Linear or Non-Linear: This is the Dilemma!

Transcription

1 Linea o Non-Linea: This is he ilemma! Paolo Pasquaiello* Sen School o Business New Yok Univesiy Mach Absac simaion o he unknown di and diusion ems o a sochasic pocess epesening he spo inees ae has eceived inceased aenion in he inancial lieaue. In his bie pape we analyze he peomance o he wo mos popula non-paameic appoaches o he poblem Sahalia s FGLS and Sanon s Taylo xpansion. An ouigh esimaion o each o he poposed models o a sample o yeas o daily obsevaions o he 7-days euodolla spo inees ae seems o sugges he exisence o non-lineaiy in boh he evesion and he volailiy o he ae pocess. Howeve he esuls o a seies o simulaions om a known pocess eveal ha none o he mehods hee examined has enough powe o ecognize a ue linea di i.e. ha he lineaiy o he inees ae di componen is ejeced oo oen. Moe accuacy is howeve ound o he esimaed diusion. We ae hen inclined o conim Sahalia s claim ha he volailiy o he spo ae appeas o be inceasing wih he level o he ae. * Ph.. candidae a he Sen School o Business. Please addess commens o he auho a he Leonad N. Sen School o Business New Yok Univesiy Kauman Managemen ducaion Cene Suie Wes 4h See New Yok NY -6 o hough ppasqua@sen.nyu.edu.

2 . Inoducion The coec speciicaion o he spo inees ae as a sochasic pocess and a pope and eicien esimaion o he esuling di and diusion ems have always been puzzling paciiones and ades picing complex deivaive secuiies beoe inally capuing he aenion and he eos o he inancial lieaue in he pas ew yeas. I has in ac been agued ha an incoec speciicaion o he poenial lucuaions o he undelying andom inees ae would induce incoec valuaion o coningen claims wien on bonds o cuencies. Thee main appoaches o he esimaion o he di and diusion em o a geneal sochasic pocess o he om: ( ) d σ ( ) dw d µ [ ] can be by now ideniied. In one case given a speciic sucue o µ() and σ() maximum-likelihood esimaos o he esuling unknown paamees have been obained by applying he Fowad Kolmogoov quaion o deive an expession o he ansiion densiy o he daa. This appoach usually known as Paameic and pioneeed by Lo(988) Peason and Sun(994) and moe ecenly Sahalia (999) povides he clea advanage o always leading o eicien esimaes o he assumed sucue bu is unable o disinguish beween alenaive sucual speciicaions o he di and diusion o he ae pocess. In he aemp o educe he numbe and signiicance o he sucual and disibuional esicions needed o ideniy he Likelihood uncion o Chan e al. (99) uie and Singleon (993) and Hansen and Scheinkman (995) popose o esimae he given µ() and σ() by maching a se o momen condiions wih he coesponding esimaes om he obseved daa. This Mehod o Momens aenuaes bu does no eliminae he isk o mis-speciicaion ha aecs any heavily sucual esimaion pocedue. Hence he Non-Paameic appoach o Sahalia (996 a b) and Sanon (997) appeaed o be paiculaily pomising. In one case Sahalia speciies an appaenly vey geneal om o µ() and σ() and shows ha none o he moe speciic sucues ha ae ypically eeed o in he lieaue om he CIR pocess o he CV diusions seems o eally i he daa. Sanon uses sochasic Taylo seies o geneae even moe geneal and sucue-less poin-wise esimaes o µ() and σ(). Boh mehods ae no aeced by he limiaions esuling om he esimaion o a coninuous pocess via disceely sampled daa bu ely heavily on non-paameic esimaion o he maginal and/o ansiional densiy o he undelying and unknown pocess o using gaussian Kenels. Unounaely no many o he iniial pomises wee ulilled. Non-paameic Kenel esimaion o densiies is highly inpecise and he song pesisence in he inees ae daa limis he amouns o eal inomaion ha any o hose mehods can exac om he available obsevaions.

3 In his pape we esimae he di and he diusion o he geneal sochasic pocess o equaion [] applying he FGLS mehod o Sahalia and he Sanon s appoach o a daa se o 555 obsevaions o he 7-day spo uodolla ae om June s 973 o Febuay 5 h 995. We hen simulae seies o inees aes o he same ime-ame using a CIR diusion and show ha boh mehods ail o ecognize he ue di and diusion ems embedded in he consuced seies unless he esimaion is epeaed seveal imes i.e. ha boh appoaches show low powe. An ou-o-sample analysis o he esimaes geneaed wih Sahalia s and Sanon s pocedues is povided. The simulaed inees ae seies in boh cases show moe vaiabiliy han wha was eecively obseved in he ou yeas ollowing he oiginal sample. In he nex secion we descibe he Sahalia s FGLS esimaion. Then in secion 3 we descibe and implemen he Sanon s esimaion. Secion 4 evaluaes he powe o he wo saegies wih epeaed simulaions. Secion 5 concludes.. The FGLS simaion Sahalia (996 b) poposes he ollowing geneal sucue o he di and he diusion ems in equaion []: β [ β β β 3 ] dw α3 d d α α α [ ] The oiginal non-paameic esimaion o he veco o paamees (α β) is caied ove by obseving ha any speciicaion o he di and he diusion like in equaion [] implies a one-o-one mapping wih he maginal and ansiional densiies o. Fo he puposes o his pape his esimaion is achieved hough he ollowing FGLS muli-sep pocedue: Regess: simae: 3 [ ] α α ε α α [ 3 ] αˆ ˆ 3 ε [ ] αˆ αˆ αˆ [ 4 ] Use Non-Linea Leas Squaes o esimae : The saing values necessay o geneae NLSQ esimaes o he beas ae obained by assuming ha iniially β 3 and hen by esimaing he ollowing uncional om (.) β β ( ) β ( ). We hen assume ha he iniial β 3 o he non-linea egession descibed in equaion [5].

4 [ ˆ β3 ] β β β u ε [ 5 ] Calculae: ˆ ˆ ˆ ˆ β3 β β β ˆ ε [ 6 ] Use he esimaed squaed esiduals o equaion [6] o esimae Σ he covaiance maix o ε. We assume ha he esiduals ae heeoscedasic bu no auocoelaed i.e. ha Σ has he om: σ Σ σˆ ˆ ε simae he alphas in equaion [3] by Feasible GLS: σ σ T [ 7 ] ( ) ( ˆ ) ' ˆ α α α ' ' Σ Σ ( d) α [ 8 ] 3 The esimaes o beas obained om equaion [6] and o alphas om equaion [8] ae epoed in Table below. Table : FGLS simaion o quaion [] Sahalia s Geneal Paameic Model FGLS simaion Paamees FGLS simaes Sandad o -Sa α α α α β β β β

5 As eviden om he -saisic values epoed in column 3 o able ew o he paamees esimaes ae saisically signiican. The compaison o hese esuls wih Sahalia s suggess ha he maching densiy appoach he designs is moe eecive in educing he degee o unceainy aound he esuling esimaed paamees. Figue plos he di and diusion uncions coesponding o he FGLS values o Table. Figue : i and iusion FGLS simaion µ() σ ().. One-ay change in spo ae % 4% 8% % 6% % 4% Spo Rae (One-ay change in spo ae)^ % 4% 8% % 6% % 4% Spo Rae When he spo ae is a away om is mean he esimaed di appeas o evese songly bu is essenially zeo o values o beween 4 and 3 % hus suggesing ha in such an ineval he inees ae pocess is a andom walk 3. This inding i conimed would explain why ove shoe ime-samples he inees ae pocess appeas no o ejec he uni oo hypohesis. Fom he diusion picue he spo ae appeas o be moe volaile ouside he same middle egion we peviously ideniied o he di. In paicula he volailiy o he spo ae appeas o be inceasing o highe levels o bu smalle o levels o below is long-em mean. Noe ha he le-hand axis ime-scale o he diusion cha depends on he sampling ineval o he daa while he pope non-paameic esimaes esuling om Sahalia s maching densiy mehod ae unaeced by he equency o he daa. 3 In he Appendix we aach a sample Limdep code o he FGLS esimaion o equaion []. 4

6 5 3. The Sanon s simaion Sanon s saegy elies on he obsevaion ha given an abiay uncion () o he pocess [] i is possible o expand is condiional expecaion: ( ) [ ] [ 9 ] as a sochasic Taylo expansion aound ( ): ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) () ()!... L O L n L L n n n σ µ [ ] Fom equaion [] a is-ode poxy o he condiional expecaion o (.) is hen: ( ) ( ) ( ) [ ] { } ( ) O L [ ] Given a suiable choice o (.) Sanon uses he esul o equaion [] o expess L(.) as a uncion o µ() and σ (). Finally esimaes o he momens o (.) can be used o calculae poin-wise esimaes o µ() and σ (). Using a second-ode appoximaion as suggesed in Sanon s pape he ollowing expessions o µ() and σ () ae obained: ( ) [ ] [ ] { } ( ) ( ) ( ) [ ] ( ) [ ] { } ( ) 4 4 O O σ µ [ ] I is clea om equaion [] ha he only emaining ask le o he economeician is o esimae he condiional momens in he squae backes. Sanon suggess o employ a non-paameic Gaussian- Kenel esimao o he maginal densiy o he undelying inees ae pocess. Moe speciically we adop he ollowing esimae o he densiy o he inees ae pocess a :

7 6 () () () () () () 5 * ˆ * * * * ˆ T h e Th h K Th T h T σ π [ 3 ] whee σ^ is he esimae o he volailiy o he undelying ae pocess ove he sampling ineval 4 and h(*) is a bandwidh 5 based on he dispesion o he obsevaions ha minimizes he asympoic mean inegaed squaed eo o he esimaed densiy uncion. Given he densiy esuling om equaion [3] we can now esimae any momen we desie om he disibuion. We will hen use he ollowing algoihm: ( ) [ ] ( ) ( ) ( ) ( ) () ( ) () () j i h K h K d d j T j T i j j j i j j j i j i j * * [ 4 ] The esimaed maginal densiy ove he inees-ae sample is epoed in Figue. Figue : Non-Paameic simaed ensiy o Ove ou sample we esimaed his value o be aound 3.59 %. 5 Suggesed by Sco(99) % 5.%.% 5.%.%

8 As eviden om he igue above alhough mos o he pobabiliy mass is allocaed o aes beween 4 and 3 % levels in he 5 o 7 % ange have some mass as well. This obsevaion is going o be elevan soon o he analysis. The nex igue shows he poin-wise esimaes o he di and he diusion o he 7-days spo euodolla inees ae esuling om equaion [] and he se o momens o equaion [4]. The di em µ() is subsanially zeo o mos o he values o beween and 3 %. No song evesion appeas o small values o conay o he indings o Sahalia in secion. Noneheless hee sill appeas o be a signiican negaive evesion push om µ() o values o above 5 %. I is also woh commening on he sho posiive spike in he di o beween 7 and 9 %. This spike happens because o he ac ha such values o have some mass in he maginal densiy o igue. In ohe ems as in ou sample hee ae moe han ew obsevaions o a hose levels a consequence o he loaing-inees-ae policy adoped by Paul Volcke a he Fed beween 979 and 98 oo a negaive di o ha ange would peven he esimaed pocess o equaion [] o i he daa. In pacice he spike is a consequence o he Sanon s echnique ove-iing he available daa. Figue 3: Non-Paameic Poin-wise simaes o µ() and σ() µ() % 5.%.% 5.%.% σ().% 5.%.% 5.%.% The diusion em appeas o be convex as suggesed by Sahalia bu no deceasing a low levels o he inees ae. Again a downwad spike appeas o beween 7 and 9 %: as long as he di em is pushing upwad o each he high levels o in he sample he volailiy dops in ode no o obsacle he i 6. Boh Sahalia and Sanon s esuls seem o sugges ha he spo inees ae behaves subsanially as a andom walk o mos o he levels o he euodolla ae obseved in he pas welve yeas ha he insananeous volailiy is inceasing wih he level o he ae and ha he di and he diusion in 6 In he Appendix we epo a sample code in C o he esimaion o di and diusion hough he Sanon s appoach. 7

9 equaion [] ae highly non-linea alhough hey disagee on he naue and eniy o hese non-lineaiies. How sensiive ae hese conclusions o he paicula esimaion saegy seleced? In ohe ems how saisically poweul ae hese conclusions? The nex secion ackles his compelling issue. 4. A CIR Simulaion The main diiculy we ace in evaluaing qualiaively he esuls o boh Sahalia s and Sanon s empiical analysis lies in ha we do no acually know how he ue undelying inees ae pocess uly behaves. Hence any o hei conclusions ae ex-ane accepable and diicul o dispove. Noneheless we would be able o say moe abou he powe o hei saisical echniques i.e. moe speciically abou hei capaciy o ideniy he ue di and diusion ems o equaion [] when we acually know ex-ane wha hese ue elemens eally ae. Founaely such an evaluaion is possible: we can geneae simulaed ime seies o inees aes om a given inees ae pocess whee we speciy exacly he om o µ() and σ(). Then we simply apply he FGLS appoach and he Taylo s xpansion Mehod o hese simulaed seies and obseve whehe any o hose echniques is able o capue he main chaaceisics o he ue di and diusion om which he daa wee acually geneaed. We selec a popula model o inees ae dynamics he CIR pocess o he om: d ( ϑ ) d σ dw k [ 5 ] We assume ha a he beginning o he sample peiod o he same lengh as he oiginal sample CIR () () and hen geneae a sample pah o om equaion [5] wih he ollowing (asympoically exac) daily Fowad-ule appoximaion: k( ϑ ) ( ) d σ G N G [ 6 ] whee G is a andom vaiable disibued as a Gaussian Nomal wih mean zeo and uni vaiance. We geneae a ime seies o independenly dawn values o G wih he Box-Mulle algoihm. Values o k ϑ and σ come om Piske (998) 7 and ae especively and (.374/5).5. We inally epea he wo pocedues descibed in secions and 3 on his simulaed inees ae seies. Le s sa wih he Feasible GLS o Sahalia. In igue 3 we epo he esimaed di esuling om he 7 I is wohwhile obseving ha Piske values ae annualized hence o he simulaion o he CIR pocess we use d.4 ( / 5). 8

10 esimaion o equaion [] on he simulaed CIR daa and wha is supposed o be he ue di em (he doed line). Figue 4: FGLS simaes o µ() o he CIR Pocess % 5% % 5% % One-ay change in spo ae % 4% 8% % 6% % 4% Spo Rae As clealy picued above Sahalia s epesenaion appeas o impose non-lineaiy ove a uly linea di o vey low and vey high levels o he inees ae while slighly oveesimaing he evesion componen o he di o values o beween 4 % and 5 %. We ague ha his happens because o he polynomial chaace o he uncional om o µ() in equaion []. I is in ac well known ha polynomial esimaes end o explode a he ails o he daa sample ove which he esimaion is un. And Sahalia ideniies song evesion exacly a he ails o sampled values o. Figue 5: FGLS simaes o σ() and σ () o he CIR Pocess σ().6.4. % 5% % 5% % (One-ay change in spo ae)^ σ () % 4% 8% % 6% % 4% Spo Rae The esimaion o he diusion em epoed in igue 5 appeas o be moe successul o mos o he values o in he ange 5 o %. Sill Sahalia s esimaion ovesaes he insananeous volailiy o he CIR pocess o low values o. Again his is a esul o he ove-shooing in he di em and he need o he diusion componen o pull he pocess away om he unaainable level o zeo. 9

11 Using he same simulaed ime seies we apply Sanon s pocedue and epo he esuls in igue 6 whee he ue di and diusion ae shown in doed line. Figue 6: Sanon s simaes o µ() and σ() o he CIR Pocess µ() σ() % 5.%.% 5.%.% % 5.%.% 5.%.% Sanon s esimae o he di seems o sugges ha alhough ee o oveshooing in he sho end his pocedue is biased owad negaive evesion o high levels o inees aes. ven in his case he diusion esimae appeas o be quie accuae o mos o he values o in he ange o 8 %. Again as in he case o FGLS he accuacy dops o values a he exemiies o he ange. The evidence pesened so a leads us o conclude ha he wo mehods appea o be biased by consucion in he measuemen o di and diusion o values o in he ails o he ange o possible values obseved in he given ime seies. Obviously his conclusion is no by isel supising as i is exacly o vey high and vey low values o ha we have ewe obsevaions available hence any ineence becomes moe poblemaic. In ode o popely evaluae he powe o hese saegies we epea he expeimen descibed above N imes hen calculae he mean poin-wise esimaes o µ() and σ() and 95 % conidence inevals. Table epos Sahalia s FGLS mean esimaed paamees values while able 3 compaes he mean esimaed values esuling om 3 simulaions o he CIR pocess (and coespondingly 3 epeiions o he FGLS pocedue) o seleced values o. Figue 3 epos he esimaes o µ() and σ() ove he ange % vesus he ue values o he di and diusion (doed line). We also epo 95 % poin-wise conidence inevals. The accuacy o µ^() and σ^() appeas o be enomously inceased especially as a as he diusion em is concened. Noneheless he FGLS appoach sill ails o popely accoun o he behavio o he di a he ails o he ange o possible values o. This means ha even wih 3 addiional yeas-long ime seies simulaed om CIR he geneal om o equaion [] keeps imposing non-linea evesion o he daa.

12 Table : FGLS simaion o N 3 Simulaed CIR Pocess Sahalia s FGLS simaion o he CIR iusion Paamees FGLS simaes Sandad o -Sa α α α α β β β β Table 3: FGLS simaion o N 3 Simulaed CIR Pocesses vesus he ue µ() and σ() FGLS simaion o he CIR iusion: Analysis o i and iusion a seleced values Raes µ(x) s.e. CIR µ(x) σ (x) s.e. CIR σ(x) % n.a. n.a % % % % Figue 6: FGLS simaes o µ() and σ() o he CIR Pocess: N 3 Simulaions µ() σ() % 5% % 5% %.6.4. % 5% % 5% % The same analysis is hen epeaed o he Sanon s pocedue wih N simulaions. Table 4 epos he esimaed values o µ^() and σ^() o seleced.

13 Table 4: Sanon s simaion o N Simulaed CIR Pocesses vesus he ue µ() and σ() Sanon simaion o he CIR iusion: Analysis o i and iusion a seleced values Raes µ(x) s.e. CIR µ(x) σ (x) s.e. CIR σ(x) % % % % % As in he FGLS case again he echnique ails o impove signiicanly is pecision a he ails o he ange. In igue 7 we epo he esimaes o µ() and σ() ove he ange % vesus he ue values o he di and diusion (doed line). The conidence inevals widen signiicanly o small and high values o i.e. exacly whee we ae mos ineesed in undesanding he behavio o he undelying and in his case known inees ae pocess. Finally igue 8 epos he esimaed maginal densiy o 8 and is conidence inevals. This igue and in paicula he widh o he uppe and lowe bands suggess he high degee o unceainy suounding he non-paameic esimaion o he ue densiy a he ails o he ange. Figue 7: Sanon s simaes o µ() and σ() o he CIR Pocess: N Simulaions µ() σ() % 5.%.% 5.%.% % 5.%.% 5.%.% Few poins deseve addiional claiicaion. The poo esuls o boh mehodologies even ae moe han N 3 epeiions o he same simulaion expeimen conim ealie indings by Piske ha FGLS and Taylo s appoach gossly oveesimae he amoun o inomaion ha is acually in he daa. In ac a 8 We emind he eade ha he ue condiional disibuion o i.e. condiional on he iniial value () is a Non- Cenal Chi-Squaed.

14 ailue o FGLS o ecognize he lineaiy o he di even ae 3 simulaions o newly dawn CIR ime seies ove a span o yeas means ha he mehod would have ailed o ideniy he linea evesion even wih moe han 66 ( imes 3) yeas o daily obsevaions. The ailue appeas even moe embaassing o he Sanon s pocedue as moe han yeas o daily inees ae daa would have no helped he saegy in inally emoving he negaive di om he ied µ(). The degee o pesisence in inees ae daa whehe eal o simulaed hough a pocess (CIR) ha embeds i limis he accuacy o hese appoaches hence hei evenual eicacy in answeing he quesions o which hey wee oiginally designed. Figue 8: Sanon s simaes o he maginal densiy o he CIR Pocess: N Simulaions % 5.%.% 5.%.% Slighly moe comoing ae he esuls o he esimaion o he ue diusion: wih jus a single simulaion i.e. jus yeas o daa he poin-wise esimaes o σ() ae aily accuae. In he appendix igue 9 we epo he esuls o an ou-o-sample analysis o he esimaed pocess [] ove he sample peiod Ou-o-sample seies o have been geneaed wih a single imeseies o he Gaussian Wiene Pocess dw o he oiginal Sahalia s Maching-ensiy RFS esimaed pocess [] he FGLS pocess [] he Sanon pocess and he CIR pocess (wih Piske s coeiciens given a he beginning o his secion). We also epo (in he dakes line) he obseved pocess o 3- monh consan-mauiy inees ae as calculaed daily by he Fedeal Reseve Bank o S. Louis unil Mach 8 h o exend Sahalia s sample and compae he peomance o each o he poposed models. Sahalia s RFS pocess shows he bigges lucuaions bu all o hem pedic a sudden incease in he inees ae beween 997 and 998 ha did no happen in he eal daa. The CIR pocess is he one ha ges close o he obseved ae on Mach 8 h. All esimaed pocesses lucuae moe han he ue inees ae. This analysis suggess ha he esimaed inees ae pocess [] especially ove a single ime-seies dawing o dw should no be used o pedic he uue behavio o inees aes as moe economically-sound models ae designed o bu simply o descibe he poenial uue dynamics o. 3

15 These dynamics ae hen undamenal in picing coningen claims wien on asses whose pice lucuaions may depend heavily on he poenial lucuaions o he spo ae isel. 5. Conclusions simaion o he unknown di and diusion ems o a sochasic pocess epesening he spo inees ae has eceived inceased aenion in he inancial lieaue. In his bie pape we analyze he peomance o he wo mos popula non-paameic appoaches o he poblem. An ouigh esimaion o each o he poposed models o a sample o yeas o daily obsevaions o he 7-days euodolla spo inees ae seems o sugges he exisence o non-lineaiy in boh he evesion and he volailiy o he ae pocess. Howeve he esuls o a seies o simulaions om a known pocess eveal ha none o he mehods hee examined has enough powe o ecognize a ue linea di i.e. ha he lineaiy o he inees ae di componen is ejeced oo oen. Moe accuacy is howeve ound o he esimaed diusion. We ae hen inclined o conim Sahalia s claim ha he volailiy o he spo ae appeas o be inceasing wih he level o he ae. 4

16 Reeences Ai-Sahalia Yacine 996 a Non-Paameic Picing o Inees Rae eivaive Secuiies conomeica Vol. 64 No. 3. Ai-Sahalia Yacine 996 b Tesing Coninuous-Time Models o he Spo Inees Rae Review o Financial Sudies Vol. 9 No.. Ai-Sahalia Yacine 999 Maximul Likelihood simaion o isceely Sampled iusions: A Closed-Fom Appoximaion Appoach unpublished manuscip. Band M. Sana-Claa P. 999 Simulaed Likelihood simaion o iusions wih an Applicaion o xchange Rae ynamics unpublished manuscip. Chapman avid Peason Neil Is he Sho Rae i Acually Nonlinea Jounal o Finance ohcoming. Hansen Las Scheinkman Jose 995 Back o he Fuue: Geneaing Momen Implicaions o Coninuous-Time Makov Pocesses conomeica Vol. 63 No. 4. Piske Ma 998 Non-Paameic ensiy simaion and Tes o Coninuous Time Inees Rae Models Review o Financial Sudies Vol. No. 3. Sanon Richad 997 A Non-Paameic Model o Tem Sucue ynamics and he Make Pice o Inees Rae Risk Jounal o Finance Vol. LII No. 5. 5

17 Appendix Figue 9: Ou-o Sample simaion o he inees ae behavio ue inees ae in black Ou-o-Sample xpeimen.5. Rae (%).5. Sahalia FGLS Sanon Sahalia RFS CIR Real aa ae 6