Estimation of Markov Regime-Switching Regression Models with Endogenous Switching

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1 Esmon of Mrkov Regme-whng Regresson Models wh Endogenous whng Chng-Jn Km * Kore Unvers nd Unvers of Wshngon Jerem Pger Unvers of Oregon Rhrd r Unvers of Wshngon Frs Drf: Mrh 003 Ths Drf: eember 007 Absr Followng Hmlon (989) esmon of Mrkov regme-swhng regressons ll reles on he ssumon h he len se vrble onrollng regme hnge s eogenous. We rel hs ssumon nd develo rsmonous model of endogenous Mrkov regme-swhng. Inferene v mmum lkelhood esmon s ossble wh relvel mnor modfons o esng reursve flers. The model ness he eogenous swhng model eldng srghforwrd ess for endogene. In Mone Crlo eermens mmum lkelhood esmes of he endogenous swhng model rmeers were que ure even n he resene of ern model mssefons. As n lon we eend he voll feedbk model of equ reurns gven n Turner r nd Nelson (989) o llow for endogenous swhng. Kewords: Endogene Regme-whng JEL Clssfon: C3 C G Km: De. of Eonoms Unvers of Wshngon Bo ele WA (hngn@u.wshngon.edu); Pger: Dermen of Eonoms 85 Unvers of Oregon Eugene OR (ger@u.oregon.edu); r: De. of Eonoms Unvers of Wshngon Bo ele WA 9895 (sr@u.wshngon.edu). r knowledges fnnl suor from he Cel nd Jne Csor Professorsh he Unvers of Wshngon. We hnk he edor n ssoe edor hree nonmous referees Brr Arnold Rober Bever Mhel Dueker Jmes Morle Chrles Nelson nd semnr rns UC nd he 003 NBER ummer Insue for helful ommens. Prs of hs er were wren whle Pger ws enor Eonoms he Federl Reserve Bnk of. Lous. The vews eressed n hs er should no be nerreed s hose of he Federl Reserve Bnk of. Lous or he Federl Reserve sem.

2 Reen dedes hve seen eensve neres n me-vrng rmeer models of mroeonom nd fnnl me seres. One noble se of models re regme-swhng regressons whh de o les Qund (958). Goldfeld nd Qund (973) nrodued rulrl useful verson of hese models referred o n he followng s Mrkov-swhng model n whh he len se vrble onrollng regme shfs follows Mrkov-hn nd s hus serll deenden. In n nfluenl rle Hmlon (989) eended Mrkov-swhng models o he se of deenden d sefll n uoregresson. The vs lerure genered b Hmlon (989) ll ssumes h he regme shfs re eogenous wh rese o ll relons of he regresson dsurbne. In hs er we work wh Mrkov-swhng regressons of he e onsdered b Hmlon (989) nd vrous eensons bu rel he eogenous swhng ssumon. We develo model of endogenous Mrkov regme-swhng bsed on rob sefon for he relon of he len se. The model s que rsmonous nd dms es for endogenous swhng s smle rmeer resron. The model rmeers n be esmed v mmum lkelhood wh relvel mnor modfons o he reursve fler n Hmlon (989). Wh re we moved o nvesge Mrkov-swhng regressons wh endogenous swhng? Mn of he model s lons re n mroeonoms or fnne n suons where s nurl o ssume he se s endogenous. As n emle s ofen he se h he esmed se vrble hs srong busness le orrelon. Ths n be seen n reen lons of he regme-swhng model o denfed moner VARs suh s ms nd Zh (006) nd Owng (00). I s no hrd o mgne h he shoks o he regresson suh s he mroeonom shoks o he VAR would be orreled wh he busness le. As noher emle some lons of he model onn rmeers h reresen he reon

3 of gens o relon of he se s s he se n he model of equ reurns gven n Turner r nd Nelson (989) (TN herefer). However s lkel h gens do no observe he se bu nsed drw nferene bsed on some nformon se he onens of whh re unknown o he eonomern. Use of he ul se o ro for hs nferene leds o regresson wh mesuremen error n he elnor vrbles nd hus endogene. In order o evlue he fne-smle erformne of mmum lkelhood esmes of he endogenous swhng model rmeers s well s ess for endogenous swhng we ondu ber of Mone Crlo eermens. These eermens sugges h: ) When he rue Mrkov-swhng roess s endogenous mmum lkelhood esmon ssumng eogenous swhng elds bsed rmeer esmes ) Mmum lkelhood esmes of he endogenous swhng model were lose o her rue vlues s were qus-mmum lkelhood esmes obned from d genered b non-gussn endogenous swhng model nd 3) The lkelhood ro es for endogenous swhng ws lose o hvng orre se. As n lon we eend he voll feedbk model of equ reurns gven n TN o llow for endogenous swhng. As dsussed bove hs model rovdes seng n whh we mgh resonbl ee he Mrkov-swhng se vrble o be endogenous. We fnd mrgnl ssl evdene of endogenous swhng n he model nd h llowng for endogene hs subsnl effes on rmeer esmes. The model of endogenous swhng develoed n hs er hs muh n ommon wh n erler lerure usng swhng regressons. Ths lerure suh s Mddl nd Nelson (975) ws ofen onerned wh endogenous swhng s he rmr lons were n lmed deenden vrble ones suh s self-seleon nd mrke dsequlbrum sengs. The model we hve resened here n be nerreed s n eenson of he Mddl

4 nd Nelson (975) roh whh ws model of ndeenden swhng o he Hmlon (989) regme-swhng model n whh he se roess s serll deenden. In he ne seon we l ou wo-regme Mrkov-swhng regresson model wh endogenous swhng nd dsuss mmum lkelhood esmon. eon 3 generles hs model o he N-regme se. eon 4 gves he resuls of Mone Crlo eermens evlung he erformne of rmeer nferene nd ess for endogenous swhng. eon 5 resens he emrl emle o he voll feedbk model of TN. eon 6 onludes.. A Two-Regme Endogenous whng Model. Model efon Consder he followng Gussn regme-swhng model for he smle h of me T seres { } : + ε (.) ( 0) ε ~.. d. N where s slr s (k ) veor of observed eogenous or redeermned elnor vrbles whh m nlude lgged vlues of nd s he se vrble. Boh nd re ssumed o be ovrne-sonr vrbles. Denoe he number of regmes b N so h. N. We begn wh he se where N. In ddon o dng nuon he wo-regme se s oulr sefon n led work. As he regme orderng s rbrr we ssume h he model n (.) s rorel normled. ee Hmlon Wggoner nd Zh (007) for deled dsusson of hs ssue. 3

5 The se vrble s unobserved nd s ssumed o evolve ordng o frs-order Mrkov hn wh rnson robbles: P ) P ( ). (.) ( In (.) he rnson robbles re nfluened b (q ) veor of ovrne-sonr eogenous or redeermned vrbles where m nlude elemens of. The Mrkov hn s ssumed o be sonr nd o evolve ndeendenl of ll observons of hose elemens of no nluded n. To model he nfluene of on he [0] rnson robbles n (.) we use rob sefon for : f f η < η + b + b (.3) ( 0) η ~.. d. N. The rnson robbles re hen: ( ) P( < + b ) Φ( b ) η (.4) + ( ) P( + b ) Φ( b ) η + everl sel ses of (.) re worh menonng. The unresred model s he me-vrng rnson robbl Mrkov-swhng model of Goldfeld nd Qund (973) Debold Lee nd Wenbh (994) nd Flrdo (994). When he rnson robbles re no nfluened b we hve he me-vrng rnson robbl ndeenden swhng model of Goldfeld nd Qund (97). When he rnson robbles re no nfluened b we hve he fed rnson robbl Mrkov-swhng model of Goldfeld nd Qund (973) nd Hmlon (989). When he rnson robbles re nfluened b neher or we hve he fed rnson robbl ndeenden swhng model of Qund (97). 4

6 where Φ s he sndrd norml umulve dsrbuon funon. 3 To model endogenous swhng ssume h he on dens funon of ε nd η s bvre norml: ε ~ η N(0 Σ) Σ (.5) where ε nd η re unorreled h 0. Regme-swhng models found n me-seres h lons nerl lws mke he ssumon h ε s ndeenden of orresonds o he resron h 0 n he model resened here. 4 h h whh. Mmum Lkelhood Esmon Le ( ) Ω nd ξ (... ) be veors onnng observons observed hrough de nd θ ( b b ) be he veor of model rmeers. The ondonl lkelhood funon for he observed d ζ s onsrued s L T ( θ ) f ( Ω ξ θ ) ( ξ ;θ ) Ω ; where: f (.6) f ( Ω ξ ; θ ) Pr( Ω ξ ; θ ). 3 Alernvel logs sefon ould be used o desrbe he rnson robbles s n Debold Lee nd Wenbh (994) or Flrdo (994). The rob sefon s used here beuse rovdes srghforwrd roh o model endogenous swhng. 4 In reen work Chb nd Dueker (004) develo non-mrkov regme swhng model n whh observble vrbles re reled o he sgn of Gussn uoregressve len se vrble he nnovons o whh re 5

7 The weghng robbl n (.6) s omued reursvel b lng Bes rule: ( Ω ξ ; θ ) P ( ) Pr( Ω ξ ; θ ) Pr (.7) ( Ω ξ ; θ ) Pr( Ω ξ ; θ ) Pr + f ( Ω ξ ; θ ) f ( Ω ξ ; θ ) Pr( Ω ξ ; θ ). To nle (.7) he usul re s o rome ( ξ ;θ ) P wh he 0 Ω 0 unondonl robbl P ( 0 ;θ ). Alernvel hs nl robbl n be reed s n ddonl rmeer o be esmed. To omlee he reurson n (.6)-(.7) we requre he regme-deenden ondonl dens funon f ( Ω ξ ;θ ) 0 hs dens funon s Gussn:. For he eogenous swhng se when f ( Ω ξ ;θ ) φ (.8) where φ s he sndrd norml robbl dens funon. However for non-ero vlues of () f ( Ω ξ ;θ ) s: 5 llowed o be orreled wh he model resdul hrough bvre norml sefon s n (.5). The uhors develo Besn roedures o esme hs model. 5 The dens (.9) belongs o he skew-norml fml of dens funons whh re ommonl reded o Aln (985). ee Arnold nd Bever (00) for surve of hs lerure. 6

8 7 ( ) ξ ;θ Ω f ) ( b φ + Φ (.9) ( ) ξ ;θ Ω f ( ) ) ( b φ + + Φ. The end rovdes dervon of (.9). When s endogenous mmum lkelhood esmon ssumng s eogenous nd hus bsed on he dsrbuon n (.8) s nonssen n generl. To see hs noe h: ( ) ( ) ( ) ( ) b b b E E ; + Φ + + < φ η ε θ ε (.0) ( ) ( ) ( ) ( ) b b b E E ; + Φ + + φ η ε θ ε. Thus when 0 he regme-deenden ondonl men of ε s non-ero mlng h mmum lkelhood esmes bsed on (.8) suffer from he ordnr roblem of omed vrbles. Anoher less obvous soure of nonssen rses beuse ) ; ( θ ξ Ω f s non-gussn when 0 s s ler from (.9). In hs se mmum lkelhood esmon bsed on (.8) s qus-mmum lkelhood esmon whh s oned ou n Cmbell (00) s nonssen for regme-swhng models n generl.

9 .3 Tesng for Endogene In he model of endogenous swhng resened bove he null hohess h s eogenous s equvlen o he slr resron 0. Thus es for eogene n be rred ou b n suble es of hs resron. One obvous hoe s bsed on he -ss: ˆ (.) se ( ˆ ) where se ( ˆ ) s n esme of he sndrd error of ˆ. Assumng he lkelhood funon s orrel sefed n rore se ( ˆ ) n be onsrued from n esme of he nverse of he nformon mr suh s h bsed on he negve of he seond dervve of he loglkelhood funon. Alernvel one ould es for endogene usng he lkelhood ro ss onsrued s: ( L( ˆ θ ) L( ˆ θ ) LR (.) R where L ( θˆ ) s he mmed vlue of he lkelhood funon nd ( ) L θˆ s he mmed vlue of he lkelhood funon under he resron h 0. If he lkelhood funon s orrel sefed boh nd LR hve her usul smo dsrbuons when 0. For furher dels see Hmlon (994). R 3. An N-Regme Endogenous whng Model 8

10 In hs seon we generle he wo-regme Gussn endogenous-swhng model resened n eon o N regmes. We begn b modfng he rob sefon of he rnson robbles gven n (.3). uose he relon of s now deermned b he ouome of ~.. d. N( 0) η s follows:.. N N f f f f N N - + b + b + b N N η η η η < < < < N + b + b + b N. (3.) The rnson robbles ) re hen gven s follows: ( ( ) Φ( ) Φ( ) (3.) where 0 N nd + b for 0 < < N. Agn o model endogenous swhng ssume h he on dens of ε nd η s bvre norml s n (.5). Le he veor of model rmeers be θ ( θ θ... θ ) where N θ ( b ). Gven f ( Ω ξ ;θ ) he lkelhood funon L ( θ ) gn be onsrued usng he reurson n (.6)-(.7). I s shown n he end h: n f ( Ω ξ ;θ ) 9

11 0 ) ( φ Φ Φ. (3.3) Fnll s wh he wo regme endogenous swhng model es of he null hohess h s eogenous s equvlen o es of he resron Mone Crlo Anlss In hs seon we rovde Mone Crlo evdene regrdng mmum lkelhood esmon of he endogenous swhng model nd ssoed ess for endogene. Gven s romnene n he led lerure we fous on he wo-regme model wh fed Mrkovswhng rnson robbles so h 0 b b. We begn b evlung he erformne of mmum lkelhood esmon when he rue model s he endogenous swhng model resened n eon wh vrng levels of. We hen nvesge he sensv of mmum lkelhood esmon bsed on he on norml ssumon n (.5) o derures from hs Gussn ssumon n he d generng roess. uh derure renders he esmor bsed on (.5) Qus-mmum lkelhood (QML) esmor whh s nonssen for Mrkov-swhng models n generl. Our Mone Crlo eermens hen rovde some lmed evdene of how bdl he QML esmor erforms n re. For eh Mone Crlo eermen 000 smuled seres re genered from he model gven n (.)-(.3). We onsder wo smle ses for he smuled seres T 00 nd T 500. For eh smulon he veor of eogenous elnor vrbles s se o [ ] * where

12 ( 0) * ~.. d. N nd he veor of regme swhng rmeers s se o ( ) 0 (.0.0 ) 0 ( ) (.0.0) nd We onsder hree dfferen ses of rnson robbles orresondng o modere erssene ( ) hgh erssene ( ) nd dfferenl erssene ( ). We lso onsder hree dfferen vlues for orresondng o hgh orrelon 0. 9 modere orrelon 0. 5 nd ero orrelon 0. We onsder wo dfferen on dens funons for ε nd η lbeled DGP nd DGP. DGP s he bvre norml dsrbuon n (.5). DGP reles hs on norml ssumon. Insed ε s genered s sndrd norml rndom vrble whle η s genered s weghed sum of ε nd -dsrbued rndom vrble wh four degrees of freedom. The weghng s lbred so h ( ε ) hs ovrne mr: η γ 4 Σ γ 4 γ 4 where γ s he vrne of -dsrbued rndom vrble wh four degrees of freedom. 4 For eh smuled me seres wo mmum lkelhood esmes re omued. 6 The frs whh we lbel he eogenous esmor ssumes h 0 nd s hus bsed on he reurson n (.6)-(.7) usng (.8) o mesure f Ω ; ). The seond ( ξ θ whh we lbel he endogenous esmor llows for 0 nd s hus bsed on he reurson n (.6)-(.7) usng (.9) o mesure f Ω ; ). Fnll we lso reord ( ξ θ 6 All omuons were erformed n GAU 8.0 usng he QNewon numerl omon kge.

13 he ouome of 5% nomnl se nd lkelhood ro ess of he null hohess 0. 7 For hose ses where 0 n he d generng roess hese ess doumen he emrl se of he 5% nomnl se ess. For hose ses where 0 we use se-dused rl vlues ken from he Mone Crlo smulons genered wh 0 o mesure he ower of he ess. Tbles -5 show he resuls of he Mone Crlo eermens nvesgng mmum lkelhood esmon of he endogenous swhng model. For he rmeers eh ble shows he men of he 000 mmum lkelhood on esmes s well s he roo men squred error (RME) of he 000 mmum lkelhood on esmes from he rue vlue of he rmeer. 8 Tble gves resuls when he d generng roess hs eogenous swhng h s 0. In hs se boh he eogenous nd endogenous esmor re e mmum lkelhood esmors bu he endogenous esmor s neffen s does no resr 0. As s ler from he ble for boh smle ses nd ll vlues of he rnson robbles he eogenous nd endogenous esmors rodue esmes of he model rmeers h re ver lose o her rue vlues. As would be eeed he endogenous esmor s less effen hn he eogenous esmor wh he RME s generll hgher for he endogenous esmor. Tbles nd 3 gve resuls when he rue d generng roess nludes endogenous swhng of he form n DGP. The bles demonsre he esmon bs h ours when he endogenous se vrble s reed s eogenous n esmon. When he eogenous esmor s used he men esmes of 0 nd 0 re fr from her rue vlues wh he bs lrger for 7 The -ess were onsrued usng sndrd error esme bsed on he seond dervve of he log-lkelhood funon. Resuls when he sndrd error esme s lernvel bsed on he ouer rodu of he grden re ver smlr nd re vlble from he uhors.

14 hgher vlues of. The men esmes of nd re lso bsed downwrd. Noe h he men esmes re nerl denl n he T 00 nd T 500 ses suggesng he bs s no smll smle henomenon. Also noe h he esmes of nd re lose o her rue vlues. The ur of hese rmeer esmes n be red o he model ssumon mnned n he Mone Crlo smles h * s ndeenden of he endogenous se vrble. Fnll Tbles nd 3 lso demonsre h he endogenous esmor rodues ver ure esmes of he endogenous swhng model. Indeed for boh smle ses nd ll vlues of he rnson robbles nd onsdered he men rmeer esmes re nerl denl o her rue vlues. Tbles 4 nd 5 resen resuls for DGP h s when he on dens beween ε nd η s non-norml. For he rulr on dens funon onsdered he romon rovded b he norml ssumon s que good. Agn for boh smle ses nd ll vlues of he rnson robbles nd onsdered he men rmeer esmes from he endogenous esmor re ver lose o her rue vlues. Whle hs resul m no generle o non-norml dsrbuons more generll s suggesve h he qul of he endogenous esmor roedure s no her-sensve o he on-norml ssumon. No surrsngl he eogenous esmor whh gnores he oenl for he se vrble o be endogenous ll ogeher onnues o rodue bsed rmeer esmes under DGP. Tble 6 reors he se nd se-dused ower of he 5% nomnl se nd lkelhood ro ess of he null hohess h 0 for he d generng roesses onsdered n Tbles -5. When he null hohess s rue he -es s somewh oversed wh reeon 8 Model esmon lso rodues esmes of he rnson robbles nd n he se of he endogenous 3

15 res rehng s hgh s 3%. However he lkelhood ro es hs roughl orre se for ll ses onsdered. When he lernve hohess s rue he -es nd lkelhood ro es hve smlr se-dused ower for mos of he lernves onsdered. The one eeon s when T 00 nd 0. 7 n whh se he lkelhood ro es hs sgnfnl hgher sedused ower hn he -es. 9 Overll he Mone Crlo eermens onfrm h mmum lkelhood esmes usng he endogenous esmor re que good for he emles onsdered whle he eogenous esmor rodues subsnll bsed rmeer esmes when he rue roess hs endogenous swhng. Also he lkelhood ro es ers o be frl relble es for endogenous swhng. In he ne seon we urn o n emrl lon of he endogenous swhng model. 5. Alon: Mesuremen Error nd Esmon of he Voll Feedbk Effe A sled f of U.. equ reurn d s h he voll of reled reurns s mevrng nd redble. Gven hs lss orfolo heor would ml h he equ rsk remum should lso be me-vrng nd resond osvel o he eeon of fuure voll. However he d sugges h reled reurns nd reled voll s mesured b squred reurns re negvel orreled. 0 One elnon for he observed d s h whle nvesors do requre n nrese n eeed reurn n ehnge for eeed fuure voll he re ofen surrsed b news bou esmor he orrelon rmeer. Alhough no reored resuls for hese rmeer esmes re qulvel smlr o hose for he ondonl men nd vrne rmeers of he regresson model. 9 The se nd ower erformne of % nd 0% nomnl se ess (no reored) ws ver smlr o h for he 5% nomnl se ess. In rulr he -es s moderel oversed n generl he lkelhood ro es hs lose o orre se n ll ses nd he ess hve smlr se-dused ower for mos of he lernves onsdered. 0 For reen dsusson of hs resul see Brnd nd Kng (004). 4

16 reled voll. Ths voll feedbk effe rees reduon n res n he erod n whh he nrese n voll s reled. If he voll feedbk effe s srong enough m ree negve onemorneous orrelon beween reled reurns nd voll n he d. The voll feedbk effe hs been nvesged eensvel n he lerure see for emle Frenh hwer nd mbugh (987) TN Cmbell nd Henshell (99) Beker nd Wu (000) nd Km Morle nd Nelson (004). TN model he voll feedbk effe wh Mrkov-swhng model: r * ( Ψ ) + θ ( E( Ψ ) E( Ψ ) ε θ + (5.) E ( 0) ε ~.. d. N where s dsree Mrkov-swhng vrble kng on vlues or wh rnson robbles rmeered s n equon (.4). For normlon we ssume > so h s he hgh voll se. The model n (5.) s moved s follows. A he begnnng of erod he rsk remum θ ( ) E Ψ s deermned bsed on he eeon of erod voll formed wh nformon vlble he end of erod -. Durng erod ddonl nformon regrdng voll s observed. B he end of erod hs nformon s olleed n he nformon se Ψ. When E( Ψ ) * E( ) * Ψ nformon bou voll reveled durng he erod hs surrsed gens. If θ < 0 surrses h revel greer robbl of he hgh-vrne se re vewed negvel b nvesors nd hus redue he onemorneous reurn. 5

17 One esmon dfful wh he model n (5.) s h here ess dsren beween he nvesors nd he eonomern s d se. In rulr whle Ψ m be summred b ll d u o - h s Ψ { r r...} he nformon se * Ψ nludes nformon h s no summred n he reserher s d se on observed reurns. Ths s beuse whle he reserher s d se s olleed dsreel he begnnng of eh erod he mrke rns onnuousl observe rdes h our durng he erod. E ( ) Ψ * To hndle hs esmon dfful TN use he ul voll s ro for. Th s he esme: r ( Ψ ) + θ ( E( Ψ ) u θ E + (5.) u ~ N( 0) * In essene hs romon reles he esmed robbl of he se P( Ψ ) one f wh nd ero oherwse. Assumng hese dffer hs nrodues lssl mesuremen error no he se vrble n he esmed equon hus renderng endogenous. The esng lerure esmes (5.) ssumng he se vrble s eogenous. However he ehnques develoed n eon n be used o esme he voll feedbk model llowng for endogene s well s o es for endogene. Here we esme (5.) usng monhl reurns for vlue-weghed orfolo of ll NYE-lsed soks n eess of he onemonh Tresur Bll re over he smle erod Jnur 95 o Deember 999 he sme d s used n Km Morle nd Nelson (004). Tble 7 summres he resuls. The frs nel of Tble 7 shows esmes when endogene s gnored. These esmes whh re smlr o hose n TN re onssen wh boh osve relonsh 6

18 beween he rsk remum nd eeed fuure voll ( θ > 0 ) nd subsnl voll feedbk effe ( θ << 0 ). The esmes lso sugges domnn voll feedbk effe h s θ s ver smll relve o θ. The seond nel shows he esmes when endogene s llowed so h he orrelon rmeer s esmed. The esme of s subsnl equlng The lkelhood ro es whh reorded relble fne smle se erformne n he Mone Crlo eermens dsussed n eon 4 rovdes mrgnl evdene gns he null hohess h 0 (-vlue 0.08). The rmr dfferene n he rmeer esmes s for he voll feedbk oeffen θ whh s esmed o be bou one-hrd smller when endogene s llowed hn when s gnored. Thus whle here s sll evdene of srong voll feedbk effe s subsnll smller hn h mled b he model wh no llowne for endogene. 6. Conluson We hve develoed model of Mrkov-swhng n whh he len se vrble onrollng he regme shfs s endogenousl deermned. The model s que rsmonous nd dms es for endogenous swhng s smle rmeer resron. The model rmeers n be esmed v mmum lkelhood wh relvel smll modfons o he reursve fler n Hmlon (989). In Mone Crlo eermens mmum lkelhood esmon of he endogenous swhng model nd he lkelhood ro es for endogene erformed que well for he d generng roesses onsdered. We l he model o es for endogenous I s worh emhsng h he vld of he lkelhood ro es for eogenous swhng reles on he orre sefon of he model lkelhood funon. Evdene n fvor of endogenous swhng should herefore be nerreed ondonl on hs mnned hohess. 7

19 swhng n he voll feedbk model of equ reurns gven n Turner r nd Nelson (989). 8

20 Aend Dervon of (.9) nd (3.3): We roeed b generlng he dervon of he unvre skew-norml dens funon gven n Arnold nd Bever (00). The rndom vrbles desrbed n (.5) n be wren s: ε η ε A ω (A.) where ω ~.. d. N(0) nd A 0 s he Cholesk deomoson of Σ so h AA Σ. From (A.): η ε + ω. (A.) We n hen wre suressng Ω ξ nd θ from he ondonng se for onvenene: f ( ) ( < ) f η ε ε f ω < (A.3) where nd re defned n eon 3. Consder he umulve robbl dsrbuon funon: 9

21 0 < < Pr ε ω ε g (A.4) < < < Pr Pr ε ω ε ε ω ε g The denomnor of (A.4) s: < Pr ε ω ε ( ) ( ) ) ( Φ Φ. (A.5) The numeror of (A.4) s: < < Pr ε ω ε g ( ) ( ) g d d f ε ω ω ε ε / / ) ( ( ) ( ) ( ) g d d f / / ε ε ε ω ω φ Φ Φ g d ε φ. (A.6) Combnng (A.5)-(A.6) nd dfferenng wh rese o g elds:

22 ) ( f (3.3) ) ( φ Φ Φ whh s he dens funon (3.3). In he se where N we hve: ( ) ) ( b f φ + Φ ( ) ( ) ) ( b f φ + + Φ whh uon renmng nd b b s he dens funon n (.9).

23 Referenes Arnold B.C. nd R.J. Bever 00 kewed Mulvre Models Reled o Hdden Trunon nd / or eleve Reorng oedd de Esds e Invesgon Oerv Aln A. 985 A Clss of Dsrbuons whh Inludes he Norml Ones ndnvn Journl of ss Beker G. nd G. Wu 000 Asmmer Voll nd Rsk n Equ Mrkes Revew of Fnnl udes 3-4. Brnd M. nd Q. Kng 004 On he Relonsh beween he Condonl Men nd Voll of ok Reurns: A Len VAR Aroh Cmbell J. Y. nd L. Henshel 99 No News s Good News: An Asmmer Model of Chngng Voll n ok Reurns Journl of Fnnl Eonoms Cmbell.D. 00 efon Tesng nd emrmer Esmon of Regme whng Models: An Emnon of he U hor Term Ineres Re Brown Unvers Dermen of Eonoms Workng Per #00-6. Chb. nd M. Dueker 004 Non-Mrkovn Regme whng wh Endogenous es nd Tme Vrng e Lenghs Federl Reserve Bnk of. Lous workng er # A. Debold F. Lee J-H. nd G. Wenbh 994 Regme whng wh Tme-Vrng Trnson Probbles n Non-sonr Tme eres Anlss nd Conegron ed. C. Hrgreves Oford Unvers Press Oford U.K. Flrdo A.J. 994 Busness-Cle Phses nd Ther Trnsonl Dnms Journl of Busness nd Eonom ss Frenh K.R. hwer G. nd R. F. mbugh 987 Eeed ok Reurns nd Voll Journl of Fnnl Eonoms Goldfeld.M. nd R.E. Qund 97 n Nonlner Mehods n Eonomers Norh Hollnd Amserdm. Goldfeld.M. nd R.E. Qund 973 A Mrkov Model for whng Regressons Journl of Eonomers 3-6. Hmlon J.D. 989 A New Aroh o he Eonom Anlss of Nonsonr Tme eres nd he Busness Cle Eonomer Hmlon J.D. Wggoner D.F. nd T. Zh 007 Normlon n Eonomers Eonomer Revews 6-5.

24 Km C.J. Morle J.C. nd C.R. Nelson 004 Is There Posve Relonsh Beween ok Mrke Voll nd he Equ Premum? Journl of Mone Cred nd Bnkng Mddl G.. nd F. Nelson 975 whng Regresson Models wh Eogenous nd Endogenous whng Proeedngs of he Amern sl Assoon Owng M. 00 Modelng Volker s Non-Absorbng e: Agnos Idenfon of Mrkov-whng VAR workng er Federl Reserve Bnk of. Lous. Qund R.E. 958 The Esmon of he Prmeers of Lner Regresson sem Obeng Two ere Regmes Journl of he Amern sl Assoon Qund R.E. 97 A New Aroh o Esmng whng Regressons Journl of he Amern sl Assoon ms C. nd T. Zh 006 Were here Regme whes n U.. Moner Pol? Amern Eonom Revew Turner C. M. r R. nd C.R. Nelson 989 A Mrkov Model of Heeroskeds Rsk nd Lernng n he ok Mrke Journl of Fnnl Eonoms

25 Tble Mone Crlo Resuls 0 (eogenous swhng) T Eog. Esmor.00 (0.04) -.00 (0.07).00 (0.0) -.00 (0.04) 0.33 (0.03) 0.66 (0.05) Endog. Esmor.00 (0.09) -.00 (0.8).00 (0.0) -.00 (0.04) 0.34 (0.03) 0.68 (0.06) Eog. Esmor.00 (0.05) -.00 (0.06).00 (0.03) -.00 (0.03) 0.3 (0.04) 0.67 (0.04) Endog. Esmor.00 (0.0) -.00 (0.07).00 (0.03) -.00 (0.03) 0.33 (0.04) 0.67 (0.04) Eog. Esmor.00 (0.04) -.00 (0.07).00 (0.0) -.00 (0.04) 0.33 (0.03) 0.66 (0.05) Endog. Esmor.00 (0.04) -.00 (0.08).00 (0.0) -.00 (0.04) 0.33 (0.03) 0.66 (0.05) T Eog. Esmor.00 (0.0) -.00 (0.04).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.03) Endog. Esmor.00 (0.05) (0.0).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.03) Eog. Esmor.00 (0.03) -.00 (0.03).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.03) Endog. Esmor.00 (0.05) -.00 (0.05).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.03) Eog. Esmor.00 (0.0) -.00 (0.04).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.03) Endog. Esmor.00 (0.03) -.00 (0.05).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.03) Noes: Ths ble onns summr resuls from 000 Mone Crlo smulons when he rue d generng roess s hrered b eogenous swhng. Eh ell onns he men of he 000 mmum lkelhood on esmes for he rmeer lsed n he olumn hedng s well s he roo men squred error of he 000 on esmes from h rmeer s rue vlue (n renheses). Eog. esmor refers o he mmum lkelhood esmor ssumng he se roess s eogenous so h 0. Endog. esmor refers o he mmum lkelhood esmor llowng he se roess o be endogenous so h (). 4

26 Tble Mone Crlo Resuls DGP 0. 5 (endogenous swhng) T Eog. Esmor 0.89 (0.) (0.4).00 (0.0) -.00 (0.03) 0.3 (0.03) 0.6 (0.07) Endog. Esmor.00 (0.07) -.0 (0.5).00 (0.0) -.00 (0.03) 0.33 (0.04) 0.68 (0.07) Eog. Esmor 0.86 (0.5) -0.9 (0.).00 (0.03) -.00 (0.03) 0.3 (0.04) 0.64 (0.05) Endog. Esmor.0 (0.09) -.00 (0.07).00 (0.0) -.00 (0.03) 0.33 (0.04) 0.67 (0.05) Eog. Esmor 0.95 (0.06) (0.4).00 (0.0) -.00 (0.03) 0.3 (0.03) 0.65 (0.05) Endog. Esmor.0 (0.04) -.0 (0.09).00 (0.0) -.00 (0.03) 0.33 (0.03) 0.67 (0.05) T Eog. Esmor 0.89 (0.) (0.4).00 (0.0) -.00 (0.0) 0.3 (0.03) 0.6 (0.06) Endog. Esmor.00 (0.05) -.00 (0.09).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.04) Eog. Esmor 0.86 (0.5) -0.9 (0.0).00 (0.0) -.00 (0.0) 0.3 (0.03) 0.65 (0.03) Endog. Esmor.00 (0.0) -.00 (0.05).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.03) Eog. Esmor 0.95 (0.06) (0.).00 (0.0) -.00 (0.0) 0.3 (0.0) 0.65 (0.03) Endog. Esmor.00 (0.0) -.00 (0.05).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.03) Noes: Ths ble onns summr resuls from 000 Mone Crlo smulons when he rue d generng roess s hrered b DGP (deled n eon 4) wh Eh ell onns he men of he 000 mmum lkelhood on esmes for he rmeer lsed n he olumn hedng s well s he roo men squred error of he 000 on esmes from h rmeer s rue vlue (n renheses). Eog. esmor refers o he mmum lkelhood esmor ssumng he se roess s eogenous so h 0. Endog. esmor refers o he mmum lkelhood esmor llowng he se roess o be endogenous so h (). 5

27 Tble 3 Mone Crlo Resuls DGP 0. 9 (endogenous swhng) T Eog. Esmor 0.80 (0.) (0.4).00 (0.0) -.00 (0.03) 0.5 (0.08) 0.5 (0.6) Endog. Esmor.00 (0.05) -.00 (0.09).00 (0.0) -.00 (0.03) 0.33 (0.03) 0.67 (0.07) Eog. Esmor 0.74 (0.6) (0.7).00 (0.0) -.00 (0.03) 0.8 (0.06) 0.60 (0.08) Endog. Esmor.0 (0.07) -.00 (0.06).00 (0.0) -.00 (0.0) 0.33 (0.04) 0.67 (0.05) Eog. Esmor 0.90 (0.0) (0.).00 (0.0) -.00 (0.04) 0.3 (0.03) 0.63 (0.06) Endog. Esmor.00 (0.04) -.0 (0.07) 0.99 (0.0) -.00 (0.03) 0.33 (0.03) 0.67 (0.05) T Eog. Esmor 0.80 (0.) (0.4).00 (0.0) -.00 (0.0) 0.5 (0.08) 0.5 (0.5) Endog. Esmor.00 (0.03) -.00 (0.05).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.04) Eog. Esmor 0.74 (0.6) (0.7).00 (0.0) -.00 (0.0) 0.9 (0.05) 0.60 (0.08) Endog. Esmor.00 (0.04) -.00 (0.04).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.03) Eog. Esmor 0.9 (0.0) (0.0).00 (0.0) -.00 (0.0) 0.3 (0.0) 0.63 (0.05) Endog. Esmor.00 (0.0) -.00 (0.04).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.03) Noes: Ths ble onns summr resuls from 000 Mone Crlo smulons when he rue d generng roess s hrered b DGP (deled n eon 4) wh Eh ell onns he men of he 000 mmum lkelhood on esmes for he rmeer lsed n he olumn hedng s well s he roo men squred error of he 000 on esmes from h rmeer s rue vlue (n renheses). Eog. esmor refers o he mmum lkelhood esmor ssumng he se roess s eogenous so h 0. Endog. esmor refers o he mmum lkelhood esmor llowng he se roess o be endogenous so h (). 6

28 Tble 4 Mone Crlo Resuls DGP 0. 5 (endogenous swhng) T Eog. Esmor 0.87 (0.3) (0.7).00 (0.0) -.00 (0.03) 0.30 (0.04) 0.6 (0.08) Endog. Esmor.00 (0.07) -.00 (0.4).00 (0.0) -.00 (0.03) 0.33 (0.04) 0.67 (0.07) Eog. Esmor 0.85 (0.6) (0.).00 (0.03) -.00 (0.03) 0.3 (0.04) 0.64 (0.05) Endog. Esmor.00 (0.09) -.00 (0.07).00 (0.03) -.00 (0.03) 0.33 (0.04) 0.67 (0.05) Eog. Esmor 0.94 (0.07) (0.4).00 (0.0) -.00 (0.04) 0.3 (0.03) 0.65 (0.05) Endog. Esmor.00 (0.04) -.00 (0.09).00 (0.0) -.00 (0.03) 0.33 (0.03) 0.67 (0.05) T Eog. Esmor 0.87 (0.3) (0.6).00 (0.0) -.00 (0.0) 0.30 (0.03) 0.6 (0.06) Endog. Esmor.00 (0.04) -.00 (0.08).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.04) Eog. Esmor 0.85 (0.5) (0.).00 (0.0) -.00 (0.0) 0.3 (0.03) 0.65 (0.03) Endog. Esmor.00 (0.05) -.00 (0.04).00 (0.0) -.00 (0.0) 0.33 (0.03) 0.67 (0.03) Eog. Esmor 0.95 (0.06) (0.).00 (0.0) -.00 (0.0) 0.3 (0.0) 0.66 (0.03) Endog. Esmor.00 (0.0) -.00 (0.05).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.03) Noes: Ths ble onns summr resuls from 000 Mone Crlo smulons when he rue d generng roess s hrered b DGP (deled n eon 4) wh Eh ell onns he men of he 000 mmum lkelhood on esmes for he rmeer lsed n he olumn hedng s well s he roo men squred error of he 000 on esmes from h rmeer s rue vlue (n renheses). Eog. esmor refers o he mmum lkelhood esmor ssumng he se roess s eogenous so h 0. Endog. esmor refers o he mmum lkelhood esmor llowng he se roess o be endogenous so h (). 7

29 Tble 5 Mone Crlo Resuls DGP 0. 9 (endogenous swhng) T Eog. Esmor 0.79 (0.) (0.4).00 (0.0) -.00 (0.03) 0.5 (0.08) 0.5 (0.6) Endog. Esmor.00 (0.04) (0.08).00 (0.0) -.00 (0.0) 0.33 (0.03) 0.67 (0.07) Eog. Esmor 0.75 (0.6) (0.8).00 (0.0) -.00 (0.03) 0.8 (0.06) 0.59 (0.08) Endog. Esmor.00 (0.06) -.00 (0.06).00 (0.0) -.00 (0.0) 0.33 (0.04) 0.67 (0.05) Eog. Esmor 0.90 (0.0) (0.).00 (0.0) -.00 (0.03) 0.3 (0.03) 0.63 (0.06) Endog. Esmor.00 (0.04) -.00 (0.07).00 (0.0) -.00 (0.03) 0.33 (0.0) 0.67 (0.05) T Eog. Esmor 0.80 (0.0) (0.43).00 (0.0) -.00 (0.0) 0.5 (0.08) 0.5 (0.6) Endog. Esmor 0.99 (0.03) (0.05).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.04) Eog. Esmor 0.75 (0.5) (0.8).00 (0.0) -.00 (0.0) 0.9 (0.04) 0.60 (0.08) Endog. Esmor.00 (0.04) -.00 (0.04).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.03) Eog. Esmor 0.90 (0.0) (0.).00 (0.0) -.00 (0.0) 0.3 (0.03) 0.63 (0.05) Endog. Esmor.00 (0.0) -.00 (0.04).00 (0.0) -.00 (0.0) 0.33 (0.0) 0.67 (0.03) Noes: Ths ble onns summr resuls from 000 Mone Crlo smulons when he rue d generng roess s hrered b DGP (deled n eon 4) wh Eh ell onns he men of he 000 mmum lkelhood on esmes for he rmeer lsed n he olumn hedng s well s he roo men squred error of he 000 on esmes from h rmeer s rue vlue (n renheses). Eog. esmor refers o he mmum lkelhood esmor ssumng he se roess s eogenous so h 0. Endog. esmor refers o he mmum lkelhood esmor llowng he se roess o be endogenous so h (). 8

30 Tble 6 Mone Crlo Resuls e nd e Adused Power of Tess of 0 e 0 Power DGP 0. 5 Power DGP 0. 9 Power DGP 0. 5 Power DGP 0. 9 T 00 LR LR LR LR LR T 500 LR LR LR LR LR Noes: Eh ell of he ble onns he erenge of 000 Mone Crlo smulons for whh he -es or lkelhood ro (LR) es desrbed n eon.3 reeed he null hohess h 0 he 5% sgnfne level. For olumns lbeled e rl vlues re bsed on he smo dsrbuon of he es-ss. For olumns lbeled Power se dused rl vlues re luled from he 000 smuled es sss from he orresondng Mone Crlo eermen for whh 0. DGP nd DGP refer o he d generng roess used o smule he Mone Crlo smles nd re deled n eon 4. 9

31 Tble 7 Mmum Lkelhood Esmes of he Turner r nd Nelson (989) Voll-Feedbk Model Prmeer Ignorng Endogene Aounng for Endogene θ 0.3 (0.0) 0.36 (0.0) θ -.55 (0.45) 0.40 (0.0) 0.75 (0.07).05 (0.0) -.07 (0.45) 0.40 (0.0) 0.74 (0.07).05 (0.7) -.09 (0.) -.6 (0.) (0.8) Log Lkelhood Noes: Ths ble reors mmum lkelhood esmes of he voll feedbk model of eess equ reurns gven n Turner r nd Nelson (989) nd deled n equon (5.). Eess reurns re mesured usng monhl reurns n eess of he one-monh Tresur Bll re genered from vlueweghed orfolo of ll NYE-lsed soks. The smle erod s Jnur 95 hrough Deember 999. The olumn lbeled Ignorng Endogene holds esmes n whh he Mrkov-swhng se vrble s ssumed eogenous of he regresson error erm. The olumn lbeled Aounng for Endogene holds esmes n whh he Mrkov-swhng se vrble s llowed o be endogenous usng he roh deled n eon. ndrd errors reored n renheses re bsed on seond dervves of he loglkelhood funon n ll ses. 30