FinanceandEconomicsDiscussionSeriesFederalReserveBoard,Washington,TheRelationshipbetweenMarketDepthandLiquidityFragilityintheTreasuryMarketAndrewMeldrum,OlegSokolinskiy2025-014Meldrum,Andrew,andOlegSokolinskiy(2025).“TheRelationshipbetweenMarketDepthandLiquidityFragilityintheTreasuryMarket,”FinanceandEconomicsDiscus-/10.17016/FEDS.2025.014.NOTE:StafworkingpapersintheFinanceandEconomicsDiscussionSeries(FEDS)arepreliminarymaterialscirculatedtostimulatediscussionandcriticalcomment.TheanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyothermembersoftheresearchstafortheBoardofGovernors.ReferencesinpublicationstotheFinanceandEconomicsDiscussionSeries(otherthanacknowledgement)shouldbeclearedwiththeauthor(s)toprotectthetentativecharacterofthesepapers.1*AndrewMeldrumtOlegSokolinskiy?AbstractAnalysisofmarketliquidityoftenfocusesonmeasuresofthecurrentcostoftrading.However,investorsandpolicy-makersalsocareaboutwhatwouldhappentoliquidityintheeventofanadverseshock.Ifliquidityweretodeteri-oraterapidlyattimeswheninvestorswereseekingtorebalanceportfolios,thiscouldamplifytheefectsofshockstothe?nancialsystemevenifliquidityishighmostofthetime.WeexaminethepotentialforsuchfragilityofliquidityintheTreasurymarket.Weshowthatareductionintheavailabilityofrest-ingorderstotrade(“marketdepth”)increasesliquidityfragility,likelybecauselowerdepthincreasesthedependenceoflowtraishmentofrestingorders.OurresultsapplytoallmajorbenchmarkTreasurysecuritiesindividually,whichenablesustoestablishanalogousconcmarket-wideliquidityfragility.Keywords:liquidity,fragility,Treasurymarket,priceimpact,volatility,mar-ketdepth,hiddenMarkovmodel21IntroductionThispaperisconcernedwithfragilityofTreasurymarketliquidity—thatis,thepossibilitythatwhileliquidityisgenerallygooditcoulddeteriorateinthefaceofshocks.UnderstandingthedriversofTreasurymarketliquidityisimportanttomarketparticipantsandpolicymakersalike.Innormalcircumstances,themarketishighlyliquid,meaningthatinvestorscantradeeasilyandcheaplyinlargevolumes.However,investorsarenotonlyconcernedwiththecurrentliquidityofthemarket;whatalsomattersishowliquidthemarketwouldbeattimeswhentheyneedtoad-justpositions.Forexample,ifliquiditywoulddeteriorateatpreciselythetimewhenaninvestorwouldneedtosellasecurity,potentiallyamplifyingmovementsinitsprice,thatsecuritymaycarryaliquiditypremiumevenifthemarketisliquidmostofthetime.Investorsthereforeneedtounderstandthefragilityofliquidity.Po-tentiallyfragileliquidityinresponsetoshocksmayalsopresenta?nancialstabilityconcernifitcouldcausethe?nancialsystemtoamplifytheefectsofshocks.Inthispaper,weshowthatwhileareductioninthevolumeofquotespostedtoelectronictradingplatforms(commonlyknownas“marketdepth”)doesnotneces-sarilymeanthattradingcostsincreaseinnormalcircumstances,itdoesmeanthatthefragilityofliquidity—theprobabilityofasuddenincreaseintradingcosts—meanthatthemarketislessliquidinnormaltimes;infact,themarketcansupporthighervolumeoftradingatlowexecutioncostwithalowerlevelofdepththaninthepastasaconsequenceoftwothings.First,marketparticimoresophisticatedexecutionalgorithms,splittinglargeordersintomultiplesmallerordersovertimetoreducetheirpriceimpact.Second,high-speedliquidityproviderscanmorequicklyreplenishquotesontheorderbookinrespoInthisenvironment,thecontinuedabilityandwillingnessofmarketmakerstore-plenishordersatthebestprices,ratherthanthelevelofquotespostedtotheorderbookatanyonetime,hasbecomethekeytothecostoftradingremaininglow.ThemicrostructureoftheTreasurymarketiskeytounderstandingwhatdrives3itsliquiditybecausethemicrostrucallyviainter-dealerbrokers(IDB)—asectorcapturingalargeproportionoftradedandquotesfromtheIDBwiththelargestmarketshare,theBrokerTecAlternativeTradingSystem(ATS).BrokerTecmatchesordersfollowingacentrallimitorderbook(CLOB)protocol:Marketparticipantscanprovideliquiditybysubmittingbuyandselllimitorderswithasmediatelymatched(e.g.,aask)andarepostedtotheCLOB.Marketparticipantscanalsoconsumeliquiditybysubmitting“marketable”limitordersthatcanbeimmediatelymatched(e.g.,aCLOB.Thegreaterthemarketdepth,thelargerthetransactionsizethatcanbeMarketmakersprovideliquiditybylevelsoftheCLOB,seekingtopro?tfromabid-askspread.Toreducetheirexposuresubmittinglargeordersthatwouldfbestprices,withtheexpectationthatmarketmakerswillrespondbyreplenishingprices.4Akeychallengewhenmodelingliquidityisitslatentnature.WhileliquiditymayappearobservableinaCLOB-basedmarkWeusethenaturalframeworkofaHiddenMarkovmodel(HMM)tocapturlatentnatureofliquidity.InourHMM,wecapturethefragilityofliquidityastheendogenousprobabilityofatransititobeinverselyrelatedtointerestratevolatilivolatility,theriskstomarketmliquiditytheyprovide.Weextendthesetofvariablesdrivicompassuncertaintyaboutvolatility,sincethisaddstotailrisk,costofmarketmaking.Wealsoconsidervolatilitypersistenincreaseinventorymanagementcostsfortraditionaldealers.Fintransitionprobabilitiesbetweenliquiditystisfundamentalforexplainingliquidityfragility.Ourpaperextendstheexistingliteratureinanumberofdirections.First,weconsiderourcontributionstothe?nancialstabilityliterature.Whilethelevelofliquiditymetricsiscommonlyseenasa?nancialstabilGovernorsoftheFederalReserveSystem(2020)andO’Hara(2004)amliquiditydeteriorationasthemorerelevantindicator,sinc5toCespaandVives(2017),whostudyliquidityfragilitywithinatheoreticalframe-workthatallowsforendogenousFoucault(2014)andRamanetal.(2020)explorethefragilitythatarisesfromilliq-thatlowmarketdepthraisestheprobabilityofaliquiditydeteriorationhasanipricessubstantially.provision.First,ourapproachisrelatedtothatofFloodetal.(2016),wholinkourframeworkintegratesexplanatoryvariablesdirectlyintotandvolatilitythroughdiferentconditionalpriceimpacasrequiringstatestocorresponthestatetransitionspeci?cation,whichprovesparticularlybene?cialforstudyingliquidityfragility.Third,ourstudyisrelatedtoHautschandHuang(2012a),whodependsonhowdeepinthecentrallimitorderbookandvolatilityathigh-frequenci6signi?cantlyafectintradayvolatility.Weshowthatatatrading-daytimescalemeasurescandirectlyafectpriceimpact,buttheydonotconsidermarketdepth.oureconometricframeworkwithexploratorittothe10-yearbenchmarkTreasurysecurity.InSection4,weresultsforotherbenchmarkTreasurysecuritiesthroughHMlatentliquiditystates.Then,weextendourresultstomarket-wideliquidityintheframeworkofanHMMwithlatentliquiditysTreasurysecurities.InSection5,weofersomeconcludingremarks.2MotivatinganHMMInthissection,weextendtheliteraturerelatingliquiditytovolatilityandestablishtherationaleformodelingthereSection2.1,weshowthattheefectofagivenchangeinvolatilityonpriceimincreaseswiththelevelofvolatility.This?ndingjusti?esanonlinearrelationshipbetweenpriceimpactandvolatility,whichcanbliquiditystates.Inasanindicatorofthefragilityofliquidity,whichmotivatestheinclusionofmarketdepthasavariablethatcanafecttheprobabilityoftransitioningbetweendiferentliquiditystatesintheHMM.72.1PriceImpactandVolatilityLiquidityisgenerallyworseattimesofhighvolatility,asnotedbyChordiaetal.(2005),amongothers.Intuitively,marketintermediariesarelesswillingtoprovideliquiditywhentheriskoflargepricemovolatilityafectsliquiditymorewhenvolatilityishigh.Speci?cally,weconsiderat=β1Vt+(β2-β1)G(Vt,1)Vt+(β3-β2)G(Vt,2)Vt+∈i,t,(1)Here,θtispriceimpactondayt.Ontheright-measureoftheswaption-impliedvolatility;Gisvolatility.Atwo-thresholdmodelisstatisticallypreferredtoalinearspeci?cationvolatilitytomotivatetheintroductionofanHMMinSection3.TreasuryNote.Speci?cally,itistheincrementalpricemoveassociaperiods.Thisestimateoestimatepriceimpact.TreasuryNote,weusetheannualizedbasispoint8obtainedfromTPICAP.ThesampleperiodrunsfromApril1,2014toDecemberTable1reportsβi(i=1,2,3)estimatesfortheoptimalvolatilitythresholds.Thepositiveandstatisticallysigni?cant,andarwhichthesensitivityisgreaterasvolatilityincreases.Table1:ThresholdregressionofpriceimpactonvolatilityModel1β10.097***(0.001)β20.137***(0.001)β30.196***(0.004)93.882***(0.744)146.806***(0.749)0.897Sources:RepoInterDealerBrokerCommunity;TPICAP,SwaptionsandInt2.2PriceImpactandMarketDepthtwopanelsbothshowpriceimprisingpriceimpactarebothassociatedwithadeteriorationinliquidityconditions.9oflocaltroughsindepthidenti?edbyAronovichetapricesmovedsharplybeforereversingquickly;theU.K.“Brexit”referendumonJune24,2016;thesharpspikeintheVIXindexofequitymarketvolatiInaddition,wepickouttheMondayfollowingtheclosureofSiliconValleyBank,impacthasrisennotabimpactanddepthappeartohavesubstantiallydiferenttime-seriesdynamics.Forexample,thesampleautocorrelastantiallymorepersistentthanpriceimpact.Wealsoobservesubstantialdifer-BrexitreferendumonJune24,2016,andtheVIXspikeonFigure1:PriceImpactandMarketDepthCOVID-19pandemicinMarch2020tooksubstantiallylongerstill.Priceimpactbackrelativelyrapidlycomparedwiththeslowerrecoveryofdepth;thisobservationisconsistentwithAronovichetal.(2021),whoshowthatbid-askspreadstendtorecoversubstantiallyfasterthanmarketdepthfollowingtheseepisodes.Aplausibleexplanationforthisisthatmarketparticipantsrapidlyadjustedtolowerdepthbytradinginsmallersizestoallowquotesontheorderboothercases—mostnotablyMarch2020andMardidnotcomeuntilseveraldaysafterthetroughindepth,suggestingthatthelowlevelofdepthmayincreasetheriskofaspikeinpriceimpact.Figure2:AutocorrelationFunctionsofPriceImpactandMarketDepthrelated,andwhetheritmattersthatdepthremainslowafterastresseventeventhoughpriceimpacthasreturnedtomorenormallevels.Givenmarketparticipants’unwillingnesstotradethroughmultiplelevelsofthebook,asnotedinHautschandprescribesplittingtheparentorderintosmallerchildorders.Hypothetically,marketFigure3:PriceImpactandMarketDepthfollowingEpisodesofKnownLiquidityStrainsDobrevandMeldrum(2020)andAronovichetal.(2021).Theimmediatehigh-speedliquidityprovision,butthisliquidityprovisionreturnsfbutpriceimpactfallsback.participants’toprovideliquidityremainshidden.However,aconjectureblinewithRamanetal.(2014)andBoardofGovernorsoftheFederalReservetrading?rmsmaybeinclinedtoscalebactroughindepth.OurHMMisnaturalframeworkforstudyingmoresystematically3HiddenMarkovModelforLatentLiquidityStates(HMM)framework,weapplyittomodelingtheliquidityofthe10-yearTreasuryHMMframework.InSection3.2,weconsideranHMMwithconstantprobabili-tiesoftransitioningbetweenthelatentliquidityshowthesensitivityofpriceiexplanatoryvariables—variesoverdiferentliquiditystates.Ourresultsshowthatthesensitivitiesofpriceimpacttovolatility,thesurprisecomponentofaboutvolatility,aswellasthesurprisecomponentofvolatilitypersistence,allvaryovertimewiththeliquiditystate.InSection,3.3,weextendtheHMMtoallowthetransitionprobabilitiestovaryovertimeasafunctionofmarketdepth.Ourresultsprovideclearevidencethatthefragilityofliquiditydecreaseswithmarketliquiditystates.3.1HiddenMarkovModel:ObservationEquationOurHMMexplainspriatesXt:t=βiXt+∈i,t,(2)i,t~N(0,σ)forthreelatentliquiditystatesi=1,2,3.3WeestimatevariousversionsoftheHMMwithdiferentvectorsofcovXt.However,wealwaysincludeameasureofinterestratevolatility.ThischoiceismotivatedbytheanalysisinSection2.1,whichdemonstratesthatpriceimpactissensitivetovolatilityasvolatilityincreases.Welabelthestatesscoe伍cientswithrespecttointerestratevolatilitythemeasurementequation,Equation(2);thus,state1isa“low-liquidity”stateinwhichpriceimpactismostsensitivetovolatility,state2isa“medium-liquidity”state,andstate3isa“high-liquidity”state.Thecharacterizationofhiddenliquiditystatesbasedonfactorsensitivitiescor-respondswelltointerpretinglatentliquidityintermsofthewillingnesvolatilityexactlywunitincreaseofvolatility.The3.2HiddenMarkovModel:ConstantTransitionProbabilitiesMatrixWestartourinvestigationwithanHMMcharacterizedbyconstantprobabilitiesofpi,j,t=pi,j,(3)wherepi,j,tdenotesthetradingday-tprobabilitythatst+1=jconditionalonst=i,thispaper)bymaximumlikelihood,usingthemethodofVisserandSpeek(2010).Table3reportstheresultityofratesasanexplestratevolatility.Theslopecoesigni?cant,meaningthathighervolatilityisassociatedwiarealsolargeandsigni?cant,inlinesensitivetovolatilitywhenliquidityislow.Table2:HMMwithFixedTransitionProbabilitiesMatrix:ParameterEstimatesThetablereportsparameterestimatesfortheHMMsdescribedbyEquation(2)and?xedtransitionprobabilitymatrixthatexplainsthe10-yearTreasuryNoteliquidity.ThesampleperiodisfromApril1,2014toDecember31,2023.βVcoe伍cientscorrespondstotheefectofswaptionimpliedvolatility,βVV–totheefectofthesurprisecomponentofthevolatilityofswaptionimpliedvolatility,andβCS–totheefectofsurprisecomponentofvolatilitypersistence.Sub-indicescorrespondtostates,i=1,2,3,rankedfromlowesttohighestliquidity.PanelBcontainsestimatesofthetransitionequationparameters:pi,jistheprobabilityoftransitioningfromlatentliquiditystateitostatej.NumbersinbracketsindicatestandarderrorsbasedontheHessian.A.,*,or**indicatestwo-sidedp-valuesoflessthan0.1,0.05,and0.01,respectively.ModelIModelIIModelIIIModelIVPanelA.Observationequationβ0.191**0.163**0.185**0.163**(0.007)(0.003)(0.005)(0.003)ββ1.92**1.829**(0.123)-0.715**(0.092)(0.131)-0.096(0.069)σ19.317**5.575**7.804**5.607**(0.547)(0.259)(0.453)(0.264)β0.12**0.115**0.12**0.117**(0.001)(0.001)(0.001)(0.001)ββ0.633**0.697**(0.067)0.011(0.017)(0.057)0.057**(0.018)σ21.886**1.631**1.883**1.639**(0.061)(0.05)(0.059)(0.054)β0.084**0.081**0.084**0.083**(0.001)(0.001)(0.001)(0.001)ββ0.194**0.285**(0.048)0.102**(0.01)(0.044)0.106**(0.01)σ31.029**0.918**1.003**0.924**(0.027)(0.029)(0.027)(0.034)p1,10.841**0.926**0.844**0.923**(0.037)(0.02)(0.036)(0.021)p1,20.155**0.071**0.152**0.073**(0.036)(0.02)(0.035)(0.021)p1,30.0040.0040.0040.004(0.007)(0.004)(0.007)(0.004)p2,10.026**0.018**0.028**0.018**(0.006)(0.005)(0.007)(0.006)p2,20.935**0.936**0.933**0.93**(0.01)(0.01)(0.011)(0.011)p2,30.039**0.045**0.039**0.052**(0.008)(0.008)(0.008)(0.009)p3,10.0010.0030.0010.004(0.002)(0.002)(0.001)(0.002)p3,20.029**0.043**0.028**0.039**(0.006)(0.008)(0.006)(0.008)p3,30.97**0.955**0.971**0.957**(0.006)(0.008)(0.006)(0.008)Sources:RepoInterDealerBrokerCommunity;TPICAP,SwaptionsandInterestRateCapsandFloorsData;authors’calculations.Lookingbeyondvolatilityasapotentialexplanatoryvariableforpriceimpact,wehypothesizethatmarketmakers’riskconsiderationsbeyondvolatility–heavy-tailsoftheratedistribution–afectsupplyofliquidity.Volatilityofvolatilityinducesheavy-tailsoftheratedistribution.Sincevolatilitytendstobevolatilewhenitslevelishigh,weisolatethesurprisecomponentfromthisrelationshiptomeasuretheincrementalimpactofheavytails.Speci?cally,weestimatethesurprisecomponentofthevolatilityofvolatilityintwosteps.First,weobtainthe?ltereddynamicconditionalvolatilitiesfromaAR(1)-GARCH(1,1)modelwithStudent-tinnovationsappliedtoswaptionvolatilities.Sonswaptionvolatilitiestoobtaintheresiduals;theseresidualsareanestimateofthesurprisecomponentofthevolatilityofvolatility.InModelII,weextendthesetofexplanatoryvariablestoincludethesurprisecomponentofthevolatilityofswaption-impliedvolatility.Theefectsofthesurprisecomponentofthevolatilityofvolatility,βiVV,arehighlysigni?cantandpositiveinallliquiditystates.Thesurprisecomponentofvolatilityofvolatilityhaslargerefectsinlowerliquiditystates:βiVV>βfori<j.Since,heavytailsandvolatilityarebothriskstomarketmakers,wecanextendthede?nitionoflowliquiditystatestostateswhereliquidityissensitivetoraterisks,ingeneral.Shockstovolatilitypersistenceafectliquiditysupplythroughthebalancesheetchannel–theriskofmaintaininganinventoryofsecuritiesovermultipletrad-ingdays.Thischannelisrelevantfortraditionaldealers,butnotprincipaltrading?rms(PTFs)–traditionaldealersoftencarryconsiderablepositionsovernight,whilePTFstendtoavoidsuchexposures.ModelIIIincludesvolatilityandthesurprisecomponentofvolatilitypersistenceasexplanatoryvariables.Again,therationaleforretainingonlythesurprisecomponentistoallowforthetypicalrelationshipbe-tweenvolatilityanditspersistence.Weestimatethesurprisecomponentofvolatilitypersistenceintwosteps.First,wecalculatethediferencebetweenthe3-monthand1-monthswaptionvolatilities–thevolatilitycalendarsprresidualsfromaregressionofthevolatilitionvolatility–theseresidualsareoursurprisecomponentofvolatilitypersistencetolastduetothecriticalroleoftheTreInsummary,thecoe伍cientestimatesβCSinModelIIIvalidateourintuition:βCSispositiveinthehighandmoderateliquiditystates,i=2,3,andnegativeinthelowestliquiditystate,i=1.Inmoderate-to-goodliquidworstliquiditystatecoincideswithsomeperiodsofpotentialmFinally,ModelIVcontainsthefullsetofconsideredcovariates:volatility,thevolatilitypersistence.Allobservationequationparameterestimatesaresigni?canttransitionprobabilitymatrix.Theobservationsarerobustacrossspeci?cations.Irrespectiveofthecurrentstate,theprobabilityofremaininginthepersistenceofliquiditystates.Theprobabilityofremaininginthesamestateinthenextperiodisgreaterforthehigherliquiditystates.Overall,transition5Aliquiditystate.Transitioningbetweenadjacentliquiditystatesismoreprobablethanmigratingdirectlybetweenworstandbestliquiditystates.3.3HiddenMarkovModelwithEndogenousTransitionProbabilitiesMatrixWeconjecturethatlowdepthpresentsavulnerabilityofmarketliquiditytoafutureshocktothewillingnessofmarketparticipantstoreplenishtheorderbook.Extend-ingthemodeltoallowforendogenoustransitionprobabilitiespermitsustoexploretheroleofmarketdepthintransitioningbetweenlatentliquiditystates.Inthiscase,wemodelthetransitionprobabilitiesusingthemultinomiallogisticspeci?cationofVisserandSpeekenbrink(2letsddenotetheliquiditystateondayd.Then,thetime-varyingprobabilityoftransitioningbetweenstatesiswherepi,j,t,withj1,denotesthetradingday-tprobabilitythatst+1=jcondi-tionalonst=i,andztisaveragemarketdepthondayt.Toensurethatprobabilitiesaddup(overj)toone,theprobabilitiesoftransitionsto“lowest-liquidity”state1Table3reportsparameterestimatesfortheHMMwithanendogenoustransi-tionprobabilitiesmatrix.Considering?rstthemeasurementequationofthemod-elsreportedinPanelA,themainresultsforthemeasurementequationsofthemodel,whichwediscussedinSection3.2,arerobusttoallowingfotime-variationinthetransitionprobabilities.Wethereforefocusonthetransitionequationsofthemodels,reportedinPanelB.Theestimatesof吖parameters,capturingtheefectofmarketdepth,arestatis-ticallysigni?cantatthe1percentlevel,withtheexceptionof吖1,3inModelsIandIIliquiditystates,吖i,j(i=1,2,j=2,3),arepositive:greatermarketdepthincreasestheprobabilityofatransitiontoabetterliquiditystateandreducestheprobabilityofatransitiontothelow-liquiditystate.吖1,3areestimatedwithgreateruncertaintybecausetherearefewtransitionsfromthelowtothehigh-liquiditystate,bypassingthemoderateliquiditystate.Inthehigh-liquiditystate3,吖3,3arepositiveforallspeci?cationsindicatingthathighermarketdepthincreasestheprobabilityofre-maininginthehigh-liquiditystate.InModelsIIandIV,吖3,2arenegativesuggestingthathighermarketdepthreducestheprobabilityofatransitiontoamoderateliq-uiditystate.InModelsIandIII,吖3,2arepositive,albeitinsigni?cant.However,theprobabilityoftransitioningtothelowliquiditystate.ComparingModelsIandIIIagainstModelsIIandIV,weseethataccountingforthesurprisecomponentofthevolatilityofswaption-impliedvolatilityintheobservationequationmoderatesthemagnitudestheefectsofdepthontransitionprobabilities.Comparisonoftheefectsofdepthonthetransitionprobabilitiessuggeststheirgreaterstrengthwhenliquidityislower,since吖i,j>吖i+1,jfori=1,2,3andj=2,3.Tofurtherillustratetheefectofdepthonthetransitionprobabilities,Figure4showsthetransitionprobabilitiesconditionalonvariouslevelsofdepth,asimpliedbyModelIV.Asdepthdecreases,theprobabilityoftransitioningtoaworseliq-uiditystateincreases.Forexample,whendepthis$10million,theprobabilityoftransitioningfrommoderateliquidity,state2,tolowliquidity,state1,isaboutnineinten;and,incontrast,whendepthis$120million,thisprobabilitydecreasestoFigure5showstheestimatedsmoothedprobabilitiesofbeinginagivenlatentliquiditystateateachpointintime,asimpliedbyModelIV,andcomputedusinganalgorithmfromVisserandSpeekenbrink(2010).ThelowerpanelplotspriceTable3:HMMwithEndogenousTransitionProbabilitiesMatrix:Pa-rameterEstimatesModelIModelIIModelIIIModelIVPanelA.Observationequationβ0.190**0.165**0.185**0.167**(0.007)(0.004)(0.006)(0.004)ββ1.675**1.512**(0.108)-0.675**(0.092)(0.132)-0.162*(0.073)σ18.949**5.712**7.825**5.776**(0.525)(0.27)(0.468)(0.299)β0.119**0.115**0.12**0.116**(0.001)(0.001)(0.001)(0.001)ββ0.427**0.463**(0.056)0.026(0.017)(0.065)0.044*(0.018)σ21.729**1.509**1.762**1.555**(0.061)(0.053)(0.064)(0.063)β0.083**0.081**0.084**0.083**(0.001)(0.001)(0.001)(0.001)ββ0.157**0.236**(0.044)0.095**(0.01)(0.045)0.096**(0.01)σ30.974**0.917**0.958**0.910**(0.033)(0.028)(0.029)(0.028)δ1,2-9.893**-8.286**-10.000**-8.988**(2.501)(1.955)(2.656)(2.701)-200.496-22.938**-284.616**(144.634)(132.53)(7.075)(91.647)0.200**0.300**0.230**(0.079)(0.056)(0.088)(0.082)3.3380.528**4.696**(2.321)(2.142)(0.148)(1.482)δ2,2-3.999**-3.023**-4.277**-3.117**(1.137)(0.935)(1.276)(1.002)δ2,3-8.317**-7.562**-8.889**-7.467**(1.379)(1.52)(1.515)(1.474)0.100**0.144**0.104**(0.027)(0.018)(0.031)(0.02)0.122**0.174**0.126**(0.029)(0.023)(0.033)(0.025)3.586*-5.4173.366.(3.357)(1.588)(3.474)(1.968)δ3,3-5.428.-0.042-7.287*0.294(2.957)(0.429)(3.128)(1.123)-0.0320.094.-0.029(0.046)(0.02)(0.05)(0.024)0.039**0.144**0.036**(0.041)(0.005)(0.046)(0.012)Sources:RepoInterDealerBrokerCommunity;TPICAP,SwaptionsandIntFigure4:HMMTransitionMatrixThe?gureshowsthetransitionprobabilitymatricesfromtheHMMconditionalonvariouslevelsofmarketdepth,asimpliedbyModelIV.Thelevelsofmarketdepthare$10million,$30million,$60million,and$120million.Ineachcase,thematrixshows