae Markov swiching process characerized by k = Q n i=1 k i 10

12 differen regimes and a single ( Q n i=1 k i) ( Q n i=1 k i) dimensional ransiion probabiliy marix P = P 1 P 2... P n. (11) In pracical mulivariae problems of even moderae size his represenaion is no, of course, feasible o use. For example, in he case wih hree variables each of whose marginal disribuion has hree saes (n =3,k i =3)he oal number of saes would be 27, involving he esimaion of 702 parameers in he ransiion probabiliy marix alone. This suggess he need for carefully considering ways for he economeric modeler o reduce he se of saes required o capure he essenial dynamics of he join disribuion. To deermine he number of saes for he join model, k, we underake an exensive specificaion search, considering values of k =1, 2, 3, 4, 5 and differen values of he auoregressive order, p. Resuls from he specificaion analysis are presened in Table 2. In all cases lineariy is very srongly rejeced no maer how many saes and lags are presen in he regime swiching model. The Hannan-Quinn informaion crierion suppors four saes. There is only weak evidence of an auoregressive componen in asse reurns. We herefore sele on a four-sae regime swiching model wihou auoregressive erms Inerpreaion of he Saes Having deermined he number of saes we nex focus on heir economic inerpreaion. Table 3 repors he parameers of he four-sae regime swiching model while Figure 4 plos he associaed smoohed sae probabiliies. For reference we also show he esimaes of a single-sae model wih no auoregressive erms. I is relaively sraighforward o inerpre he four regimes. Regime 1 is a crash sae characerized by large, negaive mean excess reurns and high volailiy. I includes he wo oil price shocks in he 1970s, he Ocober 1987 crash, he early 1990s, and he Asian 6 The number of parameers involved in our model depends on he number of asses, n, henumberof saes, k, and he number of auoregressive lags and is equal o (nk + pn 2 k + k n(n+1) 2 + k(k 1)). For he preferred model n =3, k =4, p =0, so we have 48 parameers and 1,656 daa poins for a sauraion raio (he number of daa poins per parameer) of

13 flu. Regime 2 is a low growh regime characerized by low volailiy and small posiive mean excess reurns on all asses. Regime 3 is a susained bull sae in which sock prices especially hose of he small socks grow rapidly on average. Ineres raes frequenly increase in his sae and excess reurns on long-erm bonds are negaive on average. The drawback o he high mean excess reurns on small socks is heir raher high volailiy, while large socks and bonds have less volaile reurns. Noice he big difference beween mean reurns on small and large socks in regimes 2 and 3. In sae 2 he mean reurn of large socks exceeds ha of small socks by abou 7% per annum, while his is reversed in sae 3. Regime 4 is a bounce-back regime wih srong marke rallies and high volailiy for small socks and bonds. 7 Mean excess reurns, a annualized raes of 27%, 55%, and 12%, are very large in his sae as is heir volailiy. Correlaions beween reurns also vary subsanially across regimes. The correlaion beween large and small firms reurns varies from a high of 2 in he crash sae o a low of 0.50 in he recovery sae. The correlaion beween large cap and bond reurns even changes signs across differen regimes and varies from 0.37 in he recovery sae o -0 in he crash sae. Finally, he correlaion beween small sock and bond reurns goes from -6 in he crash sae o 0.12 in he slow growh sae. Mean reurns and volailiies are greaer in absolue erms in he crash and recovery regimes, so i is perhaps unsurprising ha persisence also varies considerably across saes. The crash sae has low persisence and on average only wo monhs are spen in his regime. Ineresingly, he ransiion probabiliy marix has a very paricular form. Exis from he crash sae are almos always o he recovery sae and occur wih close o 50 percen chance suggesing ha, during volaile markes, monhs wih large, negaive mean reurns cluser wih monhs ha have high posiive reurns. The slow growh sae is far more persisen 7 The volailiy esimae may seem low for he large socks. However, i should be recalled ha, for each sae, he volailiy esimae is measured around he mean reurn for ha sae. Esimaes of he condiional volailiy saring from sae four also depend on he probabiliy of shifing o anoher sae, muliplied by he squared value of he difference beween ha sae s mean and he meanreurninsaefour,summed across saes

14 wih an average duraion of seven monhs. The bull sae is he mos persisen sae wih a sayer probabiliy of 8. On average he marke spends eigh successive monhs in his sae. Finally, he recovery sae is again no very persisen and he marke is expeced o say jus over hree monhs in his sae. The seady sae probabiliies, reflecing he average ime spen in he various regimes are 9% (sae 1), 40% (sae 2), 28% (sae 3) and 23% (sae 4). Hence, alhough he crash sae is clearly no visied as ofen as he oher saes, i is by no means an oulier sae ha only picks up exremely rare evens. I is ineresing o relae hese saes o he underlying business cycle. Correlaions beween smoohed sae probabiliies and NBER recession daes are 0.32 (sae 1), (sae 2), -1 (sae 3), and 0.18 (sae 4). Noice ha since he sae probabiliies sum o one, by consrucion if some correlaions are posiive, ohers mus be negaive. This suggess ha indeed, he high-volailiy saes - saes 1 and 4 - occur around official recession periods Mean and Variance Resricions The preferred four-sae regime swiching model is characerized by a large number of parameers so i is herefore legiimae o ask wheher a more parsimonious specificaion can be consruced by imposing furher resricions on he parameer space, as in, e.g., Ang and Bekaer (2002a, pp ). Alhough he resuls repored in Table 3 confirm ha mos of he mean excess reurns parameers in µ s are significanly differen from zero and differ from each oher, i is commonly found ha mean asse reurns are difficul o esimae precisely, suggesing ha he fi of our model would no be grealy reduced by resricing he inercep vecor µ o be idenical across regimes: y = µ + ε ε N(0, Ω s ), (12) 8 I could be argued ha he sae probabiliies backed ou from movemens in financial asse reurns should lead economic recession monhs. Indeed, he correlaion beween he sae-1 probabiliy lagged 6 monhs and he NBER recession indicaor rises o 0. 13

15 Table 4 repors he parameer esimaes from his resriced model. The imposed resricions lead o imporan changes in he ransiion dynamics. Regime 1 in he resriced model has no persisence and is bes characerized as a purely ransien sae ha leads o regime 4 ( ˆP [1, 4] = 0.99). Furhermore, regime 1 iself is likely o be accessed mosly from regime 4 ( ˆP [4, 1] = 4), so he resuling model implies a sequence of relaively calm periods (regimes 2 and 3) briefly inerspersed by a period wih highly volaile markes (regimes 1 and 4). In view of he similariy beween ˆΩ 1 and ˆΩ 4 in his model, effecively he consrained model is an overparameerized version of a much simpler hree-sae model wih regime-independen µ. The parameric resricions implied by he null hypohesis ha mean reurns do no vary across saes are srongly rejeced using a likelihood-raio es, LR = 2( ) = 36. This yields a p-value of 004. Anoher resricion naurally suggesed by he resuls in Tables 3 and 4 is ha he covariance marices are idenical in he highly volaile crash and recovery regimes. To invesigae his possibiliy, we esimaed he four-sae model (5) subjec o he resricion ˆΩ 1 = ˆΩ 4. Resuls are provided in Table 5. The resuling esimaes of he high-variance covariance marix are, as expeced, an average of he unresriced esimaes of ˆΩ 1 and ˆΩ 4. The six parameer resricions implied by he null hypohesis ha ˆΩ 1 = ˆΩ 4 were srongly rejeced by means of a likelihood-raio es, LR = 2( ) = 27.64, which implies a p-value of 001. Clearly he daa suppors correlaions and volailiies ha are differen even in he wo regimes wih he highes volailiy. 14

16 4 Addiional Predicor variables Equaion (5) can easily be exended o incorporae an m 1 vecor of addiional predicor variables, x 1.Define he (m + n) 1 vecor z =(y 0 x0 )0. Then (5) is readily exended o z = µ s px + A j,s z j + ε, (s =1,..., k) (13) µ xs j=1 ε x where µ xs =(µ x1s,..., µ xms ) 0 is he inercep vecor for x in sae s, {A j,s } p j=1 are now (n+m) (n+m) marices of auoregressive coefficiens in sae s, and (ε 0 ε 0 x) 0 MN(0, Ω s ), where Ω s is an (n + m n + m) covariance marix. In his exended model predicabiliy of reurns occurs hrough wo channels. Mos obviously, if he auoregressive erms or lagged predicor variables are significan, he condiional mean of sock and bond reurns are predicable. Even in he absence of ime-varying predicor variables or auoregressive erms, predicabiliy arises in general as long as here are wo saes, s and s 0 for which µ s 6=µ s 0. Variaion in he sae probabiliies over ime will hen lead o ime-variaion in expeced reurns. Variaions in he covariance marix across saes, will lead o furher predicabiliy in higher order momens such as volailiy, correlaions and skews. This seup is direcly relevan o he large lieraure in finance ha has repored evidence of predicabiliy in sock and bond reurns. While many predicor variables have been proposed, one of he key insrumens is he dividend yield; see, e.g., Campbell and Shiller (1988) and Fama and French (1988, 1993). Noice ha when k =1, equaion (13) simplifies o a sandard vecor auoregression. Our model hus ness as a special case he sandard linear (single-sae) model used in much of he asse allocaion lieraure; see e.g. Barberis (2000). 4.1 Empirical Resuls Again we conduced a baery of ess o deermine he bes model specificaion. To selec he lag order for he exended model we firs esimae a range of VAR(p) models,where 15

17 p is gradually augmened and informaion crieria used o evaluae he effec of including addiional lags. 9 All informaion crieria as well as a sequenial likelihood raio es poined owards a VAR(1) model. This is unsurprising given he srong persisence of he dividend yield. Turning nex o he search across differen numbers of saes, k, able 6 suggess ha, alhough he model has now been exended by an auoregressive erm, a four-sae model coninues o provide he bes rade-off beween fi and parsimony. 10 Table 7 shows he parameer esimaes for he preferred model specificaion. Resuls for a comparable single sae VAR(1) model are shown o provide a benchmark for he richer four-sae model. In he linear model he dividend yield predics reurns on small socks bu does no appear o be significan in he equaions for reurns on large socks and long-erm bonds. 11 As expeced, he dividend yield is highly persisen and he esimaed correlaion marix shows a srong posiive correlaion beween he reurns of small and large socks while sock reurns are srongly negaively correlaed wih simulaneous shocks o he dividend yield. Esimaes of he auoregressive marices, Â j, sugges ha he effec of changes in he dividend yield on asse reurns coninues o be srong in he muli-sae model. Inclusion of he dividend yield herefore does no weaken he evidence of muliple saes, nor does he presence of such saes in a framework ha allows for heeroskedasiciy remove he predicive power of he dividend yield over asse reurns. 12 As in he pure reurn regime-swiching model, he ransiion probabiliy marix coninues o have a very special srucure. Exis from saes 1 and 2 are almos always o he bull-burs 9 As suggesed by Krolzig (1997, p. 128) he auoregressive order p in a regime swiching model can convenienly be pre-seleced as he maximal lag order p obained in he single sae VAR. 10 There is clear evidence of separae regimes in he univariae dividend yield series. Independenly of he specific form of he esimaed regime swiching model, he null of lineariy was rejeced using Davies (1977) upper bounds for he p-values of likelihood raio ess in he presence of nuisance parameers. 11 A one sandard deviaion increase in he dividend yield incrases he annualized mean excess reurn on small socks by 1.2%. The corresponding figures for large socks and bonds are 3% and 5%, respecively. 12 Afer conrolling for regime swiching in a univariae model for he reurns of a value-weighed porfolio of socks, Schaller and van Norden (1997) find ha he dividend yield remains significan in a regime swiching model wih homoskedasic shocks bu is insignifican once he volailiy is allowed o be sae-dependen. 16

18 sae 4, while exis from saes 3 and 4 are predominanly o he crash sae 1. To assis wih he economic inerpreaion of he four regimes, Figure 5 plos he smoohed sae probabiliies. Regime 1 coninues o pick up marke crashes, characerized by negaive, double-digi (on an annualized basis) mean excess reurns (-38% and -49% for large and small firms and -10% for bonds). 13 The dividend yield is relaively high in his sae (4%) and volailiy is also above average. The probabiliy of regime 1 is highes around he oil price shocks of he 1970s, he recession of he early 1980s, he Ocober 1987 sock marke crash, he Kuwai Invasion in 1990 and he Asian flu. I maches he beginning of major U.S. business cycle conracions and also picks up many well-known episodes wih low reurns and high volailiy. In seady sae his regime occurs 15% of he ime alhough i has an average duraion of only wo monhs. The auoregressive coefficiens indicae subsanial predicabiliy of small and large firms reurns in his sae. Lagged bond reurns and dividend yields have he sronges predicive power and small socks reurns are also srongly serially correlaed. The dividend yield is highly persisen bu unpredicable from pas asse reurns in his sae. Regime 2 is a slow growh sae characerized by single-digi mean excess sock reurns (9.9% and 8.8% for large and small firms, respecively) and moderae volailiy. Long periods of ime was spen in his sae during he sagnaing markes of he mid-1970s and he firs half of he 1990s. This sae is highly persisen, lasing on average almos 16 monhs and occurring close o one-hird of he ime. There is less predicabiliy of reurns in his regime alhough he dividend yield sill affecs sock reurns, again wih he expeced posiive sign. Regime 3 is a bull sae in which he annualized mean excess reurn on large and small socks is 11% and 14%, respecively. This sae includes he long expansions of he 1950s and 1960s, he high growh periods of , he proraced boom of he 1980s as well as 13 The mean excess reurn in each regime (k) is esimaed as he weighed sample average of mean excess reurns: ( 1999:12 X =1954:02 ) 1 ( 1999:12 X bπ k, =1954:02 ) bπ k, E 1 [y s 1 = k] where E 1 [y s 1 = k] =ˆµ s 1 =k +  s 1 =ky 1. 17

19 some periods in he early 1990s. I is ofen accompanied by ineres rae cus and herefore by posiive mean excess reurns on long-erm bonds. A 2.8%, he mean dividend yield, on he oher hand, is low. Reurn volailiies reach inermediae levels. This regime is also highly persisen and occurs one hird of he ime, lasing on average almos 15 monhs. Reurn predicabiliy is weak in his sae alhough he dividend yield remains posiively correlaed wih sock reurns. Finally, regime 4 is again a bull-burs regime wih srong sock marke rallies accompanied by subsanial volailiy. Annualized mean excess reurns on large and small socks are 57% and 95%, respecively, while long-erm bonds have mean excess reurns of 17%. This sae hus picks up eiher he iniial and more impeuous sages of business cycle upurns or marke rebounds following crashes. Many peaks of U.S. expansions and marke booms such as , or he new economy of occurred during his sae which does no las long wih an average duraion of only 2 monhs. Neverheless, a 18%, is seadysae probabiliy is quie high. As in he firs sae, here is some predicabiliy and he dividend yield forecass reurns on small caps and long-erm bonds in he fourh sae. 4.2 Relaion o Fama-French Facors Fama and French (1993) proposed a number of facors o explain he cross-secional variaion in sock and bond reurns. For sock reurns hey considered he marke porfolio, a porfolio capuring book-o-marke effecs (HML) and a porfolio capuring size (SMB). For bond reurns hey considered a defaul premium and a erm premium facor. Alhough he analysis of Fama and French (1993) was primarily concerned wih explaining paerns in he cross-secion of reurns on sock and bond reurns by means of facors measured during he same period, while our analysis is concerned wih predicive paerns in reurns, i is ineresing o relae expeced reurns implied by our four-sae model o he five Fama-French facors. To do so, we esimae univariae predicive regressions of he expeced sock and bond reurns implied by he regime swiching model (ŷ i )onhelagged 18

20 values of he Fama-French facors: ŷ i = β 0i + β 1i HML 1 + β 2i SMB 1 + β 3i r MKT 1 + β 4i DEF 1 + β 5i TERM 1 + ε i, (14) where HML is he reurn on he Fama-French High-minus-Low Book-o-Marke sock porfolio, SMB is he reurn on he Fama-French Small-minus-Big Size sock porfolio, r MKT is he excess reurn on he marke (he value-weighed CRSP porfolio), DEF is he defaul premium (difference beween he yield on Moody s BAA and AAA corporae bonds), and TERM is he erm premium (difference beween he reurn on long-erm governmen bond yields and 30-day T-bill raes). Resuls are shown in Figure 6. The correlaion coefficiens beween expeced reurns calculaed from (14) vs. he ones implies by he four-sae regime model esimaed in Secion 3 are 53 for bonds, 73 for small caps and for large caps. Hence, here is a posiive bu weak relaionship beween he lagged Fama-French facors and expeced reurns under he regime swiching model. 5 Conclusion The join process of sock and bond reurns follows a rich and complex dynamic paern. We found evidence ha sandard linear models do no capure essenial feaures of his disribuion and ha four regimes are required o capure he ime-variaion in he mean, variance and correlaion beween large and small firms sock reurns and long-erm bond reurns. Two regimes capure periods wih high volailiy and low persisence and wo regimes are inermediae saes wih higher persisence. Furhermore, ransiions beween hese regimes ake a very special form wih exis from he highly volaile bear sae mosly being o he volaile recovery sae wih high expeced reurns, suggesing he presence of bounce-back effecs afer a period wih large negaive reurns. These conclusions do no change when we add he dividend yield as a predicor in our model. There are several exensions of his work ha would be ineresing o consider. Firs, while we used diagnosic ess and informaion crieria o choose he number of regimes in he univariae and mulivariae models, anoher possibiliy is o selec he preferred model 19

21 on he basis of is forecasing performance in an ou-of-sample experimen. I is a common finding in economics ha nonlinear models provide good in-sample fis, bu perform worse ou-of-sample. One could selec he archiecure of he regime swiching model - primarily he number of saes and he number of auoregressive erms - on he basis of is ou-ofsample forecasing performance. A second exension of our resuls is o consider heir asse allocaion implicaions. This is done in Guidolin and Timmermann (2003). I urns ou ha he regime swiching model no only affecs he opimal level of asse holdings across a range of preference specificaions, bu also affecs how he opimal asse allocaion relaes o he invesor s ime horizon, bear saes giving rise o upward sloping demand for socks while bull saes give rise o a downward sloping demand for socks as a funcion of he invesmen horizon. References [1] Ang A., and G., Bekaer, 2002a, Inernaional Asse Allocaion wih Regime Shifs, Review of Financial Sudies, 15, [2] Ang, A., and G., Bekaer, 2002b, Regime Swiches in Ineres Raes, Journal of Business and Economic Saisics, 20, [3] Barberis, N., 2000, Invesing for he Long Run When Reurns Are Predicable, Journal of Finance, 55, [4] Campbell, J., and R. Shiller, 1988, The Dividend Price Raio and Expecaions of Fuure Dividends and Discoun Facors, Review of Financial Sudies, 1, [5] Davies, R., 1977, Hypohesis Tesing When a Nuisance Parameer Is Presen Only Under he Alernaive, Biomerika, 64, [6] Engel, C., and J., Hamilon, 1990, Long Swings in he Dollar: Are They in he Daa and Do Markes Know I?, American Economic Review, 80,

22 [7] Fama, E., and K., French, 1988, Dividend Yields and Expeced Sock Reurns, Journal of Financial Economics, 22, [8] Fama, E., and K., French, 1993, Common Risk Facors in he Reurns on Socks and Bonds, Journal of Financial Economics, 33, [9] Fong, W.-M., and K.H., See, 2001, Modelling he Condiional Volailiy of Commodiy Index Fuures as a Regime Swiching Process, Journal of Applied Economerics, 16, [10] Franses, P.H. and D. van Dijk, 2000, Non-linear Time Series Models in Empirical Finance. Cambridge Universiy Press. [11] Garcia, R., 1998, Asympoic Null Disribuion of he Likelihood Raio Tes in Markov Swiching Models, Inernaional Economic Review, 39, [12] Gray, S., 1996, Modeling he Condiional Disribuion of Ineres Raes as Regime- Swiching Process, Journal of Financial Economics, 42, [13] Guidolin, M., and A., Timmermann, 2003, Sraegic Asse Allocaion under Regime Swiching, mimeo, Universiy of Virginia and UCSD. [14] Hamilon, J., 1989, A New Approach o he Economic Analysis of Nonsaionary Time Series and he Business Cycle, Economerica, 57, [15] Hamilon, J., and G., Lin, 1996, Sock Marke Volailiy and he Business Cycle, Journal of Applied Economerics, 11, [16] Hansen, B., 1992, The Likelihood Raio Tes Under Non-Sandard Condiions: Tesing he Markov Swiching Model of GNP, Journal of Applied Economerics, 7, S61-S82. [17] Kim, C.-J., C., Nelson, and R., Sarz, 1998, Tesing for Mean Reversion in Heeroskedasic Daa Based on Gibbs-Sampling-Augmened Randomizaion, Journal of Empirical Finance, 5,

23 [18] Krolzig, H.-M., 1997, Markov-Swiching Vecor Auoregressions, Berlin, Springer-Verlag. [19] Perez-Quiros, G. and A., Timmermann, 2000, Firm Size and Cyclical Variaions in Sock Reurns, Journal of Finance, 55, [20] Romer, C. and D., Romer, 1989, Does Moneary Policy Maer? A New Tes in he Spiri of Friedman and Schwarz, NBER working paper No [21] Rydén, T., T., Teräsvira, and S., Asbrink, 1998, Sylized Facs of Daily Reurn Series and he Hidden Markov Model, Journal of Applied Economerics, 13, [22] Schaller, H., and S., van Norden, 1997, Regime Swiching in Sock Marke Reurns, Applied Financial Economics, 7, [23] Schwer, G., 1989, Why Does Sock Marke Volailiy Change over Time?, Journal of Finance, 44, [24] Turner, C., R., Sarz, and C., Nelson, 1989, A Markov Model of Heeroskedasiciy, Risk, and Learning in he Sock Marke, Journal of Financial Economics, 25, [25] Whielaw, R., 2001, Sock Marke Risk and Reurn: An Equilibrium Approach, Review of Financial Sudies, 13,

24 Table 1 Univariae Regime Swiching Models for Sock and Bond Reurns This able repors esimaion resuls for he model y p i = is + i= 1 µ a y + σ u i, s i i is i, where s is governed by an unobservable, discree, firs-order Markov chain ha can assume k values (saes). u i is IIN(0,1). i =1, 2, 3 indexes excess reurns on porfolios of large and small socks and 10-year T-bonds. Daa are monhly and obained from he CRSP apes. The sample period is 1954: :12. For likelihood raio ess we repor in square brackes he p-value based on he χ 2 (r) disribuion (r is he number of resricions) and in curly brackes he p-value based on Davies (1977) upper bound. Parameer Large caps Small caps Bonds Large caps Small caps Bonds Panel A Two-Sae AR(0) Models Panel B Two-Sae AR(1) Models µ µ a 1 NA NA NA a 2 NA NA NA σ σ P P Log-likelihood Linear Log-lik LR es of lineariy [00] {00} [00] {00} [00] {00} [00] {00} [00] {00} [00] {00} Hannan-Quinn Panel C Three-Sae AR(0) Models Panel D Three-Sae AR(1) Models µ µ µ a 1 NA NA NA a 2 NA NA NA a 3 NA NA NA σ σ σ p p p p p P Log-likelihood Linear Log-lik LR es of lineariy [00] {00} [00] {00} [00] {00} [00] {00} [00] {00} [00] {00} Hannan-Quinn

25 Table 2 Model Selecion for Sock and Bond Reurns (join model) This able repors values of he log-likelihood funcion, lineariy ess and informaion crierion values for he mulivariae Markov swiching condiionally heeroskedasic VAR model: where µ s is he inercep vecor in sae s, ε = [ ε ε 2 ε3 ]'~ N( 0, Ωs ) 1 y js = µ + A r s p j= 1 js ε j + A is he marix of auoregressive coefficiens a lag j = 1 in sae s and. S is governed by a firs-order Markov chain ha can assume k values. p auoregressive erms are considered. The hree monhly reurn series comprise a porfolio of large socks (ninh and enh CRSP size decile porfolios), a porfolio of small socks (firs and second CRSP deciles), and 10-year T-bonds. Reurns are measured in excess of he 30-day T-bill rae. The daa was obained from he CRSP apes. The sample period is 1954: :12. MMSIA is shor for Mulivariae Markov Swiching wih regime-dependen Inercep and Auoregressive erms, while MMSIAH inroduces regime-dependen heeroskedasiciy. Model (k,p) Number of parameers Loglikelihood LR es for lineariy Hannan- Quinn Base model: MSIA(1,0) MMSIA(1,0) NA MMSIA(1,1) NA MMSIA(1,2) NA Base model: MSIA(2,0) MMSIA(2,0) MMSIAH(2,0) MMSIAH(2,1) MMSIAH(2,2) Base model: MSIA(3,0) MMSIA(3,0) MMSIAH(3,0) MMSIAH(3,1) MMSIAH(3,2) Base model: MSIA(4,0) MMSIA(4,0) MMSIAH(4,0) MMSIAH(4,1) MMSIAH(4,2) MMSIAH(4,3)

26 Table 2 (coninued) Model Selecion for Sock and Bond Reurns (join model) Model (k,p) Number of parameers Loglikelihood LR es for lineariy Base model: MSIA(5,0) MMSIA(5,0) MMSIAH(5,0) MMSIAH(5,1) MMSIAH(5,2) Hannan- Quinn

27 Table 3 Esimaes of Regime Swiching Model for Sock and Bond Reurns This able repors parameer esimaes for he mulivariae regime swiching model y = µ + ε, s where µ s is he inercep vecor in sae s and ε = [ ε 1 ε 2 ε3 ]'~ N( 0, Ωs ). S is governed by a firs-order Markov chain ha can assume four values. The hree monhly reurn series comprise a porfolio of large socks (ninh and enh CRSP size decile porfolios), a porfolio of small socks (firs and second CRSP deciles), and 10-year T-bonds. Reurns are measured in excess of he 30-day T-bill rae. The daa was obained from he CRSP apes. The sample is 1954: :12. The firs panel refers o he single-sae benchmark case (k = 1). Values on he diagonals of he correlaion marices are annualized volailiies. Aserisks aached o correlaion coefficiens refer o covariance esimaes. For mean coefficiens and ransiion probabiliies, sandard errors are repored in parenheses. Panel A Single Sae Model Large caps Small caps Long-erm bonds 1. Mean excess reurn 066 (018) 082 (026) 008 (009) 2. Correlaions/Volailiies Large caps *** Small caps ** *** Long-erm bonds *** Panel B Four Sae Model Large caps Small caps Long-erm bonds 1. Mean excess reurn Regime 1 (crash) -510 (146) -810 (219) -131 (047) Regime 2 (slow growh) 069 (027) 008 (033) 009 (016) Regime 3 (bull) 116 (032) 167 (048) -023 (007) Regime 4 (recovery) 226 (055) 458 (098) 098 (033) 2. Correlaions/Volailiies Regime 1 (crash): Large caps *** Small caps 233 *** 479 *** Long-erm bonds -060 * *** Regime 2 (slow growh): Large caps *** Small caps *** *** Long-erm bonds 043 *** *** Regime 3 (bull): Large caps *** Small caps 707 *** *** Long-erm bonds *** Regime 4 (recovery): Large caps *** Small caps *** 429 *** Long-erm bonds *** *** 3. Transiion probabiliies Regime 1 Regime 2 Regime 3 Regime 4 Regime 1 (crash) 940 (0.1078) 001 (001) 2409 (417) 818 Regime 2 (slow growh) 483 (233) 529 (403) 307 (110) 682 Regime 3 (bull) 439 (252) 701 (296) 822 (403) 038 Regime 4 (recovery) 616 (501) (718) 827 (498) 836 * significan a he 10% level, ** significan a he 5% level, *** significan a he 1%. level