Thisiswhattheleadingindicatorslead¤

M aximoCamachoU niversitatA utonomadeBarcelona

e-mail: mcamacho@ idea.uab.es

G abrielPerez-Q uirosEuropeanCentralBank

e-mail: gabriel.perezquiros@ ecb.int

A bstract

W eproposeanoptimal…ltertotransform theConferenceBoardCompositeL eadingIndex(CL I) intorecessionprobabilities. W ealsoanalyzetheCL I’s accuracyatanticipatingoutputgrowth. W ecom-parethepredictiveperformanceoflinear, V A R extensionsofsmoothtransition regression and switching regimes, probit, nonparametricmodelsandconcludethatacombinationoftheswitchingregimesandnonparametricforecasts is thebeststrategyatpredictingboth theN BER business cyclescheduleand G D P growth. T his con…rms theusefulnessofCL I, eveninareal-timeanalysis.

¤Comments from PedroA lvarez, A rielle Beyaert, M ichaelCreel, and seminarpartic-ipants attheU niverity A utonomadeBarcelona, U niversityofM urcia, FederalR eserveBankofN ewYorkandSeventhA nnualSymposiumoftheSocietyforN on-L inearD ynam-icsandEconometricatN ewYorkU niversity, aregratefullyacknowledged. W eareindebtedtoM ichaelBoldinforgraciouslysharingthedata. Errorsareofourownresponsibility.

1

1 IntroductionConsumption, savings andproductiondecisionsmadeby individualagentsandmonetaryand…scalpolicymadebypolicymakersarebasedonforecastsaboutthefuturedevelopmentsofmacroeconomicvariables. T hestateofthebusinesscycleisoneofthekeyelementsfortheevolutionofsuchvariables.H ence, forecastingturningpointsiscrucialfortheoptimalityoftheeconomicagents’decisions. .

A nextensiveliteratureexistswhichattemptsto…ndthebestforecastingtoolforthebusinesscycleturningpoints, from theearlyheuristicattemptsbyM itchellandBurns(19 38)tothemoresophisticatedofStockandW atson(19 8 9 ). W hateverapproachweconsider, theforecastingproblem istwofold.First, weneedtoidentifythegroupofvariablesthatmoveinandoutreces-sionsbeforetherestoftheeconomy. Second, wehaveto…ndtheappropriate…ltertoextractthesignaloutoftheseseries.

W efocusonthesecondaspectoftheforecastingproblembyattemptingto…ndanoptimalsignalextractionmethodtoanalyzethepredictivepoweroftheComposite L eadingIndex(CL I). T his series, combinationofseveralpromisingleadingvariables, is releasedbytheConferenceBoardsinceO c-tober19 9 6andbytheBureauofEconomicA nalysispriortothatdate. W eusetheCL I because, eventhoughithassu¤eredanumberofimportantre-visions, ithasbeenpublishedwithoutinterruptionsince19 68, allowingtheresearchertoanalyzethepredictivepoweroftheleadingindexwithinforma-tionavailableineachtimeperiod.1

StudiesthatanalyzetheaccuracyoftheCL I forpredictingturningpoints…nd contradictory results. D iebold and R udebusch (19 9 1) donotdetectpredictivepowerinalinearcontext. T heprobitmodelusedbyEstrellaandM ishkin(19 9 8)outlinethepoorperformanceofCL I, speciallyintheout-of-sampleanalysis. H amiltonandPerez-Q uiros(19 9 6)andKim (19 9 4)presentevidenceinfavoroftheusefulnessofCL I.Filardo(19 9 4) concludesthatlagvalues oftheCL I explains changes intheprobabilityofswitchingfrom anexpansiontoarecession. Finally, G ranger, Terä svirtaandA nderson(19 9 3)…ndthattheCL I is thedrivingfactorinaSmooth Transition R egression(ST R )model.

T herearetwomainpurposesforthis paper. First, wewanttoformallycomparetheseprevious analysis andtoproposeasetofalternative…lters.

1O nly theexperimentalleadingindex (X L I) proposed by Stockand W atson (19 8 9 )wouldallowthesamekindofanalysis. H owever, areal-timeevaluationofX L I is compli-catedsincethenumberofobservations is toosmallandthis indexonlyfacesarecessionintheearly19 9 0s. FurtherresearchshouldgointhedirectionofcomparingthepredictivepowerofX L I versusCL I.

2

O nlyFilardo(19 9 9 ), in independentwork, has triedthesamekindofap-proach. H owever, heusesadi¤erentsetofmodels, conductsonlyadescrip-tiveanalysisanddoesnotformallytestthepredictivepowerofeachmodel.Second, amoreambitiousgoalofthis paper, is tocombineallthedi¤erentapproaches toproposea…lterthattransforms thedatafortheCL I intoaprobabilityofrecessiononequarterhence.

Toourknowledge, thisisthe…rstattempttoaddressbothissues. Infact,thispaperisthe…rstformalcomparisonofhowthemostpopulartimeseries…ltersanalyzethepredictivepoweroftheCL I forforecastingturningpoints.2In particular, wecomparetheaccuracyofthepreviouslyproposed linear,M arkovswitching, andprobitmodels, avectorautoregressiveextensionofST R speci…cations, andanewnonparametric…lter. T hen, wecombinetheinformationcontainedinallofthemodels inordertoobtaina”consensus”…lterfortransformingtheCL I dataintoarecessionprobability.

Inaddition, weacknowledgethatpredictingturningpointsmaynotbetheonlygoaloftheCL I. T herefore, werepeattheapproachtoanalyzethepredictivepoweroftheCL I onG D P movements.

W econcludethatacombinationofdi¤erentmodels performs betterinandoutofsamplethaneachofthesinglemodelproposed. T hus, theCL I isusefulinanticipatingbothturningpointsandoutputgrowth, eveninreal-timeanalysis. M oreover, incontrasttoH essandIwata(19 9 7 ), we…ndthatnonlinearspeci…cationsarebetterthansimplerlinearmodelsatreproducingthebusinesscyclesfeaturesofrealG D P.

T hepaperisorganizedasfollows. Section2 describes thedata, section3 outlines thedi¤erentmodels, section 4 presents theempiricalevidence,section5 analyzesthecombinationofforecasts, andsection6concludes.

2 PreliminaryanalysisofdataForthein-samplestudy, weusehistoricallyrevisedCL I series issuedinJan-uary19 9 8. FortheG D P, weusechained-weighteddata. T hedatarunsfromthesecondquarterof19 60 tothefourthquarterof19 9 7 . W etransform themonthlyCL I series intoquarterlybychoosingthelastobservationofeachquarter. A sapreliminaryanalysis, wetestthestationaritypropertiesofourseries. T heaugmentedD ickey-Fullertestcannotrejectthenullhypothesisofaunitrootfortheloglevelsoftheseries, butitis consistentwithasta-

2StockandW atson(19 9 8) analyzeabatteryofmodelsthatincludeslinear, nonlinear,parametricnonparametricandacombination ofthem. H owever, theydonotfocus onpredictingturningpointsandtheirstudyissoextensivethattheycannotapplyaformalcomparison.

3

tionaryspeci…cationforthedi¤erencesofthelogarithms. T hus, fornowon,ourseriesofinterestwillbethegrowthratesofG D P andCL I, denotedasyandx.

Inaddition, Johansenprocedurefailstodetectevidenceofcointegration.3.O therauthorsasH amiltonandPerez-Q uiros(19 9 6)andG ranger, Terä svirtaand A nderson(19 9 3) havefoundthatprevious series oftheCL I presentedcointegrationwith G D P. H owever, as pointedoutH arveyinacommentinG rangeretal. (19 9 3), therewasnotstrongeconomicreasonforG D P andCL I tobecointegrated. T hus, theabsenceofcointegrationisanimportantcharacteristicofthelastCL I revisions.

3 M odelsdescriptionInordertoquantifytheaccuracyoftheCL I topredictbothG D P movementsandperiodsofrecessionintheU S economy, weanalyzedi¤erentlinearandnonlinear, parametricandnonparametricmodels. T his section brie‡yde-scribesthesemodels.

3.1 U nivariateandbivariatelinearmodelsL inearmodels havebeenwidelydeveloped intheearlierforecastinglitera-ture. H owever, thesemodels havebeenappliedjusttogenerateaforecastoftheexplainedvariable, let’s say, rateofgrowthofG D P, rateofgrowthofindustrialproductionorsomecoincidentindicator. Itis notcommontousethemtoforecastanon-linearphenomenasuchasaturningpoint. Intheliterature, StockandW atson(19 9 3)proposea…ltertoextractturningpointsforecasts from alinearmodel. T his is usedbyH amiltonandPerez-Q uiros(19 9 6)tosuccessfullydescribethepredictivepoweroftheCL I overthebusi-nesscycles. T heirbasic…ndingsarethefollowing. L etthelinearprocessesfortheG D P beeitheranA R (p)

yt= ¹ + a(L)(yt¡1¡¹)+ et; (1)

oraVA R (p)

yt = ¹ + a(L)(yt¡1¡¹)+ b(L)(xt¡1¡´)+ etxt = ´ + c(L)(yt¡1¡¹)+ d (L)(xt¡1¡´)+ ut; (2)

3T helikelihoodratiotestofthenullhypothesis ofnocointegrationagainstthealter-nativeofonecointegratingrelationis 14.48

4

whereL isthelag-operator:4 Errorsin(1)arei.i.d. gaussianwithzeromeanandvariance¾ 11:Errorsin(2) followtheusualassumptions:

µetut

¶s i:i:d :N

·µ00

¶;µ

¾11 ¾12¾12 ¾ 2 2

¶¸:

U ndertheprevioushypotheses, oneandtwoquarteraheadoutputgrowthpredictionsformtherandomvector

µyt+ 1=yt;yt¡1;:::;:y1yt+ 2 =yt;yt¡1;:::;:y1

¶s i:i:d :N [gt; Q ]: (3)

FortheA R case

gt=µ¹ + a(L)(yt¡¹)¹ + a2 (L)(yt¡¹)

¶;

and

Q =·

¾11 a1¾ 11a1¾ 11 (1 + a¤(1))¾11

¸:

H owever, fortheVA R case

gt=µ

¹ + a(L)(yt¡¹)+ b(L)(xt¡´)¹ + (a2 (L)+ b(L)c(L))(yt¡¹)+ (a(L)b(L)+ b(L)d (L))(xt¡´)

¶;

and

Q =·

¾ 11 a1¾ 11 + b1¾12a1¾ 11 + b1¾12 (1 + a¤(1))¾ 11 + b¤(1)¾ 2 2 + a0b¾12

¸:

W eadoptO kun’s ruleofthumb thatarecessionoccurswhenevertherealG D P fallsforatleasttwoconsecutiveperiods. T herefore, theprobabilityofbeinginrecessionatt+ 1 dependsontheactualvalueofyt:Ifyt< 0 ; theforecastedprobabilityofrecessionistheprobabilitythatyt+ 1 waslessthanzero. O ntheotherhand, theprobabilitythatthedownturnstarts att+ 1whenyt> 0 ;coincideswiththeprobabilitythatbothyt+ 2 andyt+ 1 werelessthanzero. T hus, givenobservationsuntilt, theseprobabilitiescanbeeasilycalculatedfromprobabilitytablesorM onteCarlosimulationson(3).

4Fornowon, wede…neh=(h1 ;h2;:::;hp)0, h(L )=(h1 + h2L + :::+ hpL p¡1);h2(L )=Pih2i

¡L i¡1

¢2 + 2Pi< j

hihjL i¡1 L j¡1 , andh¤(1)=(h21 + h22 + :::+ h2p);withhbeinga;b;c

andd:

5

T heresults forthelinearA R and VA R speci…cations arepresented inthe…rstandsecondrowofTable1. T heoptimalnumberoflags, applyingSchwarzcriteriais 1 inbothcases. Inaddition, as in H amiltonandPerez-Q uiros, we …nd thatlagged growth ofG D P does nothelp in forecastingneithercurrentgrowthratesofG D P norcurrentgrowthratesofCL I.

3.2 VectorSmoothTransitionR egression(V ST R )W eextendtheST R modelsproposedbyG rangerandTerä svirta(19 9 3) toaVA R context. T heseweredevelopedtocapturethefactthatmayexisttwo(ormore) datageneratingprocesses thatchangewiththestateoftheeconomy. Theprobabilityofbeingineachstateisdeterminedbythetransi-tionfunction. Tostudyhowthesemodelswork, westartfromthefollowingV ST R model:

yt = ¹ + a(L)yt¡1 + b(L)xt¡1 +he¹ + ea(L)yt¡1 + eb(L)xt¡1

iFy+ et

xt = ´ + c(L)yt¡1 + d (L)xt¡1 +he+ ec(L)yt¡1 + ed (L)xt¡1

iFx + ut (4)

wherelag-operatorsanderrorsholdthesameassumptionsthanin(2). N otethatthereareaspossibleV ST R speci…cationsasdi¤erentexplanatoryvari-ables andfunctionalforms areconsidered inthetransitionfunction F. Inordertoselectamongthem, weuselinearityandmodelselectiontestsbasedonmaximum likelihoodprinciplesasfollows. W e…rstspecifyalinearVA Randchoosetheoptimallaglength p. Second, weapplylinearitytests foreachselectedcandidatetobeexplanatoryvariableinF. T hird, foreachofthemthatrejectslinearity, wecarryoutmodelselectionteststoobtainoneofthepossibleV ST R forms. Finally, weperform in-sampleandout-of-samplemodelevaluationtechniques toselectone…nalspeci…cationfrom thesetofpossibleV ST R models.5. Fromthisanalysis, we…ndthatthebestspeci…ca-tionforytandxtisaL ogistic-V ST R withthefollowingfunctionalformforFy and Fx:

Fi(yt¡2 ) =1

1 + e¡°i(yt¡2¡gi); (5)

wherei= y;x:Variables°iandgiarecalledsmootherparameterandthresh-old, respectively.

T hesemodelsimplicitlycontaininformationaboutrecessionprobabilitiesasfollows. Forsimplicityintheexpositionassumeboth, thatparameters in

5D etails oftheselectionprocess andresults fortheestimationofeachofthepossiblemodelsconsideredcanbefoundinCamacho(19 9 8).

6

ea(L) andeb(L) arezeroandthat°y ande¹ arepositive. Inextremecontrac-tions, yt¡2 takesamuchlowervaluethanthethreshold. H ence, thehigheristhesmootherparameter, theclosertozerois thetransitionfunctionvalue.L ikewise, greatexpansionscanbeassociatedwithtransitionfunctionvaluesneartoone. H ence, thetransitionfunctionlocatesthemodeleitherneartoorfarfromrecessionsdependingonthevaluesofyt¡2 relativetothethresh-old. T hus, once(4) is estimatedwith informationuntilt, F(yt¡1) canbeinterpretedasaonequarteraheadforecastedrecessionprobability.

T heresults forthis estimationarepresented in thethirdrowofTable1. A s in thelinearcase, wegetanoptimallaglength equalto1 andwe…ndthatlaggedgrowthofG D P doesnothelptoforecastneitherynorx. Inaddition, weacceptthenullthattheconstantistheonlychangingparameter.T herefore, ea,eb;ecand ed arestatisticallyinsigni…cantonthismodel.

3.3 SwitchingregimesmodelO urstatisticalde…nitionoftheswitchingregimemodelisdescribedindetailin H amiltonandPerez-Q uiros (19 9 6). A s inthepreviouscase, tworegimesareconsidered. L etstbeanunobservedlatentvariablewhichtakesavalueequals to1 whentheeconomyis inanexpansionand2 whentheeconomyis inacontraction. Intheformercase, G D P andCL I areexpectedtogrowby amounts ¹1 and ´1: H owever, in a contractions they growata lowerrates ¹ 2 and´2:Inswitchingregimesmodels, thechangesbetweenregimesdonotfollowalogisticfunction(whichdependsuponobservablevariables).T heirlawofmotionisgovernedbytheunobservablestatevariablest; thatevolvesaccordingtoahomogeneousM arkovchainthatisindependentofpastobservations onytandxt. T his implies thattheprobabilitythatstequalssomeparticularvaluejdepends onthepastonlythroughthemostrecentvaluest¡1:

p¡st= jjst¡1 = i;st¡2 = k;:::;Ât¡1

¢= p (st= jjst¡1 = i) = pij;

whereÂt= (yt;xt;yt¡1;xt¡1;:::):T hus, ourtime-seriesmodelis

yt = ¹st+ a(L)(yt¡¹st¡1)+ b(L)(xt¡´st+ r¡1

)+ etxt = ´

st+ r+ c(L)(yt¡¹

st¡1)+ d (L)(xt¡´

st+ r¡1)+ ut; (6)

withlag-operatorsanderrorsfollowingthesameassumptionsas in(2).A ftertesting,weimposetherestrictionthattheCL IandtheG D P ”share”

7

thestateofthebusinesscycle, as inH amiltonandPerez-Q uiros(19 9 6).6 Inparticular, astheysuggest, theCL I moves r periodsbeforeG D P. T hus, theconditionalexpectationofCL I dependsonst+ r:

W iththiskindofspeci…cation, recessionsarepredictedasfollows. First,wede…nes¤tasalatentvariablewhichsummarizesthevaluesofst¡p throughst+ r; and the transition probabilities matrixP¤:Second, weestimate themodelandcalculatethevectorb»t=t;whoseith elementgivestheprobabilitythatstateioccurs, giventheobservedvaluesofyandxuntilt. A forecastofwhethertheeconomywillbeinarecessiononequarterfromnowisobtainedbysummingthoseelementsofb»t+ 1=t= P¤b»t=tcorrespondingtost= 2 :

W e…ndthesamekindofresultsasinH amiltonandPerez-Q uiros(19 9 6).Schwarzcriterionhas selected p = 1 in (6). T hehighestvalueforthelike-lihood function is reached by r = 1:Furthermore, like in previous cases,thecoe¢cientsforthelaggedG D P growtharenotsigni…cantinanyoftheequations. T hefourthpanelofTable1 presentstheresultsforthismodel.

3.4 ProbitmodelT he…fthmodelthatweanalyzefollowsthelinesoftheprobitmodelproposedinEstrellaandM ishkin(19 9 8). T heseauthorsdevelopa…lterforquantifyingthepredictiverecessionspowerofthevariablescontainedinad -dimensionalvectorz t¡1.

L etz t¡1 belaggedCL I growthrates. L etrtbeanunobservablevariablethatdeterminestheoccurrenceofarecessionattimet. T hemodelisde…nedinreferencetothetheoreticalrelation

rt= ¯ 0z t¡1 + et; (7 )

whereetfollowsastandardnormaldistribution. Sincertisunobservable, theestimationisbasedinadichotomousrecessionindicatord tthatequalsoneiftheeconomyis inrecessioninquartert, andzerootherwise. Ifthemodeliscorrect, rt shouldbegreaterthanzerowheneverd twasequaltoone. T hisimpliesthat

P(d t= 1) = P(rt> 0 ) = F ( 0z t¡1); (8)

whereF is thecumulativenormaldistributionfunction. T heestimationoftheparameteruses standardmaximum likelihoodprocedures on theloga-rithmiclikelihoodfunctionofprobitmodels.7

6T hissomehowcoincideswiththeresultthatasimilartransitionfunctionlocatesbothG D P andCL I betweenregimes inV ST R models.

7 T hereaderwill…ndin(18) anexplicitexpresion.

8

Inpractice, werelyontheN BER recessionindicatorfordeterminingd t.ToexaminetheCL I’susefulnessatpredictingrecessions, wetrycurrentandlaggedvaluesofxtin(7 ). Foreach, wecalculatethepseudoR 2

PR 2 = 1¡µlogLu

logLc

¶¡(2 =T)logLc; (9 )

where Luand Lcaretheunconstrainedmaximum valueofthe likelihoodfunction, andsuchvalueundertheconstraintthatallcoe¢cients arezeroexceptfortheconstant. Intuitively, this corresponds tothecoe¢cientofdeterminationinlinearregression. T hemaximumvalueisachievedbyxt¡1:8T heresultsforthismodelareshowninthelastrowofTable1.

3.5 N onparametricgaussiankernelSmoothingmethodsprovideapowerfulmethodologyforgaininginsightsintothedatasincetheyavoidtheproblemofspecifyingaclosedformfortheden-sityfunction. H owever, asearchfortheoptimalnon-parametricspeci…cationusingallpossiblesetofexplanatoryvariablescouldbecostly. T herefore, weusesomeresults from thepreviousanalysis. Inparticular, wehavelearnedthatallparametricmodelsshowacommoncharacteristic: CL I is aturningpointpredictorintheshortrun. Speci…cally, we…ndarelationbetweencur-rentG D P growthandcurrentrecessionswithCL I growthduringtheprevi-ousquarter. T hen, wewillusext¡1 asexplanatoryvariableinnonparametricmodels.

Inforecastinggrowth, weapproximatetherelation

yt= m (xt¡1)+ et;

usingthestandardN adaraya-W atsonestimatorinlinewithH ä rdle(19 89 ).9W efocusonthequestionofhowrecessionscanbepredictednonparamet-

rically. KeepinginmindtheO kun’sruleofthumb, weproposethefollowingmethodologyforpredictingprobabilities ofrecession inreal-time. A ssumethatyt< 0:First, weconstructtheconditionaldensityfunction, depending

8 T hisvalueis 0.280 anddeclineswithinayear.9 Speci…cally, weestimate

byt=

24

TX

j=1

Kµxt¡1 ¡xj

h

¶yj

35 =

24

TX

j=1

Kµxt¡1 ¡xj

h

¶35 ;

whereK isthegaussiankernelandhis selectedbyleave-one-outcross-validation.

9

upontheunknownvalueyt+ 1, giventhegrowthoftheCL I att, thatis

f(yt+ 1=xt) =f(yt+ 1;xt)f(xt)

: (10)

Second, wecalculatetheexpectationthatittakesvalueslessthanzero:

pt(yt+ 1 < 0 =xt) =Z

yt+ 1< 0

f(yt+ 1=xt)d yt+ 1: (11)

O ntheotherhand, whenyt> 0 ; theprobabilityattforarecessionatt+ 1coincides withtheprobabilitythatboth yt+ 1 andyt+ 2 wouldbeless thanzero. Followingthesamemethodologywepropose

pt(yt+ 2 < 0 ;yt+ 1 < 0 =xt) =Z

yt+ 2 < 0

Z

yt+ 1< 0

f(yt+ 2 ;yt+ 1=xt)d yt+ 2 d yt+ 1: (12)

Standardsmoothertechniquessu¤erfromaslightdrawbackwhenappliedtomultidimensionaldatawithlong-taileddistribution. T his ispreciselythecaseofpredictingrecessions.10 Inordertoavoidthisproblem, wehaveusedadaptivekernelestimation, whichconsistsof…ndingkernelestimatorswithbandwidthvaryingfrom onepointtoanother. Inparticular, theestimatedjointdistributionofany d -dimensionalvariable z atanypointjisgivenby

bf(z j) =1T

TX

t= 1

dY

i= 1

1h d ¸ dt

k(z j;i¡z t;ih ¸t

); (13)

whereh isthebandwidth, and¸tisthelocalbandwidthfactorattimet.T heprocedurethatweusetogeth and¸ in(13) isthefollowing. First,

wede…nethelocalbandwidthfactoras

¸t=

Ãbf(z t)g

! ¡®;

where bf(z t) is a pilotestimation of(13), with ¸ r equals toone, and thebandwidth chosen byreferencetoastandarddistribution.11 Parametergis thegeometricmeanofbf(z t):W eset® equalto1=2 ; followingA bramson(19 82).

10G D P growthobservationsareusuallynon-negative, butestimatingtheirdensitytreat-ingthemasobservationson(¡1 ,1 ). T hisleadstonoisydensityestimationintheright-handtail.

11SeeSilverman19 86, page8 7 .

10

Second, we selectthe bandwidth thatmaximizes the likelihood cross-validationfunction

LC V (h) =1T

TX

j= 1

logbf¡j(z j);

with bf¡j(z j) de…nedas in(13), andwherethesum doesnotincludevaluesoftequaltoj:12

4 EmpiricalevidenceT heabilityofanyleadingindicatortoanticipateeventsdependsonusingtheappropriatetechniquetoextracttheinformationcontainedinthepredictor.W eapplytwodi¤erentstatistics tomeasuretheaccuracyofthedi¤erentspeci…cations atforecastinggrowth and recessions. First, toanalyze theaccuracyatforecastinggrowth, weusetheM eanSquareError:

M SE =1T

TX

t= 1

(yt¡byt)2 ; (14)

whereytandbytareactualandestimatedG D P 13.O ntheotherhand, tocomparethepowerofsuchmodelsatanticipating

turningpoints, weconstructthe TurningPoints Error, ameasure ofthesquareddeviationfromtheN BER schedule:

TPE =1T

TX

t= 1

(d t¡bd t)2 ; (15)

whered tisadichotomousvariablewhichequals1 if, accordingtotheN BER ,theeconomy is in recessionattimet; and 0 otherwise. Variable bd t is theforecastedprobabilityofbeinginrecessionattimet14:

12In fact, real-timepredictions in nonparametricmodels follows thesamestrategyasin parametricmodels. A tanyperiodt;parametershand ¸ areestimated in (11) fromtherelationbetweenytandxt¡1 ,whereastheyareestimatedin(12) fromtherelationshipamongyt, yt¡1 andxt¡2. O ncethesevaluesareapproximated, weusethemforanticipatingrecessionsfort+ 1 .

13Fortheoutofsampleexercise, wede…ne, M ean SquareForecastingError(M SFE)de…nedwiththesameformula, wherebyt istheestimatedvalueforytwithinformationuptoperiodt¡1 .

14Fortheoutofsampleexercise, wede…ne TurningPointForecastingError(T PFE)with the same formula, where bdt is the estimated value fordt with information up toperiodt¡1 .

11

Inordertotestifthedi¤erencesbetweeneachpairofmodelsaresignif-icant, weusethetestproposedbyD ieboldandM ariano(19 9 5), henceforthD M .. Speci…cally, considertwodi¤erentspeci…cations, modeliandmodelj. L etfbgitgTt= 1 andfbgitg

Tt= 1 betheirrespectiveforecasts.15 L etEtbeeither

(yt¡byit)2 ¡(yt¡byjt)2 atforecastinggrowth, or³d t¡bd it

2¡

³d t¡bd jt

2at

anticipatingrecessions. Finally, letE beequalto 1T

TPt= 1

Et. U nderthenull

hypothesisofnodi¤erenceintheaccuracyofthesetwocompetingforecasts,thelarge-samplestatistic

DM =Eq2 ¼ bfd (0 )

T

; (16)

where

2 ¼ bfd (0 ) =1T

T¡1X

r= ¡(T¡1)1(

rS(T)

)TX

t= jrj+ 1(Et¡E)(Et¡jrj¡E);

theindicatorfunction

1(r

S(T))

(1 for

¯¯ rS(T)

¯¯·1

0 otherwise

isthelagwindow, andS(T) isthetruncationlag, followsaN (0 ;1)randomvariable.

T he…rsttwocolumns ofTable2 displaythe in-sampleM SE andT PEforthewholesetofmodels.16 Foreachmodel, the…rstentryrefers totheentiresample. T hesecondandthirdentriesexclusivelyrefertorecessionaryandexpansionaryperiods, accordingtotheN BER schedule. T hein-sampleresultsshowthattheM arkovswitchingmodelperformsbetterthananyotherspeci…cationinboththeG D P andturningpointsforecasts. Inaddition, mostofthegainscomefromthereductioninthemeansquareerrorinrecessions(inthecaseoftheswitchingversustheA R model, 58% ofthereductioncomesfrom recessions whereas only 26% comes from expansions). T herefore, insamplewecanconcludethattheCL I haspredictivepoweroverthebusinesscycleand G D P movements. T hesearebettercapturedwith anon-linearM arkovswitchingspeci…cationthatallowstheforecastertotakeintoaccount

15N otethatg is eithery (in 14) ord(in 15). A lsothatforecasterrors maybenon-gaussians, nonzeromean, andseriallyandcontemporaneouslycorrelated.

16M odelsCO M B andRT CO M B willbetreatedindetailinthenextsection.

12

thechangesinthedatageneratingprocessofbothG D P andCL I duetothephenomenaofexpansionsandrecessions.

U singtheD M test, Table3presentsstatisticalevidenceofthesigni…ca-tivityofthesegains. Comparinglinearmodels, theinclusionoftheCL I intheG D P equationgivesanstatisticallysigni…cantimprovementintheM SE(D M testof2.47 ) butnotin the T PE (D M testof1.02). H owever, in anonlinearcontext, werejectthenullofnogainwithrespecttotheunivariatelinearmodelinboththeM SEandT PEandwerejectalsothenullofnogainwithrespecttothemultivariateVA R .17

N evertheless, suchpromisingresults in-sampledonotnecessarily implythattheCL I is usefulforreal-timepredictions. First, itiswell-knownthatvery‡exiblenonlinearmodelshaveapoorperformanceinout-of-sampleex-ercises. Second, theCL I series isrevisedveryfrequently, andtherefore, thein-sampleanalysis contains informationnotavailableforpredictionateachperiodoftime.

T heout-of-sampleanalysispredictsinreal-time104values. T he…rstdatapointforwhichpredictionsaremadeisthesecondquarterof19 7 2.18 Foreachperiodoftimet, weestimateeachmodelwithdatafrom thebeginningofthesampleup toperiodt, usingtherevisionoftheCL I available in thatperiodoftime. T hetransformationfrommonthlytoquarterlyobservationsis doneas in the in-sampleanalysis. T hen, with thecoe¢cientestimates,aoneperiodaheadforecastis computedthroughthe…rstquarterof19 9 8.T his proceduremimicswhatastatisticalmodelwouldhavepredictedwiththeinformationavailableatanypointinthepast.19

Itis importanttomentionthatsometimesthereleaseoftheN BER de-cisionaboutthestateoftheeconomyinperiodtmaybedelayedforalmosttwoyears. T hisleadstoaseriousproblemwhenreal-timeanalysisisappliedtotheprobitmodel, since d tisusuallyunknownattimet. Tosolvethis, atanytimet;weestimate¯ in(8)withobservationsuntilt¡8;toensurethatd is available. T his estimation is thenusedtopredictaprobabilityattof

1 7 D M testcomparingthein-sampleaccuracyofSW IT CH versus V A R is 4.9 forM SEand3.2 forT PE.

18 W eselectthisdatebecausewewanttohaveenoughnumberofobservationstoestimatethedi¤erentmodelsandtocaptureintheout-of-sampleanalysistherecessionintheearly19 7 0s.

19 Inordertoforecastforquartert+ 1 withtheinformationuptoperiodt, weneedtheCL I inperiodt, whichisnotknowuntilonemonthaftertheendofquartert. H owever,this …rstnumberisusuallystronglyrevised. T hus, weusethe…rstpublishedrevisionofthisdata, madetwomonthsaftertheendoftheperiod. T herefore, forexample, toforecasttheG D P inthe…rstquarter(…guresavailableinmay), weusetheCL I indecember(therevisionpublishedinFebruary).

13

beinginrecessionatt+ 1 asfollows:

Pt(d t+ 1 = 1) = F(b0t¡8 z t):

T he lasttwocolumns ofTable 2 presenttheresults forthe real-timeanalysis. L ookingattheresults, weobservethatevenintheout-of-sampleexercise, there is stillgainfrom usingtheCL I and, again, thebestmodelis the M arkov-switching. In addition, wecan concludethatallthegainsfromusingtheCL I comefromtherecessionaryperiods. H owever, asshowninTable3, eventhoughthebivariatespeci…cations’M SFEarenumericallylowerthan inthecaseoftheunivariatelinearmodel, noteventhebestofthem is statistically signi…cantaccordingtotheD M tests.20 H ence, whiletheCL I appears usefulinforecastingG D P withinthehistoricalsample, itseems notbeas usefulinareal-timeexercise. T hereadercan…ndsimilarconclusionsatanticipatingrecessions.

5 CombinationofforecastsA s shown in Table 2, di¤erentmodels havedi¤erentpredictivepowerde-pendingonthestateofthebusinesscycle. Forexample, thenonparametricestimatorpresentsthebestT PFEinexpansionsbutitholdstheworstrecordin recession times amongthebivariate speci…cations. Filardo(19 9 9 ) also…nds thattheperformanceofthedi¤erentmodels changewiththesampleperiodconsidered. T herefore, heproposesthatthebestwaytoimprovetheirreliabilityisbycontinuouslymonitoringtheirperformance, therebylearningaboutwhentheyarelikelytopredictcorrectlyandwhentheyarelikelytofail. T hisispreciselywhatweallowbyusingencompassingmethods. H ence,wesuggestthatacombinationoftheforecastsmaydrawmoreleadinginfor-mationfromtheCL I thananyoftheindividualforecastingmodels.

Inordertocombinegrowth’sforecasts, weapplythelinearcombinationruleproposedby G rangerandR amanathan(19 84). Tocombinein-sampleforecasts, weightsareobtainedbysimplelinearsquarestechniqueson

yt= ¯0ft+ ut; (17 )

whereyt is outputgrowthatt; ¯ =¡¯ 0 ;¯ 1;:::;¯ m + 1

¢; m is thenumberof

di¤erentforecastingmethods, ft= (1;ft;1;:::;ft;m + 1);andft;i istheforecastfortimetthatcorrespondstoA R , V A R , L V ST R , SW IT CH andKER N EL ,respectively. Tocombine out-of-sample forecasts, even though individual

20T his resultis similartoD ieboldand R udebusch(19 9 1) conclusions andcontradictsH amiltonandPerez-Q uiros(19 9 6) results.

14

modelspredictgrowthfort+ 1; thedependentvariableyt+ 1 is notactuallyavailableatanytimet. W esolvethisproblem byusingreal-timecombina-tion. M orespeci…cally, we…t(17 )within-samplepredictionsuntilt;andweusetheseweighstocombinetheout-of-sampleforecastsft+ 1;i=ttoobtainanestimationofyt+ 1.

Inthecaseofforecastingrecessions, itisnotclearthatsucharulewouldimplyanoutputlyingbetweenzeroandone. Instead, inthespiritofL i andD orfman(19 9 6), weproposeanencompassingstrategybasedupondiscretechoiceanalysis. Tocombinein-sampleprobabilitiesofrecession, considerthefollowingrelation

rt= ® pt+ et;

where ® = (® 0 ;® 1;:::;® m + 1), pt = (1;pt;1;:::;pt;m + 1);rtand m havebeende…nedabove, andpt;i isthein-sampleforecastedrecessionprobabilitiesfortimetfrom A R , V A R , L V ST R , SW IT CH , P R O B IT andKER N EL models.L etd tbetheN BER indicatorvariablepresentedinsection3.4. W econcludethat

P(d t= 1) = F(® pt);

where F is the cumulative normaldistribution function as in the probitmodel. W eighs areobtainedbyapplyingmaximum likelihoodprinciples totheobjectivefunction

L(®) =TX

t= 1

fd tln[F(® pt)]+ (1¡d t)ln[1¡F(® pt)]g: (18)

Combiningforecasts inreal-time, we…ndthesameproblem thatinout-of-sampleestimationfrom theprobitmodel. T hedelayonwhich d t is knownhas been solved usingreal-timecombinationas before. T hus, tocombineforecasts foranytimet+ 1, we estimatethe ® in (18) thatcombines in-sampleforecastsuntilt¡8.21 T hen, weusesuchestimationforcombiningtheout-of-sampleprobabilitiesofrecessionfort+ 1. N otethatthereal-timecombinationuseschangingweightsforeachperiodoftime.

A sa…rstapproximation, wemadeacombinationofthesix(…ve, forfore-castinggrowth)alternativemodelsinin-sampleandout-of-sampleforecasts.A sweexpected, theseforecastsarehighlycorrelated, whichsuggeststhatthecombinationusesredundantinformation. SinceSW IT CH andKER N EL are

21A s inthecaseoftheout-of-sampleforecastsfromtheprobitmodel, weareassumingthatthedelayonthereleaseoftheN B ER decisionaboutthestateoftheeconomyisofatmosttwoyears.

15

thebestmodelswithinrecessionandexpansiondata, wetryanencompassingmethodthatcombinesthesetwospeci…cations. IntermsofPR 2 andT PE,this combinationisasgoodasthecombinationthatcontainsthewholesetofmodels. In-sampleandout-of-samplecombinations ofswitchingregimesandnonparametricforecastsarecalledCO M B andRTCO M B respectively.

T he out-ofsample results presented in Tables 2 and 3 revealone themostimportant…ndingsofthispaper. RTCO M B presentsthelowestM SFEand T PFE. M oreover, D M tests (Table 3) con…rm thatthis combinationsigni…cantlyimprovesthelinearmodel’sresults. T hisimpliesthattheCL I isusefulinanticipatingbothrecessionsandG D P growth, eveninreal-time.22Figures 1 and 2 presentthe in-sample and out-of-sample probabilities ofrecessionpredictedfrom SW ITCH andKER N EL . T heyalsoshowhowwellthein-sampleandreal-timecombinationsmimictheN BER schedule.

W earenowreadytoproposea…lterthattransforms theCL I releasesintoprobabilitiesofrecessionnextquarter. T heCL I wasoriginallydesignedas atooltopredictbusiness cycleturningpoints. H owever, everymonththeConferenceBoardonlyreleasestherateofgrowthofsuchleadingindex.A lternatively, weconstructa…lteringruletoextracttheCL I’sleadinginfor-mationaboutturningpoints, bytransformingthegrowthrateofCL I intoprobabilitiesofrecession. B asedonpreviousresults, weproposeareal-timecombinationoftheswitchingregimesmodelandthenonparametricspeci…-cation.

Toseehowitworks, wepresentthefollowingempiricalexercise. Supposeweareinthelastquarterof19 9 7 , andwewanta…lteringrulefortheCL Irelease. W esimulatethepossibleoutcomesoftheCL I growthratefrom -2%to+ 2% . andwepredictinreal-timetheprobabilityofrecessionfor19 9 8.1(belongingtoawideexpansionaryperiod). Figure3displaysthepredictedprobabilityofrecession, associatedtoeachCL Igrowthratevalue, thataretheoutcomes ofSW IT CH , KER N EL andRTCO M B . Furthermore, wepresentinFigure4 theresults ofasimilaranalysis, butwepredictprobabilityofrecessionsfor19 9 0.1 (justafterarecession)usingexclusivelytheinformationavailableat19 9 7 .4.

Sinceweprovedthatthebest…lteris thereal-timecombination, letusconcentrate in theanalysis ofRTCO M B results. A s wecan see from thepictures, thesameCL I growthratecontainsverydi¤erentinformationabouttheprobabilityofan imminentrecessiondependingontheperiodthatweconsider. Speci…cally, in19 9 0.4, aCL I growthrateof0% wouldbeassociatedtoaprobabilityofrecessionnextquarterofalmost1. H owever, in 19 9 7 .4,

22D M tests tocomparetheaccuracyofSW IT CH andRTCO M B are1.7 forM SE and1.2 forT PE.T his impliesthatthereexistssomeevidenceinfavorofRT CO M B .

16

thesameCL I growthratewouldhaveimpliedarecessionprobabilitynextperiodcloseto0. T heintuitionisclear. Inordertopredictthatarecessionis coming, weneed strongerevidence intheCL I behaviorafter9 years ofexpansionsthatjustafterarecessiontobelievethatarecessionisimminent.O ur…ltere¢cientlyusestheinformationaboutthestateoftheeconomytointerprettherateofgrowthoftheCL I ineachperiodoftime.

6 ConclusionsConferenceBoard’s CL I is releasedtoanticipateturningpoints. H owever,theability ofa predictordepends upon ourmodel’s accuracy atextract-ingits leadinginformation aboutfutureevents. T hus, wehaveevaluatedhowwellthemoststandard speci…cations predictrecessions. W eproposeamethodologytocombinedi¤erentforecastedprobabilities. W econcludethatacombinationofaswitchingVA R modelandanonparametricsystemisthebestapproachtoanticipaterecessions. T hiskindofapproachusestheCL I toreproducetheU S businesscycledatafairlywell, comparedwiththeex-postN BER schedule. H ence, we…ndthattheCL I is statisticallyusefulatanticipatingrecessions, eveninreal-timeanalysis.

W econcludethatCL I isalsousefulinforecastingU S G D P growth, eveninout-of-sampleexercise. A gain, acombinationisthebestapproachinreal-time, con…rmingthepowerofthecombinationofforecastsatextractingtheleadinginformationfromtheCL I.

T hus, weproposea…lteringruletoextracttheCL I’sleadinginformationaboutturningpoints. O urpropositiontransformstherateofgrowthoftheCL I in accordancewiththestateoftheeconomy in theperiodoftime inwhichitisreleased.

17

R eferences[1]A bramson, I., “O nBandwidthVariationinKernelEstimates -aSquare

R ootL aw,” A nnalsofStatistics, 10 (19 82), 1217 -1223.

[2]Camacho, M ., “VectorSmoothTransitionR egressionR epresentationofU S G D P andCL I,” M asterThesisD issertation(19 9 8).

[3]D iebold, F., andR .M ariano, “Comparingpredictiveaccuracy,” JournalofBusiness andEconomicStatistics 13(19 9 5), no.3: 253-263.

[4]D iebold, F., andA .R udebusch, “ScoringtheL eadingIndicators”, Jour-nalofBusiness62 (19 8 9 ), 369 -39 1.

[5]— — — – “A N onparametric Investigation ofD uration D ependence intheA mericanBusinessCycle,” JournalofPoliticalEconomy 9 8 (19 9 0),59 6-616.

[6]— — — – “ForecastingO utputW ith theCompositeCeadingIndex: aR eal-T ime A nalysis,” Journalofthe A merican StatisticalA ssociation86(19 9 1), 603-610.

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[8]Filardo, A ., “BusinessCyclePhasesandT heirTransitionalD ynamics,”JournalofBusinessandEconomicStatistics 12 (19 9 4), 27 9 -288.

[9 ]Filardo, A ., “H owR eliable A reR ecessionPrediction M odels,” FederalReserveBankofKansasCityQ uarterlyReview, secondquarter(19 9 9 ),35-55.

[10]G ranger, C. and R . R amanathan, “Improved M ethods ofCombiningForecasts,” JournalofForecasting3(19 84), 19 7 -204.

[11]G ranger, C., and T . Terä svirta, M odellingN onlinearEconomicRela-tionships (N ewYork: O xfordU niversityPress, 19 9 3).

[12]G ranger, C., T . Terä svirta, and H . A nderson, “M odellingN onlineari-tiesO vertheBusinessCycle”, InJ.StockandW .W atson(eds.), Busi-nessCyclesIndicators, andForecasting(Chicago: U niversityofChicagoPress, 19 9 3).

18

[13]H amilton, J., “A N ewA pproachtotheEconomicA nalysis ofN onsta-tionaryT imeSeriesandtheBusinessCycles,”Econometrica 57 (19 8 9 ),357 -384.

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[15]H amilton, J., and G . Pérez-Q uiros, “W hatD otheL eadingIndicatorsL ead?” JournalofBusiness 69 (19 9 6), 27 -49 .

[16]H ä rdle, W ., A ppliedN onparametricRegression (Cambridge: CambridgeU niversityPress, 19 8 9 ).

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[18]Kim, Ch., “PredictingBusinessCyclePhaseswith IndexesofL eadingandCoincidentEconomicIndicators: aM ultivariateR egime-ShiftA p-proach”, W orkingPaper.Seoul: KoreaU niversity(19 9 4).

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19

[25]— — — – , “A ComparisonofL inearandN onlinearU nivariateM odelsforForcastingM acroeconomicT imeSeries”, W orkingPaper6607 , N B ER(19 9 8).

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20

Table1. M aximum likelihoodestimatesofparameters

M odelestimation VA RCO VA R byt= 0:76

(0:0 9)+ 0:2 8(0:07)

(yt¡1¡0:76)(0:0 9)

b¾ 11 = 0:78(0:0 9)

VA Rbyt= 0:78

(0:07)+ 0:61(0:0 6)

(xt¡1¡0:2 4(0:07)

)

bxt= 0:2 4(0:07)

+ 0:4 3(0:0 5)

(xt¡1¡0:2 4(0:07)

)

b¾ 11 = 0:60(0:0 2 )

b¾ 2 2 = 0:52(0:0 2 )

b¾ 12 = 0:0 9(0:03)

L V ST R

byt= 0:91(0:12 )

bFx + 0:60(0:0 5)

xt¡1

bxt= 0:4 2(0:18)

+ ¡0:31(0:13)

bFy + 0:4 3(0:0 5)

xt¡1

bFy =·1 + exp

µ¡1:85(0:2 6)

(yt¡2 ¡0:13(0:0 1)

)¶ ¡1

bFx =·1 + exp

µ¡87:57(15:66)

(yt¡2 + 0:33(0:4 7)

)¶ ¡1

b¾ 11 = 0:55(0:0 2 )

b¾ 2 2 = 0:51(0:0 2 )

b¾ 12 = 0:11(0:03)

SW IT CH

byt= b¹ st+ 0:4 3(0:0 1)

¡xt¡1¡b¹st¡1

¢

bxt= bst+ 0:35(0:0 1)

¡xt¡1¡bst¡1

¢

b¹1 = 1:0 0(0:0 1)

; b¹ 2 = ¡0:2 3(0:0 8)

b1 = 0:4 2(0:0 1)

; b2 = ¡0:57(0:0 8)

p11 = 0:95(0:0 2 )

; p 2 2 = 0:79(0:10 )

b¾ 11 = 0:53(0:0 1)

b¾ 2 2 = 0:4 4(0:0 1)

b¾ 12 = 0:0 7(0:0 1)

P R O B IT brt= ¡0:97(0:2 0 )

¡1:19(0:2 2 )

f1g

N ote. T his estimationuses thesample19 60.2-19 9 7 .4. Variables xtandytaregrowthofG D P andCL I respectively. Variable rt determines howprobableitis thatarecessionwilloccurattimet:Parameters ¾11 and ¾ 2 2arevariancesofG D P andCL I errors, whereasparameter¾ 12 isthecovariancebetweenthem. Standarderrorsareinparentheses. N otethejointuncertaintyintheestimationofsmootherparameterandthresholdwhentheformerislarge. FollowingEstrellaandM ishkin, theprobitmodel’sstandarderrorsareestimatedbyusingtheN adaraya-W atsonestimator.

21

Table2. M SEandT PEin-sampleandout-of-sample.

M SEin

T PEin

M SFEout

T PFEout

A R0.7 81.7 30.57

0.100.570.005

0.7 92.190.49

0.110.620.008

V A R0.601.090.49

0.110.620.006

0.7 51.820.53

0.090.470.011

L V ST R0.550.880.48

0.160.300.133

0.7 01.460.54

0.170.550.09 1

SW IT CH0.480.7 20.42

0.050.220.011

0.681.560.50

0.090.270.060

P R O B IT ...0.100.400.035

...0.090.350.038

KER N EL0.601.180.47

0.110.610.002

0.7 31.800.48

0.100.570.006

CO M B0.550.9 00.47

0.030.100.009

... ...

RT CO M B ... ...0.601.440.48

0.050.240.007

N ote. ”In” refers to19 60.2-19 9 7 .4. ”O ut” refers to19 7 2.2-19 9 8.1. Foreachmodel, …rstentryhavebeencalculatedfrom theentireforecastingsample. Sec-ond and third entries only refers torecessionary and expansionary data, usingthe N BER schedule. M SE and T PE arede…ned in (14) and (15). CO M B andRT CO M B arethecombinationofKER N EL andSW IT CH asSection5 describes.

22

Table3. D ieboldandM arianotests.

A R CO M B RT CO M BM SE T PE M SE T PE M SE T PE

SW ITCHIN

O U T3.7 9 3.121.30 0.54

2.45 0.831.80 2.03

... ...1.7 1 1.19

V A RIN

O U T2.47 1.020.42 1.83

2.15 2.9 4... ...

... ...2.50 1.89

CO M BIN

O U T2.7 9 2.82... ... ... ...

RT CO M BIN

O U T... ...1.9 8 2.48 ... ...

N ote. ”In” refers to19 60.2-19 9 7 .4. ”O ut” refers to19 7 2.2-19 9 8.1. M SE andT PE arede…ned in (14) and (15). CO M B and RT CO M B arethecombinationofKER N EL and SW IT CH as Section 5 describes. A lltheentries refers totheabsolutevalueoftheD M statisticwhich is calculatedforin-rowand in-columnmodelsas(16)describes. Forexample, 3.7 9 (3.12) istheabsolutevalueoftheD Mstatisticsunderthehypothesisofnodi¤erenceintheaccuracyofmodelsSW IT CHandA R atanticipatingin-samplegrowth(recessions).

23

Graph 1. SWITCH model

1960.2 1963.2 1966.2 1969.2 1972.2 1975.2 1978.2 1981.2 1984.2 1987.2 1990.2 1993.2 1996.20

0.2

0.4

0.6

0.8

1

Time

Prob

abili

ty o

f rec

essi

on

Graph 2. KERNEL model

1960.2 1963.2 1966.2 1969.2 1972.2 1975.2 1978.2 1981.2 1984.2 1987.2 1990.2 1993.2 1996.20

0.2

0.4

0.6

0.8

1

Time

Prob

abili

ty o

f rec

essi

on

Note: Graph 1 and Graph 2 represent in-sample probabilities of recession from the switching regimes and the nonparametric specifications respectively. Graph 3 shows in-sample probabilities of recession using a combination of the first two models as Section 5 describes. "In-sample" refers to the period 1960.2-1997.4. Shaded areas correspond to the NBER recessions.

Figure 1: In-sample probabilities of recession

Graph 3. COMB model

1960.2 1963.2 1966.2 1969.2 1972.2 1975.2 1978.2 1981.2 1984.2 1987.2 1990.2 1993.2 1996.20

0.2

0.4

0.6

0.8

1

Time

Prob

abili

ty o

f rec

essi

on

Graph 2. KERNEL model

1972.2 1974.2 1976.2 1978.2 1980.2 1982.2 1984.2 1986.2 1988.2 1990.2 1992.2 1994.2 1996.20

0.2

0.4

0.6

0.8

1

Time

Prob

abili

ty o

f rec

essi

on

Graph 3. RTCOMB model

1972.2 1974.2 1976.2 1978.2 1980.2 1982.2 1984.2 1986.2 1988.2 1990.2 1992.2 1994.2 1996.20

0.2

0.4

0.6

0.8

1

Time

Prob

abili

ty o

f rec

essi

on

Figure 2: Out-of-sample probabilities of recessionGraph 1. SWITCH model

1972.2 1974.2 1976.2 1978.2 1980.2 1982.2 1984.2 1986.2 1988.2 1990.2 1992.2 1994.2 1996.20

0.2

0.4

0.6

0.8

1

Time

Prob

abili

ty o

f rec

essi

on

Note: Graph 1 and Graph 2 represent out-of-sample probabilities o f recession from the switching regimes and the nonparametric specifications respectively. Graph 3 shows out-of-sample probabilities of recession using a real-time combination of the first two models as Section 5 describes. "Out-of-sample" refers to the period 1972.2-1998.1. Shaded areas correspond to the NBER recessions.

24

Figure 3: Simulation for 1998.1Graph 1. SWITCH simulation

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 .04 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

0.2

0.4

0.6

0.8

1

CLI quarterly growth 1997.4

Prob

abili

ty o

f rec

essi

on 1

998.

1

Graph 2. KERNEL simulation

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 .04 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

0.2

0.4

0.6

0.8

1

CLI quarterly growth 1997.4

Prob

abili

ty o

f rec

essi

on 1

998.

1

Graph 3. RTCOMB simulation

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 .04 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

0.2

0.4

0.6

0.8

1

CLI quarterly growth 1997.4

Prob

abili

ty o

f rec

essi

on 1

998.

1

Note: Horizontal axes represent simulated CLI quarterly growth values for 1997.4. Vertical axes show the real-time forecasts of the probability of recession in 1998.1 from the switching regimes model (Graph 1), the nonparametric specification (Graph 2), and the real-time combination of them (Graph 3) as Section 5 describes.

Figure 4: Simulation for 1991.1Graph 1. SWITCH simulation

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 .04 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

0.2

0.4

0.6

0.8

1

CLI quarterly growth 1990.4

Prob

abili

ty o

f rec

essi

on 1

991.

1

Graph 2. KERNEL simulation

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 .04 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

0.2

0.4

0.6

0.8

1

CLI quarterly growth 1990.4

Prob

abili

ty o

f rec

essi

on 1

991.

1

Graph 3. RTCOMB simulation

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 .04 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

0.2

0.4

0.6

0.8

1

CLI quarterly growth 1990.4

Prob

abili

ty o

f rec

essi

on 1

990.

1

Note: Horizontal axes represent simulated CLI quarterly growth values for 1990.4. Vertical axes show the real-time forecasts of the probability of recession in 1991.1 from the switching regimes model (Graph 1), the nonparametric specification (Graph 2), and the real-time combination of them (Graph 3) as section 5 describes.

25