Estimation in Models of the Instantaneous Short Te

EstimationinModelsoftheInstantaneousShortTermInterestRatebyuseofaDynamicBayesianAlgorithmRamaprasadBharSchoolofBankin[e-mail]inanceandEconomicsUniversityofTechnologySydneyPOBox123,[e-mail]iMatematicaPuraedApplicataUniversit´adiPadovaViaBelzoni735131-Padova,[e-mail]29October20011AbstractThispaperconsiderstheestimationinmodelsoftheinstantaneousshortinterestratefromanewperspective.Ratherthanusingdiscretelycompoundedmarketratesasaproxyfortheinstantaneousshortrateofinterest,wesetupthestochasticdy-namicsforthediscretelycompoundedmarketobservedratesandproposeadynamicBayesianestimationalgorithm(i.e.afilteringalgorithm)foratime-discretisedver-sionoftheresultinginterestratedynamics.Thefiltersolutioniscomputedviaafurtherspatialdiscretization(quantization)andtheconvergenceofthelattertoitscontinuouscounterpartisdiscussedindetail.Themethodisappliedtosimulateddataandisfoundtogiveareasonableestimateoftheconditionaldensityfunctionandtobenottoodemandingcomputationally.1IntroductionLiteratureonestimationinmodelsoftheinstantaneousspotrateofinteresthasbur-geonedsincetheseminalcontributionofChanetal.(1992)(henceforthCKLS).Thesustainedinterestinthistopicisduetothegreatdealofactivity,bothamongstaca-demicsandpractitioners,inpricinginterestraterelatedsecurities.CKLSappliedthegeneralisedmethodofmoments(GMM)toestimatetheparametersofaonefactormodeloftheinstantaneousspotrateofinterestthatismean-revertinginthedrifttermandhasadiffusiontermthatisofconstantelasticityinthespotrate.TheirestimateforUSdataofabout1.5fortheelasticityinthediffusiontermprovokedmuchdiscussionasitwasmuchhigherthanthevalueof0.5forthepopularCox,IngersollandRoss(1985)model.SubsequentcontributionseitherextendedthebasicCKLSformulationand/orconsid-eredalternativeestimationprocedures.LongstaffandSchwartz(1992),Brenneretal.(1996),AndersenandLund(1997)andKoedijketal.(1997)invariouswaysaddvolatilitydynamicstothemodelforinterestratedynamics.Asfarasestimationmethodologyisconcerned,GMMhasremainedthework-horseformostoftheempiricalstudiescited.HoweverNowman(1997,1998)appliedtheGaus-sianestimationtechniquesdevelopedbyBergstrom(1990)forcontinuoustimestochasticdifferentialequations.IncontrasttoGMMtheGaussianestimationmethodologyhastheadvantageofproducinganexactmaximumlikelihoodestimator.Episcopos(2000)subse-quentlyappliedthismethodologytoestimatetheparametersoftheCKLSspecificationfortheshort-terminterestrateforanumberofcountries.Interestinglyheobtainedesti-matesfortheelasticityinthediffusiontermthataremuchlowerthanthoseobtainedbyCKLS.Thefactthattwowellestablishedestimationmethodologiescanyieldwidelydif-feringparameterestimatessuggestsaneedtolookattheestimationissuefromadifferentperspectiveandthatisoneofthemotivationsforthecurrentpaper.Irrespectiveoftheestimationmethodologyoneemploys,anothersignificantissuere-latestowhatdataisusedtoproxytheinstantaneousspotrateofinterestthatitselfisnotanobservedquantity.ProxyvariablesthathavebeenusedincludeUSone-monthTreasurybillrates(CKLS)andone-monthinterbankrates(Episcopos).Itseemsstrange2thattheliteraturehasnotdevelopedinthedirectionofderivingandincludingintheestimationprocedure,thestochasticdifferentialequationsthattheassumeddynamicsoftheinstantaneousspotrateimplyforthesemarketobservedrates.CertainlyChapmanetal.(1999)haveprovidedsomeevidencethatuseoftheindicatedproxyvariablesmaynotinduceagreatdealoferror.Howevergiventhatthechoiceofappropriateestimationproceduresisnotyetanentirelysettledissue,itwouldseemusefultoestablishaframe-workthatremovesentirelyanypotentialforerrorsorbiasesduetochoiceoftheproxyvariable.Thecurrentpaperprovidessuchaframework.InthispaperweassumefortheinstantaneousspotrateofinterestthestochasticdifferentialequationspecificationusedbyCKLS.WethenapplyIto’sLemmatode-rivethestochasticdifferentialequationthatthisspecificationimpliesforthediscretelycompoundedratesobservedinfinancialmarkets.Theresultisasetoftwostochasticdifferentialequations,onefortheunobservedinstantaneousspotratetheotherfortheobserveddiscretelycompoundedrate.SincewearethendealingwithapartiallyobservedsystemthepaperproposesafilteringmethodologybasedonadynamicBayesianalgo-rithmasatooltoestimatetheshortrateaswellastheparametersinthemodelfromobservationsofdiscretelycompoundedrates.Theplanofthepaperisasfollows:insection2wederivethestochasticdynamicsystemfollowedbytheinstantaneousspotrateanddiscretelycompoundedrates.Wealsoshowhowonecansetupthemodelinawaythattheshortratetakesvaluesthatarepositiveandbelongtoacompactinterval.Sincethedataareobservedindiscretetime,insection3weoutlinethewayinwhichthecontinuoustimestochasticdifferentialequationsystemisdiscretisedwithparticularattentionbeinggiventoensuringthatthediscretisedsystemgeneratesonlypositiveinterestrateswiththesamesupportasthecontinuoustimecounterpart.InthissectionwealsooutlinethedynamicBayesianalgorithmforthecalculationofthedistributionofthespotr