Estimation of Markov Regime-Switching Regression Models with Endogenous Switching

Transcription

1 Esimion of Mrkov Regime-Swiching Regression Models wih Endogenous Swiching Chng-Jin Kim Kore Universiy nd Universiy of Wshingon Jeremy Piger Federl Reserve Bnk of S. Louis Richrd Srz Universiy of Wshingon July 3 Absrc Following Hmilon (989), esimion of Mrkov regime-swiching regressions nerly lwys relies on he ssumpion h he len se vrible conrolling he regime chnge is exogenous. We relx his ssumpion nd develop wo echniques o esime Mrkov-swiching models wih endogenous swiching. The firs exends he endogenous swiching models in Mddl nd Nelson (975) o he Hmilon (989) Mrkov regime-swiching regression. The second is bsed on he inerpreion of he endogenous swiching regression s regression model wih endogenous unobserved dummy vribles. In his cse he regression cn be esimed using insrumenl vribles echniques. For boh echniques, idenificion is chieved when he rnsiion probbiliies of he regime-swiching process re influenced by observed exogenous vribles. However, even in he fixed rnsiion probbiliy cse, idenificion is chieved if he se process is serilly dependen nd lgged vlues of he se process re exogenous. This is rue even hough he lgged se is unobserved. The bis correcion echniques lso dmi srighforwrd ess for endogeneiy. Mone Crlo experimens confirm h he proposed bis correcion echniques perform quie well in prcice. We pply he procedure o he voliliy feedbck model of equiy reurns given in Turner, Srz nd Nelson (989). Keywords: Endogeneiy, Regime-Swiching, Insrumenl Vribles JEL Clssificion: C3, C, G Kim: Dep. of Economics, Kore Universiy, Anm-Dong, Seongbuk-ku, Seoul, 36-7, Kore, (cjkim@kore.c.kr); Piger: Reserch Deprmen, Federl Reserve Bnk of S. Louis, 4 Locus S., S. Louis, MO 6366, USA, (piger@sls.frb.org); Srz: Dep. of Economics, Universiy of Wshingon, Box 35333, Sele, WA 9895, USA (srz@u.wshingon.edu). We hnk Michel Dueker for helpful commens. The views expressed in his pper re solely he responsibiliy of he uhors, nd should no be inerpreed s reflecing he views of he Federl Reserve Bnk of S. Louis or of ny oher person ssocied wih he Federl Reserve Sysem.

2 There is mouning evidence h empiricl models of mny economic ime series, priculrly mcroeconomic nd finncil series, re chrcerized by prmeer insbiliy. This hs sprked n explosion of ineres in ime-vrying prmeer models. One noble se of models re swiching regressions wih unknown smple seprion, in which prmeers move discreely beween fixed number of regimes, wih he swiching conrolled by n unobserved se vrible. Swiching regressions hve rich hisory in economerics, ding bck o les Qund (958). Goldfeld nd Qund (973) inroduced priculrly useful version of hese models, referred o in he following s Mrkov-swiching model, in which he len se vrible conrolling he regime shifs follows Mrkov-chin, nd is hus serilly dependen. In n influenil ricle, Hmilon (989) exended Mrkov-swiching models o he cse of dependen d, specificlly n uoregression. The vs lierure spwned by Hmilon (989) hs ypiclly ssumed h he regime shifs re exogenous of ll relizions of he regression error. However, he erlier lierure on swiching regressions, such s Mddl nd Nelson (975), ws ofen concerned wih endogenous swiching, s he primry pplicions of swiching regression models in his lierure were in limied dependen vrible conexs such s self-selecion nd mrke disequilibrium problems. In his pper we work wih swiching regressions of he ype considered by Hmilon (989) nd vrious exensions, bu relx he exogenous swiching ssumpion. We show h filure of he exogeneiy ssumpion leds o significn bis in he coefficiens of regimeswiching regression when esimion mehods bsed on he exogeneiy ssumpion re used. We hen consider wo esimion echniques for correcing his bis. In he firs, we exend he endogenous swiching models in Mddl nd Nelson (975) o he Hmilon (989) regime

3 swiching regression, in which he swiching process is serilly dependen, specificlly Mrkovswiching. This pproch relies on join normliy ssumpion for he relionship beween he regression error erms nd he innovions o he se relizion equion. If one is unwilling o mke his ssumpion, we develop n lernive echnique, bsed on he inerpreion of he endogenous swiching regression s regression model wih endogenous unobserved dummy vribles. In his cse he regression cn be esimed using insrumenl vribles echniques. Boh bis correcion echniques lso dmi srighforwrd es for endogeneiy. For boh of hese esimion echniques, we show h for serilly dependen se processes, such s Mrkov-swiching se process, he lgged se cn provide informion necessry for idenificion, providing i is uncorreled wih he curren regression error. This is rue even hough he lgged se is unobserved. Addiionl informion is obined when he rnsiion probbiliies of he swiching process re influenced by exogenous vribles, s in he so clled ime-vrying rnsiion probbiliy cse. We use Mone Crlo experimens o confirm he efficcy of he proposed bis-correcion procedures nd find h heir performnce is quie good for empiriclly relevn d genering processes. Why re we moived o invesige Mrkov-swiching regressions wih endogenous swiching? Mny of he model s pplicions re in mcroeconomics or finnce in siuions where i would be nurl o ssume h he se is endogenous. As n exmple, in mny models he esimed se vrible hs srong business cycle correlion, ofen corresponding wih recessions. This cn be seen in recen pplicions of he regime-swiching model o idenified monery VARs, such s Sims nd Zh () nd Owyng (). I is no hrd o imgine h he shocks o he regression, such s he mcroeconomic shocks o he VAR, would be correled wih recessions. As noher exmple, some pplicions of he model conin

4 prmeers h represen he recion of gens o relizion of he se (see for exmple Turner, Srz nd Nelson (989)). However, i is likely h gens do no observe he se, bu insed drw inference bsed on some informion se, he conens of which re unknown o he economericin. Use of he cul se o proxy for his inference leds o regression wih mesuremen error in he explnory vribles, nd hus endogeneiy. In he nex secion we ly ou cnonicl Mrkov-swiching regression nd documen he bises h rise when he len se vrible is correled wih he regression error. Secion 3 develops he wo bis correcion sregies, discusses idenificion, nd presens ess for endogeneiy. Secion 4 gives he resuls of Mone Crlo experimen documening he performnce of he proposed bis correcion procedures. In Secion 5 we presen n empiricl exmple bsed on model of voliliy feedbck in equiy mrkes ken from Turner, Srz nd Nelson (989).. Endogenous Regime-Swiching nd Esimion Bis The model we consider is liner regression wih regime-swiching coefficiens nd residul vrince. Assume h he dependen vrible, y, is genered by one of wo rue regression equions: y = + x + ε, ε N(, σ ), ' S S S = ( S ) + S, S = ( S ) + S, S σ = σ ( S ) + σ S, S () We focus here on wo-regime model, however, ll of he resuls re esily generlized o he n regime cse. 3

5 where S = {, } is se vrible h indices which of he wo regression equions generes he observion y nd x is k x vecor of exogenous or predeermined explnory vribles mesured ime, which my include lgged vlues of y. We re ineresed in he cse of unobserved S nd hus require probbiliy lw governing S for esimion purposes. Here we consider regime-swiching process for S h encompsses severl populr specificions in he lierure. In priculr we ssume h he probbiliy h S = i depends on S nd on vecor of exogenous or predeermined vribles z, where z my include elemens of x. Formlly: PS ( = S =,..., S = iz, ) = PS ( = S =, z) = pz ( ) j PS ( = S =, z) = pz ( ) PS ( = S =,..., S = iz, ) = PS ( = S =, z) = qz ( ) () j PS ( = S =, z) = qz ( ) where he rnsiion probbiliies p z ) nd qz ( ) my hve eiher Probi or Logisic specificion. ( Mximum likelihood esimion of he model vi he recursive filer given in Hmilon (989) or he expecion mximizion (EM) lgorihm given in Hmilon (99) will yield consisen prmeer esimes under he ssumpion h S is uncorreled wih ε. Relxion of his ssumpion will led o bised esimion of he prmeers i nd, i =,. Noe i h, when ε is correled wih S, equion () cn be rewrien s follows: y = + x + E( ε S, S, z ) + u, ' S S u = ε E( ε S, S, z ) (3) 4

6 Th is, S inroduces predicbiliy ino ε h vries wih he regime swiching inercep. Thus, here is no informion o seprely idenify i from E( ε S = i, S, z). In generl, mximum likelihood esimion will yield bised esimes of boh i nd i. However, in he specil cse where x nd S re independen, he bis will be conined o i only. In he nex secion we presen echniques o correc he bis inroduced by he endogenous se vrible. 3. Bis Correcion in he Endogenous Swiching Model 3. Bis Correcion wih Join Normliy In his Secion, we prmeerize he rnsiion probbiliies p( z ) nd qz ( ) in equion () using he following probi specificion: S = if if S S < S = + S + z ' + z ' S + η (4) 3 η ~... iid N(,) So h: p( z ) = Pr( S = S =, z ) = Pr( η > ( ( + ) z '( + )) = Φ( ( + ) z '( + )) 3 3 qz ( ) = Pr( S= S =, z) = Pr( η z' ) =Φ( z' ) where Φ is he sndrd norml cumulive disribuion funcion. Severl specil cses of () re worh menioning. The unresriced model represens he ime-vrying rnsiion probbiliy Mrkov-swiching model (TVP-MS) of Goldfeld nd Qund (973), Diebold, Lee nd Weinbch (994) nd Filrdo (994). When = 3 =, we hve he fixed rnsiion probbiliy Mrkov-swiching model (FTP-MS) of Goldfeld nd Qund (973) nd Hmilon (989), h is 5

7 p( z ) = p, qz ( ) = q. If = = 3 = we hve he fixed rnsiion probbiliy independen swiching model (FTP-IS) of Qund (97), so h q= p. Finlly, if = 3 =, we hve he ime-vrying rnsiion probbiliy independen swiching model (TVP-IS) of Goldfeld nd Qund (97), q z ) = p( z ). ( Correlion beween ε nd S is inroduced hrough correlion beween ε nd η. Thus, he join disribuion beween ε nd η is mulivrie Norml wih he following covrince mrix: η ρσ S ~ N(, ), ε Σ Σ= ρσ σ S S (5) In his cse, he model in (), () nd (5) is closely reled o he swiching regression wih endogenous swiching considered by Mddl nd Nelson (975). The min ddiion we consider here is o llow he unobserved se process o be serilly dependen, specificlly Mrkov-swiching, s in Goldfeld nd Qund (973) nd Hmilon (989). In he following, we ouline mximum likelihood esimion procedure for his Mrkov-swiching regression wih endogenous swiching. Using he join norml ssumpion given in (5), one cn derive explici chrcerizions of he bis erms from equion (4), E( ε S = i, S, z). In priculr, noe from (5) h: 6

8 E( ε S =, S, z ) = E( ε η < S z ' z ' S ) 3 φ ( S z ' z ' S ) = ρσ = ρσ M 3 Φ ( S z ' z ' 3S ) E( ε S =, S, z) = E( ε η S z ' z ' 3S ) φ( S z ' z ' 3S ) = ρσ = ρσ M ( Φ( S z ' z ' S )) 3 (6) where φ is he sndrd norml densiy funcion. Using (6), we cn correc for bis inroduced by he endogenous se vrible by rewriing (3) s: y = + x + ρσ M + u, ' S S S S u = ε E( ε S, S, z ), (7) where M S, ( S =,), is defined in (6). The prmeers of (7) re only idenified if M nd M vry, oherwise here is no informion o idenify S from ρσ S M S, ( S, = ). From (6), we cn see his is chieved in wo wys. Firs, if he se process depends on exogenous or predeermined observble vribles z, he model is idenified. Second, even when = 3 =, s in he Mrkov-swiching model wih fixed rnsiion probbiliies, he model is sill idenified so long s. Th is, if S is serilly correled nd S is exogenous, S serves o idenify he model. Assuming idenificion, consisen esimes of he prmeers of () re chieved by mximum likelihood esimion of (7), which cn be performed compuionlly using he recursive filer given in Hmilon (989) or he expecion mximizion (EM) lgorihm discussed in Hmilon (99). As hese echniques re well known, we will no repe hem in deil here. Noe hough h compuion of he uncondiionl likelihood requires h he likelihood for (7) be compued condiionl on ech possible combinion of S nd S. This requires h we chrcerize he vrinces of u under ech se. These re given by: 7

9 vr( u S =, S, z ) = vr( ε S =, S, z ) = ρ σ ( M ( M + + S + z ' + z ' S )) + σ ( ρ ) 3 vr( u S =, S, z ) = vr( ε S =, S, z ) = ρ σ ( M ( M + + S + z ' + z ' S )) + σ ( ρ ) 3 (8) 3. Bis Correcion wihou Resoring o Join Normliy The bis correcion echnique presened in Secion 3. relied on specific disribuionl ssumpion describing he relionship beween ε nd η. In his secion, we presen n lernive bis correcion sregy, for which we do no resor o such disribuionl ssumpion. The procedure inroduced in his secion would work no only for he Probi specificion of he rnsiion probbiliies in equion (4), bu lso for he Logisic specificion of he rnsiion probbiliies of he following form: exp( + z ' b) pz ( ) = Pr( S = S =, z) = + exp( + z ' b) (9) exp( + z ' b ) qz = S= S = z = + + ( ) Pr(, ), exp( z ' b) where S is correled wih he disurbnce erm ε in equion (). In order o move owrd esimion wih bis correcion, we consider useful lernive AR() chrcerizion of S s suggesed by Hmilon (989): 8

10 S λ = c + S c = q( z ) λ = + p( z + v ) + q( z ) () where condiionl on S = : v = q( z )) wih probbiliy q z ) ( v = q z ) wih probbiliy q( z ) ( ( E( ν S = ) = Vr( ν S = ) = σ =, qz ( )( ν qz ( )) (.A) nd condiionl on S = : v = p( z )) wih probbiliy p z ) ( ( v = p z ) wih probbiliy p( z ) ( E( ν S = ) = Vr( ν S = ) = σ =, pz ( )( ν pz ( )), (.B) where he rnsiion probbiliies p z ) nd qz ( ) my hve eiher Probi or Logisic specificion. ( Wihin our frmework, he discree-vlued erm ν is correled wih ε in equion (). We ssume h while S my be endogenous, S j, j > is uncorreled wih ε. Agin, z is ssumed o be predeermined or exogenous. Under hese ssumpions, he exogenous componen of S cn be consruced s c + λ S while he endogenous componen cn be consruced s v = S c λs. 9

11 By defining ν ν =, where σ ν, S σ is he sndrd deviion of ν condiionl on ν, S S, we consider he following join disribuion of ν nd ε : ν ρ σ S ~(, Σ ), Σ = ε ρ σs σ S () A key o developing unbised esimion of he model is in rewriing he shocks using he Cholesky decomposiion of he vrince covrince mrix ( Σ ): AA' Σ = () where A = ρ σ S -ρ σ S. Then, if ω is discree rndom vrible wih men zero nd vrince one nd if ω is sndrd Norml rndom vrible independen of ω, we cn rewrie ν nd ε s follows: ν = σ ω, ν, S ρσ ε ρ σ ω ρ σ ω ν ρ σ ω S = S + S = + σν, S S (3) Using equion (3), equion () cn be rewrien s: y σ ω ' S = S + x S + ν + ρ σsω σν, S, S ν ρσ, = qz ( )( qz ( ))( S ) + pz ( )( pz ( )) S, i.i.d. N(, ), (4)

12 where ν = S c λs. In he bove rnsformed model in (4), noe h he new disurbnce erms ω is uncorreled wih ny of he explnory vribles in he men equion. This llows us o employ he usul Hmilon s (989) pproch for consisen esimion of he model. In equion (4), noe h he erm ν serves s conrol vrible for he endogenous componen of S, llowing he coefficiens nd o cpure he effec of he exogenous componen of S. Idenificion requires h we cn idenify S v = c+ λs s he exogenous componen of S. This is chieved in wo wys. Firs, if n exogenous or predeermined z exiss, s in he Mrkov-swiching models wih ime-vrying rnsiion probbiliies, he model is idenified. Second, even when = 3 =, s in he Mrkov-swiching model wih fixed rnsiion probbiliies, he model is sill idenified so long s S is serilly correled nd S is exogenous, nd S serves s n insrumen for S. This is rue even hough S is unobserved. 3.3 Tesing for n Endogenous Se Vrible Esimion of he models in (7) nd (4) provide srighforwrd es for endogeneiy in S. Given vlid insrumens, es of he null hypohesis of no endogeneiy cn be performed s es of he null hypohesis h ρ = in (7) or ρ = in (4). Noe h his cn be inerpreed s n pplicion of he Wu (973) es for endogenous regressors, where in his cse he regressor is n unobserved len se vrible. 4. Mone Crlo Assessmen of he Proposed Bis Correcion Procedure To evlue he proposed bis correcion procedures we perform series of Mone Crlo experimens. In ech Mone Crlo experimen, d ses re genered from he model

13 given in equions (), (4), nd (5). To model correlion beween S nd ε we ssume h η in equion (4) nd ε hve he join norml disribuion given in (5). Noe h his ssumpion is he sme s h used o develop he bis correcion procedure bsed on (7) described in Secion 3., mening his bis correcion procedure will be opiml. The Mone Crlo experimens re clibred s follows: Ech experimen is performed for hree vlues of he correlion prmeer ρ =.3,.6 nd.9 nd wo smple sizes, T = nd 5. The vecor z is ssumed o be sclr nd is genered s n independen sndrd norml rndom vrible. The vecor x is lso ssumed o be sclr nd is genered o be exogenous, bu correled wih S. We chieve his by genering x s sndrd norml rndom vrible dded o he exogenous pr of S, which from equion (4) is equl o + S + z ' + z ' S. This yields correlions beween x nd S rnging from.5 o 3.4, depending on he prmeerizion of equion (5) used in he Mone Crlo experimen. Equion (4) is clibred wih =, =, =, =, σ =. 5 nd σ =. We consider wo differen ses of prmeers for he se process (4). These re: DGP (Time-Vrying Trnsiion Probbiliies): =.5, =, =.5, 3 = DGP (Fixed Trnsiion Probbiliies): =.5, =, =, 3 = Bsed on his clibrion we perform hree ses of Mone Crlo experimens. In he firs, d is genered from DGP nd he model is esimed wihou bis correcion. This experimen will verify he bis in he regression prmeers discussed in secion. In he second nd hird experimens, d is genered from DGP nd DGP, nd boh of he bis correcion procedures discussed in secion 3, h bsed on equion (7) nd h bsed on equion (4), re implemened. In ech cse, he likelihood funcion for he pproprie model is consruced

14 using he filer given in Hmilon (989) nd mximized compuionlly. The resuls of he Mone Crlo experimens re conined in Tbles -4. Ech ble shows he men nd sndrd deviion of he mximum likelihood poin esimes of he prmeers of () nd (4), which will henceforh be referred o s he men nd sndrd deviion. Tble presens he resuls when d is genered ccording o DGP, nd no bis correcion procedure is employed. The ble demonsres h here is significn bis inroduced ino he regime-swiching coefficiens i nd i by he endogenous S. For exmple, when T = nd ρ =. 6, he men esimes of nd re over 3 sndrd deviions from he rue vlues. The bis is somewh smller in he esimes of nd, he men esimes of which re round one sndrd deviion from he rue vlues. As would be expeced, he bis in boh ses of prmeers is lrger when ρ =. 9 nd smller when ρ =.3. The bis is no miiged s he smple size grows when T = 5, he men esimes of i nd i re nerly he sme s for he T = cse. In Tble, we gin genere d ccording o DGP, bu now esime he model employing he bis correcion procedures from equion (7) (Tble ) nd equion (4) (Tble b). DGP supplies us wih informion o idenify he model prmeers boh from he ime vrying rnsiion probbiliies, z, nd he seril dependence of he se, S. The Mone Crlo resuls sugges h boh bis correcion procedures re very effecive. In priculr, he men esimes of i nd i re now close o he rue vlues for boh smple sizes considered. Tble 3 holds he resuls when he bis correcion echnique is pplied o DGP, in which here re no observed vribles wih which o idenify he model prmeers. Insed, only he unobserved lgged se, S, is vilble for idenificion. Tble 3 shows h for boh bis 3

15 correcion procedures, he esimes of i nd i re very close o he rue vlues. The sndrd deviions of he esimes in Tbles 3 re in generl somewh lrger hn hose in Tble. This is no surprising, s in Tble (bsed on DGP) here is more informion o idenify he prmeers, in he form of z, hn in Tbles 3. Overll, he Mone Crlo experimens confirm h ) endogenous regime swiching cn inroduce significn bis ino he mximum likelihood esimes of Mrkov regime-swiching model nd ) he bis correcion procedures proposed in Secion 3 re quie effecive in elimining his bis. In he nex secion we urn o n empiricl pplicion of he bis correcion procedures. 5. An Applicion: Mesuremen Error nd Esimion of he Voliliy Feedbck Effec A sylized fc of U.S. equiy reurn d is h he voliliy of relized reurns is imevrying nd predicble. Given his, clssic porfolio heory would imply h he equiy risk premium, he expeced reurn of he mrke porfolio bove he risk-free re, should lso be ime-vrying nd respond posiively o he expecion of fuure voliliy. However, he d sugges h relized reurns nd relized voliliy, s mesured by squred reurns, re negively correled. One explnion for he observed d is h while invesors do require n increse in expeced reurn in exchnge for expeced fuure voliliy, hey re lso ofen surprised by news bou relized voliliy. This voliliy feedbck effec crees reducion in prices in he period in which he increse in voliliy is relized. If he voliliy feedbck effec is srong enough, i my cree negive conemporneous correlion beween relized reurns nd voliliy in he d. The voliliy feedbck effec hs been invesiged exensively in he 4

16 lierure, see for exmple French, Schwer nd Smbugh (987), Turner, Srz nd Nelson (989), Cmpbell nd Henschell (99), Beker nd Wu () nd Kim, Morley nd Nelson (). One pproch o modeling ime-vrying voliliy is hrough he use of regimeswiching model. For exmple, Turner, Srz nd Nelson (989), henceforh TSN, model he excess of equiy reurns over he risk free re, r, s rising from norml disribuion wih ime dependen expecion µ nd vrince: σ r, = σ r, ( S ) + σ r, S, S = (,) σ r, > σ r, (5) S follows firs-order Mrkov-swiching process wih fixed rnsiion probbiliies, h is, he process in (4) wih =. The rnsiion probbiliies re hen given by: 3 = P( S P( S = S = ) = q = P( η ( ) = Φ( ) = S = ) = p = P( η > ( ( + ))) = Φ( ( + )) In his model, excess reurns re drwn from eiher high vrince disribuion ( S = ) or low vrince disribuion ( S = ). Also, since he se vrible S is persisen, deviions of he reurn vrince bove is uncondiionl men re predicble. We would hus expec risk premium h vries wih he predicble componen of he vrince. This cn be modeled s follows: µ = θ + θ PS ( Ψ ) (6) 5

17 Here, P ) mesures he probbiliy of he high vrince se ime given n ( S Ψ informion se ded ime -. TSN ssume h Ψ includes ll ps reurns, r...r nd P ( ) is hus equivlen o he filered probbiliy of he se. In (6), θ is equl o he S Ψ risk premium when here is no probbiliy plced on he occurrence of he high-vrince se. Assuming h θ is posiive, he risk premium is incresing in he probbiliy of he highvrince se. The model bove does no incorpore voliliy feedbck in deermining he cul excess reurn. TSN incorpore voliliy feedbck by modeling r s follows: r = µ + θ ( P( S Ψ ) P( S Ψ )) + ζ ~ N(, σς, ) σ ς, = σ ς, ( S ) + σ ς, S ζ (7) The model in (7) cn be moived s follows. A he beginning of period, he risk premium µ is deermined bsed on he expecion of he high-vrince se in period bsed on - informion. During period ddiionl informion regrding he incidence of he high-vrince se is observed. By he end of period, his informion cn be colleced in he informion se Ψ. When P S Ψ ) P( S ), informion reveled during he period hs surprised ( Ψ gens regrding he occurrence of he high-vrince se. If θ <, surprises h revel higher probbiliy of he high-vrince se re viewed negively by invesors, nd hus reduce he conemporneous reurn. One difficuly in esiming he bove model of voliliy feedbck is h here exiss discrepncy beween he invesors informion se nd he economericin s d se. In 6

18 priculr, while Ψ my be summrized by ll d up o -, h is Ψ = { r, r,...}, he informion se Ψ includes informion h is no summrized in he resercher s d se on observed reurns. This is becuse, while he resercher s d se is colleced discreely he beginning of ech period, he mrke pricipns coninuously observe rdes h occur during he period. This is priculr problem when he period is relively long, s is he cse for he monhly d se used by TSN. To hndle his esimion difficuly, TSN use he cul se, S, s proxy for P( Ψ ). Th is, hey esime: S r = µ + θ ( S P( S Ψ )) + ζ ~ N(, σ ) ς, σ ς, = σ ( S, ) + ς σ S ς, ζ (8) This pproch inroduces mesuremen error ino he explnory vribles of he esimed equion, s S is noisy mesure of PS Ψ. In priculr, noe h he residul in (8) is ( ) ζ = ζ + θ ( P( S Ψ ) S ) so h S will be correled wih he residul ζ in equion (8). The resuls from secion sugges h mximum likelihood esimes of he prmeers of equion (8) will hen be bised. The bis correcion echniques inroduced in Secion 3 cn be used o correc for his bis. We show his here using he echnique of Secion 3., in which join normliy ssumpion is employed. As here re no ime-vrying rnsiion probbiliies in he TSN model, h is z =, here is single insrumen in he model, S. Thus idenificion requires h S be Mrkov- 7

19 swiching, rher hn independen swiching. Formlly, equion (7) for he model in (8) becomes: r = µ + θ ( S P( S ψ )) + ω = ς ρσ ς,m ( S ) ρσ ρσ ς,m ( S ) + ς,m S ρσ ς,m S + ω (9) We esime he model in (9) using monhly reurns for vlue-weighed porfolio of ll NYSE-lised socks in excess of he one-monh Tresury Bill re, our mesure of he risk free re, over he smple period 95-. This is he sme d s used in Kim, Morley nd Nelson (). The firs pnel of Tble 4 shows hese esimes when no bis correcion is employed, h is ρ =. These esimes, which re similr o hose in TSN, re consisen wih boh posiive relionship beween he risk premium nd expeced fuure voliliy ( θ > nd θ > ) nd subsnil voliliy feedbck effec ( θ << ). The esimes re lso consisen wih dominn voliliy feedbck effec, h is θ is very smll relevn o θ. The second pnel shows he esimes when bis correcion is employed, so h ρ is esimed. There is subsnil evidence of endogeneiy, s ρ is esimed o be lrge in bsolue vlue nd sisiclly significn. Alhough he prmeer esimes re very imprecise, correcing for his endogeneiy does yield subsnil chnges in he poin esimes of he model prmeers. Firs, θ is esimed o be nerly en imes he esime for he model wih no bis correcion, suggesing risk premium h is much more responsive o forecsed fuure voliliy. Second, θ is esimed o be hlf he esime for he model wih no bis correcion, suggesing srong voliliy feedbck effec, bu less so hn for he model wih no bis correcion. 6. Conclusion 8

20 We hve explored he implicions of relxing he ssumpion of exogeneiy for he len se vrible in Mrkov regime-swiching regression. We show h endogeneiy of he se vrible cn led o significn esimion bis nd hve considered wo echniques o correc for his esimion bis. The firs exends he endogenous swiching models in Mddl nd Nelson (975) o he Hmilon (989) regime swiching regression, in which he swiching process is serilly dependen. This pproch relies on join normliy ssumpion for he relionship beween he regression error erms nd he innovions o he se relizion equion. The second echnique is bsed on he inerpreion of he endogenous swiching regression s regression model wih endogenous unobserved dummy vribles. In his cse he regression cn be esimed using insrumenl vribles echniques. For boh echniques, idenificion cn be chieved using informion from exernl, exogenous vribles h ffec he rnsiion probbiliies of he regime-swiching process. However, even in he fixed rnsiion probbiliy cse, we hve shown h idenificion is chieved if he se process is serilly dependen, nd lgged vlues of he se re exogenous. This is rue even hough he lgged se is unobserved. Mone Crlo experimens confirm h he proposed bis-correcion procedure is quie good for empiriclly relevn d genering processes. We pply he echnique o he voliliy feedbck model of equiy reurns given in Turner, Srz nd Nelson (989). 9

21 References Beker, G. nd G. Wu,, Asymmeric Voliliy nd Risk in Equiy Mrkes, Review of Finncil Sudies, 3, -4. Cmpbell, J. Y. nd L. Henschel, 99, No News is Good News: An Asymmeric Model of Chnging Voliliy in Sock Reurns, Journl of Finncil Economics, 3, Diebold, F., Lee, J-H. nd G. Weinbch, 994, Regime Swiching wih Time-Vrying Trnsiion Probbiliies, in Non-sionry Time Series Anlysis nd Coinegrion, ed. C. Hrgreves, Oxford Universiy Press, Oxford, U.K. Filrdo, A.J., 994, Business-Cycle Phses nd Their Trnsiionl Dynmics, Journl of Business nd Economic Sisics,, French, K.R., Schwer, G. nd R. F. Smbugh, 987, Expeced Sock Reurns nd Voliliy, Journl of Finncil Economics, 9, 3-9. Goldfeld, S.M. nd R.E. Qund, 97, in Nonliner Mehods in Economerics, Norh Hollnd, Amserdm. Goldfeld, S.M. nd R.E. Qund, 973, A Mrkov Model for Swiching Regressions, Journl of Economerics,, 3-6. Hmilon, J.D., 989, A New Approch o he Economic Anlysis of Nonsionry Time Series nd he Business Cycle, Economeric, 57, Hmilon, J.D., 99, Anlysis of Time Series Subjec o Chnges in Regime, Journl of Economerics, 45, Kim, C.J., Morley, J.C. nd C.R. Nelson,, Is There Posiive Relionship Beween Sock Mrke Voliliy nd he Equiy Premium?, Journl of Money, Credi nd Bnking, forhcoming. Mddl, G.S. nd F. Nelson, 975, Swiching Regression Models wih Exogenous nd Endogenous Swiching, Proceedings of he Americn Sisicl Associion, Mddl, G.S., 984, Disequilibrium, Self-Selecion nd Swiching Models, in Hndbook of Economerics, eds. Z. Griliches nd M.D. Inrilligor, Norh Hollnd, Amserdm. Owyng, M.,, Modeling Volcker s Non-Absorbing Se: Agnosic Idenificion of Mrkov-Swiching VAR, working pper, Federl Reserve Bnk of S. Louis. Qund, R.E., 958, The Esimion of he Prmeers of Liner Regression Sysem Obeying Two Sepre Regimes, Journl of he Americn Sisicl Associion, 53,

22 Qund, R.E., 97, A New Approch o Esiming Swichign Regressions, Journl of he Americn Sisicl Associion, 67, Sims, C. nd T. Zh,, Mcroeconomic Swiching, mimeo. Turner, C. M., Srz, R. nd C.R. Nelson, 989, A Mrkov Model of Heeroskedsiciy, Risk, nd Lerning in he Sock Mrke, Journl of Finncil Economics, 5, 3-. Wu, D., 973, Alernive Tess of Independence Beween Sochsic Regressors nd Disurbnces, Economeric, 4,

23 Tble Mone Crlo : Mximum Likelihood Esimion of Time-Vrying Trnsiion Probbiliy Mrkov-Swiching Regression Model (TVP-MS) No Bis Correcion Prmeer T= ρ =.3 ρ =. 6 ρ =. 9 Prmeer Esimes: Men / Sd. Dev. Prmeer Esimes: Men / Sd. Dev. Prmeer Esimes: Men / Sd. Dev. =.9 /.5.83 /.5.74 /.5 = -.77 / / /. =.97 /.4.95 /.4.93 /.4 = -.6 /.7 -. / /.6 = / / /.6 =.3 /.8.6 /.8.7 /.9 =.5.54 /.9.56 /.8.56 /.9 3 =.3 /.5.3 /.53. /.5 T=5 =.9 /.4.8 /.3.74 /.3 = -.78 / / /.6 =.98 /..95 /..93 /. = -.6 /.4 -. / /.4 = /. -.5 /. -.5 /. =. /.7.3 /.7.4 /.7 =.5.5 /..53 /..54 /. 3 =.3 /.5.4 /.7.5 /.9 Noes: Ech column conins he men nd sndrd deviion of he mximum likelihood poin esimes from he Mone Crlo experimen.

24 Tble Mone Crlo : Mximum Likelihood Esimion of Time-Vrying Trnsiion Probbiliy Mrkov-Swiching Regression Model (TVP-MS) Bis Correcion from Equion (7) Prmeer T= ρ =.3 ρ =. 6 ρ =. 9 Prmeer Esimes: Men / Sd. Dev. Prmeer Esimes: Men / Sd. Dev. Prmeer Esimes: Men / Sd. Dev. =. /.9. /.9. /.7 = -.98 / / /.6 =. /.4.99 /.4.99 /.4 = -. /.9 -. /.8 -. /.7 = / / /.4 =. /.9. /.7.3 /. =.5.53 /.8.5 /.7.5 /.4 3 =. /.48. /.44.9 /.37 T=5 =.99 /.6.99 /.5. /.4 = -.98 / / /. =. /.3. /.. /. = -. /.5 -. /.5 -. /.4 = / / /.8 =. /.7. /.6. /.3 =.5.5 /..5 /..5 /.8 3 =.4 /.5.3 /.4. /.9 Noes: Ech column conins he men nd sndrd deviion of he mximum likelihood poin esimes from he Mone Crlo experimen. 3

25 Tble b Mone Crlo : Mximum Likelihood Esimion of Time-Vrying Trnsiion Probbiliy Mrkov-Swiching Regression Model (TVP-MS) Bis Correcion from Equion (4) Prmeer T= ρ =.3 ρ =. 6 ρ =. 9 Prmeer Esimes: Men / Sd. Dev. Prmeer Esimes: Men / Sd. Dev. Prmeer Esimes: Men / Sd. Dev. =.99 /..99 /.. /. = -.98 / / /.3 =. /.4.99 /.4.99 /.4 = -. /.9 -. / /.8 = / / /.5 =. /.8. /.7.97 /.4 =.5.5 /.8.53 /.7.5 /.5 3 =. /.47.5 /.44.9 /.4 T=5 =.98 /.6.98 /.6.99 /.6 = -.96 / / /.5 =. /.3.99 /.3.99 /. = -. / / /.5 = / / /.9 =. /.7.98 /.5.94 /.4 =.5.5 /..5 /..48 /.9 3 =.4 /.7.97 /.3.9 /. Noes: Ech column conins he men nd sndrd deviion of he mximum likelihood poin esimes from he Mone Crlo experimen. 4

26 Tble 3 Mone Crlo 3: Mximum Likelihood Esimion of Fixed Trnsiion Probbiliy Mrkov- Swiching Regression Model (FTP-MS) Bis Correcion from Equion (7) Prmeer T= ρ =.3 ρ =. 6 ρ =. 9 Prmeer Esimes: Men / Sd. Dev. Prmeer Esimes: Men / Sd. Dev. Prmeer Esimes: Men / Sd. Dev. =. /.4. /.3. /.9 = -. /.8 -. / /.7 =.99 /.5. /.5.99 /.4 = -. /. -. /.9 -. /.7 = / / /.4 =.98 /..99 /.. /. = = T=5 =. /.8. /.8. /.6 = -.99 / / /. =. /.3.99 /.3.99 /. = -. /.6 -. /.6 -. /.5 = / / /.9 =. /.4.99 /.4.99 /.3 = = Noes: Ech column conins he men nd sndrd deviion of he mximum likelihood poin esimes from he Mone Crlo experimen. 5

27 Tble 3b Mone Crlo 3: Mximum Likelihood Esimion of Fixed Trnsiion Probbiliy Mrkov- Swiching Regression Model (FTP-MS) Bis Correcion from Equion (4) Prmeer T= ρ =.3 ρ =. 6 ρ =. 9 Prmeer Esimes: Men / Sd. Dev. Prmeer Esimes: Men / Sd. Dev. Prmeer Esimes: Men / Sd. Dev. =. /.5. /.5.6 /.6 = -. / /.3 -. /.3 =.99 /.5.99 /.5.99 /.4 = -. /. -. /. -. /.8 = / / /.5 =.99 /..98 /..95 /. = = T=5 =. /.9. /.9.4 /.9 = -. /.7 -. / /.7 =. /.3.99 /.3.99 /. = -. /.6 -. /.6 -. /.5 = / / /. =.98 /.4.98 /.4.96 /.4 = = Noes: Ech column conins he men nd sndrd deviion of he mximum likelihood poin esimes from he Mone Crlo experimen. 6

28 Tble 4 Mximum Likelihood Esimes of Voliliy-Feedbck Model from Turner, Srz nd Nelson (989) (sndrd errors in prenhesis) Prmeer Wihou Bis Correcion Wih Bis Correcion θ.6 (.6) θ. (.3) θ -. (.5) σ ς,.38 (.) σ ς,.67 (.4) -. (.5) 3.5 (.7).4 (.7).9 (.34) -. (.49).4 (.).74 (.6) -.6 (.6) 3.6 (.3) ρ (.9) 7