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
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.
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
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
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
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
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
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
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
and predce quales of he Ohlson model. able.3. --- 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
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.3. --- Oerall Esmae Abl Comparson of Varous Models Model oal Regress R-square R-square AIC BIC OFM --- OLR.84.84 OFM --- AR.756.856 59 596 OFM --- AR.756.857 594 597 log rans of OFM AR.768.33 653 7 sqr rans of OFM --- AR.776.6 83 86 cur rans of OFM --- AR.7733.46 6 76 n rans of OFM --- AR.63.43-35734 -35685 log rans of OFM AR.768.34 656 7 sqr rans of OFM --- AR.776. 86 87 cur rans of OFM --- AR.7733.48 3 86 n rans of OFM --- AR.64.43-35756 -357 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.
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.3.3 --- General Indusral Classfcaon GIC Dsrbuon GIC Value 3 4 5 6 Class Oerall Indusral Ul ransporaon Bans/Sangs and Loan Insurance o. of Frms Oher Fnancal 39 84 4 5 9 8 3 oe: when GIC equals means hs group ncludes all 39 frms.
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.3.4 --- Esmaed Parameers from Base Daa GIC Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR 5.7.86 4.48 3.437 -.89.456 LOG-rans of OFM AR 3.75.88.49.47.634 -. CUR-rans of OFM --- AR.8435.79.47.76.34 -.764 GIC Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR 5.93.899 6. 5.5564.6.988 LOG-rans of OFM AR 3.464.39.43.678.79 -. CUR-rans of OFM --- AR.8.37.94.74.4.76 GIC Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR 3.65.646.7959 -.3845.4846.983 LOG-rans of OFM AR.959.9.645.787.355 -.9 CUR-rans of OFM --- AR.738.6.68 -.56 7.643 -.838 GIC 3 Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR.44.989 3.4738.48.68 -.3959 LOG-rans of OFM AR.6.34.38.4. -.45 CUR-rans of OFM --- AR.363.343.8.9.65 -.44 GIC 4 Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR 8.678.35 8.7575-4.6-7.456.74 LOG-rans of OFM AR 3.66.36.9 -.5 -.9 -.9 CUR-rans of OFM --- AR.755.36.49 -.8 -.87 -.449 GIC 5 Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR 3.975.665-3.7.7879 -.5.56 LOG-rans of OFM AR 3.4.47 -.59.356 -.68 -.856 CUR-rans of OFM --- AR 3.393.66 -.739.45 -.4 -.654 GIC 6 Model Bea Bea Bea3 Bea4 Bea5 Bea6 OFM --- AR 3.5.633 5.9995 5.49-6.83 -.3595 LOG-rans of OFM AR 3.7.94.4457.45 -.69 -.58 CUR-rans of OFM --- AR 3.8.99.474.58.86 -.536 3
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.3.5 --- Quanles of R and umber of onnegae/egae R s --- Orgnal Scale --- ransformed Scale o. of Frms 39 84 4 3 5 4 9 5 8 6 3 Model Mn Q Q Q3 Ma o. o. - OFM --- AR -.49.73.4.936 5.73 35 76 log rans of OFM AR- -.564 -.38.347.795 3.37 84 7 log rans of OFM AR- -.96 -..88.9 6.875 84 7 cur rans of OFM AR- -.54 -.4.36.84 3.99 9 99 cur rans of OFM AR- -.7..9.9.466 93 98 OFM --- AR -.458.58.44.59 5.77 5 59 log rans of OFM AR- -.53 -.9.365.879.66 8 76 log rans of OFM AR- -.79 -.8.98.4 6.78 8 76 cur rans of OFM AR- -.57 -.3.46.899 3.496 74 cur rans of OFM AR- -.9 -.3.4.4.439 74 OFM --- AR -.97.68.8.4 9.35 35 7 log rans of OFM AR- -.34..74.6 7.4 3 log rans of OFM AR-...49.84. 3 cur rans of OFM AR- -.84.4.8.85 8.4 33 9 cur rans of OFM AR- -.4.6.66.99.84 34 8 OFM --- AR.4.95.359.474.574 5 log rans of OFM AR-.6.8.8.357.368 5 log rans of OFM AR-.4.5.56.93.5 5 cur rans of OFM AR-..8.4.47.43 5 cur rans of OFM AR-.37.58.76..8 5 OFM --- AR -.5.7.36.48.394 5 4 log rans of OFM AR- -.33.54.85.4.94 4 5 log rans of OFM AR- -..5.5.3.99 4 5 cur rans of OFM AR- -.3.89..44.5 4 5 cur rans of OFM AR- -..3.7.6.3 4 5 OFM --- AR -.33..4.39 3.663 5 3 log rans of OFM AR- -.359.47.9.3 3.357 4 4 log rans of OFM AR- -.9.3.47.76.645 4 4 cur rans of OFM AR- -.347.66.97.34 3.435 4 4 cur rans of OFM AR-.3.3.63.99.644 4 4 OFM --- AR -.45.95.348.847 4.76 log rans of OFM AR- -.5.43.66.8 3.56 log rans of OFM AR- -.3.37.7.74.87 cur rans of OFM AR-.75.37.466.66 4.8 3 cur rans of OFM AR-.6..38.35.76 3 4
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
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
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
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
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
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.
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
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.3.3...3.3 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..
.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
able.5.a --- Wh Saonar Resrcon -Orgnal Scale -ransformed Scale GIC o. of Frms 39 84 4 3 5 4 9 5 8 6 3 Mehod Mn Q Q Q3 Ma o. o. - LB UB no rans -5.4 -.9.7.89 9.9 36 55.579.68 log rans- -.963 -..69.74 3.8 37 54.58.63 log rans- -7.75 -.3..75 3.94 37 54.58.63 cur rans- -.94 -.5.65.69 4.54 35 56.576.66 cur rans- -.455 -.35..84.497 37 54.58.63 no rans -5.4 -.4.77.38 9.9 7.577.635 log rans- -.963 -.5.75.89 3.8 73.58.638 log rans- -7.75 -.34..79 3.94 73.58.638 cur rans- -.94 -..7.85.493 7 3.573.63 cur rans- -.455 -.37.4.88.497 73.58.638 no rans -.948 -.5.7.37.7 3 9.47.64 log rans- -.938 -.47.7. 5.54.447.6 log rans- -.89 -.48.5.6.6.447.6 cur rans- -.8 -.5.7..854.447.6 cur rans- -.64 -.5.7.67.344.447.6 no rans -.9 -.5.89.5.39 4.6.979 log rans- -.65 -.4.7.7.99 3.38.89 log rans- -.35 -.3..58.45 3.38.89 cur rans- -.75 -.4.73.6.653 3.38.89 cur rans- -.37 -.3.5.67.386 3.38.89 no rans -.4 -.5.8.43.56 9.567.743 log rans- -.95 -.43.88.5 4.336 9.567.743 log rans- -.754 -.3.5.66.485 9.567.743 cur rans- -.897 -.49.8.4 3.8 9.567.743 cur rans- -.53 -.5.8.76.65 9.567.743 no rans -.587 -.6.9.78 4.695 6.556.778 log rans- -.89 -.48.89.7 5.43 6.556.778 log rans- -.437 -.4.4.67.795 6.556.778 cur rans- -.759 -.53.86.7 4.86 6.556.778 cur rans- -.377 -.7.9.84.84 6.556.778 no rans -8.584 -.39.5.44 7.7 7 6.4.677 log rans- -.8 -.9.37. 5.635 7 6.4.677 log rans- -.88 -.36..6.957 7 6.4.677 cur rans- -.994 -.6.3.9 4.96 7 6.4.677 cur rans- -.88 -.43..69.84 7 6.4.677 4
GIC able.5.b --- Whou Saonar Resrcon -Orgnal Scale -ransformed Scale o. of Frms 39 84 4 3 5 4 9 5 8 6 3 Mehod Mn Q Q Q3 Ma o. o. - LB UB no rans -6.845 -.4.6.73 6.443 3 6.563.63 log rans- -.987 -.3.59.6 43. 3 6.566.66 log rans- -5.938 -.36.7.7.744 3 6.566.66 cur rans- -.834 -.9.54.55 6.583 8 63.558.68 cur rans- -.94 -.4.9.8.6 3 6.563.63 no rans -6.75 -..6.63 7.86 68 6.56.6 log rans- -.975 -.7.63.59 33.557 7 4.57.68 log rans- -5.938 -.38.9.75.744 69 5.566.64 cur rans- -.834 -..6.69 5.583 67 7.559.67 cur rans- -.94 -.4..84.6 68 6.56.6 no rans -4.539 -..5.45 4.784 4 8.495.648 log rans- -.86 -..6.64 9.3 5 7.59.67 log rans- -3.64 -.53..57.956.43.577 cur rans- -.39 -.65.5.98 9.44.43.577 cur rans- -.339 -.57.3.63.73 9.459.64 no rans -6.94 -.374.38.39 64.95 3.38.89 log rans- -.998 -.99.35.8 58.857 3.38.89 log rans- -.66 -.67.9.54.57 3.38.89 cur rans- -.638 -.5.66.93.7 3.38.89 cur rans- -.86 -.6.3.6.455 3.38.89 no rans -4.497 -.97 -.85.33 4.389 8.53.7 log rans- -.86 -.98.8.33 9.66 8.53.7 log rans- -.467 -.6.4.67.795 9.567.743 cur rans- -.97 -.57.78.38.799 9.567.743 cur rans- -.546 -.8.6.75.56 9.567.743 no rans -4.9 -.37.83.3 4.5 7.496.76 log rans- -.994 -..77.99.33 7.496.76 log rans- -.387 -.8..65.63 6.556.778 cur rans- -.836 -.69.74.58 3.97 7.496.76 cur rans- -.45 -..5.8.67 7.496.76 no rans -5.89 -.99.44.86 4.36 7 6.4.677 log rans- -.85 -.5.44.6 9.485 7 6.4.677 log rans- -.774 -.47.3.56.773 7 6.4.677 cur rans- -.938 -.64..9 3.79 7 6.4.677 cur rans- -.65 -.57..6.678 7 6.4.677 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
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
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.5. --- 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 4 5 6 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
able.5.3 --- Aerage Lengh of Credble Ineral GIC o. of Frms 39 84 4 3 5 4 9 5 8 6 3 Mehod Wh S-Resrcon o S-Resrcon Ae. CI Sd Ae. CI Sd Lengh De Lengh De no rans 4.699 3.3 4.786 3.573 log rans- 4.76.87 4.9.839 log rans-.949.46.966.47 cur rans- 6.58 3.5 6.69 3.7 cur rans-.958.45.975.434 no rans 5.7 3.69 5.73 3.547 log rans- 4.4.4 4.436.66 log rans-.3.486..499 cur rans- 7.455 4.95 7.577 4.496 cur rans-.6.456.3.466 no rans 7.537.44 7.76.64 log rans- 6.45.38 6.44.334 log rans-.9.464.98.469 cur rans-.679 8.734.737 9.5 cur rans-.866.37.878.36 no rans.75.773 7.649 5. log rans- 99.76 68.683 67. 94.9 log rans-.699.95.73.3 cur rans- 9.46 6.83 9.944 7.35 cur rans-.68.34.75.47 no rans 5.364 8.994 5.895 9.3 log rans- 3.89.478 4.87 5.5 log rans-.65.84.674.97 cur rans-.7 9.685.554 9.876 cur rans-.686.3.77.37 no rans 34..96 34.5. log rans- 9.73. 8.75.4 log rans-.764.73.83.5 cur rans- 3.86.99 3.7.43 cur rans-.883.6.93.49 no rans 35.6 4.488 35.659 4.45 log rans- 7.7.793 7.485.737 log rans-.94.95.95.3 cur rans- 3.7 3.85 3.73.87 cur rans-..36.3.39 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
able.5.4 --- CPO GIC o. of Frms 39 84 4 3 5 4 9 5 8 6 3 GIC o. of Frms 39 84 4 3 5 4 9 5 8 6 3 Wh Saonar Resrcon Mehod Mn Q Q Q3 Ma Mean no rans -5.6-3.8-3.46-3.66 -.89-3.444 log rans -5.55-3.779-3.449-3.87 -.43-3.43 cur rans -4.76 -.55 -.7 -.98 -.39 -.67 no rans -5.578-3.774-3.44-3.75 -.6-3.43 log rans -4.74-3.769-3.445-3.79 -.4-3.47 cur rans -3.677 -.583 -.9 -.995 -.39 -.85 no rans -4.74-3.789-3.54-3.56 -.338-3.44 log rans -4. -3.79-3.578-3.94 -.9-3.44 cur rans -.995 -.468 -. -.969 -.373 -.4 no rans -8.45-6.47-6.77-4.88-3.97-5.79 log rans -6.963-6.65-4.399-3.976-3.756-5.44 cur rans -.78 -.8 -.4 -.75 -.674 -.978 no rans -4.6-3.778-3.567-3.33 -.353-3.494 log rans -5.68-3.844-3.598-3.67 -.59-3.63 cur rans -.94 -.99 -.4 -.757 -.39 -.4 no rans -5.83-3.9-3.73-3.56 -.977-3.794 log rans -5.8-4.34-3.767-3.4-3. -3.874 cur rans -.87 -.57 -.39 -.5 -.765 -.339 no rans -5.33-3.995-3.853-3.43 -.97-3.83 log rans -5.38-3.987-3.736-3.335-3.6-3.833 cur rans -4.76 -.75 -.488 -.49 -.8 -.639 Whou Saonar Resrcon Mehod Mn Q Q Q3 Ma Mean no rans -5.7-3.78-3.489-3.9 -.44-3.456 log rans -5.94-3.8-3.47-3.96 -.6-3.435 cur rans -4.6 -.66 -.36 -.7 -.35 -.3 no rans -5.4-3.753-3.48-3. -.44-3.44 log rans -5.46-3.759-3.474-3.9 -.44-3.44 cur rans -.75 -.456 -.46 -.978 -.36 -.85 no rans -4.86-3.855-3.57-3.3 -.47-3.43 log rans -5.349-3.99-3.558-3.8 -.3-3.5 cur rans -.75 -.456 -.46 -.978 -.36 -.85 no rans -9.9-6.379-4.343-4.73-4.78-5.63 log rans -7.4-5.53-4.54-3.863-3.595-4.98 cur rans -.37 -.64 -.93 -.88 -.6 -.97 no rans -4.68-3.95-3.65-3.43 -.38-3.53 log rans -4.46-3.964-3.574-3.53 -.63-3.546 cur rans -3.58 -.53 -.93 -.869 -.377 -.77 no rans -4.84-4.5-3.686-3.45-3.5-3.696 log rans -5.453-4.9-3.799-3.49-3.3-3.85 cur rans -.887 -.545 -.367 -.96 -.86 -.34 no rans -4.85-4.3-3.93-3.573-3.8-3.836 log rans -5.7-4.46-3.95-3.75-3.55-3.765 cur rans -3.8 -.669 -.449 -.78 -.63 -.446 9
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 -5.79 GIC 3 o -3.44 GIC and -3.444 oerall GIC. Under log ransformaon he mean alue of CPO ranges from -5.44 GIC 3 o -3.47 GIC and -3.43 oerall GIC. Under cubc roo ransformaon he mean alue of CPO ranges from -.639 GIC 6 o -.978 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
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
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
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. 3..3 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 3..4 33
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
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. 3.3. M Leng be a sequence of range M from he Gbbs sampler an esmaor of f s 35
M h M f h f ˆ. 3.3. 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 3.3.3 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
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 3.5. --- Quanles of Rao & umbers of onnegae/egae Raos GIC o. of Frms 39 84 4 3 5 4 9 5 8 6 3 Wh Saonar Resrcon Mehod Mn Q Q Q3 Ma o. o. - LB UB log rans -.78 -.74.4.83. 9.486.537 cur rans -.8 -.73.8.9.386 6 85.5.55 log rans -.678 -.7..76.9 4 4.47.53 cur rans -.899 -.73.4.85.79 46 38.484.544 log rans -.5 -.73 -.7.6.689.399.553 cur rans -.499 -.78 -.7.67.65.399.553 log rans -.7 -.98.8.8.64 3.38.89 cur rans -.78 -.93.4.3.8 3.38.89 log rans -.83 -.96...58 5 4.44.6 cur rans -.88 -.3.3.3.38 5 4.44.6 log rans -.84 -.75.3.4. 8.438.673 cur rans -.9 -.76.3.4.74 8.438.673 log rans -.67 -.58..6.7 7 6.4.677 cur rans -.678 -.66.6..89 7 6.4.677 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
able 3.5. --- 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 3.5.3 --- Aerage Lengh of Credble Ineral GIC o. of Frms 39 84 4 3 5 4 9 5 8 6 3 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 4.4.7 cur rans 3.753.95 log rans.988.88 cur rans.994.9 log rans.88.89 cur rans.94.98 log rans 3.446.6 cur rans 3.43.64 able 3.5.3 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 3.753 GIC 3 and.5 oerall GIC he sandard deaon ranges from.7 GIC o.9 GIC 4 and.8 oerall GIC. 38
For each GIC group he aerage lengh of CI s slghl smaller under cubc roo ransformaon han under log ransformaon. able 3.5.4 --- Condonal Predce Ordnae each group has s own parameers GIC o. of Frms 39 84 4 3 5 4 9 5 8 6 3 Wh Saonar Resrcon Mehod Mn Q Q Q3 Ma Mean log rans -6.44-4.33-3.64-3.86 -.36-3.858 cur rans -5.338 -.7 -.387 -.87 -.794 -.663 log rans -5.46-3.99-3.63-3.33 -.36-3.88 cur rans -4.6 -.673 -.358 -.57 -.798 -.6 log rans -6.35-3.755-3.498-3.4 -.37-3.549 cur rans -4.889 -.43 -.67 -.7 -.73 -.37 log rans -7.589-5.85-4.86-4.67-4.46-5.59 cur rans -6.7-3.7-3.66-3.53-3.34-4.65 log rans -8.444-4.873-4.485-3.88-3.444-4.646 cur rans -7.53-3.36-3.44 -.88 -.783-3.43 log rans -6.974-4.438-4.37-4.99-3.348-4.53 cur rans -5.74-3.45 -.986 -.85 -.679-3.84 log rans -9.95-5.355-4.866-4.46-3.95-5.46 cur rans -7.768-3.854-3.587-3.475-3.99-3.94 able 3.5.4 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 -5.46 GIC 3 o -3.84 GIC and -3.858 oerall GIC. Under cubc roo ransformaon he mean alue of CPO ranges from -4.65 GIC 3 o -.37 GIC and -.663 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 3.5.5 from whch he followng conclusons can be drawn. 39
Under log ransformaon he mean alue of CPO ranges from -4.49 GIC o -3.4 GIC 4. Under cubc roo ransformaon he mean alue of CPO ranges from -3.83 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 3.5.5 --- Condonal Predce Ordnae all companes hae he same parameers GIC o. of Frms 84 4 3 5 4 9 5 8 6 3 Wh Saonar Resrcon Mehod Mn Q Q Q3 Ma Mean log rans -6.44-4. -3.65-3.79 -.36-3.87 cur rans -5.338 -.69 -.383 -.9 -.794 -.659 log rans -9.95-4.99-3.998-3.56 -.48-4.49 cur rans -7.94-3.598 -.63 -.338 -.85-3.83 log rans -4.657-4.456-3.8-3.65 -.579-3.74 cur rans -3.33-3.8 -.568 -. -.875 -.599 log rans -4.33-3.783-3.36-3.6 -.495-3.4 cur rans -.473 -.37 -.5 -.3 -.95 -.76 log rans -4.437-4.6-3.739-3.568-3.39-3.84 cur rans -.943 -.568 -.4 -.65 -.53 -.434 log rans -4.738-4.8-3.855-3.453 -.95-3.8 cur rans -3.36 -.76 -.498 -.49 -.3 -.58 able 3.5.6 prodes he comparson of he mean of CPO from able 3.5.4 and able 3.5.5. 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
able 3.5.6 --- Comparson of able 3.5.3 o 3.5.4 log rans of OFM AR- GIC Mean-3.5.3 Mean-3.5.4 df df/mean-3.5.3-3.88-3.87 -.9. -3.549-4.49 -.88.48 3-5.59-3.74.545.94 4-4.646-3.4.45.68 5-4.53-3.84.77.58 6-5.46-3.8.34.57 cur rans of OFM AR- GIC Mean-3.5.3 Mean-3.5.4 df df/mean-3.5.3 -.6 -.659 -.37.4 -.37-3.83 -.8.34 3-4.65 -.599.466.36 4-3.43 -.76.55.366 5-3.84 -.434.85.59 6-3.94 -.58.434.364 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
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
Frs descrbe he obseraon b he parameers } : { where s he number of companes 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 4.. Second assumng ndependence a all leels assgn a pror dsrbuon o each parameer K φ ψ π θ π / I Γ γ π τ ν π 4..3 where } { γ θ are unnown hperparameers and } : { τ ν φ ψ are nown hperparameers whch are se as follows. P4..5 j j j j B X B X ψ φ ψ. P4. j j j j j j B X B X B X B X ν τ ν where X X X X B. hs paper does no pool s because he me seres of some companes are saonar whle some are no. I does no pool s because he sar alues can be er dfferen from he res and s dffcul o pool hem. In order o use he nformaon across 43
companes we pool s adapel. As for s we ncorporae heerogene bu we belee he come from he same populaon and allow poolng hem. hrd assgn a hper-pror dsrbuon o each unnown hperparameer n } { γ θ. P4.3 θ θ θ P ν ν P Wshar where P p SS X X X X X E ν θ X. P4.4. γ γ p p Assume ha all he parameers } : { are ndependen of each oher he jon dsrbuon of he parameers can be epressed as ]. / [ : : Γ P I p γ τ ν φ ψ θ γ θ 4..4 Snce we hae assumed ha all he unnown hperparameers } { γ θ are ndependen of each oher he jon dsrbuon for he unnown hperparameers can be epressed as. γ ν ν θ θ γ θ Wshar p P 4..5 Fnall from he lelhood funcon n 4.. and jon pror dsrbuon n 4..4 as well as n 4..5 we can ge he poseror dsrbuon of all he parameers b Baes rule: : : : p p p γ θ γ θ γ θ 44
: : p p γ θ : : p p p γ θ γ θ. ] / [ Γ P P P Wshar I γ ν ν θ θ γ τ ν φ ψ θ 4..6 4. Gbbs Samplng he arge here s o mae nferences on he parameers } : { and he unnown hperparameers } { γ θ gen he daa. he complee condonal dsrbuons are as follows. Frs ΛΣ Λ Λ I P θ γ θ where Σ Σ Σ Σ Σ Σ Σ Σ Λ. 45
Second Φ Φ Φ ψ γ θ where φ φ Φ. hrd B B B A B τ τ τ τ ν γ θ where B A. Fourh Γ S I γ γ θ where S. Ffh where : Σ m P γ θ. Σ Σ θ m Sh where : H h Wshar P γ θ. H h θ θ ν ν Seenh. hen le < < τ τ τ τ. ep : τ τ γ γ γ θ τ Γ Eghh. : Γ I θ γ 46
he Gbbs samplng s o carr ou he followng seps and erael. hese wo seps conss of sub-seps ha are carred ou sequenall for a sngle chan. he Maro chan can also be replcaed b drawng ndependen nal alues of he parameers and hperparameers. Sep : Updae he parameers { : } gen he hperparameers and daa. he followng sub-seps - o -4 are eecued ndependenl for each. - Draw drecl from he mularae normal dsrbuon descrbed n 4a; - Draw drecl from he unarae normal dsrbuon descrbed n 4b; -3 Use Deroe 986 mehod o draw descrbed n 4c; -4 Draw from he unarae normal dsrbuon drecl from he nerse gamma dsrbuon descrbed n 4d. Sep : Generae he hperparameers { θ γ} gen he alues of he parameers n sep. - Draw θ from he mularae normal dsrbuon descrbed n 4e; - Draw from he P-dmensonal Wshar dsrbuon descrbed n 4f; -3 Use a grd o draw from he complee condonal dsrbuon descrbed n 4g; -4 Drawγ from he nerse Gamma dsrbuon descrbed n 4h. 4.3 Forecasng 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. 47
Leng { } he predcons can be sampled from he poseror predce dsrbuon f f π d. 4.3. Leng M be a sequence of range M from he Gbbs sampler an esmaor of f s M h M f h f ˆ. 4.3. o ge samples of we use daa argumenaon o fll n o each h h M o ge h M from he normal dsrbuon descrbed below: h. 4.3.3 he 95% predce credble neral for can be compued from he.5% and h 97.5% emprcal quanles of he alues h M. 4.4 Condonal Predce Ordnae In hs chaper he condonal predce ordnae s defned as p ϖ h h p M h 4.4. where denoes he random fuure obseraon of compan a perod denoes he obseraons of compan from perod o denoes he h h draw of he parameers from he Gbbs Sampler and h ϖ h M f f f f h h h h h M. 48
4.5 Emprcal Resuls of Adape Baesan Analss b Poolng Informaon Across Frms As n Chaper or 3 he same crera are used for he model aluaon n hs chaper. able 4.5. --- Quanles of Rao & umbers of onnegae/egae Raos GIC o. of Frms 39 84 4 3 5 4 9 5 8 6 3 Wh Saonar Resrcon Mehod Mn Q Q Q3 Ma o. log rans -8.35 -.449.69.85 9.655 6 65.553.63 cur rans -3.5 -.46.64.868 3.5 7 64.556.66 log rans -8.35 -.489.58.79 9.655 63.545.63 cur rans -3.5 -.54.6.838 3.5 63.545.63 log rans -5.459 -.49.8.387 4.3 5 7.59.67 cur rans -3.397 -.43.88.337 3. 5 7.59.67 log rans -.484 -.43.649.749.8667 3.38.89 cur rans -.78 -.385.74.95.98 3.38.89 log rans -.63 -.39.66.475.633 7.495.678 cur rans -.6576 -.7.669.6.689 7.495.678 log rans -.939 -.53.63.636.97 9.439.66 cur rans -.456 -.4.77.793.74 9.439.66 log rans -.8886 -.33.67.735.4346 8 5.48.75 cur rans -.9548 -.358.63.786.3 8 5.48.75 o. - LB UB able 4.5. shows he quanles of rao under he ransformed scale as well as numbers of pose raos and negae raos aeraged n each GIC group wh saonar resrcon from whch we can draw he followng conclusons. Based on Q he rao alue ranges from 5.8% GIC o 8.% GIC and 6.9% oerall GIC under log ransformaon and from 6% GIC o 8.8% GIC and 6.4% oerall GIC under cubc roo ransformaon. Under boh ransformaons GIC has he smalles rao and GIC 5 has he larges rao. Based on Q for he same GIC group he rao under log ransformaon s smaller han under cubc roo ransformaon. When GIC s or.5 s no beween LB and UB; when GIC s 3 4 5 or 6.5 s beween LB and UB. hs ndcaes ha boh ransformaons oeresmae he soc prce for large GIC groups and do no oeresmae he soc prce for small GIC groups. 49
able 4.5. --- 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% Adape Poolng log rans 5.8% 8.% 6.9% Analss cur rans 6% 8.8% 6.4% able 4.5.3 --- Aerage Lengh of Credble Ineral GIC o. of Frms 39 84 4 3 5 4 9 5 8 6 3 Wh S-Resrcon Mehod Ae. CI Lengh Sd De log rans 4.85.453 cur rans 3.873.45 log rans 4.64.458 cur rans 3.854.455 log rans 4.9.45 cur rans 4.56.444 log rans 4.3.484 cur rans 3.99.573 log rans 3.868.369 cur rans 3.668.367 log rans 4.33.33 cur rans 4.5.358 log rans 4.4.357 cur rans 3.856.367 able 4.5. ges mnmum mamum and oerall alues of R under he ransformed scale from boh classcal sascal analss and adape poolng Baesan analss. I shows he noable mproemen of usng adape poolng Baesan approach o he Ohlson model compared o he classcal mehod. able 4.5.3 ges he aerage lengh of credble nerals and he correspondng sandard deaons for boh log ransformaon and cubc roo ransformaon under he ransformed scale wh saonar resrcon from whch we can draw he followng conclusons. 5
Under log ransformaon he aerage lengh of CI ranges from 3.868 GIC 4 o 4.9 GIC and 4.85 oerall GIC he sandard deaon ranges from.33 GIC 5 o.573 GIC 3 and.453 oerall GIC. Under cubc roo ransformaon he aerage lengh of CI ranges from 3.668 GIC 4 o 4.56 GIC and 3.873 oerall GIC he sandard deaon ranges from.358 GIC 5 o.573 GIC 3. For each GIC group he aerage lengh of CI s sgnfcanl smaller under cubc roo ransformaon han under log ransformaon. able 4.5.4 --- Condonal Predce Ordnae GIC o. of Frms 39 84 4 3 5 4 9 5 8 6 3 Wh Saonar Resrcon Mehod Mn Q Q Q3 Ma Mean log rans -.558 -.387 -.74 -.3 -.54 -.36 cube roo rans -.38 -.337 -.5 -.4 -.995 -.5 log rans -.558 -.358 -.68 -. -.63 -.96 cube roo rans -.38 -.3 -.6 -.35.995 -.4 log rans -.75 -.497 -.397 -.4 -.84 -.384 cube roo rans -.687 -.43 -.34 -.73 -.6 -.3 log rans -.638 -.586 -.478 -.5 -.6 -.43 cube roo rans -.586 -.56 -.43 -.64 -. -.36 log rans -.66 -.75 -.6 -.44 -.54 -.8 cube roo rans -.6 -.3 -.64 -.86 -.995 -.73 log rans -.675 -.483 -.378 -.7 -.6 -.389 cube roo rans -.68 -.46 -.37 -. -.93 -.338 log rans -.479 -.353 -.7 -. -.34 -.8 cube roo rans -.44 -.34 -.97 -.55 -.59 -.35 able 4.5.4 ges he quanles and mean of CPO for boh log ransformaon and cubc roo ransformaon under he ransformed scale wh saonar resrcon from whch we can draw he followng conclusons. Under log ransformaon he mean alue of CPO ranges from -.43 GIC 3 o -.8 GIC 4 and -.36 oerall GIC. Under cubc roo ransformaon he mean alue of CPO ranges from -.36 GIC 3 o -.73 GIC 4 and -.5 oerall GIC. Under boh ransformaons group 4 has he larges CPO and group 3 has he smalles CPO. 5
For each GIC group he mean of CPO s que larger under cubc roo ransformaon han under log ransformaon. hs ndcaes ha cubc roo ransformaon does a beer job han log ransformaon for he approach of adape poolng nformaon across companes. In all he adape Baesan analss b poolng nformaon across companes n hs chaper sll mproes he predce abl of he Ohlson model comparng o he classcal analss n Chaper. I oeresmaes he soc prce under boh ransformaons wh saonar resrcon when he sze of GIC group s large and doesn oeresmae he soc prce when he sze s small. Cubc roo ransformaon s beer han log ransformaon n hs case. 5
Chaper 5 Comparson of hree Baesan Models and Oerall Conclusons In hs las chaper we presen he conclusons of our research wor on forecasng soc prces a he Ohlson model afer comparng he hree Baesan models. 5. Comparson of hree Baesan Models hs secon frs compares he hree Baesan models ha are used he former hree chapers based on he mnmum mamum and oerall alues of R aerage lengh of credble neral and log condonal predce ordnae CPO whch are gahered n able 5... All hese alues are on he ransformed scale ecep ha CPO s are calculaed on he orgnal scale. he followng conclusons can be drawn from able 5... Based on R groupng analss has he smalles R alues less han % and adape poolng analss has he larges R alues around 6.5%. he nddual analss resuls n er small R alues oo around %. Based on aerage lengh of credble neral nddual analss has he shores lenghs around and adape poolng analss has he longes lenghs around 4. he aerage lengh alues go from groupng analss are no large around.. Based on CPO nddual analss has he larges alues greaer han -.5 and adape poolng analss has comparable alues wh groupng analss mosl less han -. 53
heorecall adape poolng analss s supposed o achee he bes predce resuls. Bu usng he crera aboe adape poolng analss does no show an adanages comparng o he oher wo mehods. able 5.. Mehod R ransformaon mn ma oerall Inddual log rans.5%.5% % Analss cur rans.7%.9%.% Groupng log rans -.7%.3%.4% Analss cur rans -.7% 3%.8% Adape Poolng log rans 5.8% 8.% 6.9% Analss cur rans 6% 8.8% 6.4% Mehod Ae Lengh of CI ransformaon mn ma oerall Inddual log rans.65.3.949 Analss cur rans.68..958 Groupng log rans.7 4.4.4 Analss cur rans.59 3.753.5 Adape Poolng log rans 3.868 4.9 4.85 Analss cur rans 3.668 4.56 3.873 Mehod CPO ransformaon mn ma oerall Inddual log rans -5.44-3.47-3.43 Analss cur rans -.639 -.978 -.67 Groupng log rans -5.46-3.84-3.858 Analss cur rans -4.65 -.37 -.663 Adape Poolng log rans -6.46-3.45-4.737 Analss cur rans -4.33 -.8-3.49 For he adape Baesan analss wo ransformaons are also compared based on he same crera as aboe. hs paper concludes ha log ransformaon and cubc ransformaon are comparable een hough he resuls show ha he cubc roo ransformaon does a slghl beer job. In he former hree chapers LB and UB ha are calculaed b he number of nonnegae R s and he number of negae R s are used o chec he sae of oeresmaon. he concluson s boh nddual analss and adape poolng analss oeresmae he soc prces bu he groupng analss does no. o hae an oerall loo see able 5... 54
able 5.. --- Oeresmaon Chec Mehod ransformaon LB UB Inddual log rans.58.63 Analss cur rans.58.63 Groupng log rans.486.537 Analss cur rans.5.55 Adape Poolng log rans.553.63 Analss cur rans.556.66 he hsogram and QQ- plos of he resduals n he adape poolng analss see Fgure 5.. Fgure 5..4 are also used o chec he model fng. Usng he log ransformaon under he ransformed scale he resdual s normall dsrbued wh mean -.45 and sandard deaon.385. Usng he cubc roo ransformaon under he ransformed scale he resdual s normall dsrbued wh mean -.43 and sandard deaon.37. Fgure 5..5 shows he scaer plo of sandardzed resdual ersus predcon for all companes under log ransformaon n he adape poolng analss Fgure 5..6 shows he scaer plo for each GIC group. I s dffcul o access hese plos because he sandard deaons are so large ha he deleed sandard resdual plos are los. Fgure 5..7 shows he lnes plos of predced soc prces ersus real soc prces for all he companes correspondng o he hree Baesan mehods. We sll canno see an adanages of he adape poolng mehod. 55
Fgure 5.. 7. 5 5.. 5 P e r c e n. 7. 5 5.. 5 -. 75 -. 875 -. 575 -. 75 -. 975 -. 675 -. 375 -. 75. 5. 55. 85. 5. 45 res dual Qunf ang 3APR5 Fgure 5.. 56
Fgure 5..3 7. 5 5.. 5 P e r c e n. 7. 5 5.. 5 -. 75 -. 875 -. 575 -. 75 -. 975 -. 675 -. 375 -. 75. 5. 55. 85. 5. 45 res dual Fgure 5..4 Qunf ang 3APR5 r e s d u a l - - -3-4 - 4 ormal Quan l es Qunf ang 3APR5 57
Fgure 5..5 Fgure 5..6 58
Fgure 5..7 59
5. Esmaed Regresson Coeffcens from Adape Baesan Analss In Chaper he classcal approach ges he esmaed regresson coeffcens for each GIC group and concludes ha he nercep BPS and he frs wo followng quarers abnormal earnngs per share 4 are generall sgnfcan n he Ohlson Model. o compare wh hs concluson he poseror dsrbuon of θ n he Baesan model for adape poolng analss for each GIC group are colleced n able 5.. whch shows ha onl he nercep BPS and he frs followng quarers abnormal earnngs per share 3 are sgnfcan. A reduced model wh onl hese hree predcors s analzed b he adape poolng Baesan approach. hose hree ems urn ou o be all sgnfcan n he reduced model see able 5... able 5.. --- Summares of he Poseror Dsrbuon of θ for he Full Model log rans wh saonar resrcon Parameers Mean SD SE C5 C975 hea 3.899.787.53 3.697 3.3557 hea.597.68.7..396 hea3.98.97.886.58.3836 hea4.83.7958.44 -.576.849 hea5.3.798.63 -.466.85 hea6 -.458.75.49 -.4558.457 cur rans wh saonar resrcon Parameers Mean SD SE C5 C975 hea.89697.6865.438.7654 3.369 hea.877.64..6.433 hea3.7.8863.88.396.39566 hea4.34.7454.37 -.397.83 hea5.3.78.577 -.4944.6467 hea6 -.48.693.46 -.44.478 oe ha SD denoes he poseror sandard deaon SE denoes he numercal sandard error C5 and C975 denoe he lower bound and upper bound of he 95% credble neral separael. 6
able 5.. --- Summares of he Poseror Dsrbuon of θ for he Full Model log rans wh saonar resrcon Parameers Mean SD SE C5 C975 hea 3.44.637.58 3.797 3.3686 hea.736.539.49.648.3735 hea3.38.594.453.947.35 cur rans wh saonar resrcon Parameers Mean SD SE C5 C975 hea.8894.6..768 3.89 hea.3.53.43.99.3958 hea3.43.573.49.3454.3568 5.3 Oerall Conclusons Oerall four approaches are appled o nerpre he Ohlson model n hs paper. he are classcal sascal analss n Chaper nddual Baesan analss n Chaper groupng Baesan analss n Chaper 3 and adape poolng Baesan analss n Chaper 4. he frs wo chapers proe ha boh he log ransformaon and cubc roo ransformaon can enhance he predce abl of he Ohlson model. he analss n Chaper resuls o use saonar resrcon and ransformed measuremen scale. All hree Baesan approaches are compared o he classcal mehod based on he mnmum mamum and oerall alues of R and he are also compared wh each oher based on he same crera. he conclusons are: Each Baesan approach used n hs paper does beer job han he classcal mehod Log ransformaon and cubc ransformaon hae comparable effcenc n enhancng he predce abl. Boo alue per share and he abnormal earnng per share for he frs followng perod affec he soc prces sgnfcanl he las hree abnormal earnngs are no mporan. hs paper epecs o reach he concluson ha he adape poolng Baesan mehod s beer han nddual or groupng Baesan approach. Bu all he four crera relae dfference R aerage lengh of credble neral lower bound LB and upper bound 6
UB of he 95% credble neral of he esmaed probabl of nonnegae R and log condonal predce ordnae CPO all fal o ge hs sensble concluson. Furher nesgaons are necessar. 6
Appendces 63
A. Orgnaon of he Ohlson 995 Valuaon Model For more accurae undersandng hs paper spls he commonl called Ohlson 995 model no wo: he Ohlson 995 Valuon Model OVM and he Ohlson 995 Appromaon Model OAM. he Ohlson 995 Valuon Model s a specal case of he Resdual Income Valuaon Model RIM whch was deeloped from he radonal Ddend Dscoun Model DDM. In economcs and fnance he radonal approach o he problem of soc aluaon based on a sngle frm has focused on he Ddend Dscoun Model DDM of Rubnsen 976. I defnes he alue of a frm as he presen alue of he epeced fuure ddends. ha s under he assumpon of no arbrage here ess a prcng ernel π r such ha he prce of a soc P s relaed o s ddends d b P E π d E r d A where r denoes he dscoun rae durng me perod E [.] denoes he epecaons operaor condoned on he dae nformaon. oe ha he arbrage s a suaon n whch he compan can mae mone b eplong he effcenc of he mare. he dea of DDM mples ha one should forecas ddends n order o esmae he soc prces. Snce ddends are arbrarl decded b managemen ma be hard o esmae a ddend process n small samples Ang & Lu 998. Moreoer mare parcpans end o focus on accounng nformaon especall earnngs. he fundamenal relaon beween boo alue of equb earnngs and ddends d s descrbed n he Clean Surplus Accounng Model b equaonb b d.e. he change n boo alue beween wo daes equals earnngs mnus ddends. hs relaons s called he Clean Surplus Relaon CSR whch can also be wren as d b b. A 64
Subsung equaon A o equaon A hereb elmnang ddends elds he Resdual Income Valuaon Model RIM descrbed n Peasnell 98. I s a funcon of onl accounng arables namel: P b r r b. A3 hs prncple shows ha he heorecal alue of he frm s equal o he openng boo alue of equ plus he presen alue of s resdual ncome or ecess earnng and wll no be affeced b accounng choces. he aenon o he relaonshp beween heorecal frm alue and he resdual ncome sream has araced consderable praconer neres and resuled n a number of proprear models beng mareed Gregor Saleh & ucer 4. he Ohlson 995 Valuaon Model OVM one specal case of he general class of RIM s ealuaed as a major breahrough Bernard 995 and landmar wors n fnancal accounng Lundholm 995. he parcular nnoaon n he Ohlson model s he emplomen of an AR lnear nformaon dnamc LID whch s comprsed of abnormal earnngs and a arable represenng oher nformaon whose source s uncorrelaed wh accounng nformaon. We can ew he abnormal earnngs as a conracon of aboe normal earnngs. he ermnolog s moaed b he concep ha normal earnngs should relae o he normal reurn on he capal nesed a he begnnng a he perod ha s ne boo alue a dae mulpled b he neres rae r. hus one nerpres a as earnngs mnus a charge for he use of capal r b as n he followng equaon:. A4 a r b he LID assumpon can be wren formall as a γ ϖ ε a ε --- AR Lnear Dnamc A5 65
where ϖ s perssence parameer of abnormal earnngs ε ε s whe nose error erms wh zero mean γ s perssence parameer of oher nformaon. a Subsung equaons A4 and A5 o RIM elds he Ohlson Valuaon Model: A6 where ϖ r. r r r ϖ a P b See Append B for he deraon process. he Ohlson model saes ha he soc prce s a lnear funcon of boo alue of he equ b curren abnormal earnngs and an nercep erm whch aes he followng regresson form: P a b ε A7 66
B. Deraon of he Ohlson Valuaon Model he followng s he sraghforward deraon process of equaon A7: ] [ ] [ ] [ ] [ ] [ ] [ a a a a E r r b E E r E r r b E r b E r b P ϖ ε ε ϖ ε ϖ ] [ a E r r r b ϖ ] [ ] [ ] [ a a a a b P r r r b E r r E r r b E r r r b ϖ ϖ ϖ ϖ ϖ a a a f r r r r b P r r r r r r b P R r r r b P ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ 67
C. he Ohlson Appromaon 995 Model Ohlson 995 presens us wh he lcense o brea wh he radonal focus on eplanng prce behaor and o shf ha focus o predcng earnngs. he e les n he followng appromaon. I saes ha he alue of he frm can be well appromaed een oer a fne horzon b a funcon of forecased earnngs boo alue and dscoun raes. he onl assumpon requred s ha hese forecass be conssen wh clean surplus relaon. We begn b defnng a arable oe ha he amoun V as follows: r τ V b r E[ τ r b τ ]. C r τ V s a funcon of fuure earnngs and boo alues measured oer a fne horzon. Howeer despe he lmed horzon V appromaes he alue of he frm so long as he horzon s long enough whch can be descrbed as he followng equaon: lmv P. C Equaons C and C mpl ha he abl o predc earnngs and boo alue --- een oer a fne horzon --- s anamoun o he abl o appromae curren alue. For a specal emprcal applcaon of he Ohlson 995 appromang model we use he fuure earnngs and boo alue forecased oer a horzon of 4 perods. ha s we le 4. Equaon C urns ou o be 4 4 4 r τ V b [ ]. 4 r E τ r b τ C3 r τ If 4 V prodes a good appromaon of frm alue we should be able o eplan a large fracon of he araon n soc prces wh he arables on he rgh-hand sde of equaon C3. hs suggess he followng regresson model: 4 τ τ τ τ P b E [ rb ]. C4 Model C4 s he Ohlson Model used n hs paper. 68
D. Defnons of Daa Iems All he defnons are for ndusrals. DOW JOES IDUSRY GROUP represens he ndusr classfcaon assgned b Dow Jones based on he compan s lnes of busness. GEERAL ISUSY CLASSIFICAIO represens a compans general ndusr classfcaon. Indusral; Ul; 3 ransporaon; 4 Bans/Sangs and Loan; 5 Insurance; 6 Oher Fnancal DOW JOE MARKE SECORS are a sandardzed seres of dgs ha are used o caegorze mare segmens ssued b Dow Jones. he Global Indusr Classfcaon Sandard GICS was deeloped b Morgan Sanle Capal Inernaonal MSCI a premer ndependen proder of global ndces and benchmar-relaed producs and serces and Sandard & Poor s S&P an ndependen nernaonal fnancal daa and nesmen serces compan and a leadng proder of global equ ndces. he GICS classfcaons am o enhance he nesmen research and asse managemen process for fnancal professonals worldwde. I s he resul of numerous dscussons wh asse owners porfolo managers and nesmen analss around he world and s desgned o respond o he global fnancal commun s need for an accurae complee and sandard ndusr defnon. GICSSECOR means GICS Codes. he followng webses are abou he secor defnons. hp://www.msc.com/equ/gics_secor_defnons_5.pdf COMPAY IDEIY KEY ID 69
PRICE CLOSE means he las prce an ssue raded a for ha da. For he quarerl daa se of PrceClose all he alues are he las prce for he las da of correspondng quarer. OAL ASSES represen he sum of oal curren asses long erm receables nesmen n unconsoldaed subsdares oher nesmens ne proper plan and equpmen and oher asses. OAL LIABILIIES represen all shor and long erm oblgaons epeced o be sasfed b he compan. I ncludes: Curren Lables Long erm Deb 3 Proson for Rs and Charges non-u.s. corporaons 4 Deferred aes 5 Deferred ncome 6 Oher lables 7 Deferred a labl n unaed reseres non-u.s. corporaons 8 Unrealzed gan/loss on mareable secures nsurance companes 9 Penson/Pos reremen benefs Secures purchased under resale agreemens bans I ecludes: Mnor Ineres Preferred soc equ 3 Common soc equ 4 on-equ reseres PREFERRED SOCK represens a clam pror o he common shareholders on he earnngs of a compan and on he asses n he een of lqudaon. For U.S. corporaons s alue s shown a he oal nolunar lqudaon alue of he number of preferred shares ousandng a ear end. If preferred soc s redeemable a 7
anme b he shareholder s shown a redempon alue or f he compan carres a a hgher alue han he nolunar lqudaon alue he saed alue. Preferred soc of subsdar and premum on preferred soc s ncluded n preferred soc. I ecludes mnor neres n preferred soc. For on-u.s. Corporaons he saed alue of preferred soc s shown and ncludes all preferred soc relaed accouns. For on-u.s. Corporaons preference soc whch parcpaes wh he common/ordnar shares n he profs of he compan s ncluded n common equ. COMMO SHARES OUSADIG represen he number of shares ousandng a he compans ear end. I s he dfference beween ssued shares and reasur shares. For companes wh more han one pe of common/ordnar share common shares ousandng represens he combned shares adjused o reflec he par alue of he share pe denfed n feld 65 - pe of Share. BOOK VALUE PER SHARE represens he boo alue proporoned common equ dded b ousandng shares a he compans fscal ear end for non-u.s. corporaons and a he end of he las calendar quarer for U.S. corporaons. Preference soc has been ncluded n equ and he calculaon of boo alue per share where parcpaes wh common/ordnar shares n he profs of he compan. I s ecluded n all oher cases deduced a lqudaon alue for U.S. companes and a par alue for all ohers. EARIGS PER SHARE means he poron of a compans prof allocaed o each ousandng share of common soc. EPSMeanQR s he mean alue n a se of EPS esmaes of he frs fscal quarer for a compan; EPSMeanQR s he mean alue n a se of EPS esmaes of he second fscal quarer for a compan. Same deas can be adoped b EPSMeanQR and EPSMeanQR3. See able for more deals. 7
EPSConsensusForecasPerodQR s he perod monh for whch an EPS esmae s beng forecased. Same deas can be adoped o EPSConsensusForecasPerodQR3-4. See able 3 for more deals. able EPSMeanQR parl Ke: Feb-98 Mar-98 Apr-98 Ma-98 Jun-98 Jul-98 C34.5.5.5.53.46.46 C48.39.38.37.37.37.34 C8595.54.54.54.56.56.56 C6.6.6.6.6.7.7 C85......4 C68.4.3.3.4.3. C4 -.7 -.9 -.8 -.8 -.9 -.8 C.7.7.7...9 C.93.9.84.94.94.9 C9653.7.7.7.7.7.8 C5.9.9.9.9.9.8 able 3 EPSConsensusForecasPerodQR parl Ke: Feb-98 Mar-98 Apr-98 Ma-98 Jun-98 Jul-98 C34 Mar998 Mar998 Mar998 Jun998 Jun998 Jun998 C48 Mar998 Mar998 Jun998 Jun998 Jun998 Sep998 C8595 Mar998 Mar998 Mar998 Jun998 Jun998 Jun998 C6 Apr998 Apr998 Apr998 Apr998 Jul998 Jul998 C85 Feb998 Feb998 Ma998 Ma998 Ma998 Aug998 C68 Mar998 Mar998 Mar998 Mar998 Jun998 Jun998 C4 Mar998 Mar998 Jun998 Jun998 Jun998 Sep998 C Mar998 Mar998 Mar998 Jun998 Jun998 Jun998 C Mar998 Mar998 Mar998 Jun998 Jun998 Jun998 C9653 Mar998 Mar998 Mar998 Mar998 Mar998 Jun998 C5 Mar998 Mar998 Mar998 Jun998 Jun998 Jun998 reasur blls are shor-erm deb nsrumens used b he U.S. Goernmen o fnance her deb. Commonl called -blls he come n denomnaons of 3 monhs 6 monhs and ear. Each reasur bll has a correspondng neres rae.e. 3-monh -bll rae - ear -bll rae. 7
E. How o reree daa b homsom OE Analcs Ecel Add-n For nsance we wan o reree he daa se of PrceClose.. Mae sure ou hae Mcrosof Ecel nsalled n our compuer.. Download and nsall he sofware homson OE Analcs Ecel Add-n. Open he homson OE Baner Analcs Web page. Clc on ools a he op of he homson One page. 3 Clc on he Offce ools ab on he ools page. 4 See he secon homson OE Analcs for Offce Verson.. 5 Clc on he Download ln n hs secon and follow he drecons o sae he add-n nsallaon fle o our compuer. 6 here are bref nsallaon drecons on hs page and sep-b-sep nsallaon nsrucons also can be downloaded from hs secon. Durng he nsallaon process our homson Analcs username and password mus be enered. Once nsalled he homson Analcs oolbar wll be sble n Ecel. 3. Upload S&P5 no homson Analcs call 8-66-7878-- for help. 4. In a blan Ecel shee choose he frs cell A. 5. On he homson Analcs oolbar Clc on Wzards. Choose Repor Wzards. 3 In Sep clc Add Enes. 4 On he pop-up wndow clc Download. 5 Choose SP5 and clc OK. 6 On he reurned wndow clc OK agan. 7 In Sep clc Add/Ed ems. 8 Under Daa Iems For clc on he pull-down buon choose Daasream. 9 Under Search for Iem npu prceclose double clc prceclose n he wndow below. hen clc OK. In Sep 3 under Sarng Perod clc on he buon wh an con of calendar. Frs choose Quarerl hen choose Q of 998 ; under Endng Perod use he same wa o choose Q of 4. 73
Clc e. Clc Fnsh. 3 On he pop-up wndow choose o. You can ge he PrceClose daa se of SP5 n a spreadshee. 4 Sae he fle. 74
F. How o erac quarerl daa ou of monhl daa Companes forecas her epeced earnngs eer monh for he followng four fscal quarers. oe ha he fscal quarers ma hae dfferen sarng and endng monhs from he calendar quarers. Dfferen companes ma hae dfferen defnons of her fscal quarers. hs paper uses he laes forecas alue for each quarer o represen he correspondng quarer alue. Based on hs dea here are 4 seps o ge quarerl daa ou of monhl daa. For nsance we wan o ge he quarerl daa of EPSMeanQR n able. able 4 ransformed EPSConsensusForecasPerodQR parl ID Feb-98 Mar-98 Apr-98 Ma-98 Jun-98 Jul-98 34 3 3 3 48 3 3 3 4 8595 3 3 3 6 3 3 85 3 68 3 3 4 3 3 3 4 3 3 3 3 3 3 9653 3 5 3 3 3. Replacng he alues n each row n able 3 EPSConsensusForecasPerodQR wh o 4 b ncreasng each me from he earles dae o he laes dae whch resuls n able 4.. Fnd ou he las forecas monh for each fscal quarer and represen wh. All oher forecas monhs are se o be. hs can be done b he command dff n MaLab. he resul can be shown n Mar. 3. Consruc a mar Mar conanng he alues of EPSMeanQR n able 3. 4. Fnd ou he epeced earnng correspondng o each quarer. hs can be done b do mulplng.* Mar o Mar whch resuls n Mar 3. 75
5. Erac he quarerl epeced earnng b deleng all he from Mar 3. hs can be done b command BAA> where A refers o he resed form Mar 3 B s he quarerl daa mar. oe ha hs command onl wors when each row of A has he same amoun of pose alues. In applcaon A needs o be adjused o sasf hs requremen. Also noe ha some epeced earnngs are negae and some are zeros. In order no o mss hs alues we can add a relae large pose alue sa o each em n order o mae hem pose. Afer geng he quarerl daa mar can be subraced o ge he real alues. Mar Mar.5.5.5.53.46.46.39.38.37.37.37.34.54.54.54.56.56.56.6.6.6.6.7.7......4.4.3.3.4.3. -.7 -.9 -.8 -.8 -.9 -.8.7.7.7...9.93.9.84.94.94.9.7.7.7.7.7.8.9.9.9.9.9.8 76
Mar 3.5.46.38.37.54.56.6...4 -.9 -.9.7.84.9.7.8.9.8 77
G. Deraon of CPO d p d p p π M f f f f f f f f f f f f f f f f f f f f f p f p p M p M d d p d p M M M M M M M / / / / π π π π ϖ π π π π π π π π ϖ ϖ π π π π π π π π π π π π π 78
References Ohlson J. 995. Earnng boo alue and ddends n secur aluaon. Conemporar Accounng Research : 66-687. Bernard V. 995. he Felham-Ohlson framewor: mplcaons for emprcss. Conemporar Accounng Research : 733-747. Felham G.& Ohlson J. 995. Valuaon and clean surplus accounng for operang and fnancal aces. Conemporar Accounng Research : 684-73. Callen. J. & Morel M.. Lnear accounng aluaon when abnormal earnngs are AR. Reew of Quanae Fnance and Accounng 6: 9-3. Oa K.. A es of he Ohlson 995 model: emprcal edence from Japan. he Inernaonal Journal of Accounng 37: 57-8. andram B Peruccell J. 997. A Baesan analss of auoregresse me seres panel daa. Journal of Busness & Economc Sascs 5: 38-334. Yng & Kuo 3. Forecasng soc prces usng a herarchcal Baesan approach. Unpublshed Research Paper. Alber J Chb S. 993. Baesan analss a Gbbs samplng of auoregresse me seres subjec o Maro mean and arance shfs. Journal of Busness and Economc Sascs : -5. Gelfand AE Hlls S Racne-Poon A Smh AFM. 99. Illusraon of Baesan nference n normal daa models usng Gbbs samplng. Journal of Amercan Sascal Assocaon 85: 97-985. 79
Rosenberg M Andrews R Len P. 999. A herarchcal Baesan model for predcng he rae of non-accepable n-paen hospal ulzaon. Journal of Busness and Economc Sascs 7: -8. Waefeld J Smh AFM Racne-Poon A Gelfand AE. 994. Baesan analss of lnear and non-lnear populaon models b usng he Gbbs sampler. Appled Sascan 43: -. 8