DO BUBBLES AND TIME-VARYING RISK PREMIUMS AFFECT STOCK PRICES? A KALMAN FILTER APPROACH a

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1 DO BUBBS AD TIM-VARYIG RISK RMIUMS AFFCT STOCK RICS? A KAMA FITR AROACH a Dr. -Tarn Cen, Acaema Snca, Tawan Dr. C. James Hueng, Unversy o Alabama, USA Dr. Cen-u Je n, aonal Tawan Unversy, Tawan ABSTRACT Ts paper separaes e valy o e speccaon o e unamenal soc prce moel rom e mplcaons o bubbles. Te me-varyng rs premum moel oerba an Summers, 986 s use o explcly erve e msspeccaon componen. We consruc a sae-space moel an use Kalman Fler o esmae e relaonsps beween e observable prce/ven an e unobservable bubbles/msspeccaon. Te moel s apple o CRS an S& 5 aa. Te resuls sow a e unamenal prce moel oes no escrbe e mare prces well. Te me-varyng rs premum s mporan n explanng soc prce movemens. o sgncan evence o bubbles s oun. I. ITRODUCTIO Ts paper sues weer speculave bubbles an moel-msspeccaon exs n e soc prces. We separae e valy o e speccaon o e unamenal soc prce moel rom e mplcaons o a bubble componen o soc prces. Te unamenal prce moel s consruce by ucas 978. He proposes a, base on e raonal expecaons ypoess, e soc prce soul relec e scoune values o uure vens an soc prces. Ts eal speccaon, owever, s no suppore by many emprcal sues [e.g. Sller 98; eroy an orer 98; Blancar an Wason 98]. Te alure o e unamenal prce moel as oen been consere as evence o e exsence o raonal bubbles. A raonal bubble relecs a sel-ulllng bele a an asse s prce epens on varables a are no pars o mare unamenals. Te exsence o bubbles causes screpances beween an asse s prce an s unamenal value. However, sues suc as Floo an Garber 98 an Hamlon an Weman 985 argue a e evence presene n avor o e clam o selulllng speculave bubbles mg nsea ave arsen rom raonal agens responng o mare unamenals a e researcers canno observe. One possbly or e unobservable unamenals o exs s a e unamenal prce moel s msspece. Tess or bubbles a ae correc moelspeccaon as gven an nerpre evaons rom e unamenal prce as evence o bubbles may no be juse because ey rejec e jon ypoess o no bubble an correc moel-speccaon.

2 xplcly expressng e msspeccaon n e moel, owever, s rarely seen n e leraure. An excepon s e suy or e Cagan ypernlaon moel by Durlau an Hooer 994. In er moel, e prce seres s ecompose no componens corresponng o unamenals, bubbles, an moel-msspeccaon. To be able o separae e moel-msspeccaon rom a bubble componen, ey propose wo consrans, low an soc, o uncover moel noses. By esng e orogonaly o ese consrans agans e normaon se, ey are able o nrecly es wc nose componen, bubbles or msspeccaon, s responsble or nonzero noses. Tey n a e emprcal evaons rom e unamenal prce soluon are prmarly by vrue o msspeccaon raer an bubbles. Unle Durlau an Hooer 994, s paper recly moels e msspeccaon componen an reans e explc reerence o e moelmsspeccaon an bubbles. Te msspeccaon s moele by usng e me-varyng rs premum ramewor propose by oerba an Summers 986. We en express e moel n a sae-space orm, reang e bubbles an e moel-msspeccaon as unobservable sae varables. Te Kalman lerng algorm s apple o esmae e sae-space moel. Te Kalman ler s an algorm or sequenally upang a lnear projecon or e sae-space moel. I also opens e way o e maxmum leloo esmaon o e unnown parameers n e moel. In aon, e Kalman Fler s esgne o wor w nonsaonary aa, because e ler prouces srbuons o e sae varables a are cononal on e prevous realzaon o e saes. Tereore, nonsaonary n sel presens no problem [Bomo 99]. Te res o s paper s organze as ollows. Te nex secon emonsraes e eorecal moel an s sae-space orm. Secon 3 scusses e emprcal resuls. Te nal secon conclues e paper. II. THORTICA MOD Ts secon buls e eorecal ounaon an e ramewor or eecng e bubbles an e moel-msspeccaon conane n e soc prces. We explan ow e speculave bubbles an moel-msspeccaon are generae n e soc prce moel. Speccally, uner e assumpons o no-arbrage an raonal expecaons, ucas 978 sows a e curren soc prce s equal o e scoune value o e sum o e expece soc prce an e ven nex pero: g g [ ], g were s e soc prce a pero,, s e scoun rae, s e ven a pero, s e cononal expecaon gven e normaon se avalable a.

3 Subsung orwar or an mposng e ransversaly conon yels e unamenal prce o e soc, [ ]. Ts s e amous asseron a e unamenal prce o e soc s equal o e sum o e scoune low o uure vens. oe a s only one o e soluons o. Ang o any process B a sases e conon B B 3 g sll solves e equaon. Tereore, e soc prce can be parameerze by g B, were B s calle a raonal bubble. ow we urn o e moel-msspeccaon aspec o e soc prces. oerba an Summers 986 argue a e soc prces are oo volale o be explane by e unamenal prce moel. Canges n rs are responsble or a sgncan par o e soc prce volaly. Tereore, ey sugges an alernave ypoess o me-varyng rs premums. Conser e ollowng me-varyng rs premum moel: ' ϕ, 4 were ϕ Π r j, r s e consan rs-ree neres rae, an j j s e rs premum. Assume a e mean value o e rs premum s a consan. Usng e rs-orer Taylor seres expanson o expan aroun yels ', 5 r were r. Te above ervaon r ollows closely o a n oerba an Summers 986.

4 Assume e scoun rae r -. Ten e rs erm on e rgan se o 5 becomes e unamenal prce, an e secon erm s e moel-msspeccaon cause by e me-varyng rs premum, wc s an unobservable mare unamenal. Denoe e moel-msspeccaon componen as. 6 By ang e speculave bubble B o 5, e soc prce can be parameerze by B. 7 ex we use e ervaon n Hansen an Sargen 98 o approxmae e unamenal prce. Assume a e srucure o e socasc process o ven can be caracerze by an ARq process: q q e. 8 As sown n oe, e raonal expecaons ypoess mples a e unamenal prce can en be expresse as: q q, were j s are uncons o an j j,,,..., q. Tereore, equaon 7 relaes e observable an o e unobservable B an. To express s moel n a sae-space orm, we nee o n e ynamcs o e unobservable varables. Manpulang on 6 yels, 9 were.

5 By raonal expecaons,. Furermore, ollowng oerba an Summers 986, assume a ollows an AR process:, were. Ten equaon 9 becomes. In aon, ollowng Wu 995, assume a e bubble erm B a sases 3 ollows an AR process: B - B b, were b. Ten e me-varyng rs premum moel can be expresse as a sae-space moel w me-varyng parameers: [ ], B. b B B 3 quaon s nown as e observaon equaon, wc relaes e observable varables o e unobservable varables. quaon 3, nown as e sae equaon, escrbes e ynamcs o e unobservable varables. Te sae vecor an e resual vecor are assume o be mulvarae Gaussan w. ~ b b, σ σ σ σ σ 4 Here we assume a b s uncorrelae w an. Snce bubbles are ene as e componen a canno be explane by e mare unamenals suc as e vens an e rs premum, s assumpon seems o be reasonable enoug. On e oer an, can be sown a an are correlae. 3

6 III. TH MIRICA RSUTS Te sae-space moel an 3 s esmae by e Kalman ler. Te Kalman ler s an algorm or sequenally upang a lnear projecon or e sae-space moel. In aon, allows us o use e maxmum leloo esmaon o esmae e unnown parameers n e moel. Aer e esmaes are obane, we erve e smooe esmaes o e sae vecor an s error covarance marx, wc are e esmae values base on e ull se o aa collece. For eals abou e Kalman ler an smooer, see Anerson an Moore 979 an Hamlon 994. Te moel s apple o wo ses o soc prce aa. Te rs s e CRS Cener or Researc n Soc rces monly aa rom 96: o 997:, w a oal o 864 observaons. Te YS/AMX/ASDAQ Value-Wege Mare Inex aa are use. Te soc prce an ven are consruce by usng e oal reurn Seres RT an e capal apprecaon reurn Seres RTX. Te secon aa se s e quarerly S& 5 nex aa rom 935: o 997:4, w a oal o 5 observaons. Te aa are rom e 998 ssue o e Sanar & oor's sascal servce: secury prce nex recor. All prces an vens are ve by e Consumer rce Inex an seasonally ajuse. We rs esmae equaon 8, e auoregressve process o e ven. Te lag leng s cosen o be e sores lag a reners e error erm we nose. Te jung-box Q es s use o es e seral correlaon n e resuals rom e OS regresson. Te lag leng o reen s cosen o avo e rso-wel orer auocorrelaon or e monly CRS aa, an ree or e quarerly S& 5 aa o avo rs-o-sx orer auocorrelaon n e resuals. Aer e lag lengs beng selece, e lagge varables w nsgncan esmae coecens are roppe rom e regressons. Te AR processes are en re-esmae an e seral correlaon n e resuals s cece agan. Aer s sep, e unamenal prce s only a uncon o. Aer e socasc process o e ven s eermne, e sae-space moel s esmae by maxmum leloo. 4 Te esmae parameers are sown n Table. Mos o e esmaes are sgncanly eren rom zero a e raonal sgncance level. As expece, e esmae o s wn s eorecal range,,, n bo aa se. Tey are also bo very close o one, wc mples a rs, e me preerence s no very srong n e soc mare an secon, e bubble s very perssen. In aon, e esmae o, e rs-orer auocorrelaon coecen o e rs premum, s abou. an e sanar error s less an % o e esmae, wc sases e assumpon a e rs premum s saonary. Aer e esmaes are obane by maxmum leloo esmaon, we erve e smooe esmaes o e sae vecor. Fgure plos e mare prces, e esmae unamenal prces, e smooe esmaes o e bubbles, an e smooe esmaes o e msspeccaon componen or e CRS aa. Fgure plos ose or e S& 5 aa.

7 For e CRS aa se Fgure, e volaly n e soc prces s mosly cause by me-varyng rs premum, wc can be seen rom e ac a n Fgures a an, e mare prces an e moel-msspeccaon componen generally move ogeer. Te unamenal prce Fgure b s relavely sable. Te sanar evaon o e esmae unamenal prce s 4.36, wle e mare prce an e msspeccaon componen ave a sanar evaon o an33.65, respecvely. On e oer an, e bubble componen Fgure c as a ownwar ren urng e sample pero, bu s magnue s very small compare o e oer componens. Inee, e roo mean square orecasng errors o bubbles are all muc bgger an e esmaes o e bubbles, wc ncaes a ere s no sgncan evence o bubbles exsng n e CRS soc prces. 5 For e S& 5 aa se Fgure, e unamenal prce s more volale compare o a n e CRS aa. However, s sll no as mporan as e moel-msspeccaon componen n explanng e volaly n soc prces. Te sanar evaon o e esmae unamenal prce s abou 4, wle e mare prce an e msspeccaon componen ave a sanar evaon o 9. an 63.98, respecvely. Te bubble componen n e S& 5 aa se as an upwar ren, bu agan s very small an nsgncan. Anoer neresng resul rom Fgures an s a, wle e soc prce as been rsng sarply snce 99, e unamenal prce was relavely sable urng s pero. Ta s, e recen boomng soc prce s mosly cause by e rs premum componen, no e canges n vens. Ts s conssen w e argumen a nvesors n recen years ave begun o unersan a e socs are no as rsy as ey oug [Glassman an Hasse 999]. Tereore, ey requre lower rs premum. As a resul, ey ave b up soc prces. IV. COCUSIO Te raonal ypoess es on soc bubbles s base on e assumpon a e unamenal prce moel s well spece. As a resul, e nng o bubbles coul be a resul o moel msspeccaon. Ts paper sues weer bo speculave bubbles an moel-msspeccaon exs n e soc prces. Te me-varyng rs premum moel n oerba an Summers 986 s use o reveal e msspeccaon aspec n e soc prces. A sae-space moel an e Kalman Fler are use o erve e unamenal prces an e unobserve bubbles an msspeccaon conane n e soc prces. Te moel s apple o e CRS monly aa rom 96 o 997 an S& 5 quarerly aa rom 935 o 997. Te resuls sow a e unamenal prce moel canno explan e volaly o e soc prces. Mos o e movemens n e soc prces are cause by e me-varyng rs premum. In aon, ere s no sgncan evence o bubbles n bo aa ses. revous nngs o bubbles may be a resul o a me-varyng rs premum.

8 Table : Te esmaon resuls o e parameers n e sae-space moel CRS Daa: arameer: smae: S. rror: arameer: σ b l m n smae: S. rror: og-eloo: S& 5 Daa: arameer: 3 smae: S. rror: arameer: σ b l m n smae: S. rror: og-eloo: -7.8 oe: Te parameers s are e coecens n e ven generang process: qq e, were e nsgncan lags are roppe rom e regresson; s e scoun rae; s e rs-orer auocorrelaon coecen o e rs premum; an l, m, an n are e parameers n e ollowng Colesy ecomposon: l l σ σ. Ts ecomposon s use o guaranee m n m n σ σ a e varance-covarance marces n e Maxmum eloo esmaon are all posve ene. Te varance/covarance parameers σ s are ene n equaon 4. Ts paper proves a sarng pon or explcly moelng e msspeccaon o e unamenal prce moel. Te cause o e msspeccaon s lme o e me-varyng rs premum ere. xensons o s suy nclue oer speccaons or bubbles an moel-msspeccaon componen.

9 Fgure : CRS Daa a. rce Inex b. Funamenal rce c. Bubble Componen. Msspeccaon Componen Fgure : S& 5 Daa a. rce Inex b. Funamenal rce c. Bubble Componen. Msspeccaon Componen a. We are graeul o Mary Fs, Dav VanHoose, ree anonymous reerees, an sesson parcpans a e Busness an conomcs Socey Inernaonal Conerence n os Angeles or elpul commens. Cen receve researc suppor rom aonal Scence Councl, Tawan, gran #SC86-45-H--.

10 DOTS. Te Kalman ler algorm was use by Burmeser an Wall 98 o es prce-level bubbles o e German ypernlaon, an by Wu 995 o es or excange-rae bubbles.. Rewre e socasc process o ven n a marx orm: were, e ΓD D., e e, D q q an 3 M M O O O M M M Γ Tereore, D D Γ. Ten can be wren as D I g D g gd Γ Γ Γ q q, were [ ] g an j s are uncons o an j s j,,..., q. 3. Speccally, [ ]. σ

11 Tereore,, we ge [ ]. 4. Ieally, e coecens n e ven processes soul be jonly esmae w oer parameers n e maxmum leloo esmaon. However, s woul resul n a ba o e moel. Tereore, snce e OS esmaon o e ven process s conssen, we aop e curren wo-sep esmaon. In e rs sep we oban e esmaes o s an express e unamenal prce as a uncon o only. In e secon sep, s esmae, w e oer parameers n e moel, by e maxmum leloo esmaon. 5. Te roo mean square precon errors o e bubbles an msspeccaon are oo bg o be ploe clearly w e esmaes. Tereore, we o no sow em n e paper. Tey are avalable rom e auors upon requess.

12 RFRCS Anerson, B.D.O. an Moore, J.B. 979 Opmal Flerng, rence-hall: nglewoo Cls. Blancar, O.J. an Wason, M.W. 98 Bubbles, Raonal xpecaons, an Fnancal Mares, n aul Wacel, e., Crses n e conomc an Fnancal Srucure, exngon Boos, Bomo,. J. 99 Sably o Velocy n e Major Inusral Counres: A Kalman Fler Approac, IMF Sa Worng apers, Vol. 38, o. 3, Burmeser,. an Wall, K.D. 98 Kalman Flerng smaon o Unobserve Raonal xpecaons w an Applcaon on e German Hypernlaon, Journal o conomercs,, Durlau, S.. an Hooer, M.A. 994 Msspeccaon versus Bubbles n e Cagan Hypernlaon Moel, n Coln Hargeaves e., onsaonary Tme Seres Analyss an Conegraon, Oxor Unversy ress. Floo, R.. an Garber,.M. 98 Mare Funamenals versus rce-evel Bubbles: Te Frs Tess, Journal o olcal conomy, 88, Glassman, J.K. an Hasse, K.A. 999 DOW 36,, Te ew Sraegy or rong rom e Comng Rse n e Soc Mare, Tmes Busness, ew Yor. Hamlon, J.D. 994 Tme Seres Analyss, rnceon Unversy ress. Hamlon, J.D. an Weman C.H. 985 Te Observable Implcaons o Sel-ulllng xpecaons, Journal o Moneary conomcs, 6, Hansen,.. an Sargen, T.J. 98 Formulang an smang Dynamc near Raonal xpecaon Moels, Journal o conomc Dynamcs an Conrol,, eroy, S.F. an orer, R.D. 98 Te resen-value Relaon: Tess Base on Imple Varance Bouns, conomerca, 49, ucas, R Asse rces n an xcange conomy, conomerca, 46, oerba, J.M. an Summers,.H. 986 Te erssence o Volaly an Soc Mare Flucuaons, Amercan conomc Revew, 76, 4-5.

13 Sller, R.J. 98 Do Soc rces Move Too Muc o be Juse by Subsequen Canges n Dvens? Amercan conomc Revew, 7, Wu, Y. 995 Are ere Raonal Bubbles n Foregn xcange Mares? vence rom an Alernave Tes, Journal o Inernaonal Money an Fnance 4, 7-46.