Bayesian Forecasting of Stock Prices Via the Ohlson Model

Transcription

1 Baesan Forecasng of Soc Prces Va he Ohlson Model B Qunfang Flora Lu A hess Submed o he Facul of WORCESER POLYECHIC ISIUE n paral fulfllmen of he requremens for he Degree of Maser of Scence n Appled Sascs b Ma 5 APPROVED: Balgobn andram Professor and major adsor Huong Hggns Assocae Professor and co-adsor Bogdan Vernescu Professor and Deparmen Head

2 Absrac Oer he pas decade of accounng and fnance research he Ohlson 995 model has been wdel adoped as a framewor for soc prce predcon. Whle usng he accounng daa of 39 companes from SP5 n hs paper Baesan sascal echnques are adoped o enhance boh he esmae and predce quales of he Ohlson model comparng o he classcal approaches. Specfcall he classcal mehods are used for he eploraor daa analss and hen he Baesan sraeges are appled usng Maro chan Mone Carlo mehod n hree sages: nddual analss for each compan groupng analss for each group and adape analss b poolng nformaon across companes. he base daa whch conss of quarers obseraons sarng from he frs quarer of 998 are used o mae nferences for he regresson coeffcens or parameers ealuae he model adequac and predc he soc prce for he frs quarer of 4 when he real obseraons are se as he es daa o ealuae he predce abl of he Ohlson model. he resuls are aeraged whn each specfed group caegorzed a he general ndusral classfcaon GIC. he emprcal resuls show ha classcal models resul n larger soc prce predcon errors more poselbased predcons and hae much smaller eplanaor powers han Baesan models. A few ransformaons of boh classcal and Baesan models are also performed n hs paper howeer ransformaons of he classcal models do no ouwegh he usefulness of applng Baesan sascs.

3 Acnowledgemens I would le o epress m deep graude o he followng people for her academc fnancal or sprual suppor n fnshng hs hess. Adsors: Balgobn andram and Huong Hggns. Whou her help and gudance here would no hae been such a pece of wor. here are no words ha can epress he apprecaon from he boom of m hear o hem. And I promse hs s no he end of m deoon o he research on hs opc. I wll eep n ouch wh hem whereer I am. Professors n he Mah Deparmen of WPI: Joseph D. Peruccell Jason Wlbur Andrew Swf Carlos Morales and Chrsopher J. Larsen. her courses hae srenghened m nowledge n Appled Sascs and Appled Mah. Also han o Maer Hum Wllam W. Farr John Goule Daln ang Peer R. Chrsopher and Wllam J. Marn for her gudance n m wo ears of worng eperence as a eachng asssan. Secreares n he Mah Deparmen of WPI: Colleen Lews Ellen Macn Deborah Rel. he are he nces mos paen and helpful secreares ha I e eer seen. Bes wshes o hem foreer! M parens and broher: Zhongao Lu Rongzh Lu Hanlang Lu. he are alwas here when I m n need; he accep me when I m rejeced; he are sll proud of me een when I do no feel an good abou a sngle par of mself. M fancé and hs faml: John Caca Behlnne Vanella and Jennfer Graham ec. he gae me a warm home when I was n he hardes me n USA and hae been supporng me een snce hen. he le Chnese I loe hem. WPI Ballroom Dance eam: Bors Moss and Mles Schofeld ec. he helped me deelop a errfc hobb --- dancng whch s m lfe m soul and m bes jo n he free me. hans for all hose ecng pracce lessons and dance pares. 3

4 Colleagues and classmaes: Fang Huang Yan Ba Guochun Lu Alna Ursan Shnj Uemura Gregor Mahews Shawn Hallnan Ashle Moras Rajesh Kondapanen Jasraj Kohl Danel Onofre Bjaa Padh Elzabeh eera and Sco Lane ec. I had fun dscussng homewor problems wh hem. Somemes we also aled abou lfe worres and fuure dreams. he are nce smar and er frendl. I wll eep hem n m memor. Frends: Jean Lu Mar Berolna Aln Srbu Congme Ma Ch Hu Pong Pang Fang Lu Yucong Huang Qunwe A Chao Ku and Xaoln an ec. he are no man bu enough o mae me a rch and confden woman. Las bu no leas o hose who doub m nellgence wllpower and poenal. I wll fan he flame of dssasfacon and mae a mracle o hem o mself and o hs wonderful world. 4

5 Chaper he Ohlson 995 Model and Daa of S&P 5. he Ohlson 995 Model Oer he pas wo decades n fnance and accounng area consderable aenon has been pad o he relaonshp beween accounng numbers boo alues earnngs ec. and he frm alue. he Ohlson 995 approach o he problem of soc aluaon relaes secures prces o accounng daa and prodes a srucure for applcable modelng. he Ohlson 995 Valuaon Model has been wdel adoped b researchers and praconers on profabl analss as a framewor for he fundamenal aluaon of eques. I also has been deeloped no seeral ersons e.g. Felham-Ohlson 995 Valuaon Model Bernard s 995 Ohlson Appromaon Model Lu-Ohlson Valuaon Model and Callen s Ohlson AR Valuaon Model. For a hsorcal deelopmen process of he Ohlson model see Append A. hs paper ealuaes he Ohlson 995 Forecasng Model OFM or brefl he Ohlson 995 model and uses o forecas soc prces. OFM s a praccable case of Bernard s 995 Ohlson Appromaon Model see Append C. For a sngle frm OFM saes: he soc prce per share s a lnear funcon of he compan s boo alue per share and abnormal earnngs per share for he followng four perods wh normall dsrbued nnoaon erms whch represens oher nformaon whose source s uncorrelaed wh abnormal earnngs. In mahemacal form can be epressed as 4 b 3 4 a.. where denoes he soc prce per share a me b s he boo alue per share a me represens he abnormal earnng a me a 6 s he ecor of 5

6 nercep and slope coeffcens of he predcors a a a a b 3 4 s he ecor of nercep and predcors and s he nnoaon error or resdual erm. o undersand he abnormal earnng erm we can ew as a conracon of aboe normal earnng. Ohlson 995 proposes he abnormal earnng as.. a r b where s he earnng per share a me for a compan r s he dscoun rae a me. a a a a Snce he alues of he followng four perods 3 4 are used o forecas he soc prce hs paper uses he epeced earnngs o replace n... ha s...3 a E[ ] r b For he nnoaon erm Ohlson 995 assumes has a frs order auoregresse srucure AR. hs assumpon can be descrbed as ε ε..4 where s he correlaon coeffcen of me seres ε s he whe nose s he arance of he whe nose. oe ha f < he AR process s saonar. From.. and..4 we can ge ε ε ε. 6

7 Bu when ε here ess unobsered alues and hs paper les combned as ε. herefore epressons.. and..4 can be d ε ε where and are hree unnown parameers besdes he nercep and regresson coeffcens n....5 Epresson..5 s he complee form of he Ohlson 995 Forecasng Model herenafer he Ohlson model ha s used n hs paper.. Rereng Daa of S&P 5 from homson OE Analcs Yearl or quarerl daa from arous sources hae been appled o es he Ohlson model. For nsances besdes man ess ha use US daa Bao & Chow 999 es he usefulness of he Ohlson model usng daa from lsed companes n he People s Republc of Chna; McCrae & lsson compare he dfference beween Swedsh and US frms b usng daa from a Swedsh busness magazne Bonner-Fndaa daabase and I/B/E/S daabase; Oa uses emprcal edence from Japan ec. hs paper apples quarerl daa of S&P 5 from hosmon OE Analcs o he Ohlson model. S&P 5 s one of he mos wdel used measures of U.S. soc mare performance and s consdered o be a bellweher for he U.S. econom. S&P refers o Sandard & Poor s whch s a dson of he McGraw-Hll Companes Inc. 5 companes are seleced among he leaders n he major ndusres drng U.S. econom b he S&P Inde 7

8 Commee for mare sze lqud and secor represenaon. A small number of nernaonal companes ha are wdel raded n he U.S are ncluded. he needed daa of S&P 5 can be rereed from homson OE Analcs b s Ecel Add-n sofware proded b he homson Corporaon whch s a global leader n prodng alue-added nformaon sofware applcaons and ools n he felds of law a accounng fnancal serces and corporae ranng and assessmen ec. homson OE Analcs s a web based applcaon ha allows users o research nformaon abou dfferen companes and mares ncludng curren soc prces olume raded EPS epeced earnng per share and so on. he homson OE Analcs Ecel Add-n s one of he mos aluable feaures ha homson OE Analcs offers s users. Usng he Add-n fnancal analss can pull daa drecl no Ecel from a wealh of fnancal daabases such as Worldscope Compusa U.S. Prcng I/B/E/S and I/B/E/S Hsor and Eel b usng he powerful PFDL Premer Fnancal Daabase Language. Iems n he rereed daa are: oal Asses oal Lables Preferred Soc Common Shares Ousandng from daabase Worldscope ; EPSmeanQR-4 and EPSConsensusForecasPerodQR-4 from daabase I/B/E/S Hsor noe ha hese are monhl daa; Dow Jones Indusr Group DJIC General Indusr Classfcaon GIC Dow Jones Mare Secor DJMS and GICSSECOR from homson Fnancal ; PrceClose and 3-monh -bll reasur bll rae from Daasream. Boo alue per common share BPS can be calculaed b he frs four ems n followng formula: BPS oal Asses oal Lables Preferred Soc/Common Shares Ousandng. Companes forecas her epeced earnngs eer monh for he followng four fscal quarers. hs paper uses he laes forecas alue for each quarer o represen he correspondng quarer alue. he quarerl EPS are eraced from he monhl daa of EPSmeanQR-4 and EPSConsensusForecasPerodQR-4. For easer use alues 8

9 of Dow Jones Indusr Group Dow Jones Mare Secor and he Compan Iden Kes are ransformed no negers. For eample use 34 nsead of he orgnal alue C34. o undersand hese fnancal / accounng erms please see Append D. Append E eplans how o use Ecel Add-n. Append F eplans how o erac quarerl daa ou of monhl daa. Afer deleng all he mssng and ncomplee daa pons and he daa pons ha cause programmng errors 39 companes are seleced. hs fnal quarerl daa se hae pons for each compan coerng 5 quarers from he frs quarer of 998 and he frs quarer of 4. I s formaed no 6 ems whch are all numercal alues and conan 4 secors DJIC GIC DJMS and GICSSECOR compan den e ID me PrceClose BPCS-3 EPS-4 and R 3-monh -bll rae..3 Eploraor Daa Analss b Classcal Approaches Whle consderng he deas n arous ersons of he Ohlson model hs paper scs o he man frame of he Ohlson model n..5 and ses up dfferen models whch are descrbed n able.3. for he eploraor analss. hese models can be classfed no hree groups b dsngushng he assumpon of he nnoaon erm: ndependen errors among me perods whch belongs o he ordnar lnear regresson srucure OLR AR srucure for he error and AR srucure for he error. he man pon of hs classfcaon s o chec wheher he AR assumpon s proper for he nnoaon erm of he Ohlson model. Besdes hs four nds of ransformaon o he erm of soc prce per share are appled o he model under eher AR or AR assumpon for he nnoaon erm: logarhmc ransformaon log rans square roo ransformaon sqr rans cubc roo ransformaon cur rans and nerse ransformaon n rans. wo relael beer ransformaons are o be seleced b he classcal sascal analss. hs paper assumes her prores o be adoped n furher research b he nnoae mehods for he reason of her beng more f o he daa. he purpose of usng ransformaons s o mproe boh he esmae 9

10 and predce quales of he Ohlson model. able Varous Models ame Equaon OFM --- OLR 4 ε ε b d a OFM --- AR 4 ε ε b d a OFM --- AR 4 ε ε b d a log rans of OFM AR log 4 ε ε b d a sqr rans of OFM --- AR 4 ε ε b d a cur rans of OFM --- AR 4 3 ε ε b d a n rans of OFM --- AR / 4 ε ε b d a log rans of OFM AR log 4 ε ε b d a sqr rans of OFM --- AR 4 ε ε b d a cur rans of OFM --- AR 4 3 ε ε b d a n rans of OFM --- AR / 4 ε ε b d a wo procedures PROC REG and PROC AUOREG n SAS are he classcal mehods ha are used o he whole daa se o es he esmae abl of he models. Specfcall PROC REG s onl used o model OFM---OLR and PROC AUOREG s used o he models wh ARp srucures for he nnoaon erm. hree nds of crera are used o compare her esmae ables: R-squares oal R-square and Regress R- square Aae Informaon Creron AIC and Baesan Informaon Creron BIC. See able.3. for he emprcal resuls. oe ha R-square s he coeffcen of deermnaon regresson sum of squares dded b oal sum of squares. oal R- square s R-square and regress R-square s R-square adjused for addonal coaraes. he are nearl he same n PROC REG procedure bu can be er dfferen n PROC AUOREG procedure especall when he nnoaon erms are hghl correlaed among

11 me perods. BIC s a quan proporonal o he negae log lelhood afer all parameers are negraed ou. AIC s a deance measure.e. dfference beween obsered and fed models. Models wh small AIC and BIC alues are preferred. able Oerall Esmae Abl Comparson of Varous Models Model oal Regress R-square R-square AIC BIC OFM --- OLR OFM --- AR OFM --- AR log rans of OFM AR sqr rans of OFM --- AR cur rans of OFM --- AR n rans of OFM --- AR log rans of OFM AR sqr rans of OFM --- AR cur rans of OFM --- AR n rans of OFM --- AR he followng conclusons can be drawn b comparng he R-squares AIC s and BIC s n able.3.. Usng PROC REG o model OFM---OLR R-square urns ou o be er small.84. Usng PROC AUOREG o he oher models he oal R-square alues are oer.75 o all ecep n he cases of usng nerse ransformaon. For he models wh ARp srucure o he nnoaon erm he resuls show bg dfference beween he oal R-square >.75 and he Regress R-square <.. All hese resuls ndcae ha he assumpon of ndependence of he nnoaon erms among dfferen me perods canno sand. In oher words seng an AR p srucure o he nnoaon erm can be a sound assumpon. AR srucure s no beer han AR for he hae eremel close R-square alues. hs s n lne wh he concluson drawn b Callen. he oal R-square alue.63 under he nerse ransformaon s much less han whou a ransformaon.756 whle he oal R-square alues under he oher hree ransformaons are slghl bgger han whou a ransformaon. hs concludes ha he nerse ransformaon canno enhance he esmae abl whle he oher hree can slghl enhance he esmae abl.

12 Based on oal R-square alue cubc roo ransformaon enhances he esmae abl he mos.7733 hen he square roo ransformaon.776 and hen he log ransformaon.768. Bu he dfferences among hem are er small. Based on AIC and BIC cubc ransformaon has he smalles alue 6 and 76 hen he log ransformaon 653 and 7. he square roo ransformaon has much larger AIC and BIC 83 and 86. herefore cubc roo ransformaon and log ransformaon are relae beer han he ohers. Afer comparng he esmae ables of he models hs paper proceeds furher eploraor analss b concenrang on 3 models: OFM --- AR log rans of OFM AR and cur rans of OFM --- AR. In order o es he predce ables of he models he rereed daa of S&P 5 are dded no wo pars for each compan. he frs par conans he frs perods of daa whch wll be used as base daa o esmae regresson coeffcens; he second par has he s perod of daa whch wll be used as es daa o compare wh he predcons for hs perod from he base daa. he same base daa and es daa as n hs dson are also used n he followng chapers. Afer usng PROC AUOREG o he daa n each GIC group see able.3.3 for he General Indusral Classfcaon Dsrbuon he esmaed regresson coeffcens for hree models are colleced n able.3.4. he resuls show ha he nercep BPS and abnormal earnngs per share of he frs wo followng quarers are generall sgnfcan n he Ohlson Model. he alues n bold are sgnfcan ohers are nsgnfcan. able General Indusral Classfcaon GIC Dsrbuon GIC Value Class Oerall Indusral Ul ransporaon Bans/Sangs and Loan Insurance o. of Frms Oher Fnancal oe: when GIC equals means hs group ncludes all 39 frms.

13 he PROC AUOREG procedure also ges he predced alues of he s perod usng he base daa. o compare he predce ables among he hree seleced models for dfferen GIC groups he creron s defned as R ˆ / where R s he relae dfference of predced soc prce oer real soc prce for a.3. compan ŷ s he predced soc prce of a compan for he s perod s he real soc prce of a compan for he s perod. able Esmaed Parameers from Base Daa GIC Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR LOG-rans of OFM AR CUR-rans of OFM --- AR GIC Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR LOG-rans of OFM AR CUR-rans of OFM --- AR GIC Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR LOG-rans of OFM AR CUR-rans of OFM --- AR GIC 3 Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR LOG-rans of OFM AR CUR-rans of OFM --- AR GIC 4 Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR LOG-rans of OFM AR CUR-rans of OFM --- AR GIC 5 Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR LOG-rans of OFM AR CUR-rans of OFM --- AR GIC 6 Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR LOG-rans of OFM AR CUR-rans of OFM --- AR

14 he followng conclusons can be drawn from able.3.5 where he quanles of R he number of nonnegae R s o. and he number of negae R s o. - are colleced. he dgal and afer he names of ransformaons are o dsngush dfferen scales of measuremen. denoes usng he orgnal scale denoes usng he ransformed scale. GIC able Quanles of R and umber of onnegae/egae R s --- Orgnal Scale --- ransformed Scale o. of Frms Model Mn Q Q Q3 Ma o. o. - OFM --- AR log rans of OFM AR log rans of OFM AR cur rans of OFM AR cur rans of OFM AR OFM --- AR log rans of OFM AR log rans of OFM AR cur rans of OFM AR cur rans of OFM AR OFM --- AR log rans of OFM AR log rans of OFM AR cur rans of OFM AR cur rans of OFM AR OFM --- AR log rans of OFM AR log rans of OFM AR cur rans of OFM AR cur rans of OFM AR OFM --- AR log rans of OFM AR log rans of OFM AR cur rans of OFM AR cur rans of OFM AR OFM --- AR log rans of OFM AR log rans of OFM AR cur rans of OFM AR cur rans of OFM AR OFM --- AR log rans of OFM AR log rans of OFM AR cur rans of OFM AR cur rans of OFM AR

15 he emprcal resuls show ha he dsrbuons of R are asmmercal wh long als whch suggess he 5% quanle Q of R as a major creron. From able.3.5 he followng conclusons can be drawn. Based on Q alues n orgnal scale he rao alue ranges from 4.% GIC 5 o 44% GIC and 4% oerall GIC under no ransformaon from 7.4% GIC o 36.5% GIC and 34.7% oerall GIC under log ransformaon and from 9.7% GIC 5 o 46.6% GIC 6 and 36% oerall GIC under cubc roo ransformaon. Based on Q alues n ransformed scale he rao alue ranges from 4.7% GIC 5 o 9.8% GIC and 8.8% oerall GIC under log ransformaon and from 6.3% GIC 5 o 3.8% GIC 6 and.9% oerall GIC under cubc roo ransformaon. hese conclude ha he log ransformaon mproes he predce abl more han he cubc roo ransformaon does whle usng he classcal mehod. In all cases he number nonnegae R s s much larger han he number of negae R s whch shows he hgh oeresmaon b he classcal mehod. he oo large magnude of R and he eremel hgh oeresmaon sae ha usng he classcal mehod he PROC AUOREG procedure o nerpre he Ohlson model s no effcen enough n forecasng soc prces. A beer approach s desred o mproe boh he esmae and predce ables of he Ohlson model. Summarl he eploraor daa analss b PROC REG/AUOREG confrms he AR assumpon of he nnoaon erm n he Ohlson model and he promsng effec of adopng logarhmc ransformaon as well as cubc roo ransformaon. I suggess ha he remanng wor focus on 3 models: OFM --- AR log rans of OFM AR and cur rans of OFM --- AR. Snce he Ohlson model s no able o predc he soc prce effcenl b he classcal means hs paper apples an nnoae sascal mehod Baesan sascal analss o he 3 chosen models n he remanng wor..4 An Oulne of Baesan Sascal Analss In he followng hree chapers of hs paper Baesan approaches are used for he purpose of sasfng he requremen of mprong boh he esmae and predce 5

16 quales of he Ohlson model comparng o he classcal mehods. In deal Chaper uses he mos basc Baesan echnques o each compan whch s he case ha dfferen companes hae dfferen regresson coeffcens; Chaper 3 apples he Baesan mehod b leng all he companes n each group share he same regresson coeffcens. Whle Chaper represens he nddual analss Chaper 3 represens he groupng analss. And Chaper 4 ends up o be he adape analss b poolng nformaon across companes. ha s dfferen companes hae dfferen regresson coeffcens n Chaper 4 and n he mean me he are pooled ogeher. Bascall Chaper 4 compromses he deas n Chaper and he ones n Chaper 3. For each Baesan approach n followng hree chapers he man ass are o mae nferences for he regresson coeffcens or parameers ealuae he model adequac and es he predce abl of he Ohlson model. Chaper 5 concludes all he wor n hs paper whch ncludes he comparson among he hree Baesan approaches as well as he comparson of he bes Baesan approach o he classcal mehod. 6

17 Chaper Baesan Sascal Analss for Inddual Frm. Baesan Verson of he Ohlson Model for a Sngle Frm As an ereme case hs chaper assumes all he companes are ndependen of each oher and hae her own regresson coeffcens n he Ohlson model. A he er begnnng of applng he Baesan sascal analss o each compan a Baesan erson of he Ohlson model s se up n he followng hree seps. Frs for a specfc compan descrbe he obseraon b he parameers } {. Under he assumpon ha he obseraons are condonall ndependen among he me perods we can ge he lelhood funcon from epresson..5: p... Second assgn a pror dsrbuon o each unnown parameer. he pror dsrbuon represens a populaon of possble parameer alues from whch he parameer of curren neres has been drawn. he gudng prncple s o epress he nowledge and unceran abou he parameer as f s alue could be hough of as a random realzaon from he pror dsrbuon. In order o ge he praccal adanage of beng nerpreable as addonal daa and compuaonal conenence hs paper assgns he conjugae pror dsrbuons as follows:. b a I U K π π π θ π Γ.. 7

18 he hperparameers { θ } n.. are se as follows. P. θ B X X X. he dea of seng θ s o use he esmaon of n he ordnar lnear regresson model P. ε ε OLM- d b he mehod of leas squares. oe ha X coaraes ogeher wh an nercep s he mar of all a a a a b 3 4 s he regresson coeffcen ecor conssng of he and predcors and s he number of me perods. SS E X X where SS E B X s he sum of squares of he P errors of OLM- P s he number of regresson coeffcens ncludng he SS nercep E P s he esmaon of he n OLM- b he mehod of leas SSE squares and X X s he esmaon of he coarance mar of P n OLM-. Mulplng he esmaon of he coarance mar b s o add more arabl. P.3 B. From ε ε n epresson..5 we can ge he esmaon of whch s pon ha s ˆ B. ang each obseraon can as sarng ε ε. OLM- he dea of seng s o use he aeraged esmaon of n OLM-. SS P.4 E s he esmaon of arance n OLM- b he mehod of leas P 8

19 squares. P.5. b a. hese wo hperparameers are chosen b conenon or eperence n Baesan sascal analss. Assume ha all he parameers are ndependen of each oher he jon pror dsrbuon of he parameers can be epressed as. b a I U p K θ Γ..3 Fnall from he lelhood funcon n.. and jon pror dsrbuon n..3 we can ge he poseror dsrbuon of he parameers gen he daa usng Baes rule:. Γ K b a I U p p p θ..4. Gbbs Samplng he Gbbs sampler s an erae Mone Carlo algorhm desgned o erac he poseror dsrbuon from he racable complee condonal dsrbuons raher han drecl from he nracable jon poseror dsrbuon whch s dffcul o acqure n eplc form. In hs chaper he arge s o mae nferences on he parameers } { gen he daa. We consder he complee condonal dsrbuons. π. π. π and. π respecel. Here he condonng argumen denoes he obseraon and he remanng parameers. From he poseror dsrbuon n..4 we can dere he complee condonal dsrbuons. 9

20 Frs ΛΣ Λ Λ θ I K where Σ Σ Σ Σ Σ Σ Σ Σ Λ. Second Φ Φ Φ where. Φ hrd U. Fourh. b a gamma In he Gbbs sampler s mplemened usng he followng s seps. Sep oban sarng alues } {. Sep draw from π. Sep 3 draw from π. Sep 4 draw from π. Sep 5 draw from π. Sep 6 repea man man man mes.

21 hs paper chooses he sarng pons { } as follows. Frs X X X. SS Second SS B ae B ae where SS SS SS ae B ae SS B ae hrd B. SS Fourh E. p B. B ae and he deas of seng and are he same as he ones of seng he hper- parameers { θ } n.. whch s usng he esmaon of parameers from he OLM- and OLM-. he dea of seng s ang as he auocorrelaon of me seres ε n an AR srucure. hs chaper deelops wo algorhms usng Maro-chan Mone Carlo mehods a resrced algorhm ha enforces saonar condon b leng < on he seres and an unresrced algorhm ha does no..3 Forecas Afer geng he poseror dsrbuon of he parameers we can use o predc he fuure soc prces. In hs paper we wsh o forecas he soc prce a me perod

22 denoed b gen he daa. Leng { } he predcon can be sampled from he poseror predce dsrbuon f f π d..3. Leng M be a sequence of range M from he Gbbs sampler an esmaor of f s M h M f h f ˆ..3. o ge samples of we use daa argumenaon o fll n h o each h M o ge h h M from he normal dsrbuon n he 95% predce credble neral for can be compued from he.5% and 97.5% h emprcal quanles of he alues h M..4 Condonal Predce Ordnae We wan o assess he goodness of f of he Ohlson Model o he daa. One procedure s o calculae he log condonal predce ordnae log p wh M h h h p ϖ p.4. where denoes he random fuure obseraon a perod h denoes he obseraons from perod o denoes he h h draw of he parameers from he Gbbs sampler and ϖ h M f f f h f h h h h M. See Append G for he deraon of.4..

23 .5 Emprcal Resuls of Inddual Baesan Analss Afer geng he Baesan erson of he Ohlson model for each frm we f o he daa correspondng o each compan n he base daa se. eraons are run n he Gbbs sampler he frs draws are hrown awa and fnall draws are colleced b pcng one draw eer paces. Snce here are oo man companes 39 he resuls are aeraged for each GIC group. Besdes mang conclusons from he emprcal resuls hs chaper also res o decde whch models from {OFM --- AR log rans of OFM AR cur rans of OFM AR} wll be used for furher Baesan analss wheher he saonar resrcon s needed and whch measuremen scale o use orgnal one or he ransformed one. Four crera are used for he model aluaon: he relae dfference of he predced soc prce oer he real soc prce R numbers of nonnegae raos and negae raos o. and o.- lengh of 95% credble nerals and log condonal predce ordnae CPO. he rao of resdual s defned n he same wa as.3. n Chaper. Bu n hs chaper and he followng wo chapers o. and o.- denoe he rounded numbers of nonnegae and negae raos dded b respecel. he quanles of R as well as o. and o.- are colleced n able.5.a whle usng he saonar resrcon and n able.5. b for he case whou he saonar resrcon. he LB and UB n hese wo ables are calculaed from o. and o.- b formulas: LB pˆ pˆ pˆ / UB pˆ pˆ pˆ / where p ˆ o. / o. o.. he are he lower bound LB and upper bound UB of he 95% confdence neral of pˆ whch are used o chec he sae of oeresmaon. If.5 s beween LB and UB hen he mehod does no oeresmae he soc prces and ce ersa. 3

24 able.5.a --- Wh Saonar Resrcon -Orgnal Scale -ransformed Scale GIC o. of Frms Mehod Mn Q Q Q3 Ma o. o. - LB UB no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans

25 GIC able.5.b --- Whou Saonar Resrcon -Orgnal Scale -ransformed Scale o. of Frms Mehod Mn Q Q Q3 Ma o. o. - LB UB no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans Smlar o Chaper he emprcal resuls show ha he dsrbuons of R are asmmercal wh long als. hs chaper also uses he 5% quanle Q of R as a major creron. he followng conclusons can be drawn from able.5.a. Based on Q alues n orgnal scale he rao alue ranges from.5% GIC 6 o 9% GIC 5 and 7% oerall GIC under no ransformaon from.7% GIC o 8.9% GIC 5 and 6.9% oerall GIC under log ransformaon and from.7% GIC o 8.6% GIC 5 and 6.5% oerall GIC under cubc roo ransformaon. Based on Q alues n ransformed 5

26 scale he rao alue ranges from.5% GIC o.5% GIC 4 and % oerall GIC under log ransformaon and from.7% GIC o.9% GIC 5 and.% oerall GIC under cubc roo ransformaon. hese conclude ha wh saonar resrcon boh he log ransformaon and he cubc roo ransformaon mproe he predce abl comparng o he mehod whou usng an ransformaon When GIC s or 4.5 s no beween LB and UB; when GIC s 3 or 6.5 s no beween LB and UB; when GIC s.5 s beween LB and UB ecep n he case of usng log ransformaon under he orgnal scale; when GIC s 5.5 s beween LB and UB ecep n he case of usng log ransformaon under he ransformed scale. Snce.5 s no beween LB and UB for large groups we conclude ha usng Baesan mehod o each compan b resrcng saonar oeresmaes he soc prces. able.5.b ges he followng conclusons. Based on Q alues n orgnal scale he rao alue ranges from 3.8% GIC 3 o -8.5% GIC 4 and 6% oerall GIC under no ransformaon from 3.5% GIC 3 o 8% GIC 4 and 5.9% oerall GIC under log ransformaon and from.% GIC 6 o 7.8% GIC 4 and 5.4% oerall GIC under cubc roo ransformaon. Based on Q alues n ransformed scale he rao alue ranges from.% GIC o.4% GIC 4 and.7% oerall GIC under log ransformaon and from.% GIC 6 o.6% GIC 4 and.9% oerall GIC under cubc roo ransformaon. hese conclude ha boh he log ransformaon and he cubc roo ransformaon also mproe he predce abl comparng o he mehod whou usng an ransformaon whou saonar resrcon. When GIC s or 4.5 s no beween LB and UB; when GIC s 3 or 6.5 s no beween LB and UB; when GIC s.5 s beween LB and UB ecep n he case of usng log ransformaon under he orgnal scale; when GIC s 5.5 s beween LB and UB ecep n he case of usng log ransformaon under he ransformed scale. Snce.5 s no beween LB and UB for large groups we 6

27 conclude ha usng Baesan mehod o each compan b resrcng saonar oeresmaes he soc prces. Comparng he conclusons from able.5.a o he ones from able.5.b here es some slgh dfferences beween hem bu hs paper consders ha hose dfferences are mnor. here are wo hngs n common. Frs usng boh ransformaons can enhance he predce abl. Second he Baesan approach o each compan oeresmaes he soc prces for mos companes. able Mn Ma and Oerall Values of R Mehod ransformaon mn ma oerall Classcal no rans 4.% 44% 4% Sascal log rans 7.4% 36.5% 34.7% Analss cur rans 9.7% 46.6% 36% Inddual no rans.5% 9% 7% Baesan log rans.7% 8.9% 6.9% Analss cur rans.7% 8.6% 6.5% hs paper uses he mnmum mamum and oerall alues of R o compare he Baesan approaches wh he classcal mehod. able.5. ges hose alues under he orgnal scale from boh classcal sascal analss and nddual Baesan analss. I shows he huge mproemen of usng nddual Baesan approach o he Ohlson model compared o he classcal mehod. he aerage lenghs of credble neral CI are n able.5.3 from whch s eas o see ha he are shorer under saonar resrcon han whou saonar resrcon. In all case GIC 3 has he longes lengh GIC and hae he shores lengh. GIC hae smlar lengh. I seems ha he more companes a GIC group has he shorer he CI s. hs hns ha poolng nformaon across companes ma mproe he predce abl of he Ohlson model. Generall he aerage lenghs of CI s are que wde under he orgnal scale and eremel smaller under he ransformed scale. he sandard deaons are er bg under he orgnal scale and much smaller under he ransformed scale. hs mples ha he resuls under he ransformed scale mae more sense whch can also been ndcaed b able.5.a and b. 7

28 able Aerage Lengh of Credble Ineral GIC o. of Frms Mehod Wh S-Resrcon o S-Resrcon Ae. CI Sd Ae. CI Sd Lengh De Lengh De no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans no rans log rans log rans cur rans cur rans he log condonal predce ordnae CPO s o ealuae he model fng adequac. I alwas has negae alues and s calculaed under he orgnal scale n hs chaper. he bgger CPO s he beer he model fs he daa. able.5.4 ges he quanles and mean of CPO n each case and shows ha he CPO alues are smaller under saonar resrcon han whou saonar resrcon. 8

29 able CPO GIC o. of Frms GIC o. of Frms Wh Saonar Resrcon Mehod Mn Q Q Q3 Ma Mean no rans log rans cur rans no rans log rans cur rans no rans log rans cur rans no rans log rans cur rans no rans log rans cur rans no rans log rans cur rans no rans log rans cur rans Whou Saonar Resrcon Mehod Mn Q Q Q3 Ma Mean no rans log rans cur rans no rans log rans cur rans no rans log rans cur rans no rans log rans cur rans no rans log rans cur rans no rans log rans cur rans no rans log rans cur rans

30 Snce he dsrbuons of CPO are que smmercal whou long als hs chaper uses he mean alue as a major creron o analze CPO. he followng conclusons are for he case wh saonar resrcon. Under no ransformaon he mean alue of CPO ranges from GIC 3 o GIC and oerall GIC. Under log ransformaon he mean alue of CPO ranges from GIC 3 o GIC and oerall GIC. Under cubc roo ransformaon he mean alue of CPO ranges from GIC 6 o GIC 3 and -.67 oerall GIC. For each GIC group he mean alues of CPO are much larger under cubc roo ransformaon han under log ransformaon. Also he hae smaller alues under no ransformaon han under eher of he wo ransformaons. hese facs pu a lo wegh on usng boh log and cubc roo ransformaons n he Baesan analss of he Ohlson model. Afer analzng he emprcal resuls of he four crera hs paper goes bac o he decsons ha need o mae. Snce all crera show he mproemen of usng log ransformaon and cubc roo ransformaon hs paper decdes o use wo models for furher Baesan analss whch are log rans of OFM AR and cur rans of OFM AR. Snce usng he ransformed scale shows more sensble resuls hs paper decde o eep and sop usng he orgnal scale n he followng wo chapers. Abou he resrcon saonar andram & Peruccell 997 saes ha resrcng saonar seres o be saonar prodes no new nformaon bu resrcng nonsaonar seres o be saonar leads o subsanal dfferences from he unresrced case. In hs paper he resuls of boh aerage lenghs of he credble nerals and CPO show he benefs of resrcng saonar. Besdes he me plos of he me seres for he companes show ha mos seres loo saonar and a few do no. herefore hs paper decdes o use he saonar resrcon n furher Baesan analss. 3

31 In summar he nddual Baesan analss n hs chaper srongl mproes he predce abl of he Ohlson model comparng o he classcal analss n Chaper. he groupng analss n Chaper 3 and adape analss b poolng nformaon across companes n Chaper 4 wll be appled o he wo ransformed models wh saonar resrcon. Mos furher resuls wll be colleced under he ransformed scale. 3

32 Chaper 3 Baesan Daa Analss whn Each GIC Group 3. Baesan Verson of he Ohlson Model for a GIC Group As anoher ereme case hs chaper assumes ha all he companes n he same GIC group share he same parameers } { and he GIC groups are ndependen of each oher hang her own regresson coeffcens n he Ohlson model. he same srucure as n Chaper s used n hs chaper. Frs of all a Baesan erson of he Ohlson model for he groupng analss s se up n he followng hree seps. Sep descrbe he obseraon b he parameers } { where s he number of companes n he GIC group and s he number of me perods. Under he assumpon ha he obseraons are ndependen among he companes we can ge he followng lelhood funcon:. p 3.. Sep assgn a pror dsrbuon o each parameer. b a I U K π π π θ π Γ 3.. where P3. X X X B θ where X. 3

33 P3. SSE X X where SS E P B X s he number of companes s he number of obseraons and P s he number of regresson coeffcens ncludng he nercep. P3.3 B. SS P3.4 E. p P3.5 a b.. j j j he deas n choosng hose hperparameers { θ } are he same as he deas n choosng he hperparameers n Chaper. ha s use he esmaon of parameers from wo ordnar lnear regresson models: OLM-3 and OLM-4. ε ε ;. OLM-3 d ε ε ;. OLM-4 Assume ha all he parameers are ndependen of each oher he jon dsrbuon of he parameers can be epressed as p K θ U IΓ a b Sep 3 from he lelhood funcon n 3.. and jon pror dsrbuon n 3..3 we can ge he poseror dsrbuon of he parameers gen he daa b Baes rule: p P θ U IΓ a b

34 3. Gbbs Samplng he process of applng Gbbs sampler n hs chaper s he same as n Chaper. Whou repeang he seps hs secon onl specfes he complee condonal dsrbuons for he parameers and he sarng pons for each parameer. he complee condonal dsrbuons for he parameers are as follows. Frs ΛΣ Λ Λ θ I P where. Σ Σ Σ Σ Σ Σ Σ Λ Σ Second Φ Φ Φ where Φ. hrd U Fourh Γ S b a I where S. hs chaper chooses he sarng pons as follows. Frs X X X. 34

35 SS Second SS B ae B ae where SS SS SS SS ae B ae B ae B ae B. Fourh B. SS Ffh E. p he deas of seng and are he same as he ones of seng he hper- parameers { θ } n 3.. whch s usng he esmaon of parameers from he OLM-3 and OLM-4. he dea of seng s ang as he auocorrelaon of me seres ε n an AR srucure. 3.3 Forecas Afer geng he poseror dsrbuon of he parameers we can use o predc he fuure soc prces a perod for each frm n he group gen he daa. Leng { } he predcons can be sampled from he poseror predce dsrbuon f f π d M Leng be a sequence of range M from he Gbbs sampler an esmaor of f s 35

36 M h M f h f ˆ o ge samples of we use daa argumenaon o fll n h o each h M o ge h M from he normal dsrbuon descrbed below. h he 95% predce credble neral for can be compued from he.5% and h 97.5% emprcal quanles of he alues h M. 3.4 Condonal Predce Ordnae In hs chaper he condonal predce ordnae s defned as p ϖ h h p M h 3.4. where denoes he random fuure obseraon of compan a perod denoes he obseraons of compan from perod o h denoes he h h draw of he parameers from he Gbbs sampler and ϖ h M f f f h f h h h h M. 3.5 Emprcal Resuls of Groupng Baesan Analss he same crera as n Chaper are used for he model aluaon n hs chaper. able 3.5. shows he quanles of R under he ransformed scale as well as numbers of pose raos and negae raos aeraged n each GIC group wh saonar resrcon from 36

37 whch we can draw he followng conclusons. In order o be conssen wh Chaper Q s used as a major creron n analzng R. Based on Q he rao alue ranges from -.7% GIC o.3% GIC 5 and.4% oer all GIC under log ransformaon and from -.7% GIC o 3% GIC 5 and.8% oer all GIC under cubc roo ransformaon. Based on Q for he same GIC group he rao under log ransformaon s no bgger han under cubc roo ransformaon. hs mples ha log ransformaon s beer for group analss. Under boh ransformaons he numbers of pose raos and negae raos for each group are er close and.5 s beween LB and UB whch ndcaes ha boh ransformaons do no oeresmae he soc prces. he onl ecepon s n he case wh GIC equal o under cubc roo ransformaon. able Quanles of Rao & umbers of onnegae/egae Raos GIC o. of Frms Wh Saonar Resrcon Mehod Mn Q Q Q3 Ma o. o. - LB UB log rans cur rans log rans cur rans log rans cur rans log rans cur rans log rans cur rans log rans cur rans log rans cur rans able 3.5. ges mnmum mamum and oerall alues of R under he ransformed scale from boh classcal sascal analss and groupng Baesan analss. I shows he magnfcen mproemen of usng groupng Baesan approach o he Ohlson model compared o he classcal mehod. 37

38 able Mn Ma and Oerall Values of R Mehod ransformaon mn ma oerall Classcal log rans 4.7% 9.8% 8.8% Analss cur rans 6.3% 3.8%.9% Groupng log rans -.7%.3%.4% Analss cur rans -.7% 3%.8% able Aerage Lengh of Credble Ineral GIC o. of Frms Wh S-Resrcon Mehod Ae Lengh Sd De log rans.4.35 cur rans.5.8 log rans.68.6 cur rans.64. log rans.7.7 cur rans.59.7 log rans cur rans log rans cur rans log rans cur rans log rans cur rans able ges he aerage lengh of credble nerals and he correspondng sandard deaons for boh log ransformaon and cubc roo ransformaon under he ransformed scale and wh he saonar resrcon from whch we can draw he followng conclusons. Under log ransformaon he aerage lengh of CI ranges from.7 GIC o 4.4 GIC 3 and.4 oerall GIC he sandard deaon ranges from.7 GIC o.88 GIC 4 and.35 oerall GIC. Under cubc roo ransformaon he aerage lengh of CI ranges from.59 GIC o GIC 3 and.5 oerall GIC he sandard deaon ranges from.7 GIC o.9 GIC 4 and.8 oerall GIC. 38

39 For each GIC group he aerage lengh of CI s slghl smaller under cubc roo ransformaon han under log ransformaon. able Condonal Predce Ordnae each group has s own parameers GIC o. of Frms Wh Saonar Resrcon Mehod Mn Q Q Q3 Ma Mean log rans cur rans log rans cur rans log rans cur rans log rans cur rans log rans cur rans log rans cur rans log rans cur rans able ges he quanles and mean of CPO for boh log ransformaon and cubc roo ransformaon under he orgnal scale wh saonar resrcon from whch we can draw he followng conclusons. As n Chaper he mean alue s used as a major creron o analze CPO n hs chaper. Under log ransformaon he mean alue of CPO ranges from GIC 3 o GIC and oerall GIC. Under cubc roo ransformaon he mean alue of CPO ranges from GIC 3 o -.37 GIC and oerall GIC. For each GIC group he mean of CPO s much larger under cubc roo ransformaon han under log ransformaon. hs ndcaes ha cubc roo ransformaon does a beer job for groupng analss. For a follow-up analss we gaher he aerage CPO s for each group n he case ha all companes share he same parameers. We call hs oerall analss. he resuls are n able from whch he followng conclusons can be drawn. 39

40 Under log ransformaon he mean alue of CPO ranges from GIC o -3.4 GIC 4. Under cubc roo ransformaon he mean alue of CPO ranges from GIC o -.76 GIC 4. For each GIC group he mean of CPO s much larger under cubc roo ransformaon han under log ransformaon. hs ndcaes ha cubc roo ransformaon does a beer job han log ransformaon for he oerall analss. able Condonal Predce Ordnae all companes hae he same parameers GIC o. of Frms Wh Saonar Resrcon Mehod Mn Q Q Q3 Ma Mean log rans cur rans log rans cur rans log rans cur rans log rans cur rans log rans cur rans log rans cur rans able prodes he comparson of he mean of CPO from able and able hs s he comparson beween groupng analss and oerall analss from whch we can conclude ha: under boh ransformaons groupng analss and oerall analss hae almos he same predce abl whn GIC whch has he larges number of companes. Group analss does a beer job han oerall analss n GIC and a worse job for he lef GIC groups. 4

41 able Comparson of able o log rans of OFM AR- GIC Mean Mean df df/mean cur rans of OFM AR- GIC Mean Mean df df/mean Summarl he groupng Baesan analss n hs chaper also greal mproes he predce abl of he Ohlson model comparng o he classcal analss n Chaper. I does no oeresmae he soc prces under boh ransformaons wh saonar resrcon and cubc roo ransformaon s beer han log ransformaon n hs case. he cubc roo ransformaon s especall applcable for hose GIC groups whch hae large amoun of companes. 4

42 Chaper 4 Baesan Daa Analss b Adape Poolng Informaon Across Frms 4. Baesan Herarchcal Model he Baesan approaches n Chaper and Chaper 3 represens wo ereme cases. Chaper reas each compan ndduall and does no borrow nformaon across companes a all. Chaper oeresmaes he soc prces. Chaper 3 supposes all he companes n he same GIC group follow he same rules of predcng soc prces whch brngs more nformaon n he nesgaon. he mproemen n Chaper 3 s correcng he bas bu he model s oo smple. Furhermore hese wo ereme cases are barel seen n he real soc leraure where on one sde he companes hae her own specfc characerscs and on he oher sde he are affeced b he same economc facors and herefore hae some hngs n common. As a combnaon of Chaper and Chaper 3 also n order o be closer o he real hs chaper deelops a herarchcal Baesan approach s o smulaneousl esmae he unnown coeffcens for each compan b adapel poolng nformaon across frms. Consderng ha all he frms wll be ncluded n he Ohlson model s requred o add he compan nde no epresson..5 whch urns ou o be ε d u ε n. ε n. n. 4.. he Baesan erson of he Ohlson Model n 4.. for a GIC group s se up n four seps. 4