The Ins and Outs of Forecasting Unemployment: Using Labor Force Flows to Forecast the Labor Market

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. The Ins and Outs of Forecasting Unemployment: Using Labor Force Flows to Forecast the Labor Market Regis Barnichon and Christopher J. Nekarda 2013-19 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

The Ins and Outs of Forecasting Unemployment: Using Labor Force Flows to Forecast the Labor Market RegisBarnichon CREI,UniversitatPompeuFabra,andBarcelonaGSE ChristopherJ.Nekarda BoardofGovernorsoftheFederalReserveSystem November21,2012 Abstract Thispaperpresentsaforecastingmodelofunemploymentbasedonlaborforceflows datathat,inrealtime,dramaticallyoutperformstheSurveyofProfessionalForecasters,historicalforecastsfromtheFederalReserveBoard’sGreenbook,andbasictimeseries models. Our model’s forecast has a root-mean-squared error about 30 percent belowthatofthenext-bestforecastintheneartermandperformsespeciallywellsurroundinglargerecessionsandcyclicalturningpoints. Further,becauseourmodeluses information on labor force flows that is likely not incorporated by other forecasts, a combinedforecastincludingourmodel’sforecastandtheSPFforecastyieldsanimprovementoverthelatteraloneofabout35percentforcurrent-quarterforecasts,and 15percentfornext-quarterforecasts,aswellasimprovementsatlongerhorizons. TheviewsinthispaperarethoseoftheauthorsanddonotnecessarilyrepresenttheviewsorpoliciesoftheBoardofGovernorsoftheFederalReserveSystemoritsstaff. Wewouldliketothank WouterdenHaan,JanHatzius,BartHobijn,ÒscarJordà,BarbaraPetrongolo,BarbaraRossi,Tara Sinclair,HermanStekler,PaoloSurico,theeditors,andparticipantsattheBrookingsPanelconference. RegisBarnichonacknowledgesfinancialsupportfromtheSpanishMinisteriodeEconomíay Competitividad(grant ECO2011-23188), theGeneralitatde Catalunya(grant2009SGR1157), and theBarcelonaGSEResearchNetwork.

Forecasting the unemployment rate is an important and difficult task for policymakers, especiallysurroundingeconomicdownturns.Typically,unemploymentrateforecastsaremadeusingone oftwoapproaches. Thefirstisbasedonthehistoricaltime-seriespropertiesoftheunemployment rateand,perhaps,near-termindicatorsofthelabormarket. Thesecondisbasedontherelationship betweenoutputgrowthandunemploymentchangesknownasOkun’slaw. Inthispaperwedevelop anewapproachthatincorporatesinformationonlaborforceflowsasdirectedbyeconomictheory toformforecastsfortheunemploymentrate. Asimpleanalogyhelpsexplainthemainideabehindourapproach. Unemploymentatagiven timecanbethoughtofastheamountofwaterinabathtub,astock. Givenaninitialwaterlevel,the levelatsomefuturetimeisdeterminedbytherateatwhichwaterflowsintothetubfromthefaucet andtherateatwhichwaterflowsoutofthetubthroughthedrain. Whentheinflowrateequalsthe outflowrate, theamountofwaterinthetubremainsconstant. Butiftheinflowrateincreasesand theoutflowratedoesnot,weknowthatthewaterlevelwillbehigherinthefuture. Inotherwords, theinflowrateandtheoutflowratetogetherprovideinformationaboutthefuturewaterlevel—orin thiscase,ofunemployment. Thisinsightformsoneoftwocornerstonesofourapproach: weexploitthisconvergenceproperty whereby the actual unemployment rate converges toward the rate implied by the labor force flows. Theotherisforecastingthoseunderlyingflowsdirectly. Becausetheindividualflowshave differenttime-seriespropertiesandtheircontributionschangeoverthebusinesscycle,focusingon theflowsallowsustobettercapturetheasymmetricnatureofunemploymentmovements. For near-termforecasts, ourmodel dramatically outperformsthe Surveyof Professional Forecasters (SPF), historical forecasts from the Federal Reserve Board’s Greenbook, and basic timeseriesmodels,achievingaroot-mean-squarederror(RMSE)about30percentbelowthatofthebest alternative forecast. Moreover, our model has the highest predictive ability surrounding business cycleturningpointsandlargerecessions. Inaddition,ourmodelisaparticularlyusefuladditiontotheexistingsetofforecastingmodels becauseitusesinformation—dataonlaborforceflows—thatislikelynotincorporatedbyotherforecasts.Thus,combiningtheforecastsfromourmodel,theSPF,andasimpletime-seriesmodelyields areductioninRMSE,relativetotheSPFforecastalone,ofabout35percentforthecurrent-quarter forecast,15percentforthenext-quarterforecast,andsmallerimprovementsatlongerhorizons. Finally, our model can also be used to forecast the labor force participation rate. Here the improvement in forecast performance is more modest than for the unemployment rate: our model improvesontheGreenbookonlyforthecurrent-quarterforecast. Nevertheless,combiningtheforecastsofourmodelandtheGreenbookyieldsasizablereductioninRMSE. I Using Labor Force Flows to Forecast Unemployment We incorporate information from labor force flows through the concept of the conditional steadystateunemploymentrate,whichistherateofunemploymentthatwouldeventuallyprevailwerethe 1

Figure1. UnemploymentRate,ActualandSteady-State,1951–2012 Percent PercentagPeerpcoeinntt Unemployment rate 10 Steady−state* 9 8 7 6 5 4 3 Deviation (right scale) 1.0 0.5 0.0 −0.5 1950 1960 1970 1980 1990 2000 2010 Source:BureauofLaborStatisticsdataandauthors’calculations. Notes: Quarterlyaverageofmonthlydata. ShadedareasrepresentperiodsofbusinessrecessionasdeterminedbytheNationalBureauofEconomicResearch. *Definedasu∗=s/(s+ f),wheresand f areflowratesintoandoutofunemployment;seeequation3. flowsintoandoutofunemploymenttoremainattheircurrentrates. Insteadystatetheseflowsare balanced. However,iftheinflowrateweretojump,astendstohappenattheonsetofarecession, thentheconditionalsteady-stateunemploymentratewouldalsojump. Withnoadditionalshocksto eitherflow,theunemploymentratewouldprogressivelyrisetowardthisnewsteadystate.Moreover, becausethisconvergenceprocessoccurstypicallytakes3monthsormore, theconditionalsteadystaterateprovidesinformationabouttheunemploymentrateinthenearfuture. Figure1showsthetight,leadingrelationshipbetweenthesteady-stateunemploymentrate,u∗, andthe actualunemployment rate, u. As shownbythe deviation(u∗ −u)plotted atthe bottomof the figure, in periods when u∗ is above the actual rate, the unemployment rate tends to rise, and, conversely,whenu∗liesbelowu,theunemploymentratetendstofall. However,relyingsolelyoncurrentlaborforceflowsconstrainsourapproachtoverynearterm forecasts, because the steady state to which the actual unemployment rate converges also changes overtimeastheunderlyingflowschange. Thus,weforecasttheunderlyinglaborforceflowsusing 2

a time-series model and feed those forecasts into a law of motion relating these flows to the unemployment rate to generate unemployment forecasts at longer horizons. Directly forecasting the flowsintoandoutofunemploymentratherthantheunemploymentstockitself,asiscustomary,is anotherreasonthatourmodeloutperformsotherapproaches. Itallowsourmodeltobettercapture thedynamicsofunemployment,becausetheunemploymentstockisdrivenbyflowswithdifferent time-series properties, and because the contributions of the different flows change throughout the cycle(Barnichon,2012). Anadditionaladvantageoffocusingonlaborforceflowsisthatitallowsustocapturetheasymmetricnatureofunemploymentmovements—inparticular,thefactthatincreasestendtobesteeper than decreases.1 Although our model is not explicitly asymmetric, it relies on the unemployment inflowsthatareresponsiblefortheasymmetryofunemployment(Barnichon,2012). Byusingsuch information as inputs in the forecasts, our model can incorporate the impulses that generate this asymmetry.2 Thanks to this property, we find that our model outperforms a baseline time-series model around turning points and large recessions (Montgomery et al., 1998; Baghestani, 2008). Thisisparticularlyusefulsincethesearethetimeswhenaccurateunemploymentforecastsaremost valuable. ThispaperbuildsontheinfluentialworkofMontgomeryetal.(1998)andextendsthegrowing literature aimed at improving the performance of unemployment forecasting models.3 We draw especiallyontherecentliteratureonlaborforceflows,whichhasbeenoverlookedbytheforecasting literaturebuthasbeenthesubjectofnumerousstudiesaimedatunderstandingthedeterminantsof labormarketfluctuations.4 Wealsoweighinonadebateinthisliteratureovertherelativeimportance ofinflowsandoutflows;usingpredictiveabilityasametric,wefindthatbothareequallyimportant forforecastingunemployment. II The Model Our model is built on two elements: a law of motion describing how the unemployment rate converges to its steady-state value, and a forecast of the labor force flows determining steady-state unemployment and the speed at which actual unemployment converges to steady state. We first 1. A long literature discusses the apparent asymmetry of unemployment: for example, Mitchell (1927), Neftçi(1984)(correctedbySichel,1989),Rothman(1991),andmorerecently,Hamilton(2005),Sichel(2007), andRothman(2008). 2. Moreover,unlikestandardtime-seriesmodelsusedtocaptureasymmetries(suchasthresholdmodelsor Markovswitchingmodels), ourmodeldoesnotrelyonathreshold(whichmaychange)tointroduceasymmetry.AcomprehensivestudybyStockandWatson(1999)concludedthatlinearmodelsgenerallydominated nonlinearmodelsforout-of-sampleforecastsofmostmacroeconomictimeseries. Theunemploymentrateis anexception;see,forexample,MilasandRothman(2008). 3. See,forexample,Rothman(1998),GolanandPerloff(2004),BrownandMoshiri(2004),andMilasand Rothman(2008). 4.SeeShimer(2012);PetrongoloandPissarides(2008);Elsby,MichaelsandSolon(2009);Nekarda(2009); Barnichon(2012);Elsby,HobijnandS¸ahin(2011);andFujitaandRamey(Forthcoming),amongothers. 3

present a model with only two labor force states and then expand it to the more general case with threelaborforcestates. WediscussfurthertheintuitionbehindthemodelinsectionIV.E. II.A TheLaborMarketwithTwoStates Thetwo-stateversionofourmodelconsidersonlyemploymentandunemployment. Thatis,weexplicitlyassumethattherearenomovementsintooroutofthelaborforce.Thisapproachisconsistent with recent literature showing that a two-state model does a good job of capturing unemployment fluctuations. In addition, it provides a simpler framework for understanding the basic flow-based accountingoftheconditionalsteady-stateunemploymentrate, anditcanbeestimatedoveralong periodusingdurationdata. However,insectionII.Bwegeneralizeourapproachtothreestatesand allowformovementsintoandoutofthelaborforce. II.A.1 TheLawofMotionforUnemployment Denote u t+τ as the unemployment rate at instant t + τ with t indexing months and τ ∈ [0,1) a continuous measure of time within a month. Assume that between month t and month t + 1 all unemployedpersonsaresubjecttofindingajobaccordingtoaPoissonprocesswithconstantarrival rate f t+1 ,andallemployedworkersaresubjecttolosingorleavingtheirjobaccordingtoaPoisson processwithconstantarrivalrates t+1 .5 Theunemploymentratethenproceedsaccordingto (1) du d t τ +τ = s t+1 (1−u t+τ )− f t+1 u t+τ , as changes in unemployment are given by the difference between the inflows and the outflows. Solvingequation1yields (2) u t+τ =β t+1 (τ)u∗ t+1 +(cid:2) 1−β t+1 (τ) (cid:3) u t , where (3) u∗ ≡ s t+1 t+1 s t+1 + f t+1 denotestheconditionalsteady-stateunemploymentrate,andβ t+1 (τ) ≡ 1−e−τ(st+1 +ft+1) istherateof convergencetothatsteadystate. Equation2relatesvariationintheunemploymentstocku t+τ overthecourseofamonthtovariationintheunderlyingflowhazards, f t+1 ands t+1 . A1-month-aheadforecastfortheunemployment rate,uˆ t+1|t ,canthusbeobtainedfrom (cid:16) (cid:17) (4) uˆ t+1|t =βˆ t+1 u∗ t+1 + 1−βˆ t+1 u t , 5. Weadoptthistimingconventiontoreflectdataavailability,asthehazardrateisobservedonlyinmonth t+1. 4

whereβˆ t+1 isthemonthtforecastofβ t+1 ,theconvergencespeedbetweenmonthtandmontht+1. Over 1951–2011, the sum of monthly unemployment inflow and outflow rates averaged 0.62, implyingthatthehalf-lifedeviationofunemploymentfromitssteadystateisslightlymorethana month. As a result, 90 percent of the gap between unemployment and its conditional steady-state valueisclosedinabout4months,onaverage. However,asthelowerpaneloffigure2shows,this convergencespeedvariesconsiderablyoverthebusinesscycle,asinflowandoutflowratesfluctuate. Asaresult,thetimeneededtoclose90percentofthegaprangesfromabout3monthsintightlabor marketstoabout5monthsinslackmarkets.Inthe2008–09recession,thedropintheunemployment exitratewassodramaticthatthefigureincreasedtoanunprecedented9months. Ithassinceedged lower,tojustunder8monthsinthethirdquarterof2012. Partoftheexceptionalincreaseinthelastrecessionowestoadramaticdeclineinthejobfinding rate,theresultofexceptionallylowjobcreationandlowmatchingefficiency(BarnichonandFigura, 2010).Moreover,Elsbyetal.(2011)showthatthisexceptionaldeclinewasalsoinpartanartifactof measurementerror,becausenotallpersonsflowingintounemploymenthadadurationoflessthan 5 weeks. This phenomenon became much more prevalent in the recent recession, where a larger fraction of persons moving from employment to unemployment reported a duration of more than 5weeks,leadingtheduration-basedmeasureoftheunemploymentexitratetosufferfromalarger downwardbias. AswediscussinsectionVII,totheextentthatthebiasisstrongerthaninprevious recessions,forecastingperformancecoulddeteriorate. II.A.2 ForecastingLaborForceFlows Becauseequation4canforecasttheunemploymentrateonly1monthaheadgiventhecurrentvalues ofthehazardrates,forecastingtheunemploymentrateatlongerhorizonsrequiresmakingforecasts of the hazard rates. A simple approach is to assume that the hazard rates remain constant at their last observed value over the forecast horizon. However, in real time a forecaster does not observe s t+1 and f t+1 ,butonlys t and f t ,becauseatmonthtonecanobservelaborforceflowsonlyfromt−1 tot.6 Thus,the j-period-aheadforecastoftheunemploymentratecanbeformedfromthemontht valuesofsand f asfollows:7 (cid:104) (cid:105) (5) uˆ t+j|t = 1−e−j(ft +st) u∗ t +e−j(ft +st)u t . Ifthehazardratesarepersistentenough,equation5willprovidereasonableforecasts.8 However,as figure 2 shows, the hazard rates do change, and with them the conditional steady-state unemploy- 6. Aconcreteexamplehelpsclarifythispoint. LastSeptember’semploymentreport(publishedonOctober 5, 2012) provided information on the stock of unemployment in September and the average unemployment inflowandoutflowratesbetweenAugustandSeptember(s and f).Lookingbackatequation4,thisallowsus t t tomeasureβ t ,u∗ t andu t .Thus,forecastinguˆ t+1|t (theunemploymentrateinOctober),requiresforecastsof fˆ t+1|t andsˆ t+1|t (thatis,theflowsfromSeptembertoOctober). 7. ThislawofmotionformsthebasisofElsby,HobijnandS¸ahin’s(2011)strategytogeneralizeShimer’s (2012)unemploymentdecompositiontoincorporateout-of-steady-statedynamics. 8.InsectionIVweconsideraforecastbasedonlyontheconvergencetothesteady-stateunemploymentrate. 5

Figure 2. Unemployment Inflow and Outflow Hazard Rates and Convergence to Steady-StateUnemploymentRate,1951–2012 Unemployment inflows and outlflows Log points Log points Inflow (left scale) 1.6 4.6 Outflow (right scale) 1.5 4.4 1.4 4.2 1.3 4.0 1.2 3.8 1.1 3.6 1.0 3.4 0.9 3.2 1950 1960 1970 1980 1990 2000 2010 Time to convergence* Months 9 8 7 6 5 4 3 2 1 1950 1960 1970 1980 1990 2000 2010 Source:Authors’calculationsbasedonBureauofLaborStatisticsdata. Notes: Quarterlyaverageofmonthlydata. ShadedareasrepresentperiodsofbusinessrecessionasdeterminedbytheNationalBureauofEconomicResearch. *Timeneededtoclose90percentofthegapbetweentheactualandthesteady-stateunemploymentrate. mentrateandthespeedofconvergence. Becausethehazardratesarenotsufficientlypersistent,weuseavectorautoregression(VAR)to forecasttheinflowandoutflowrates. Wealsoincludetwoleadingindicatorsoflaborforceflows: vacancypostingandinitialclaimsforunemploymentinsurance. Specifically,let y =(lns,ln f,∆lnu,lnuic,∆lnhwi)(cid:48), t t t t t t where uic is the monthly average of weekly initial claims for unemployment insurance and hwi is Barnichon’s (2010) composite help-wanted index. Note that, given our timing convention for the 6

flows,thehazardrateseffectivelyentertheVARlaggedby1month. WeestimatetheVAR (6) y =c+Φ y +Φ y +ε t 1 t−1 2 t−2 t overafifteen-yearrollingwindow.9 Aftergeneratingforecastsofthehazardrates,weobtain j-period-aheadforecastsofunemploymentbyiteratingonthefollowing:10 (cid:16) (cid:17) (7) uˆ t+j|t =βˆ t+j uˆ∗ t+j + 1−βˆ t+j uˆ t+j−1|t , with (8) uˆ∗ = sˆ t+j|t t+j sˆ t+j|t + fˆ t+j|t and (9) βˆ t+j =1−e−(sˆt+j|t +fˆ t+j|t ) . Withmontht+jforecastsoftheflowratesinhand,wecancalculatethemontht+jvaluesofu∗ andβ. Themontht+ junemploymentforecastisthenobtainedbytakingaweightedaverageofthe previous-period(montht+j−1)unemploymentrateandthecurrent-period(montht+j)conditional steady-stateunemploymentrate,withtheweightsdeterminedbythespeedofconvergencetosteady state. II.B TheLaborMarketwithThreeStates Sofar,wehaveconsideredalabormarketwithonlytwostates:employedorunemployed.However, notallthosewithoutjobsareunemployed. Indeed,flowsintoandoutofthelaborforcedwarfthose intoandoutofunemployment.11 Thissectionconsidersamodelthatincorporatesflowsamongall threelaborforcestates. An important advantage of the three-state model is its ability to capture more accurately the flowstakingplaceinthelabormarket. Forinstance,theunemploymentinflowratecomprisesboth 9. We found that, in real time, a rolling window (in which the model is estimated over the previous K months)yieldedmoreaccurateforecaststhanarecursivewindow(inwhichthemodelisestimatedoverthe entireobservedhistory),likelybecauseofthelow-frequencypatterns; windowsof15yearsweresuperiorto 10-and20-yearwindows.Wealsoconsideredlaglengthsbetween1and12. 10. Becauseuˆ t+j|t isanonlinearfunctionof fˆ t+j|t and sˆ t+j|t ,Jensen’sinequality,intheory,prohibitsusfrom directly forecasting the unemployment rate from equation 7 and forecasts of f t+j|t and s t+j|t . To avoid this problem,weuseMonteCarlosimulationandsamplewithreplacementfromtheVARresiduals{(cid:15)f,(cid:15)s}andform forecasts(equations8and9)usingthesamplingdistributionof(cid:15)f and(cid:15)s. Inpractice,giventhemagnitudeof theinflowsandoutflows,theunemploymentrateforecastsobtainedbyMonteCarlosimulationarenotdifferent fromthoseformedfromequation7.Forsimplicity,weusethatapproximation. 11.SeeBlanchardandDiamond(1990)fortheseminalstudyofgrossflows. 7

thoselosingorleavingjobsandthoseenteringorreenteringthelaborforce. Sincethesetwoflows (andinfactallsixflows)displaydifferenttime-seriesproperties, athree-statemodelmayproduce betterforecaststhanatwo-statemodel.12 Inaddition,thethree-statemodelcanbeusedtoforecast thelaborforceparticipationrate. Togeneralizeourtwo-stateframeworktothreestates,weneedtospecifyandsolvethesystem ofdifferentialequationsgoverningthenumberofpeopleinunemployment, U; inemployment, E; oroutofthelaborforce,N. Betweenmontht andmontht+1,individualscantransitfromstatea ∈ {E,U,N}tostateb ∈ {E,U,N}accordingtoaPoissonprocesswithconstantarrivalrateλab . Thestocksofunemployed, t+1 employed,andpersonsnotinthelaborforcesatisfythefollowingsystem:13  (10)   N U E ˙ ˙ ˙ t t t + + + τ τ τ = = = λ λ λ t E t E U t + + + N U 1 E 1 1 E E U t t + t + + τ τ τ + + + λ λ λ U t t N + t N + + N 1 U 1 E 1 U N N t t t + + + τ τ τ − − − ( ( ( λ λ λ t N U t t E + + + E U 1 E 1 1 + + + λ λ λ t N U t t E + + + U 1 N N 1 1 ) ) ) N E U t t t + + + τ τ τ . Forinstance,asthefirstlineshows,changesinunemploymentaregivenbythedifferencebetween theinflowstounemployment(workerslosingorquittingtheirjobsandseekingwork, andpersons joining or rejoining the labor force) and the outflows from unemployment (unemployed persons findingajobordroppingoutofthelaborforce). Then,usingtheinitialandterminalconditions,the1-step-aheadforecastsofthethreestockscan besolvedasfunctionsofthetransitionprobabilities(theλabsinequation10). Theappendixgives the details of the solution. We then use the solution to these equations to generate 1-period-ahead forecastsoftheunemploymentrateandthelaborforceparticipationratefrom (11) uˆ t+1|t = Uˆ t+1 U | ˆ t t + +1 E |t ˆ t+1|t and (12) l(cid:100)fpr = Uˆ t+1|t +Eˆ t+1|t . t+1|t Eˆ t+1|t +Uˆ t+1|t +Nˆ t+1|t Notethat,ineffect,whatweassumeforpopulationgrowthdoesnotaffectourforecasts,becausewe forecastpopulationshares. Aswiththetwo-statemodel,toconstructforecastsbeyondoneperiodahead,weuseaVARto 12.SeeBarnichonandFigura(2010)formoreonthepropertiesofthedifferentflows. 13. Equation 10 assumes that P is constant within a month and that inflows and outflows to the civilian t noninstitutionalpopulationagedsixteenandolderarenegligible. Thisassumptionisreasonablegiventhatthe working-agepopulationincreasesbyabout150,000permonth,anorderofmagnitudeortwolargerthanthe flowsintoandoutofE,U,orN. 8

forecastthesixtransitionprobabilities. Specifically,weestimate (13) y =c+Φ y +Φ y +Φ y +ε, t 1 t−1 2 t−2 3 t−3 t overa10-yearrollingwindow,where (cid:16) (cid:17)(cid:48) y =ln λEU,λUE,λEN,λNE,λNU,λUN,u,uic,hwi . t t t t t t t t t t Note that in this specification, the unemployment rate and the help-wanted index enter in levels, ratherthaninchanges,becausethisyieldedbetterforecasts. III Data Weconstructmeasuresofthetransitionratesinthetwo-stateandthree-statemodelsusingdifferent approaches: an indirect one (using information on the stocks of unemployment and of short-term unemploymenttoinferthetransitionrates)forthetwo-statemodel,andadirectone(usingmeasures oflaborforceflows)forthethree-statemodel. Forthetwo-statemodel,wefollowShimer(2012)anduseinformationonthenumberofpersons unemployed, U, and those unemployed for less than 5 weeks, US, to infer job finding and job t t separationhazardrates.14 Specifically,wecalculatetheunemploymentoutflowprobability,F,from F t+1 =1− Ut+1 U − t U t S +1, with f t+1 = −ln(1−F t+1 ) the hazard rate. The unemployment inflow rate, s, is then obtained by solvingequation1forwardover[t,t+1)andfindingthevalueofs t+1 thatsolves (cid:104) (cid:105) U t+1 = 1−e f t − + ( 1 ft+ + 1 + s st t + + 1 1 ) s t+1 (U t +E t )+e−(ft+1 +st+1)U t . Notethatinthisaccounting,givenavaluefortheunemploymentoutflowrate(whichalsocaptures movementsoutofthelaborforce)andthestockofunemployedpersons,theinflowrateistherate thatexplainstheobservedstockofunemployedpersonsinthenextmonth. Asaresult, theinflow rateincorporatesallmovementsinunemploymentnotaccountedforbytheunemploymentoutflow rate. For the three-state model, we use aggregate labor market transition probabilities among employment, unemployment, and nonparticipation calculated from longitudinally matched Current Population Survey (CPS) micro data. We construct the transition rates from labor market flows as λab ≡ ab/a , where A is the number of persons observed in state a in month t − 1, and t t t−1 t−1 AB is the number who transitioned from state a in month t −1 to state b in month t. (The time t seriesofthe ABsarecollectivelyreferredtoas“grossflows.”) Forexample,UE isthenumberof t 14.SeealsothepioneeringworkbyPerry(1972). 9

persons who moved from unemployed to employed, and UE/U is the corresponding transition t t−1 probability. TheBLSpublishesaresearchseriesofgrossflowsthatbeginsinFebruary1990. For contemporaryforecasts,thepublisheddatahaveasufficientlylonghistorytoallowestimationofthe model. However,toevaluatehistoricalmodelforecastsbeforeFebruary2000,weneeddatawitha longerhistory. Thus, weconstructmeasuresofgrossflowsthatcoverJune1967toJanuary1990, allowingustobeginourevaluationwith1976. From1976to1990weconstructgrossflowsfrom Nekarda’s(2012)LongitudinalPopulationDatabase. Before1976weusegrossflowstabulatedby JoeRitter(seeShimer,2012). Finally,weeklyinitialclaimsforunemploymentinsurancearepublishedbytheDepartmentof Labor’s Employment and Training Administration. Our measure of vacancy posting is, as noted before,thecompositehelp-wantedindexpresentedinBarnichon(2010). IV Forecasting Performance Weevaluatetheperformanceofourflowsmodelsbycomparingtheirunemploymentrateforecasts withalternativeforecastsalongtwodimensions. First,weassesstheRMSEsofout-of-sampleforecasts. Second,becauseitishardertoforecasttheunemploymentratearoundrecessions,weassess our model’s performance relativeto a baseline time-series model over thebusiness cycle. In what followswerefertothetwo-statemodelas“SSUR-2”andthethree-statemodelas“SSUR-3.” IV.A Real-TimeForecasts Ourobjectiveinthissectionistoevaluateourmodels’forecastsagainstthebestalternativeforecasts, boththoseofprofessionalforecastersandthosederivedfromothertime-seriesmodels. Weconsider fivealternativeforecastsoftheunemploymentrate. Thefirsttwoarefromprofessionalforecasters. WeconsiderhistoricalforecastsfromtheGreenbook,whichtheliteraturehasgenerallyshowntobe thebenchmarkforecast,andthemedianforecastfromtheSPF.15 Theotherthreealternativeforecastsareourownestimatesfromtime-seriesmodelsandareintended to disentangle the mechanisms behind the performance of our model. We consider a basic univariatetime-seriesmodel,intendedasa“naive”forecastthattakesintoaccountnootherinformationbutthepastunemploymentrate.FollowingMontgomeryetal.,weuseanARIMA(2,0,1)model fortheunemploymentrate. WealsoconsidertheunemploymentrateforecastfromaVARthatincludesthelaborforceflowsandthetwoleadingindicators,initialclaimsandthehelp-wantedindex. BycomparingourSSURmodels’forecastsagainstthoseoftheVAR,wecandirectlyevaluatethe nonlinearrelationshipimpliedbythetheorycomparedwithanatheoreticallineartime-seriesmodel usingthesameinformationset. Ourlastalternativeistheunemploymentrateforecastderivedfrom thelawofmotionforunemploymentrate(equation5),holdingtheinflowandoutflowratesconstant at their last known value. We call this the u∗ model. Shutting down the variation in the hazard 15.See,forexample,RomerandRomer(2000),Sims(2002),FaustandWright(2007),andTulip(2009). 10

ratesisolatesthecontributionofthecurrentconditionalsteady-stateunemploymentrate. Thethree alternativetime-seriesmodelsareestimatedovera15-yearrollingwindow. Toreflecttheenvironmentwithinwhichforecastersmustoperate,weestimateourmodelsand generateforecastsusingreal-timedata, exceptforinitialclaimsandthehelp-wantedindex.16 The historicalforecastswerenecessarilymadewithonlythedatainhandatthetime. Someeconomic data, suchasrealGDPandpayrollemployment, aresubjecttosubstantialrevisionovertime. For these variables, the current-vintage data may differ substantially from the historical data used to make those forecasts. In the case of the unemployment rate, however, the revisions are relatively minor. ThelaborforcedataobtainedfromtheCPSarerevisedonlytoreflectupdatedestimatesof seasonalfluctuations.Indeed,thenot-seasonally-adjustedstocksofemploymentandunemployment are never revised, reflecting their origin from a point-in-time survey of households. Nevertheless, eventhesmallrevisionstoseasonalfactorsmayhaveimportantconsequencesforourmodels’performance. Toconstructreal-timeestimatesofthehazardrates,webeginwithmonthlyvintagesofthepublishedseasonally-adjustedstocksofemployed,unemployed,andshort-termunemployedworkers.17 Foreachmonth,weestimatethetimeseriesoftheinflowandoutflowhazardratesfromthereal-time stocksasdescribedinsectionIII.TheseseriesarethenusedintheVARtoforecastthehazardrates. Real-timedataforinitialclaimsandthehelp-wantedindexarenotavailable,socurrent-vintagedata areusedintheVAR.Aswiththeunemploymentrate,revisionstotheseseriesarerelativelysmall.18 Nonetheless,insectionIV.Cweassesstheimplicationsofthislimitation. The SPF sends out its survey questionnaire sometime in the first month of a quarter, and the surveyparticipantsareaskedtomailbackthecompletedquestionnairebythemiddleofthesecond monthofthequarter. Thus,theforecastsincludedintheSPFincorporatelabormarketdatafromthe first month of each quarter. To make forecasts comparable, our model forecasts are made treating thefirstmonth’sunemploymentrateasdata. TheinformationsetforthehistoricalGreenbookforecastsismoreirregular. Becausethepublication of the forecast is dictated by the date of the upcoming Federal Open Market Committee meeting, Greenbook forecasts are made at different points in a quarter; as a result, some forecasts havenomonthlylabormarketdataforthecurrentquarterwhereasothershave2monthsofdata. For example,atthetimetheMarch2004Greenbookwaspublished,theunemploymentratewasknown throughFebruary2004,andthusthequartert+0forecastwasmadewithdataforthefirst2months of the quarter. However, when the April 2004 Greenbook was published, the unemployment rate wasknownonlythroughMarch2004,andthusthequartert+0forecast(forthesecondquarterof 16. Real-time vintages were obtained from the Federal Reserve Bank of St. Louis’s “ArchivaL Federal ReserveEconomicData.” 17. Analternativeapproachthatsidestepstheissueofseasonalrevisionsaltogetheristoforecastthenotseasonally-adjustedunemploymentratefromnot-seasonally-adjustedCPSdata. Thatmodelperformedsimilarlytothetwo-statemodelwepresenthere. 18. There are no revisions to the print help-wanted index. Real-time data for initial claims are available beginning in June 2009. Over the 39 months for which real-time data are available, the maximum absolute variationinthemonthly-averagelevelofweeklyinitialclaimsoverthisperiodwastiny(about3percent). 11

Table 1. Comparing Unemployment Rate Forecasts: SSUR-2 Model versus ProfessionalForecasters,1976–2006 Root-mean-squarederror(percentagepoint) Forecasthorizon(quarters) Model t+0 t+1 t+2 t+3 t+4 SSUR-2 0.12 0.35 0.54 0.70 0.86 Greenbook 0.16∗∗∗ 0.39 0.54 0.65 0.78 (0.00) (0.31) (0.95) (0.63) (0.50) SPF 0.17∗∗∗ 0.37 0.53 0.66 0.82 (0.00) (0.48) (0.96) (0.67) (0.65) No. ofobservations 89 89 89 89 89 Source:Authors’calculationsusingdatafromtheBureauofLaborStatistics,theEmploymentandTraining Administration, the Federal Reserve Bank of Philadelphia, the Board of Governors of the Federal Reserve System,andBarnichon(2010). Notes:Calculatedfrom89forecastsover1976–2006thatshareacommoninformationsetwiththehistorical Greenbook and SPF forecasts. t+0 denotes the current quarter at the time of the forecast. p values of the Giacomini-Whitestatisticofequalpredictiveabilityarereportedinparentheses.Asterisksindicatestatistically significantimprovementoftheSSUR-2forecastovertheindicatedforecastatthe*10percent,**5percent,or ***1percentlevel. 2004)wasmadewithoutanypublisheddataforthequarter. Wearecarefultomimictheinformation setforeachGreenbookforecast. Finally, becausetheGreenbookforecastsaremadepublicwitha 5-yearlag,ourcomparisonusingtheGreenbookendsin2006. IV.B ForecastErrors Table 1 reports the RMSEs of real-time forecasts for quarterly unemployment rates over a 1-year horizon(aswellasaforecastofthecurrentquarter,t+0). Toevaluatethestatisticalsignificanceof ourresults,wereportthe pvaluesoftheGiacominiandWhite(2006)unconditionalteststatisticof equalpredictiveabilitybetweenourSSUR-2forecastandthecomparisonforecast.19 TheSSUR-2modeldramaticallyoutperformstheGreenbookandtheSPFforshort-termforecasts. Asthefirsttworowsoftable1show,theRMSEoftheSSUR-2modelforecastforthecurrent quarter is about 0.05 percentage point (more than 30 percent) smaller than those of the other two forecasts. Thisimprovementisstatisticallysignificantatthe1percentlevelagainstboth. Although the improvement in the next-quarter forecast is of similar magnitude, the RMSE is now only 10 percent smaller than those of the alternatives, and the difference is not statistically significant. At longer horizons the improvement over the SPF and the Greenbook disappears: SSUR-2 performs 19.WeusetheGiacominiandWhite(2006)predictiveabilitytest,becauseunliketheDieboldandMariano (1995)test,itisrobusttobothnon-nestedandnestedmodelssuchasourVAR,u∗,andSSURmodels. 12

Table 2. Comparing Unemployment Rate Forecasts: SSUR-2 Model versus Time- SeriesModels,1976–2011 Root-mean-squarederror(percentagepoint) Forecasthorizon(quarters) Model t+0 t+1 t+2 t+3 t+4 SSUR-2 0.15 0.38 0.60 0.83 1.06 ARIMA 0.22∗∗∗ 0.52∗∗ 0.84∗ 1.14 1.41∗ (0.00) (0.05) (0.08) (0.11) (0.08) VAR 0.19∗∗∗ 0.47∗∗∗ 0.73∗∗ 1.03∗ 1.30 (0.00) (0.00) (0.02) (0.07) (0.14) u∗† 0.20∗∗∗ 0.48∗∗∗ 0.70∗∗ 0.92 1.11 (0.00) (0.00) (0.01) (0.16) (0.41) SSUR-2-s‡ 0.22∗∗∗ 0.52∗∗∗ 0.75∗∗∗ 0.97∗∗∗ 1.16 (0.00) (0.00) (0.00) (0.03) (0.18) SSUR-2-f§ 0.22∗∗∗ 0.52∗∗∗ 0.77∗∗∗ 1.00∗∗∗ 1.23∗∗ (0.00) (0.00) (0.00) (0.00) (0.01) No. ofobservations 432 432 432 429 426 Source:Authors’calculationsusingdatafromtheBureauofLaborStatistics,theEmploymentandTraining Administration, the Federal Reserve Bank of Philadelphia, the Board of Governors of the Federal Reserve System,andBarnichon(2010). Notes: CalculatedfromforecastsmadeeverymonthfromJanuary1976toDecember2011. Seethetextfor detailsofthetime-seriesmodels. t+0denotesthecurrentquarteratthetimeoftheforecast. pvaluesofthe Giacomini-Whitestatisticofequalpredictiveabilityarereportedinparentheses.Asterisksindicatestatistically significantimprovementoftheSSUR-2forecastovertheindicatedforecastatthe*10percent,**5percent,or ***1percentlevel. †Forecastderivedfromequation7,holdingtheinflowandoutflowratesconstantattheirlastknownvalue. ‡VariantofSSUR-2inwhichtheunemploymentinflowrateisallowedtovaryandtheoutflowrateisheld constant. §VariantofSSUR-2inwhichtheunemploymentoutflowrateisallowedtovaryandtheinflowrateisheld constant. about0.05percentagepointworsethaneitherprofessionalforecastonaverage. Nevertheless, itis striking that our model performs only slightly worse than the Greenbook and the SPF at a 1-year horizon,consideringthattheirforecastsarebasedonanarrayofeconomicdataandmodelsofthe broadereconomy,whereasSSUR-2isastatisticalmodelthatincorporatesonlynear-terminformationaboutthelabormarket. Table 2 reports the performance of SSUR-2 against the time-series models. The univariate ARIMAmodelperformsworsethanSSUR-2atallhorizons. ThesameistrueoftheVAR,showing that the nonlinearity of our model is an important feature. Finally, the contribution of forecasting the flows is evident from the fourth row, which reports the performance of a forecast based only 13

onconvergencetotheconditionalsteady-stateunemploymentrate(u∗). Thismodel,too,performs worsethanSSUR-2atallhorizons,indicatingthattimevariationintheflowratesisalsoanimportant elementofourmodel. Itisremarkablethataforecastfromthetheoreticallawofmotion(equation 5) that relies on only the last known value of u∗ performs as well as or better than both estimated time-seriesmodels.20 Finally, we weigh in on the debate over the relative importance of inflows and outflows to unemploymentfluctuations,usingpredictiveabilityasametric.Todoso,weevaluatetheforecasting performance of variants of the SSUR-2 model in which only one of the hazard rates is allowed to vary. Forexample,toevaluatethecontributionoftheinflowrate,weholdtheoutflowrateconstant atitslastknownvalueandallowtheinflowratetovaryaccordingtotheVARforecast. Wecallthis variant“SSUR-2-s”anditscounterpartthatallowsonlyoutflowstovary“SSUR-2-f.” Thebottom two rows of table 2 show that SSUR-2-s and SSUR-2-f perform about the same at all horizons. However,bothmodelshavelargerRMSEsthantheu∗model,whichholdsbothhazardratesconstant. Thus,forforecastingunemployment,inflowsandoutflowsareaboutequallyimportant,inlinewith therecentconsensusintheliterature.21 IV.C Quasi-Real-TimeForecastsandSSUR-3 Aswenotedearlier,ourpreferredVARspecification—whichincludesinitialclaimsforunemploymentinsuranceandthehelp-wantedindex—cannotbeestimatedintruerealtimebecausevintages of these two leading indicators are not available. In work not reported in detail here, we assess whetherthisgivesourmodelanunfairadvantageoverthehistoricalprofessionalforecastersbyestimatingtheVARfortheSSUR-2modelwithoutuicand∆hwi—atruereal-timeexercise. Overthe sampleusedintheupperpaneloftable1,thismodelstillhasanRMSEalmost20percentsmaller thantheGreenbook’satquartert+0andessentiallythesameatquartert+1. Wealsocomparethe performanceofthetruereal-timeforecastsofSSUR-2withthemodel’s“quasi-real-time”forecasts, whereweusethesamerollingestimationandforecastingprocedureasinthereal-timeexercisebut usethecurrent-vintagedataatallpoints; thatis, weomitallvariationassociatedwithrevisionsto the seasonal factors. Over the same sample (results not reported), the truly real-time forecasts are actuallyslightlybetterthanthequasi-real-timeforecastsatallbutthecurrent-quarterhorizon,where they perform equally well. This suggests that evaluating the models in quasi-real time likely does notsignificantlyalterthespiritofthereal-timeexercise. With this in mind, we assess the quasi-real-time forecasts of the SSUR-3 model. (Because historicalrecordsofseasonalrevisionstogrossflowsarenotavailable,wecannotevaluatetheperformance of the SSUR-3 model in true real time.) Table 3 evaluates the performance of SSUR-3 against that of SSUR-2 in quasi-real time. Over the period from 1976 to 2006 (upper panel), the 20.ForeshadowingsectionVonforecastcombination,thisresultsuggeststhatcombiningamodelbasedon thesteady-stateunemploymentratewithanestimatedtime-seriesmodelmayyieldfurtherimprovements. 21. Elsby,MichaelsandSolon(2009);FujitaandRamey(2009);Nekarda(2009);Elsby,HobijnandS¸ahin (2011);Barnichon(2012). 14

Table 3. Comparing Quasi-Real-Time Unemployment Rate Forecasts of the SSUR Models,1976–2006 RootMeanSquaredForecastError(percentagepoint) Forecasthorizon(quarters) Model t+0 t+1 t+2 t+3 t+4 Greenbook/SPFsample,1976–2006 SSUR-2 0.13 0.32 0.50 0.67 0.84 SSUR-3 0.15∗∗∗ 0.37∗ 0.61∗∗ 0.85∗∗ 1.09∗∗ (0.01) (0.08) (0.02) (0.01) (0.02) Monthlyforecasts,2000–06 SSUR-2 0.10 0.23 0.30 0.44 0.54 SSUR-3 0.11∗ 0.23 0.37∗∗ 0.57∗∗ 0.77∗∗ (0.07) (0.39) (0.04) (0.06) (0.04) Source:Authors’calculationsusingdatafromtheBureauofLaborStatistics,theEmploymentandTraining Administration, the Federal Reserve Bank of Philadelphia, the Board of Governors of the Federal Reserve System,andBarnichon(2010). Notes: Upper panel calculated from 89 forecasts over 1976–2006 that share a common information set withthehistoricalGreenbookandSPFforecasts;lowerpanelcalculatedfrom83forecastsmadeeverymonth fromFebruary2000toDecember2006. t+0denotescurrentquarteratthetimeoftheforecast. pvaluesof Giacomini-Whiteteststatisticarereportedinparentheses.Asterisksindicatestatisticallysignificantdifference betweentheSSUR-3andSSUR-2forecastsatthe*10percent,**5percent,or***1percentlevel. three-statemodelperformsabitworsethanSSUR-2inthecurrent-,next-,and2-quarter-aheadforecasts,andatlongerhorizonsitperformsappreciablyworse. However,thegrossflowswecalculate from the CPS micro data (from before 1990) are noisier than duration-based hazard rates, in part becauseofspurioustransitionsbetweenunemploymentandnonparticipation. Ifwerestrictthetime periodtouseonlyforecaststhatwereestimatedusingthepublishedgrossflowsdata(lowerpanel), thedifferencesaresmallathorizonsofupto2quartersahead. Aswiththelongersample,SSUR-3 performsappreciablyworsethanSSUR-2atforecasthorizonsoft+3andbeyond. IV.D ForecastingPerformanceovertheBusinessCycle Theunemploymentstockisdrivenbyflowswithdifferenttime-seriesproperties,andthecontributionsofthedifferentflowschangethroughoutthecycle.22 Forinstance,inflowsareresponsiblefor the sharp increase in unemployment at the onset of recessions, but outflows are the main driving 22. At a quarterly frequency, the autocorrelation of the outflow rate is 0.91, but that of the inflow rate is only0.73(Shimer,2012). Further,whereasthedistributionofthe(detrended)inflowrateispositivelyskewed and highly kurtotic, the distribution of the (detrended) outflow rate exhibits no skewness and low kurtosis (Barnichon,2012). 15

forceofunemploymentinnormaltimes. This property suggests that the performance of our flows model may vary over the business cycle. Forinstance,becausetheSSUR-2modelincorporatestheunemploymentinflowrate,which is responsible for the asymmetry of unemployment, it may better capture the asymmetric nature ofunemploymentthanstandardmodels. Thus, itmayperformbetterduringrecessions, especially comparedwithstandardmodels,whichdonotincludelaborforceflows. TotestthisideaandevaluatewhetherSSUR-2performsdifferentlyoverthecourseofthebusinesscycle,weuse(GiacominiandRossi,2010)predictiveabilitytestinunstableenvironments.The testdevelopsameasureoftherelativelocalforecastingperformanceoftwomodelsandisidealfor testingwhethertheperformanceofourmodelvariesoverthecyclerelativetothatofabenchmark model. WeuseasthebenchmarktheARIMAmodelpresentedintable1. Weevaluatelocalforecasting performance over a 5-year window using monthly forecasts spanning November 1968 to February2012. Figure3plotstheGiacominiandRossifluctuationteststatisticforcurrent-quarterand1-quarteraheadforecasts;thecorresponding5percentcriticalvalueisalsoshown.Theunitoftheyaxisisthe (standardized) rolling difference in mean-squared-error between the two models, a measure of the relativeperformance,definedsuchthatapositivevalueindicatesasuperiorperformanceofSSUR-2. TheSSUR-2forecastsarealmostalwaysmoreaccuratethanthoseofthebenchmarkmodel,but SSUR-2 performs especially well around recessions—and particularly during the deep recessions of1973–75,theearly1980s,and2008–09—andduringtimesoflargeandswiftmovementsinthe inflowrate. Inotherwords,SSUR-2yieldsthegreatestimprovementoveranaivebaselinearound turningpoints,preciselywhenaccurateunemploymentforecastsaremostvaluable. IV.E IntuitionfortheModel’sPerformance Ourmodel’sperformanceisparticularlystrikingintworespects: themodelimprovessignificantly uponprofessionalforecastsinthenearterm,anditperformsespeciallywellduringrecessions. This sectiondiscussestheintuitionbehindthisperformance. IV.E.1 AveragePerformance Our approach to forecasting unemployment rests on the convergence of unemployment toward its steady-statevalueimpliedbythelaborforceflows. Intuitively,ifconvergencetakesplacewithina coupleofmonths,knowingthecurrentvaluesoftheflowsprovidesinformationonthefuturelevel ofunemployment,andtheSSURmodelwillperformwelloverthenextcoupleofmonths. In this subsection we develop this intuition further and discuss the reasons behind the quantitativeperformanceofthemodel. Specifically,ourabilitytoimproveforecastingaccuracydepends onthreeparameters: theleveloftheflows,thepersistenceoftheflows,andwhetherdifferentflows havedifferenttime-seriesproperties. First,asequation2makesclear,theleveloftheflowsgovernsthespeedofconvergencetosteady 16

Figure 3. Giacomini-Rossi Fluctuation Test Statistic for Comparison of the SSUR-2 andARIMAModelForecasts,1970–2012 Test statistic Same quarter 10 One quarter ahead 9 8 7 6 5 4 3 2 1 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 Source:Authors’calculationsbasedondatafromtheBureauofLaborStatistics,theEmploymentandTrainingAdministration,theFederalReserveBankofPhiladelphia,andBarnichon(2010). Notes: TheGiacomini-RossistatisticisusedheretomeasuretherelativeperformanceoftheSSUR-2and ARIMA(2,0,1)modelsasthe5-yearrollingdifferenceinthemeansquarederroroftheirforecastsspanning November1968toFebruary2012.ApositivevalueindicatesasuperiorperformanceofSSUR-2.Dashedhorizontallineindicates5percentcriticalvalue.Shadedareasrepresentperiodsofbusinessrecessionasdetermined bytheNationalBureauofEconomicResearch. state.SincetheseflowsarerelativelylargeintheUnitedStates,convergenceoccursrelativelyrapidly (in3to5months),andthecurrentflowshelpforecastunemploymentinthenearterm. Thisexplains why,withU.S.data,SSUR-2performsespeciallywellforcurrent-andnext-quarterforecasts. Ifthe flowswerelarger,thespeedofconvergencewouldbeevenhigher,andthemodelwouldperformbest at an even shorter horizon. At the extreme, if convergence were instantaneous, the current values of the flows would not help forecast unemployment. In contrast, if the flows were much smaller, convergencewouldoccurmuchmoreslowlyandperformancemightbebestatlongerhorizons. Ifflowswereverysmall,couldtheircurrentvalueshelpusforecastunemploymentinthevery longrun? Thereasonwhythisisunlikelycomesfromthesecondcrucialparameter,thepersistence of the flows. The performance of SSUR depends on the interaction between the speed of conver- 17

gence (the levels of the flows) and the persistence of the flows. The model is good at forecasting unemployment only if the flows are sufficiently persistent that their values can be well predicted overthetimeneededtoconvergetosteadystate.FortheU.S.labormarket,themodelperformswell intheneartermbecausethepersistenceoftheflowsissufficientlyhigh. The third important characteristic behind the performance of SSUR stems from our focus on forecastingtheflowsratherthanthestock. Amodelofthestock(suchastheARIMAmodelpreviously described) cannot perform as well as SSUR, because the flows differ in their time-series properties and because the contributions of the different flows change throughout the cycle. A modelofthestockcancapturetheaveragetimeseriespropertiesofthestock,butitcannotallowfor differenttimeseriespropertiesatdifferentstagesofthecycle. IV.E.2 PerformanceovertheBusinessCycle The SSUR model performs especially well during recessions and around turning points because theunemploymentratedisplayssteepnessasymmetry: increasestendtobesteeperthandecreases. This asymmetry manifests itself most forcefully during recessions, but a model of the stock is ill equippedtocaptureitunlessoneimposesanarbitrarythresholdtointroduceasymmetry,asisdone in threshold autoregressive models. Although SSUR is not explicitly asymmetric, it relies on the laborforceflowsthatareresponsiblefortheasymmetryofunemployment. Indeed,itisthedifferent behavioroftheinflowsandoutflowsoverthecyclethatisresponsibleforthesteepnessasymmetry ofunemployment(Barnichon,2012).Byincorporatingflowinformationandusingitasinputsinthe forecasts,ourmodelcanincorporatetheimpulsesresponsiblefortheasymmetryofunemployment. Specifically,thebeginningofarecessionistypicallymarkedbyasharpincreaseintheinflow rate: figure 4 plots the impulse responses from our estimated VAR to a shock to the inflow rate.23 Whereas the inflow rate itself displays a sharp increase with fairly rapid reversion to the mean, the outflow rate displays a delayed, U-shaped response with much slower mean reversion. These different impulse responses give rise to the steepness asymmetry of unemployment: following the initialshock,theinflowraterevertsrelativelyquicklytoitsmean,buttheoutflowratetakeslonger to return to steady state and thus prevents the unemployment rate from decreasing as rapidly as it increased. By relying on a VAR forecast of the flow rates, following an initial disturbance to the inflow rate at the onset of a recession, SSUR can propagate the cyclical behavior of the flows and thuscapturethesteepnessasymmetryofunemployment. V Combining Forecasts The array of forecasting models we have considered reflect different information sets. The SSUR models’ forecasts rely mainly on labor force flows data and other labor market indicators but not on information from outside the labor market. In contrast, the professional (SPF and Greenbook) 23.SeeFujita(2011)formoreevidenceontheresponseoftheinflowandoutflowratestoaggregateshocks. 18

Figure4. ImpulseResponseofaShocktotheUnemploymentInflowRate Inflow Outflow Unemployment rate Standard deviations 2 2 2 1 1 1 0 0 0 −1 −1 −1 −2 −2 −2 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Quarters Quarters Quarters Source:Authors’calculationsusingdatafromBureauofLaborStatistics. Notes:Impulseresponsefunctiontoa1-standard-deviationshocktotheinflowrate,calculatedfromaVARof y =ln(s,f,ur)(cid:48)with2lagsthatwasestimatedusingquarterlydataover1951–2007.Shadedarearepresents t t t t 95-percentconfidenceinterval. forecastsarebasedonanarrayofeconomicdataandmodelsbeyondthelabormarket,buttheymay ignoreinformationonlaborforceflows. TheARIMAmodelforecastsunemploymentfromitspast behavior. Giventhesedifferentinformationsets,anaturalquestioniswhethertheseunemploymentforecastscanbefurtherimprovedbycombiningourflowsmodels’forecastswithaprofessionalforecast such as the SPF and with a simple time-series model such as the ARIMA, to exploit the differences in correlation among the forecast errors (see Granger and Newbold, 1986). We construct suchacombinedforecastbytakingaweightedaverageofforecastsfromSSUR-2,theSPF,andthe ARIMA model. The weights are determined by ordinary least squares regression, with a constant included to account for any systematic biases in the estimate. We estimate weights separately for each forecast horizon. The evaluation is a real-time exercise where, for each forecast at a given time, the weights are determined using available history only. These weights allow us to evaluate themarginalcontributionsofeachmodelovertheSPFforecast. IftheSSUR-2modelforecastcon- 19

Table4. OptimalCombinedUnemploymentRateForecasts,1976–2006 Forecasthorizon(quarters) Forecast t+0 t+1 t+2 t+3 t+4 Rootmeansquaredforecasterror(percentagepoint)† SSUR-2 0.11 0.33 0.50 0.68 0.82 SPF 0.16 0.35 0.49 0.63 0.77 ARIMA 0.14 0.37 0.59 0.80 0.98 Combined 0.11∗∗∗ 0.30∗∗ 0.45 0.61 0.74 (0.00) (0.03) (0.15) (0.35) (0.37) Optimalweights‡ SSUR-2 0.80 0.48 0.42 0.35 0.40 (0.10) (0.14) (0.14) (0.15) (0.16) SPF 0.15 0.42 0.55 0.65 0.63 (0.08) (0.11) (0.12) (0.13) (0.14) ARIMA 0.06 0.11 0.04 0.01 −0.04 (0.11) (0.12) (0.12) (0.12) (0.13) Constant 0.00 0.00 0.00 0.00 0.00 (0.00) (0.00) (0.00) (0.00) (0.00) Source:Authors’calculationsusingdatafromtheBureauofLaborStatistics,theEmploymentandTraining Administration,theFederalReserveBankofPhiladelphia,theBoardofGovernorsoftheFederalReserveSystem,andBarnichon(2010). Notes: Calculatedfrom124forecastsover1976–2006. t+0denotesthecurrentquarteratthetimeofthe forecast. † pvaluesoftheGiacomini-Whiteteststatisticofequalpredictiveabilityarereportedinparentheses. AsterisksindicatestatisticallysignificantimprovementofthecombinedforecastovertheSPFatthe*10percent, **5percent,or***1percentlevel. ‡Optimalweightsaredeterminedfromthefollowingordinaryleast-squaresregression:u =β +β uˆSSUR+ t 0 1 t β uˆSPF+β uˆARIMA+ν.Standarderrorsarereportedinparentheses. 2 t 3 t t tributes no incremental benefit over the SPF forecast, the weight on the SSUR-2 forecast will be zero. As table 4 shows, this is not the case: combining the SSUR-2 model with the SPF and the ARIMAimprovesforecastingperformancesignificantlyathorizonsupto2quartersahead. ComparedwiththebaselineSPFforecast,thecombinedforecastachievesareductioninRMSEofabout 35percentforcurrent-quarterforecasts,15percentfor1-quarter-aheadforecasts,andalmost10percent for 2-quarter-ahead forecasts, with smaller improvements at longer horizons. The combined current-quarterand1-quarter-aheadforecastsarestatisticallysignificantlybetterthantheSPFforecastalone. The optimal weights reflect the contribution of the SSUR-2 model for short-term forecasting, 20

andthecombinedforecastputsmuchmoreweightonSSUR-2atshort-termhorizons.24 Importantly, the fact that the combined forecast performs significantly better than any single forecast indicates thattheflowsmodelbringsrelevantinformationnotcontainedintheSPForARIMAforecasts. In other words, because the forecast errors of the models are not strongly correlated, the combined forecastperformssubstantiallybetter. VI Forecasting Labor Force Participation Compared with the unemployment rate, the labor force participation rate has less of a systematic aggregate relationship with output growth. In fact, aggregate labor force participation has been thoughttobeacyclical,withchangesintheparticipationrateonlyweaklyrelatedtooutputgrowth.25 As a result, forecasting the labor force participation rate was often seen as less of a priority than forecastingtheunemploymentrate. The large and unexpected decline in labor force participation during and after the 2008–09 recession challenged that conventional wisdom and highlighted the importance of forecasting the laborforceparticipationrate.However,giventhehistoricalabsenceofastrongrelationshipbetween outputandlaborforceparticipation,forecastershavefewmodelstoturnto. Thus,oneadvantageofthethree-stateSSURmodeloverthetwo-statemodelisthatitalsogeneratesforecastsofthelaborforceparticipationrate(and,byextension,theemployment-population ratio). Table5evaluatestheperformanceoftheSSUR-3modelforecastsagainstthatoftheGreenbook forecasts.26 SSUR-3 improves on the Greenbook forecast for the current-quarter forecast, althoughthereductioninRMSEisnotstatisticallysignificant. Atlongerforecasthorizons,SSUR-3 performsmarkedlylesswellthantheGreenbook. However, SSUR-3 forecast errors need not be correlated with Greenbook forecast errors, so that, again, a combined forecast may generate significant improvement. The third row of table 5 confirmsthisintuition. TheoptimallaborforceparticipationrateforecastcombiningtheGreenbook andSSUR-3forecastsperformssignificantlybetterthateitheraloneatallhorizonsconsidered,and especially at longer horizons. The reduction in RMSE is trivial for current-quarter forecasts, but growstobetween0.1and0.3percentagepointatlongerhorizons. ThelargeweightonSSUR-3at allhorizonsreflectsthesuperiorperformancesofSSUR-3comparedwiththeGreenbook.27 24.Thereportedoptimalweightsaretheweightsestimatedoverthefullsample. 25.Nonethelessseveralpapershavefoundanimportantrolefordemographicsindeterminingtheaggregate participationrate.Aaronsonetal.(2006)andFallickandPingle(2007)usecohort-basedmodelstohelpisolate demographic and other structural factors from cyclical variation in the participation rate. They find that the apparentacyclicalityofaggregateparticipationistheresultofmoderatelycyclicalparticipationwithincertain demographicgroupsthatroughlyoffsetswhenaggregated. 26. ThehistoricalGreenbookforecastsincludesquarterlyforecastsfortheparticipationratebeginningonly in2000. 27. ThisisnotevidentinthedirectcomparisonbetweenSSUR-3andtheGreenbook,becauseSSUR-3has alargerbiasthantheGreenbook.Theconstantintheoptimalforecastaccountsforthissystematicbias. 21

Table5. LaborForceParticipationRateForecastsandOptimalCombinations,2000– 2006 Forecasthorizon(quarters) Forecast t+0 t+1 t+2 t+3 t+4 Rootmeansquaredforecasterror(percentagepoint)† SSUR-3 0.13 0.26 0.38 0.46 0.56 Greenbook 0.14 0.23∗∗ 0.31∗∗∗ 0.39∗ 0.47 (0.21) (0.03) (0.00) (0.06) (0.14) Combined 0.10∗∗∗ 0.15∗∗∗ 0.17∗∗∗ 0.18∗∗∗ 0.19∗∗∗ (0.00) (0.00) (0.00) (0.00) (0.00) Optimalweights‡ SSUR-3 0.40 0.36 0.67 0.85 0.69 (0.16) (0.16) (0.14) (0.12) (0.10) Greenbook 0.51 0.49 0.23 0.15 0.16 (0.14) (0.14) (0.11) (0.09) (0.07) Constant 0.06 0.10 0.07 0.00 0.09 (0.02) (0.04) (0.05) (0.05) (0.06) Source:Authors’calculationsusingdatafromtheBureauofLaborStatistics,theEmploymentandTraining Administration,theFederalReserveBankofPhiladelphia,theBoardofGovernorsoftheFederalReserveSystem,andBarnichon(2010). Notes:Calculatedfrom56forecastsoverJanuary2000toDecember2006thatshareacommoninformation setwiththehistoricalGreenbookforecast.t+0denotesthecurrentquarteratthetimeoftheforecast. †pvaluesoftheGiacomini-Whiteteststatisticofequalpredictiveabilityarereportedinparentheses.AsterisksindicatestatisticallysignificantimprovementofSSUR-3orcombinedforecastovertheGreenbookforecast atthe*10percent,**5percent,or***1percentlevel. ‡ Optimal weights are determined from the following ordinary least-squares regression: lfpr = β + t 0 β l(cid:100)fpr SSUR+β l(cid:100)fpr SPF+β l(cid:100)fpr ARIMA+ν.Standarderrorsarereportedinparentheses. 1 t 2 t 3 t t VII Recent Performance and Near-Term Prospects Thusfarthesampleperiodusedinourevaluationagainstprofessionalforecastshasendedin2006, shortlybeforetheGreatRecession. Acrucialquestion,however,ishowtheSSURmodelsperform during that recession and the ongoing recovery. In particular, can the models capture the steep increase in unemployment and the lack of rapid decline following this latest recession compared withotherdeeprecessions? Beyond2006,wecannolongercompareourmodelswiththeGreenbook;wethereforeusethe SPFasthebenchmarkinstead. Astable1showed,theSPFandGreenbookforecastshaveroughly similarRMSEsovera1-yearforecasthorizon. Table6reportstheRMSEsforforecastsofbothSSURmodelsandtheSPFstartinginFebruary 2007 and ending in February 2012. Although the SPF’s current-quarter forecast error is roughly 22

Table 6. Performance of Unemployment Rate Forecasts of the SSUR Model and the SPF,2007–12 Root-mean-squarederror(percentagepoint) Forecasthorizon(quarters) Model t+0 t+1 t+2 t+3 t+4 SPF 0.17 0.46 0.77 1.17 1.58 SSUR-2 0.16 0.58 0.96 1.47 1.94 SSUR-3 0.13 0.45 0.93 1.50 2.11 No. ofobservations 23 22 21 20 19 Source:Authors’calculationsusingdatafromtheBureauofLaborStatistics,theEmploymentandTraining Administration,andBarnichon(2010). Notes:Calculatedfromforecastsmadeonceperquarterover2007:Q1to2012:Q3.t+0denotesthecurrent quarteratthetimeoftheforecast. comparabletothatfortheearlierperiod,forecasterrorsatlongerhorizonsare0.1to0.8percentage pointlargerduringthisperiodthanover1976–2006. Theflowsmodels’forecasterrorsaresimilarly larger. In particular, the two-state model’s forecast for the current quarter—by far its comparative advantage—worsens appreciably. Whereas it outperformed the SPF by 25 percent in the earlier period,SSUR-2nowperformsonlyslightlybetterthantheSPF. Incontrast,thethree-statemodel,whichperformedworsethaneitherthetwo-statemodelorthe SPFover1976–2006atallhorizons,nowoutperformstheSPFbyalmost30percentinthecurrent quarter,anditsRMSEisevenabitsmaller1quarterahead. Wepointtothreefactorstoexplainthis strikingdifference. First,thegrossflowsaremuchbettermeasuredinthepublisheddatathaninthehistoricaltabulations. Becauseformuchofthe1976–2006samplethemodelwasestimatedusingthetransitions wecalculatedfromthemicrodata(ratherthanfromthepublisheddata),thethree-statemodelperforms worse than the two-state model. Indeed, table 3 showed that SSUR-3 performed about the sameasthetwo-statemodelwhenestimatedusingonlythepublishedgrossflowsdata. Second,thetwo-statemodelusescross-sectionaldataonunemploymenttoinfertheoutflowrate and backs out the inflow rate using an unemployment accounting identity. As Elsby et al. (2011) note,akeyassumptionneededtoderivethehazardsappearstohavebrokendownstartingin2009. They show that, historically, the two measures of unemployment outflows moved closely together overthebusinesscycle, butthatsince2009, Shimer’s(2012)measurehasexhibitedamuchlarger decline than the flow from the unemployment to employment. They show that the discrepancy is beingdrivenbythelargeincreaseintheunemploymentdurationofpersonsflowingintounemployment, whereas Shimer’s calculation assumes that all unemployment inflows have a duration of 5 weeksorless. Third, and most important, the two-state model by design abstracts from movements into and outofthelaborforce. Historically,thelaborforceparticipationratehasnotexhibitedmuchcycli- 23

cality. However,intherecentrecessionandrecovery,theparticipationratehasfallen21/ 2percentage points. TheCongressionalBudgetOffice’sAugust2012estimateofthetrendlaborforcesuggests that about 1 percentage point of this decline can be accounted for by declining trend participation (primarily due to aging of the population). The remainder likely reflects an unusually large cyclical decline. The two-state model cannot account for the cyclical decline and thus projects an employment-populationratiothatissystematicallytoohigh. Incontrast,theflowsinthethree-state modelreflectthedecliningparticipationrate. Figure 5 shows the time pattern of recent unemployment rate forecast misses by the SPF and the two SSUR models for both the current-quarter forecast (2 months of forecast; upper panel) and the next-quarter forecast (5 months of forecast; lower panel). A positive value indicates that the unemployment rate was higher than the model predicted. As the unemployment rate starts to increase, rising from 4.5 percent in the first quarter of 2007 to 4.8 percent at the end of 2007, the SPFandtheSSURmodelsshowrelativelysmallupsidesurprisesatboththet+0andt+1horizons (theunemploymentratewashigherthanprojected). Theunemploymentratethenacceleratesover 2008 and the first half of 2009, rising from 5 percent to 91/ 2 percent. In 2008 the SSUR models and the SPF alike show modestly surprises to the upside in their current-quarter forecasts, with missesofabout1/ 4percentagepoint. However,from2009forwardthethree-statemodelconsistently outperformstheSPFandthetwo-statemodelinthecurrentquarter,withmissestoboththeupside andthedownside. Atthe1-quarter-aheadhorizon,thetwo-statemodeldoesnoticeablyworsethan theSPForthethree-statemodelfrom2009on. What do the SSUR models say about the outlook for unemployment 6 months hence as of thiswritinginNovember2012? Table7presentstheirforecastsoftheunemploymentrateandthe laborforceparticipationrateforNovember2012throughApril2013. Thetwo-statemodelprojects the unemployment rate to decline over the rest of 2012 and into 2013, from its reported level of 7.9 percent in October to 7.8 percent in December and 7.5 percent in April. This decline reflects a projected increase in the unemployment outflow hazard and a projection of little change in the inflowhazard(notshown). Thethree-statemodelprojectstheunemploymentratetoremainat7.9 percent over the same 6 months, reflecting a slow increase in the job finding rate for unemployed workersandaslightriseinthelaborforceparticipationrate. VIII Conclusion Although the unemployment rate is typically considered a lagging indicator of the business cycle, increases in the unemployment rate have preceded the last three recessions. Recent research by Fleischman and Roberts (2011) finds that the unemployment rate provides the best single signal about the state of the business cycle in real time. Nevertheless, despite extensive research on the topic,forecastersandpolicymakersoftenrelyonOkun’slaworbasictime-seriesmodelstoforecast theunemploymentrate. Thispaperhaspresentedanonlinearunemploymentrateforecastingmodelbasedonlaborforce 24

Figure5. UnemploymentForecastErrorsoftheSSURModelsandtheSPF,2007–12 Current quarter Percentage points 1.2 SPF SSUR−2 SSUR−3 0.8 0.4 0.0 −0.4 −0.8 Q1Q2Q3Q4 Q1Q2Q3Q4 Q1Q2Q3Q4 Q1Q2Q3Q4 Q1Q2Q3Q4 Q1Q2Q3Q4 2007 2008 2009 2010 2011 2012 One quarter ahead Percentage points 1.2 0.8 0.4 0.0 −0.4 −0.8 Q1Q2Q3Q4 Q1Q2Q3Q4 Q1Q2Q3Q4 Q1Q2Q3Q4 Q1Q2Q3Q4 Q1Q2Q3Q4 2007 2008 2009 2010 2011 2012 Source: Authors’ calculations based on data from the Bureau of Labor Statistics, the Employment and TrainingAdministration,andBarnichon(2010) flowsthat,inrealtime,dramaticallyoutperformsbasictime-seriesmodels,theSPF,andtheFederal Reserve Board’s Greenbook forecast at short horizons. Our model is built on two elements: a nonlinearlawofmotiondescribinghowtheunemploymentrateconvergestoitsconditionalsteady state(therateofunemploymentimpliedbytheflowsintoandoutofunemployment),andforecastsof theselaborforceflows. Themodel’sperformance,inturn,stemsfromtwofactors: theconvergence ofunemploymenttoitsconditionalsteadystatewithalagof3to5months,andthefactthatflows intoandoutofunemploymenthavedifferenttime-seriespropertiesthanthestock. Empirically,thetwo-statemodelhasaroot-mean-squaredforecasterrorabout30percentsmaller than the next-best forecast for the current quarter, and 10 percent smaller for the next-quarter forecast. Moreover, our model has the highest predictive ability of those we analyze surrounding businesscycleturningpointsandlargerecessions. Andbecausethemodelbringsnewinformation totheforecast,acombinationofourmodel’sforecastandtheSPFforecastyieldsanimprovement ofabout35percentforthecurrent-quarterforecastandof25percentforthenext-quarterforecast, withsmallerimprovementsatlongerhorizons. Thetwonewmodelsthatweproposehavebothadvantagesanddisadvantagesrelativetoeach 25

Table7. SSURModelForecastsforNovember2012throughApril2013 Percent Quarterly Monthlyforecast average 2012 2012 2013 2013 2013 2013 2012 2013 Model Nov. Dec. Jan. Feb. Mar. Apr. Q4 Q1 Unemploymentrate SSUR-2 7.9 7.8 7.7 7.7 7.6 7.5 7.8 7.7 SSUR-3 7.9 7.9 7.9 7.9 7.9 7.9 7.9 7.9 Laborforceparticipationrate SSUR-3 64.0 64.0 64.1 64.1 64.2 64.2 63.9 64.1 Source: Authors’calculationsusingdatafromBureauofLaborStatistics,DepartmentofLabor,andBarnichon(2010). Notes:ForecastwasmadeNovember2,2012(withdatathroughOctober). other. The two-state model is easier to understand conceptually and to implement. The durationbasedunemploymentinflowandoutflowhazardrateshavealongerhistoryandaresomewhatless noisy. However, thesehazardratesarenotdirectlymeasuredbutratherinferredfromatheoretical model. More important, a key assumption for deriving the hazards appears to have broken down startingin2009. The three-state model is a more realistic characterization of the labor market and produces internally consistent forecasts for the unemployment rate, the labor force participation rate, and theemployment-populationratio. Althoughthethree-statemodel’sunemploymentrateforecastsat longerhorizonstendtobeworsethanthoseofthetwo-statemodel,since2007thethree-statemodel outperformsthetwo-statemodel—aswellastheSPF,inthenearterm—inpartbecauseitaccounts forthelargedeclineinlaborforceparticipationduringthiscycle. Appendix SolvingtheThree-StateModel DenotingY t+τ =(U t+τ ,E t+τ ,N t+τ )(cid:48),wecanrewriteequation10as (A.1) Y˙ t+τ =A t Y t+τ , 26

with   A t =  −λU t λ λ E U U t − N E λU t N −λ t EU λ λ t E E − U N λ t EN −λN λ λ E t N t N − U E λNU  . t t t t Since the columns of A sum to zero, A has one eigenvalue equal to zero. Denoting Q the t t t matrixofeigenvectorsofA correspondingtotheeigenvalues[r ,r ,0],wecanwriteasolutionto t 1t 2t equationA.1as   (A.2) Y t+τ =Q t  c c 2 1 e e τ τ ( ( r r 2 1 t t ) )  , c 3 with c , c and c the constants of integration. The two nonzero eigenvalues are negative and are 1 2 3 functionsofthehazardrates: r ≈−β ≡λUE +λUN 1t 1t t t (A.3) r ≈−β ≡λEU +λEN +λNE +λNU. 2t 2t t t t t Tofindthevaluesofc ,c andc ,weuseinitialconditionsY =(U,E,N)(cid:48)andterminalcon- 1 2 3 t t t t ditionsY −→ (cid:0) U∗,E∗,N∗(cid:1)(cid:48),thevectorofthesteady-statenumbersofunemployed(U∗),employed t t t t t t→∞ (E∗),andnonparticipants(N∗). Thesteady-statestocksaregivenby t t U∗ =k s t+1 t s t+1 + f t+1 +o t+1 E∗ =k f t+1 t s t+1 + f t+1 +o t+1 N∗ =k o t+1 , t s t+1 + f t+1 +o t+1 wherekisaconstantsetsothatU∗, E∗,and N∗ sumto P,theworkingagepopulationinmontht, t t t t ands t+1 , f t+1 ,ando t+1 aredefinedby s t+1 =λ t E + N 1 λ t N + U 1 +λ t N + E 1 λ t E + U 1 +λ t N + U 1 λ t E + U 1 f t+1 =λU t+ N 1 λ t N + E 1 +λ t N + U 1 λU t+ E 1 +λ t N + E 1 λU t+ E 1 o t+1 =λ t E + U 1 λU t+ N 1 +λU t+ E 1 λ t E + N 1 +λU t+ N 1 λ t E + N 1 . Somealgebrayieldsthe1-month-aheadforecastsofunemployment,employment,andnonpar- 27

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