Forecasting Stock Prices Using a Hierarchical Bayesian Approach

Transcription

1 Joural of Forecasg J. Forecas. 4, (005) Publshed ole Wle IerScece ( DOI: 0.00/for.933 Forecasg Sock Prces Usg a Herarchcal Baesa Approach JUN YING, LYNN KUO * AND GIM S. SEOW Idaa Uvers School of Medce, USA Uvers of Coeccu, USA ABSTRACT The Ohlso model s evaluaed usg quarerl daa from socks he Dow Joes Idex. A herarchcal Baesa approach s developed o smulaeousl esmae he ukow coeffces he me seres regresso model for each compa b poolg formao across frms. Boh esmao ad predco are carred ou b he Markov cha Moe Carlo (MCMC) mehod. Our emprcal resuls show ha our forecas based o he herarchcal Baes mehod s geerall adequae for fuure predco, ad mproves upo he classcal mehod. Coprgh 005 Joh Wle & Sos, Ld. ke words auoregresso; herarchcal mxure prors; MCMC; predco; smulaeous esmao INTRODUCTION Rece developme he secur valuao leraure has provded a model ha relaes he sock prce o s book value ad expeced fuure eargs. I cludes he work of Berard (995), Felham ad Ohlso (995), Lag ad Ludholm (996) ad Ohlso (99, 995). These sudes develop a logcall cosse framework for hkg abou equ valuao usg accoug daa. The prmar objecves of hs paper are o emprcall evaluae he adequac of he secur valuao model ad o use o forecas sock prces. The secur valuao model has bee developed based o a sgle frm. The emprcal leraure boh accoug ad face s based prmarl o classcal sascal echques. I hs paper, we appl a ovave sascal mehod, a herarchcal Baesa (HB) approach ha allows mproved esmao of he regresso coeffces b sharg formao across frms. Usg 4 ears of quarerl sock prce daa, accoug book values ad expeced fuure eargs for 8 compaes cluded he Dow Joes Idusral Average, we show ha he forecas based o he HB model s cossel superor o hose obaed usg he classcal approach. The orgal Ohlso model proposes ha he sock prce s a lear fuco of he compa s book value per share ad expeced excess eargs per share for he followg four perods wh ormall dsrbued ovao erms. Each compa has s ow coeffces; we use b = (b,..., * Correspodece o: L Kuo, Deparme of Sascs, Uvers of Coeccu, Sorrs, CT , USA. E-mal: l@sa.uco.edu Coprgh 005 Joh Wle & Sos, Ld.

2 40 J. Yg, L. Kuo ad G. S. Seow b 6 ) =(b,,..., b,6 ) o deoe he regresso coeffces of he ercep, book value, each of he expeced excess eargs for he followg four perods for he h compa wh =,...,. The model ca be descrbed as follows for all = 0,..., T; = + v + Ê ˆ b, b, k+ w + k u ËÂ b,, + 4 k= () where deoes he h compa s sock prce per share a me ; v deoes he book value per share of sock a me ; w,+k deoes he expeced excess eargs per share of sock he kh perod afer me ; ad u s he ovao erm for. I fac, Ohlso (99) proposes he expeced excess eargs as w = E[ s -rv ], + k, + k, + ( k-) () where s,+k deoes he eargs per share of sock he ( + k)h perod for k =,..., 4 ad r s he dscou rae a me. The expeced excess eargs are, however, o avalable erms of facal accoug daa. We use w, + k = E( s, + k) - rv, + ( k-) (3) sead of (). I hs paper, we use x = (x,..., x 6 ) =(, v, w,+, w,+, w,+3, w,+4 ) o deoe he vecor of he predcors of he h compa a me. We emplo a HB approach o fereces he Ohlso model. A geeral form of HB model s preseed Ldle ad Smh (97). I hs paper, we exed he HB approach wo dsc aspecs. Frs, each regresso coeffce b s modelled as a mxure of ormal dsrbuos wh ukow hperparameers. Thus, he herarchcal seup has wo ses of parameers o be esmaed. Oe s he se of parameers of eres ad he oher s he se of hperparameers ha model he parameers. Secod, he ovao erms u are modelled he followg was: (a) u has a frsorder auoregressve srucure (AR()) for each compa, ha s, corr(u,, u,- ) = r for =,..., T; ad (b) u are allowed o have heerogeeous varace amog he compaes, ha s, var(u ) = s /( - r ) for =,..., ad for all. The ukow parameers (s, r ) are modelled wh kow proper prors. I fac, he AR() srucure ca easl be exeded o a more geeral auoregressve movg average (ARMA) srucure. A wdel used HB srucure for he regresso coeffces s o assume ha b are..d. from a ormal dsrbuo wh s hperparameers beg modelled b he hperprors. The use of hs herarchcal seup ca borrow sregh across he dffere dvduals ad brg shrkage effecs o he poseror mea of he regresso coeffces. However, our daa aalss shows ha he esmaes of he regresso coeffces some compaes are que far awa from he major of oher compaes. The use of a ormal dsrbuo for he..d. regresso coeffces s o adequae ad wll cause over-shrkage o hese esmaes. Thus, we eed a model ha s flexble eough o accommodae hese oulers whle a he same me adjusg he esmaes for shrkage effecs. A aural choce would be a mxure of wo ormal dsrbuos. The frs compoe of he mxure reas he compa s dvdual mea; ad he secod compoe of he mxure shares he commo mea amog dffere compaes. Müller ad Roser (997) provde a dealed dscusso o herarchcal mxure prors. Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

3 Forecasg Sock Prces Usg a Herarchcal Baesa Approach 4 Baesa HB mehodolog has bee appled o he aalss of varace covarace marces of he ovao erms u. For example, Gelfad e al. (990), Wakefeld e al. (994) ad Roseberg e al. (999) model u as..d. radom varables over ad. Gelfad ad Sfrds (996) ad Kasm ad Raudebush (998) exed he homogeeous varace models o he heerogeeous varace models, whle keepg he codoal depede srucure of he ovao erms wh each compa. Kasm ad Raudebush (998) relax he codoal depedece assumpo ad assume a compoud smmer (cosa correlao) srucure for he ovao erms wh each compa. The varaces are assumed o be dffere, bu he correlao coeffces are he same amog he compaes. I order o model he ovao erms as a fuco of mes, Alber ad Chb (993) ad Chb (993) appl a Baesa framework o he auoregressve models her daa aalss wh homogee of varace. Chb ad Greeberg (995) exed he HB mehod from he auoregressve process o he ARMA( p, q) process. Prelmar daa aalss (o preseed here) shows ha (a) he varace dffers wdel amog he 8 compaes, ad (b) for each compa he sock prces follow a AR() process. Moreover, he auocorrelao coeffces var subsaall amog he compaes. Therefore, we exed he mehodologes Alber ad Chb (993) ad Chb (993) o accommodae he heerogee of varaces ad auocorrelaos. The advaages of he herarchcal formulao ca be summarzed. (a) We rea he frmspecfc feaure of he whole assembl. Each frm s specfc regresso coeffces wll provde us wh formao o how sesve he frm s sock prce s o he chages of book values ad expeced excess eargs. (b) We provde a more formao-effce modellg for he daa ad a full shess o all he daa for he regresso coeffces. Because he regresso coeffces for he h compa are o ol affeced b he daa from he h compa, bu also affeced b oher compaes daa (perhaps o a lesser degree), hs full shess provdes a more sasfacor soluo o modellg our daa. (c) Ths full shess s kow o provde esmaes ha have smaller mea squared errors ha he usual leas squared or classcal esmaes (Se, 955). O he mehodolog for ferece, we wll emplo he MCMC algorhm (Gelfad ad Smh, 990; Che e al., 000) o provde po ad erval esmaes for he ukow parameers. The MCMC algorhm smulaes radom parameers (or blocks of parameers) from her full codoal dsrbuo gve he daa b cosrucg a Markov cha, so he saoar dsrbuo of he Markov cha s he desrable poseror dsrbuo. The samplg-based approach crcumves he dffcules evaluag muldmesoal egrals eeded Baesa ferece. I allows us o oba a feaures, such as mea, varace, quales ad hsograms, of he poseror dsrbuo. I addo o esmao, we are also cocered wh he ssues of model adequac ad model seleco. We compare he ou-of-sample performace of our mehod o ha of he classcal approach for wo meframes: oe s from he 40h quarer o he 54h quarer, he oher s he 5d quarer (he 3rd quarer of 997). The crera we use for comparso clude he predcve mea squared error (PMSE), he predcve mea absolue error (PMAE), he predcve mea absolue relave devao (PMARD) ad he coverage probabl for he forecas ervals. Our resuls show ha he HB mehod based o he mxure prors for b has beer forecasg power ha he HB mehod wh a sgle ormal pror, ha furher mproves upo he classcal mehod for boh meframes. The mproveme s more dramac for he 5d quarer. The wo ses of prors he HB mehod are also compared b usg he prequeal pseudo-baes facor (PPBF). The resul suppors he mxure pror assumpo. Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

4 4 J. Yg, L. Kuo ad G. S. Seow MODEL The Ohlso model assumes ha he error erms are codoall depede amog compaes ad a AR() srucure me perods wh each compa. We ca express he Ohlso model usg he followg expressos: u e = x b + u = ru + e, - ~ ( 0, s ) for =,..., T..d. N (4) where s he observao of compa a me, x = (x,..., x K ) s he vecor of K predcors for he h compa a he h perod, wh x =, b = (b,..., b K ) s he vecor of ercep ad slope coeffces of he predcors. The AR() srucure s descrbed b he ovao erm. The resdue e s depede of e for π. The observao a me 0 s cosdered o be saoar, ha s, for =,...,, s Ê 0 ~ N x 0b, Ë - r ˆ (5) Codog o r, s eas o appl a chage of varable echque o (4). Le * = (*,..., * T ), X* = (x*,..., x* T ) ad e = (e,..., e T ), where * = - r,- ad x* = x - r x,- for =,..., T. Equao (4) ca be wre as * = X* b + e e ~ N ( 0, s I) T (6) We use he oao N T (m, S) o deoe a T-varae ormal dsrbuo wh mea m ad varace covarace marx S. We use I o deoe he de marx of rak T. To cosruc HB esmaors, we use he followg herarchcal model. A he frs sage, he observaos * ad 0 are descrbed b he parameers {b, s, r ) for each =,...,. Ths relaoshp ca be expressed he lkelhood fuco L for {*, 0 ; =,..., }. Uder he assumpo ha he observaos are codoall depede amog he compaes, we have È Ê L( b, s, r *, 0 ) = N T( * X* b, s I) N 0 x 0b, Ë = = ÎÍ s - r ˆ (7) A he secod sage, we provde he pror dsrbuos for he parameers {b, s, r ; =,..., }, ad we le b, s ad r be codoall depede. I parcular, we model b as a mxure of wo ormal dsrbuos: p( b m, S, h, W, w) = wn K( b m, S) + ( -w) N K( bh, W) s (8) Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

5 Forecasg Sock Prces Usg a Herarchcal Baesa Approach 43 where {m, S, w ; =,..., } ad {h, W} are he ukow hperparameers. The frs compoe ha allows dvdual meas brgs more dsperso amog {b ; =,..., }, ad hus also preves over-shrkage of he esmaes for he oulers. The secod compoe ha shares he same mea provdes a mechasm for sharg formao across frms, as s usuall doe he HB mehod. The pror dsrbuo for s s assumed o be a verse gamma, ha s, s ~ IG(a, b ) b b wh mea ad varace. Boh a ad b are kow. We assume a rucaed a - ( a -) ( a - ) ormal for he correlao coeffce r, ha s, r ~ N(r 0, s r ) I (- r ), where he hperparameers r 0 ad s r are kow. I order o make fereces o r prmarl from he daa raher ha from he pror, we usuall se s r suffcel large. The pror a he secod sage ca be wre as = [ p b u, S, h, W, w p s a, b p r r0, s ]= = ( ) ( ) ( ) {[ wn K( b m, S) + ( -w) N K( b h, W) ] IG( a, b) N ( r0, s ) ( - r ) } r r (9) The ukow hperparameers {m, S, w ; =,...,} ad {h, W} are modelled a he hrd sage. We assume he are codoall depede of each oher ad le m ~ N K (m, M), h ~ N K (h, H), S - ~ W K (u, V) (Wshar dsrbuo wh u degree of freedom ad scale marx V), W - ~ W K (q, Q) ad w ~ Be (r, s) (Bea dsrbuo) for =,...,. The parameers {m, M, h, H, u, V, q, Q, r, s} are all assumed o be kow. A schemac dagram for he h frm s gve Fgure. Uder he crcumsace ha w = 0 for all, we have he o-mxure model whch all b ( =,..., ) share he same mea ad varace across frms. s r b a b r s 0 S w h W r m m M v V r s h H q Q Fgure. Schemac dagram for he h frm ( =,..., ). The crcles coa he ukow parameers ad hperparameers, he squares coa he kow parameers. All oher frms addo o he h frm share he same parameers {h, W, m, M,..., q, Q} Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

6 44 J. Yg, L. Kuo ad G. S. Seow GIBBS SAMPLING We frs summarze he full codoal dsrbuos for he parameers {b, s, r ; =,..., }. (a) Noe where wh A B È * S S * + ( - ) * - X X r x 0x 0 = Í + ÎÍ s Ê S* ˆ = Á Ë S Ê W* ˆ = Á Ë W ( ) + ( - ) ( ) b.~ w* N b m*, S* w* N b h*, W* K K [ ms m m S m ] Ï exp Ì- - * * * Ó - - [ h W h h W h ] Ï exp Ì- * - * * Ó Thus, b s updaed b frs geerag a Beroull radom varable w ~ B (, w* ), ad he geerag b ~ N K (b m*, S* ) f w = or geerag b ~ N K (b h*, W* ) f w = 0. (b) ( ) s I Ga, b.~ * * ( ) ( - ) + - ( - ) T where a* = a + + b* b * * * * r, = + -X b X b 0 x * 0 b. Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005) - È * * + ( - ) * = S* - X r x m ÍS m + ÎÍ s È * * + ( - ) W* - X X r x 0x 0 = ÍW + ÎÍ s È X* * + ( -r ) x h* - = W* ÍW h+ ÎÍ s wa w* = w A + ( -w ) B (0) ()

7 Forecasg Sock Prces Usg a Herarchcal Baesa Approach 45 (c) Smulag he correlao coeffces r. Le u = (u,..., u T ) ad u 0 = (u 0,..., u,t- ), he full codoal dsrbuo of r ca be wre as Ï Ô pr (. ) µ ( -r ) expì- r -r* Ô 0 (- r ) ÓÔ s * r Ô Ï Ô = expì log( -r ) - r -r* Ô 0 (- r ) ÓÔ s * r Ô ( ) ( ) () where m0m0 m0 s * r = È + - Í s r s Î r0 m0m0 r* 0 = s * È r + Í Îs r s We cao oba a smple form of he codoal poseror des of r from whch he sample value ca be geeraed drecl. However, s eas o show ha p(r.) s log-cocave ad hece we ca use he adapve rejeco samplg mehod (Glks ad Wld, 99) o updae r. We ex descrbe a daa augmeao sep he Gbbs sampler o faclae he geerao of he hperparameers. (d) Daa augmeao sep of smulag he auxlar varables z. The pror mxures of b (8) usuall make dffcul o sample he ukow hperparameers {m, S, w ; =,..., } ad {h, W}. Therefore, we roduce a lae varable z as De e al. (995). The lae varable z ca be smulaed depedel from he Beroull dsrbuo B(, p ) where - w N ( b m, S ) w m, ( w ) ( h, W) K p = N K( b S) + - N K b (3) We he cosder he jo des of (b, z ) ha s useful dervg he codoal des of he hperparameers. Fall, we provde he full codoal dsrbuos for he hperparameers {m, S, w ; =,..., }, {h, W} based o her prors ad he jo des of (b, z ). (e) ( ) m N K m M.~ *, * (4) where M* = (M - + z S ī ) - ad m* = M* (M - m + z S ī b ). (f) h. ~ N K ( h*, H* ) (5) Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

8 46 J. Yg, L. Kuo ad G. S. Seow where H* = (H - + ( - z + )W - ) - È - -Ê +, h* = H* H h+ ( - ) ˆ ad z = Â z. Î Í W ËÂ z b (g) S W K u V -.~ ( *, *) = = (6) where u* = u + z ad V* = [V - + z (b - m )(b - m ) ] -. (h) W -. ~ W K ( q*, Q* ) (7) where q* = q + z + È - ad Q* = Q + ( - )( - )( - ). Î Í Â z b h b h () - - ( ) w B er s.~ *, * (8) where r* = r + z ad s* = s + - z. The dealed procedures for dervg he above full codoal dsrbuos are gve Yg e al. (00). All he dsrbuos excep (c) are sadard form ad herefore s sraghforward o geerae he radom varaes. As descrbed below, he MCMC procedure cosss of seps hrough 3, performed eravel. The seps ad 3 coss of subseps ha are carred ou sequeall for a sgle cha. We ca also replcae he Markov cha b drawg depede al values of he parameers ad hperparameers.. Updae he parameers {b, s, r ; =,..., } gve he hperparameers ad daa b carrg ou he followg subseps. o.3 depedel for each =,...,... Geerae b from he codoal mxure ormal des fucos (0) wh he mehod dscussed (a)... Geerae s drecl from he verse gamma dsrbuo descrbed ()..3. Updae r as () b applg he adapve rejeco samplg mehod.. Geerae he lae varable z from a Beroull dsrbuo dscussed (d). 3. Geerae he hperparameers {m, S, w ; =,..., } ad {h, W} gve he values of he parameers seps ad : 3.. Geerae w from a Bea dsrbuo gve (8). 3.. Geerae m ad S from he full codoal dsrbuos descrbed (4) ad (6), respecvel Geerae h ad W - from he full codoal dsrbuos descrbed (5) ad (7), respecvel. Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

9 Forecasg Sock Prces Usg a Herarchcal Baesa Approach 47 The HB mehod wh sgle pror o b ca be cosdered as a specal case of he HB mehod wh mxure pror b leg boh w (hece w* ) ad z be fxed a 0. I s sraghforward o see ha he full codoal dsrbuos of b (0), h (5) ad W - (7) wll be adjused wh w* = 0 (0) ad z = 0 (5) ad (7). Therefore, he MCMC seps o he sgle pror assumpo are gve as above b usg he adjused codoal deses (0), (5) ad (7) ad skppg seps, 3. ad 3.. FORECAST Havg obaed he poseror dsrbuo of he parameers, we ca use o predc fuure sock prces. We wll llusrae hs b he oe-sep forecas. Le Y + = (Y,+,..., Y,+ ) deoe he radom fuure observaos a perod + for he compaes, ad D ad q deoe all he observed daa ad he parameers respecvel from - T + o (a wdow of daa of legh T perods). Our predco for he ( + )h perod follows from he predcve des Ú f( Y+ D ) = f( Y+ D, q ) p( q D ) dq (9) The predcve des (9) ca be approxmaed b he followg Moe Carlo egrao from he Gbbs sampler: R L ˆ (, f Y+ D f Y+ D, lr ) ( ) = Â Â ( ) RL q r= l=îl/ + (0) where q (l,r) deoes he sample draw he lh erao ad he rh replcao of he MCMC gve he daa se D ad Î deoes he floor operaor, he larges eger less ha or equal o he argume. Sce he sock prces a perod + are depede amog he compaes gve q ad D, we have f( Y + D, q) = f( Y, + D,, q, ) = where q, = (b,, r,, s,). I parcular, he des fuco f (Y,+ D,, q, ) s jus he ormal des fuco of Y,+ wh mea x,+ b + r (, - x, b ) ad varace s,. Le (b (l,r), r (l,r),, s (l,r), ) deoe he sampled parameers from he lh erao ad he rh replcao of he MCMC, he we ca rewre (0) as R L ˆ lr, lr, f( Y + D) Ï ( ) ( ) = Â Â Ì [ N ( Y, + m,, s, )] RL Ó r= l=îl/ + = () where m (l,r), = x,+ b (l,r) + r (l,r) (, - x, b (l,r) ). Hece, he sampled sock prce for compa a me +, deoed b (l,r),+, ca easl be draw from he ormal dsrbuo fuco descrbed above Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

10 48 J. Yg, L. Kuo ad G. S. Seow gve he observed daa (,, x, ) ad he sampled parameers (b (l,r), r (l,r),, s (l,r), ). The mea of (l,r),+ over replcao r ad erao L, afer he frs half (L/) eraos beg bured, deoed b ŷ,+, ca be obaed b R L ( lr, ) ˆ, + = Â Â, + RL r= l=îl/ + () The 95% predcve erval for Y,+ ca be compued from he.5% ad 97.5% emprcal quales of he values (l,r),+, r =,..., R ad l =ÎL/ +,..., L. We he forecas Y + b movg he daa wdow up o D + (daa from perods - T + o + ). Smlarl, we repea hs process ul we oba all he forecass from ŷ,t+ o ŷ,t, where T s he edg perod of he daa se. MODEL VALIDATION AND MODEL CHOICE Boh model adequac ad model seleco ssues are dscussed hs seco. Model adequac s checked b comparg he observed,+ o s 95% predcve erval for Y,+ based o he resuls from he prevous seco. A model s judged o be adequae for each compa f abou 95% of he ervals for = T +,..., T coa he acual observed values for each compa. We use he coverage probabl, he ucoverage probabl ad he average 95% predcve erval legh as crera for model seleco. The are all compued as summar sascs based o all compaes ad all perods from T + o T. The coverage (ucoverage) probabl s he probabl ha he acual values are covered (ucovered) b he 95% predcve ervals; ad he average 95% predcve erval legh s he average erval legh of hese ervals. We prefer a model wh hgh coverage probabl ad low average legh. We also appl PMSE, PMAE, PMARD, PPBF for model seleco. The are expressed as follows: PMSE = T ( - T) PMAE = T ( - T) PMARD = T ( - T) = T T - ÂÂ = = T T - ÂÂ T - ) PPBF = f( Y + D) = = T ÂÂ = = T, +, + ( - ˆ ) T - - ˆ, +, + - ˆ, +, +, + (3) where ˆf (Y + D ) ad ŷ,+ are defed () ad (), respecvel. The bes model s he oe wh he smalles PMSE PMAE, PMARD or he bgges PPBF. Noe he PPBF s dffere from he pseudo Baes facor where he cross-valdao dea s used. I fac, he PPBF evaluaes he codoal jo predcve des of he daa T+,..., T, gve he wdow of legh T of,..., T. Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

11 Forecasg Sock Prces Usg a Herarchcal Baesa Approach 49 All he above comparsos are focused o he predcve performace of our forecas from he perod T + o T. We ca also sgle ou a parcular perod for comparso. For example, we wll focus o he 5d perod where he forecas dffereces bewee he HB ad he classcal mehod are more proouced. NUMERICAL EXAMPLES We ow appl he HB approach o he Ohlso model wh a real daa se. The prelmar daa se coas 30 compaes ha make up he Dow Joes Idex o Jul, 998 ad are recorded from he hrd quarer of 984 o he frs quarer of 998. We obaed he daa from wo dffere sources, Value Le ad he Federal Reserve Bak D.C. The sock prce, expeced eargs per share ad book value are all from Value Le. We use he -ear Treasur bll as he eres rae, ad s provded b he Federal Reserve Bak D.C. Value Le records he sock prce ad he expeced eargs per share for he ex four quarers. We selec he daa from he edg moh of each quarer he aalss ( fac, oher mohs he quarer would also be approprae). The eres rae s also colleced o a mohl bass, ad we selec he eres rae from he las moh of each quarer as he quarerl daa. Oce we have obaed he quarerl daa for expeced eargs per share, eres rae ad book value, we ca calculae he expeced excess eargs per share for he ex four quarers as descrbed (3). We also exclude wo compaes, Traveler Group ad Goodear Tre, from our daa se sce Value Le does o have complee expeced eargs per share for hese wo compaes durg he perod of our aalss. I summar, our daa se cosss of 8 compaes ad 54 quarers for each compa. As a geeral rule, we recere ad rescale he covaraes o reduce he correlaos amog he covaraes he lkelhood surface. Moreover, we use he auocorrelao fuco, paral auocorrelao fuco ad verse auocorrelao fuco o check he srucure of he ovao erms. The resuls show ha for mos of he compaes durg he perod of aalss, he ovao erm has a AR() srucure. The Durb Waso es for AR() also suppors our cocluso. Thus, we use he AR() srucure our umercal aalss. The al sud of he daa usg he classcal mehod based o he Ohlso model shows ha he esmaes of s ad r for each frm var wdel across frms. For example, he resuls from he daa of he frs 39 quarers show ha he rage of ˆr amog frms s from 0.45 o 0.977, ad he rage ) of s s from o Ths suggess ha we eed o corporae heerogeeous varaces ad separae correlao coeffces for each compa. The esmaed regresso coeffces for he expeced excess eargs per share four quarers show some oulers amog he 8 compaes. A sgle pror o b s wll cause he oulers o be shruk oo much. We cosder a mxure of wo ormal dsrbuos for he pror of b s, whch allows dvdual compaes ha share he same mea o borrow sregh from each oher bu also o keep her dvdual properes. The classcal mehod uses leas square esmaes o make fereces o he parameers a AR() lear model. Mos of he sascal sofware packages, for example PROC AUTOREG SAS, provde esmaes wh he classcal mehod. These esmaes provde useful formao for he pror choces for he parameers ad hperparameers he HB mehod. I fac, we le m= h= Â ˆb Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005) =

12 50 J. Yg, L. Kuo ad G. S. Seow ad M= H= Â Ŝ, where ˆb s he vecor of he classcal esmaes of he regresso coeffces = ad Ŝ s he classcal esmae of he varace covarace marx of he regresso coeffces of he h compa. We ca also oba ŝ from he classcal mehod. If we le he pror varace of s be 00, large eough ha he pror wll o drve he cocluso, he we ca derve a = + ( ŝ ) ad b = ŝ 00 (a - ) from he mea ad varace expresso of he verse gamma des. The mea of he pror of r s se o he esmae of he correlao coeffce ˆr ; ha s, r 0 = ˆr ; ad we le s r = 0 for all =,...,. For he kow parameers he pror of he Wshar dsrbuo, we le v = q = K = 6 ad V = Q = I K. Fall, we choose r = s =, so ha he pror of w s uforml dsrbued bewee 0 ad. We also cosder alerave prors for he pror sesv aalses. I parcular, we assume m = h = 0, M = H = 00I K ; a =.0 ad b = 0 (o make he varace of s he pror as large b b b 3 ACF ACF ACF Lag Lag Lag b 4 b 5 b 6 ACF ACF ACF Lag Lag Lag Fgure. Auocorrelao plos. The auocorrelao plos of he Gbbs sampler of regresso coeffces for Alcoa Ic. from he HB mxure pror mehod are gve Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

13 Forecasg Sock Prces Usg a Herarchcal Baesa Approach 5 as 39,); r = s =.5 (o make he Bea pror of w have more wegh owards he cere ha he uform pror); ad V = Q = 00I K (o make he pror fluece of S ī ad W - eve weaker). The resuls (o preseed here) show ha our aalss s que sesve o he dffere pror choces. All he parameer esmaes are compued from he MCMC wh 5000 eraos ad wo replcaos. Covergece of he Gbbs sampler s also assessed b he dagosc procedures from CODA (Bes e al., 995) ad Cowles ad Carl (996). The aalses based o he daa from he frs 39 quarers (ha s from he hrd quarer of 984 o he secod quarer of 993) are dsplaed Fgures o 7. Fgure shows he auocorrelao plos of he Gbbs sampler of regresso coeffces for Alcoa Ic. from he HB mxure pror mehod. The auocorrelao drops o 0 quckl, suggesg a effce Gbbs sampler. The same paer exss for oher compaes. We compare he HB esmaes of he regresso coeffces usg he mxure (sgle) pror o he classcal esmaes Fgure 3 (Fgure 4). We see he regresso coeffces of he HB mehod boh fgures shrk o each oher as compared o hose he classcal mehod b borrowg sregh from each oher. However, he HB mehod usg he sgle pror eds o over-shrk he regresso coeffces, as show Fgure b b b b 4 b 5 b 6 Fgure 3. Shrkage effec. The comparso bewee he HB mehod wh mxure pror ad he classcal mehod for he po esmae of he regresso coeffces for all he compaes s gve. The sold crcles are from he classcal mehod, he uflled crcles are from he HB mehod wh mxure pror Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

14 5 J. Yg, L. Kuo ad G. S. Seow b b b b 4 b 5 b 6 Fgure 4. Shrkage effec. The comparso bewee he HB mehod wh sgle pror ad he classcal mehod for he po esmae of he regresso coeffces for all he compaes s gve. The sold crcles are from he classcal mehod, he uflled crcles are from he HB mehod wh sgle pror We def he oulg esmae of b 3 (correspodg o he expeced s quarer excess eargs per share of sock) o be Coca Cola Co s. (KO) coeffce, ad he oulg esmae of b 6 (correspodg o he expeced 4h quarer excess eargs per share of sock) o be Procer & Gamble s (PG). Durg he frs 39 quarers, he chages of sock prces from he lowes o he hghes were $56.97 for KO ad $46.03 for PG, much hgher ha he average of he 8 compaes ($7.66). A he same me, he chages of he expeced s quarer excess eargs per share for KO ($3.) ad he expeced 4h quarer excess eargs per share for PG ($3.34) were ver close o he average of he 8 compaes ($3.3 ad $3.6, respecvel). Therefore, he regresso coeffce esmaes (b 3 KO ad b 6 PG) ur ou o be much larger ha ohers whe he classcal mehod s appled. Moreover, Fgure 5 shows ha he regresso coeffces of hese wo compaes are flueced b he oulg pos wh hgher sock prces ad relavel lower excess eargs per share, as we regress he sock prces o he correspodg covarae for each compa. The slope esmaes become smaller as he oulers are excluded from he esmao. The shrkage effec from he HB mehod wll brg smlar effecs b pug less weghs o he oulg compaes as expeced. Smlarl, he effec of he oulg compa of ChevroTexaco Corp. (CVX) for b 5 ca be smoohed ou b he HB mehod. Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

15 Forecasg Sock Prces Usg a Herarchcal Baesa Approach 53 COCA COLA CO PROCTER & GAMBLE Sock Prce Sock Prce Expeced s Qr Excess EPS 0 Expeced 4h Qr Excess EPS Fgure 5. Oulg pos. The sold les are he smple regresso les whe all he pos are cluded esmao. The dashed les are he smple regresso les whe he oulg pos (sold pos) are excluded from esmao The 95% credble ervals of he regresso coeffces wh he HB mehod usg he mxure pror as compared o ha wh he classcal mehod for each compa are show Fgure 6. We coclude: (a) he HB mehod brgs gher credble ervals ha he classcal mehod; (b) he credble ervals he HB mehod flucuae less ha hose of he classcal ervals across he compaes. Ths s aoher dcao of he effec of borrowg sregh from each oher he HB mehod. Smlarl, Fgure 7 compares he HB esmaes based o he sgle pror o ha of he classcal resuls. We see he credble ervals ed o be shruk owards he same erval, o adapve o he local varao of he regresso coeffces. Oe of he goals of our aalss s o compare dffere approaches o forecas he fuure sock prces. We produce he oe-sep-ahead predco rule based o he esmaes obaed respecvel from HB ad classcal mehods. I parcular, we sar wh he frs 39 quarers ad predc he sock prce he 40h quarer. The we move o he ex wdow, he secod 39 quarers (from he d quarer o he 40h quarer), esmae he parameers ad use hem o predc he 4s quarer. As we move he wdows of 39 quarers up b each quarer, we oba 5 quarers of predced sock prces for each compa. Fgure 8 (Fgure 9) provdes he 95% predcve ervals from he HB mehod usg he mxure (sgle) pror ad he classcal mehod for sx radoml pcked compaes. I ca be see ha he predcve ervals made b he HB mehod are gher ha hose from he classcal mehod. Moreover, we oce ha some quarers, he classcal mehod fals o forecas he real sock prce wh s 95% credble erval, whle he HB mehod s able o coa he real fuure Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

16 54 J. Yg, L. Kuo ad G. S. Seow b b b b 4 b 5 b 6 Fgure 6. Credble ervals. 95% credble ervals for he regresso coeffces from he HB mehod wh he mxure pror ad he classcal mehod for all he compaes are gve. The sold crcles are from he classcal mehod ad he uflled crcles are from he HB mehod wh he mxure pror observaos s ervals. Ths dcaes ha he HB mehod s more accurae predco ha he classcal mehod hese suaos. Table I exhbs he predcve performace amog he classcal mehod, he HB mehod wh he sgle pror ad he HB mehod wh he mxure pror for he me perod of he 40h quarer o he 54h quarer. Table II performs he same comparsos excep ol for he 5d quarer. Boh ables show HB mehods are more accurae ha he classcal mehod predco. From Table I, amog a oal of 40 predcos, he HB mehod usg he mxure pror coas all he real sock prces wh s 95% predcve ervals, he HB mehod usg he sgle pror msses fve of he real sock prces from s 95% predcve ervals, ad he classcal mehod fals o coa 0 of he real sock prces. From Table II, he classcal ervals fal o coa hree ou of 8 compaes ad oe for a of he HB mehods. Moreover, he average leghs of he classcal ervals from boh ables are loger ha hose of he HB mxure mehod. The laer ervals are also loger ha hose of he HB sgle mehod. We use PMSE, PMAE ad PMARD for model seleco. Boh ables show he HB mxure pror mehod s he bes amog hree mehods. The comparso of log(ppbf) bewee HB mxure ad sgle pror mehods also suggess ha he former s he preferred mehod. Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

17 Forecasg Sock Prces Usg a Herarchcal Baesa Approach 55 Table I. Model comparso Classcal HB sgle HB mxure Ucov (0/40) 0.0 (5/40) 0 (0/40) Ave. Lg PMSE PMAE PMARD (%) log(ppbf) b b b b 4 b 5 b 6 Fgure 7. Credble ervals. 95% credble ervals for he regresso coeffces from he HB mehod wh he sgle pror ad he classcal mehod for all he compaes are gve. The sold crcles are from he classcal mehod ad he uflled crcles are from he HB mehod wh he sgle pror CONCLUSIONS The HB mehod descrbed hs arcle has provded superor esmaes o he classcal mehod whe appled o he Ohlso model. The mproveme was obaed b poolg formao across compaes ad borrowg sregh from each oher. The mxure of wo ormal dsrbuos o he pror for he regresso coeffces helps us o avod he udesrable overshrkage problem. Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

18 56 J. Yg, L. Kuo ad G. S. Seow Table II. Model comparso for he 3rd quarer of 997 Classcal HB sgle HB mxure Ucov (3/8) 0 (0/8) 0 (0/8) Ave. Lg PMSE PMAE PMARD (%) log(ppbf) Boeg Co Chevro Corp Merck & Co Procer & Gamble Sears Roebuck Exxo Mobl Fgure 8. Forecasg. 95% predcve ervals based upo oe-sep-ahead forecasg for sx compaes 5 quarers from he hrd quarer 994 o he frs quarer 998 are ploed. The sold crcles are from he classcal mehod, he uflled crcles are from he HB mehod wh he mxure pror, ad he cross sgs dcae he real sock prces Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

19 Forecasg Sock Prces Usg a Herarchcal Baesa Approach 57 Boeg Co Chevro Corp Merck & Co Procer & Gamble Sears Roebuck Exxo Mobl Fgure 9. Forecasg. 95% predcve ervals based upo oe-sep-ahead forecasg for sx compaes 5 quarers from he hrd quarer 994 o he frs quarer 998 are ploed. The sold crcles are from he classcal mehod, he uflled crcles are from he HB mehod wh he sgle pror, ad he cross sgs dcae he real sock prces Cosequel, he HB mehod elds mproved predco as compared o he classcal mehod; ha s he 95% predcve ervals from he HB mehod are gher o average whle a he same me coag more real sock prces ha he classcal mehod. O he oher had, he HB mehod wh he sgle pror eds o over-shrk he regresso coeffces ad causes predcve credble ervals oo gh o coa he real sock prces. Hece, loses he predcve power o some exe as compared o he HB mehod wh he mxure pror. The HB mehod wh he mxure pror seems o pla a compromse bewee he classcal ad he HB sgle pror mehods. We are hopeful ha hs HB mehodolog wh he mxure dsrbuo srucure o he regresso coeffces wll provde a applcable example for facal researchers o pool formao from ma frms o oba more useful esmaes. REFERENCES Alber J, Chb S Baesa aalss va Gbbs samplg of auoregressve me seres subjec o Markov mea ad varace shfs. Joural of Busess ad Ecoomc Sascs : 5. Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

20 58 J. Yg, L. Kuo ad G. S. Seow Berard V The Felham Ohlso framework: mplcaos for emprcss. Coemporar Accoug Research : Bes NG, Cowles MK, Ves SK CODA: Covergece Dagoss ad Oupu Aalss Sofware for Gbbs Samplg Oupu, Verso 0.3. MRC Bosascs U: Cambrdge. Che M-H, Shao Q-M, Ibrahm JG Moe Carlo Mehods Baesa Compuao. Sprger: New York. Chb S Baes regresso wh auoregressve errors. Joural of Ecoomercs 58: Chb S, Greeberg E Herarchcal aalss of SUR models wh exesos o correlaed seral errors ad me-varg parameer models. Joural of Ecoomercs 68: Cowles MK, Carl BP Markov cha Moe Carlo covergece dagoscs: a comparave revew. Joural of he Amerca Sascal Assocao 9: De D, Kuo L, Sahu S A Baesa predcve approach o deermg he umber of compoes a mxure dsrbuo. Sascs ad Compug 5: Felham G, Ohlso J Valuao ad clea surplus accoug for operag ad facal acves. Coemporar Accoug Research : Gelfad AE, Hlls S, Race-Poo A, Smh AFM Illusrao of 6 Baesa ferece ormal daa models usg Gbbs samplg. Joural of he Amerca Sascal Assocao 85: Gelfad AE, Sfrds J Baesa aalss of facal eve sud daa. Advaces Ecoomercs : 5 6. Gelfad AE, Smh AFM Samplg based approaches o calculag margal deses. Joural of he Amerca Sascal Assocao 85: Glks R, Wld P. 99. Adapve rejeco samplg for Gbbs samplg. Appled Sascs 4: Kasm R, Raudebush S Applcao of Gbbs samplg o esed varace compoes models wh heerogeeous wh-group varace. Joural of Educaoal ad Behavoral Sascs 3: Lag M, Ludholm R The relao bewee secur reurs, frm eargs, ad dusr eargs. Coemporar Accoug Research 3: Ldle D, Smh AFM. 97. Baes esmaes for he lear model. Joural of he Roal Sascal Soce, Seres B 34: 4. Müller P, Roser G Semparamerc Baesa populao model wh herarchcal mxure prors. Joural of he Amerca Sascal Assocao 9: Ohlso J. 99. The heor of value ad eargs, ad a roduco o he Bell Brow aalss. Coemporar Accoug Research 8: 9. Ohlso J Earg, book value ad dvdeds secur valuao. Coemporar Accoug Research : Roseberg M, Adrews R, Lek P A herarchcal Baesa model for predcg he rae of o-accepable -pae hospal ulzao. Joural of Busess ad Ecoomc Sascs 7: 8. Se C Iadmssbl of he usual esmaor for he mea of a mulvarae ormal dsrbuo. I Proceedgs of he Berkele Smposum o Mahemacal Sascs ad Probabl. Uvers of Calfora Press: Berkele, CA. Wakefeld J, Smh AFM, Race-Poo A, Gelfad AE Baesa aalss of lear ad o-lear populao models b usg he Gbbs sampler. Appled Sasca 43: 0. Yg J, Kuo L, Seow G. 00. Forecasg sock prces usg a herarchcal Baesa approach. Techcal Repor TR0-, Uvers of Coeccu, Deparme of Sascs. Auhors bographes: Ju Yg s a asssa professor he Deparme of Medce, Dvso of Bosascs a Idaa Uvers School of Medce. Hs curre research eress clude Baesa compuao, Baesa ad profle lkelhood chage po models, ad sochasc volal models. L Kuo s a professor he Deparme of Sascs a he Uvers of Coeccu. She receved her PhD from he Uvers of Calfora a Los Ageles. Her curre research eress clude Baesa compuao, dscree choce models, eve hsor aalss, sofware relabl, sochasc volal models ad boformacs. Gm Seow s a assocae professor he Deparme of Accoug, School of Busess, Uvers of Coeccu. He receved hs PhD from he Uvers of Orego. Hs research eress clude accoug sadard- Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)

21 Forecasg Sock Prces Usg a Herarchcal Baesa Approach 59 seg ad secures regulao, accoug ad valuao of facal dervaves, ad he effecs of corporae rsk maageme polces o frm value. Auhors addresses: Ju Yg, Deparme of Medce, Dvso of Bosascs, Idaa Uvers School of Medce, Idaapols, IN , USA. L Kuo, Deparme of Sascs, Uvers of Coeccu, Sorrs, CT , USA. Gm S. Seow, Deparme of Accoug, Uvers of Coeccu, Sorrs, CT , USA. Coprgh 005 Joh Wle & Sos, Ld. J. Forecas. 4, (005)