Nowcasting Business Cycles: a Bayesian Approach to Dynamic Heterogeneous Factor Models

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Nowcasting Business Cycles: a Bayesian Approach to Dynamic Heterogeneous Factor Models Antonello D’Agostino, Domenico Giannone, Michele Lenza, and Michele Modugno 2015-066 Please cite this paper as: D’Agostino, Antonello, Domenico Giannone, Michele Lenza, and Michele Modugno (2015). “NowcastingBusinessCycles: aBayesianApproachtoDynamicHeterogeneousFactorModels,” Finance and Economics Discussion Series 2015-066. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2015.066. 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.

Nowcasting Business Cycles: a Bayesian Approach to Dynamic Heterogeneous Factor Models Antonello D’Agostino, European Stability Mechanism Domenico Giannone, Federal Reserve Bank of New York, CEPR, ECARES, LUISS Michele Lenza, European Central Bank and ECARES Michele Modugno, Federal Reserve Board Abstract We develop a framework for measuring and monitoring business cycles in real time. Following a long tradition in macroeconometrics, inference is based on a variety of indicators of economic activity, treated as imperfect measures of an underlying index of business cycle conditions. We extendexistingapproachesbypermittingforheterogenouslead-lagpatternsofthevariousindicators along the business cycles. The framework is well suited for high-frequency monitoring of current economicconditionsinrealtime-nowcasting-sinceinferencecanbeconductedinpresenceofmixed frequency data and irregular patterns of data availability. Our assessment of the underlying index of business cycle conditions is accurate and more timely than popular alternatives, including the ChicagoFedNationalActivityIndex(CFNAI).Aformalreal-timeforecastingevaluationshowsthat theframeworkproduceswell-calibratedprobabilitynowcaststhatresembletheconsensusassessment of the Survey of Professional Forecasters. JEL Classification: C11, C32, C38, E32, E38. Keywords: CurrentEconomicConditions,DynamicFactorModels,DynamicHeterogeneity,Business Cycles, Real Time, Nowcasting. The views expressed are those of the authors and do not necessarily reflect those of the European Central Bank, the Eurosystem, the European Stability Mechanism, the Federal Reserve Bank of New York, the Board of Governors of Federal Reserve System. This work was partly supported by the research contracts ARC-AUWB/2010-15/ULB-11 and IAPP7/06StUDys(DG). 1

1 Introduction Macroeconomic and financial variables are characterized by a strong correlation, which is possible only if the bulk of their fluctuations is driven by few common sources. Dynamic factor models (DFM) build on this basic fact to provide a parsimonious and, yet, suitable representation of macroeconomic and financial dynamics. The model assumes that a fewunobserveddynamicfactorsdrivethecomovementofmanyobservedvariables,whilethefeaturesthatarespecificto individualseries,suchasmeasurementerror,arecapturedbyidiosyncraticdisturbances. DynamicfactormodelswereinitiallyproposedbyGeweke(1977)andSargentandSims(1977)asatimeseriesextension ofthefactormodelspreviouslydevelopedforcross-sectionaldatainpsychometrics(seeLawleyandMaxwell,1963,fora comprehensiveanalysisoffactormodelsforseriallyuncorrelateddata). Overtheyears,factormodelshavebeensuccessfullyusedinmacroeconometricsforstructuralanalysisandforecasting(seeStockandWatson,2011,foracomprehensive survey). Dynamicfactormodelshavebeenintensivelyusedinmanycontexts,rangingfromforecasting(StockandWatson,2002)andnowcasting(Giannoneetal.,2008)totheempiricalvalidationofDynamicStochasticGeneralEquilibrium models(BoivinandGiannoni,2006;Giannoneetal.,2006),andprovideareliablestatisticalframeworkfortheestimation ofsyntheticindexesofbusinesscycleconditions(StockandWatson,1992). The main feature of business cycle fluctuations is their pervasiveness across the economy.1 Hence, variables measuring different aspects of the economy can be considered as imperfect measures of a latent common business cycle factor. Formally,thedynamicfactormodelrepresentationforasetofstationaryvariables,xi,t,i=1,...,n,iswrittenasfollows: s (cid:88) xi,t= λ i,h f t−h +ei,t i=1,...,n (1) h=0 whereftisthecommonfactorsummarizingthestateoftheeconomyandei,t,1=1,...,naretheidiosyncraticdisturbances. Themodelisidentifiedbyassumingthatcomovementamongvariablesarisesonlyfromasinglesource,thecommonfactor. Thisamountsatassumingthate1t,...,ent,ft areorthogonalatallleadsandlags. FollowingStockandWatson(1992),wewillrefertoft asthesyntheticindexofbusinesscycleconditions. Summarizing business cycle condition using a synthetic index rather than observable measures can enhance timeliness and precision. For example, GDP provides a very comprehensive measure of economic activity and summarizes well the business cycle fluctuations, as shown by the fact that recessions roughly correspond to its decline for two consecutive quarters (see Harding and Pagan, 2002). However, GDP is released with a delay, it is subject to revisions and it is characterized by measurementerror. Aggregatingtheinformationprovidedbydifferentvariablesrepresentsasortofinsuranceagainstthe measurement error2 and, in the assessment of the business cycle conditions, it allows to exploit the different sampling 1Intheirpioneeringwork,BurnsandMitchell(1946,p.3)definethebusinesscyclesasthe“typeoffluctuationfound in the aggregate economic activity of nations that organize their work mainly in business enterprises: a cycle consists of expansions occurring at about the same time in many economic activities, followed by similarly general recessions [...].” Pervasiveness is central also in the definition of the NBER dating committee: “During a recession, a significant declineineconomicactivityspreadsacrosstheeconomyandcanlastfromafewmonthstomorethanayear. Similarly, duringanexpansion,economicactivityrisessubstantially,spreadsacrosstheeconomy,andusuallylastsforseveralyears” (www.nber.org/cycles/general_statement.html). 2See,forexample,thestatementoftheCEPRdatingcommittee: “Toreducethechancethatdatarevisionsmightlead theCommitteetoreconsideritschoiceofturningpointsinthefuture,theCommitteeexaminesawidearrayofeconomic datainadditiontoGDP,suchastheindividualcomponentsofoutputandlabormarketdata.” 2

frequencyandthedifferenttimelinessofmacroeconomicdatareleases.3 Thedynamicfactormodelistypicallyspecifiedbyassumings=0inequation(1).4 Werefertothissetofrestrictionsas dynamic homogeneity. Thisassumptioncanrepresentastraightjacketsinceitimposesthatdifferentindicatorshavethe thesamelead-lagpatternalongthebusinesscyclesandproportionalimpulseresponsefunctionstoacommonshock,i.e., toanexogenousshocktothesyntheticindex. Forthesereasons,inferenceistypicallyperformedonpre-selectedeconomic indicatorsthatarejudgedtobecoincidentalongthebusinesscycle,i.e.,indicatorsthat“havebeentolerablyconsistentin theirtiminginrelationtobusinesscyclerevivalsandthatatthesametimeareofsufficientlygeneralinteresttowarrant someattentionbystudentsofcurrenteconomicconditions”(seeMitchellandBurns,1938;Moore,1983). Inthispaper,werelaxtheassumptionofdynamichomogeneityandaccommodateforheterogeneousdynamicsbyincluding alargenumberoflags(s>>0)inequation(1). Themoregeneralstructurereducestheriskofmodelmiss-specification, enabling to extract more efficiently the information from economic indicators characterized by a significant degree of dynamic heterogeneity. However, the high level of generality comes at the cost of parameters proliferation. This could increase estimation uncertainty and induce overfitting, which, in turn, could offset the potential benefits of reduced misspecification,andjeopardizethereal-timeperformanceofthemodel. Inordertocounterbalancetheseperverseeffects, wecombinesampleinformationwithapriorbeliefthatlaggedeffectsofthecommonfactorarelessimportantthelonger thedelay.5 We conduct inference using data on real GDP and popular US coincident indices of business cycles. Following a fast growing literature on Bayesian factor models, we estimate the full set of posterior densities for the model’s parameters andfortheunobservedindexofbusinesscycleconditionsusingMontecarloMarkovChaintechniques(MCMC).6 Our framework encompasses the traditional approaches to the construction of business cycle indicators. In particular, principalcomponentsareproportionaltotheposteriormodeoftheunobservedfactorassociatedtoastaticfactormodel (s = 0 and serially uncorrelated factors ft and idiosyncratic components ei,t,i = 1,...,n) with homogenous signal-tonoise ratio and a flat prior. If the factor loadings are also assumed to be the same (λi(L) = λ¯) principal components become simple cross-sectional averages. Allowing for serial correlation, but keeping the model dynamically homogenous (s=0),weobtaintheIndexofCoincidentEconomicIndicatorsofStockandWatson(1992). In-sampleinference,basedonthemostrecentavailabledata,showsthatthefactorprovidesanaccuratecharacterization of the business cycle dynamics in the United States and suggests that dynamic heterogeneity is an important feature of the data. Indeed, the posterior distribution of the common synthetic index provides a more timely account of peaks andtroughswhencomparedwithalternativeindicatorsbasedondynamicfactormodels,liketheChicagoFEDNational ActivityIndex. Inaddition,theimpulseresponsesofdifferentindicatorstoacommonshockdisplayarelevantdegreeof heterogeneity. 3The dynamic factor model can be cast in a state space form, which provides a natural environment to deal with missingdataandmixedfrequencies;itisthenasuitabletoolfortheassessmentofeconomicconditionsinrealtime. See Giannoneetal.(2008);Aruobaetal.(2009);CamachoandPerez-Quiros(2010);Jungbackeretal.(2011);Ban´buraand Modugno(2014),andforsurveysBan´buraetal.(2011,2013). 4See, for example, Stock and Watson (1992); Kim and Nelson (1999); Mariano and Murasawa (2003); Aruoba et al. (2009);CamachoandPerez-Quiros(2010);Ban´buraetal.(2013). 5ThisisthesamelogicofthepriorbeliefspopularizedbyLitterman(1979)forBayesianVectorAutoregressions. 6SeeKimandNelson(1999);DelNegro(2002);Koseetal.(2003);Justiniano(2004);Bernankeetal.(2005);DelNegro andOtrok(2008);Mackowiaketal.(2009);Moench(2012). 3

As stressed above, factor models have proved to be successful not only in the extraction of synthetic indicators, but also for nowcasting in real time. We evaluate the accuracy of our model also along this dimension. This is important alsobecauseitrevealswhetherthein-samplepropertieswejustdescribedaregenuinefeaturesofthedataandnotonly an artifact due to overfitting. More in details, we study the properties of the model-based predictive distributions for GDP growth and compare them with the consensus probability assessments of the Survey of Professional Forecasters (SPF). In order to meaningfully compare the two sets of nowcasts, we take a fully real-time perspective, i.e., we collect the real-time vintages for our variables, which were available at the time the SPF was conducted. Results indicate that the predictive densities are correctly specified - well calibrated - since they cannot be statistically distinguished from the true unconditional data densities. Predictive scores reveal that the predictions of the model are, on average, more accuratethanthoseobtainedusingaunivariateautoregressivebenchmarkandcomparewellwiththeSPF.Overall,the out-of-sample evaluation indicates that dynamic heterogeneity is a genuine and salient feature of the data, and not just theresultofoverfitting. Therestofthepaperisstructuredasfollows. Section2describesthemodelandthereal-timedatabase. Section3studies thein-samplepropertiesofourindexofbusinesscycleconditions. Section4carriesoutaformalout-of-sampleevaluation ofthedensitynowcastsofourmodel. Section5concludes. 2 The model and the database 2.1 The dynamic factor model Weassumethatasetofvariablesxi,t,withi=1,...,n,ischaracterizedbythefollowingequations:7 xi,t=λi(L)ft+ei,t, i=1,...,n (2) whereλi(L)=λi,0+λi,1L+...+λi,sLs. Theprocessforthecommonfactorft andtheidiosyncraticcomponentsei,t, i=1,...,nareapproximatedbyfiniteautoregressive(AR)models: - a(L)ft=ut, ut∼i.i.d.N(0,1); - φi(L)ei,t=vi,t, vi,t∼i.i.d.N(0,σ i 2), i=1,...,n, wherea(L)=1−a1L−...−apf Lpf andφi(L)=1−φi,1L−...−φi,pe Lpe. Thecommonshocksut areassumedtobeorthogonaltotheidiosyncraticshocksvi,t,i=1...,n,atallleadsandlags. In addition, the idiosyncratic shocks are assumed to be mutually orthogonal at all leads and lags. Under this assumption, themodelisknownas“exact”sinceitimpliesthatthecross-correlationsamongobservablesareonlyduetothecommon factor. Although this assumption may be very restrictive, Doz et al. (2012) have shown that the model is robust to 7Weassume,withoutlossofgenerality,thatthevariablesaredemeanedandstandardized. Inpractice,wewillestimate themodelondemeanedandstandardizeddataandwewillre-attributemeanandstandarddeviationafterestimation,as itiscommonpracticeinthefactormodelliterature. 4

nonGaussianityandtoweakcorrelationamongidiosyncraticcomponents,providedthatestimationiscarriedoutwitha sufficientlylargenumberofhighlycollinearvariables. Thanks to the rich dynamics allowed by the polynomials λi(L) = (cid:80)q s=0 λisLs, the model can account for complex heterogeneity in the dynamic effects of the common factors on the observable variables. However, the generality of the model is obtained at the cost of the proliferation in the number of parameters to be estimated. This is the reason why themodelistypicallyestimatedwiths=0. Themostcommonlyusedsyntheticindexesareobtainedasposteriormodes ofthefollowingconstrainedmodels,whenaflatpriorisused: • Principalcomponents: staticfactormodel(s=pe=p f =0)withsphericalidiosyncraticcomponent,(σ i 2=σ¯2); • Cross-sectionalaverages: staticfactormodel(s=pe=p f =0)withsphericalidiosyncraticcomponent(σ i 2=σ¯2) andhomogenousloadings: λi(L)=λ¯ 0; • IndexofCoincidentEconomicIndicatorsofStockandWatson(1992): Homogenous(s=0)dynamic(pe=p f =2) factormodel. WewilldefinethismodelasDFM. Therestrictions=0impliesstronghomogeneityonthepropagationofthecommonshocksonthevariables. Inparticular, animplicationofthisassumptionisthatthefluctuationsofallvariablesareperfectlycoincidentoverthebusinesscycles.8 We retain the flexibility of the model, relaxing the homogeneity restriction, and we control for the over-fitting due to parametersproliferationbyshrinkingthemodelparameterstowardsthoseofasimplena¨ıvemodel,throughtheimposition ofpriors. Thepriordistributionsforallthecoefficientsarecenteredonzero,withstrongertightnessforhigher-orderlags, sothatposteriorcoefficientsofhigh-orderlagsofthefactorsaresufficientlyawayfromzeroonlyifthedatastronglyfavor non-zerovalues. Formalbayesianinferenceallowsustocombinetheinformationfromthedataandtheprior. Anequivalentrepresentationofthemodelisobtainedbypre-multiplyingbothsidesofequation(2)byφi(L): φi(L)xi,t=θi(L)ft+vi,t, vi,t∼i.i.d.N(0,σ i 2), i=1,...,n, a(L)ft=ut, ut∼i.i.d.N(0,1), whereθi(L)=φi(L)λi(L). Sinceλi(L)andφi(L)areunrestricted,wecanestimatethemodelreparameterizedinθi(L) andφi(L).9 Thedynamiceffectsofthecommonshocksonxi,t canberetrievedbytakingtheratioλi(L)= φ θi i ( ( L L ) ) . Our priorsspecifiedasfollows: • σ2∼IG(1,3), i • θ ∼N(0,τ 1 ), i,h (h+1)2 • φ ∼N(0,τ 1 ), i,h h2 • a ∼N(0,τ 1 ). h h2 8Dynamic heterogeneity can be taken into account in the context of principal components by including additional newfactors,withoutmodelingexplicitlythattheyarelaggedversionsofeachother(seeStockandWatson,2002). This approach is suitable for forecasting but not for measuring business cycles conditions since it delivers estimated factors thatarelinearcombinationsofcontemporaneousandlaggedvaluesoftheindexofeconomicactivity. 9QuahandSargent(1993)followthesamestrategy. 5

wherehindicatesthelagofthefactororthevariabletowhichthecoefficientisassociated. Thepriorcovarianceamong coefficientsassociatedtodifferentvariablesanddifferentlagsissettozero. Noticethatthevarianceofthepriorislower for the coefficients associated with more distant lags. The hyperparameter τ controls the scale of all the variances and effectivelygovernstheoveralllevelofshrinkage. Wefixthisparametertotheconventionalvalueof0.2.10 Thesepriors, includingthechoiceofthedegreeofoverallshrinkage,aresimilartothoseproposedbyLitterman(1979)inthecontext of Bayesian Vector Autoregressive models.11 Our dynamic factor model with unrestricted dynamics is shortly defined asHeterogenousDynamicFactorModel(HDFM).Inordertobeabletocaptureverygeneraldynamics, wespecifythe modelinordertoincludetwelvelagsoftheobservables,thecontemporaneousvalueandtwelvelagsofthefactorsinthe equationsoftheobservablesandtwelvelagsofthefactorsintheequationsdescribingthedynamicsofthefactors. Asstressedintheintroduction,weconductinferenceusingGibbssamplingtechniques. Ifalldataandalsothecommon factor were observed, drawing from the posterior of the parameters is simple since the prior is conjugate. Conditionally ontheparametersandtheobservabledata,thenthecommonfactorsandthemissingdatacanbedrawnusingsimulation smoothers (Carter and Kohn, 1994; de Jong and Shephard, 1995; Durbin and Koopman, 2002).12 In other words, the Gibbssamplerconsistsinalternatingthefollowingtwosteps: • givenadrawoftheparameters,drawthemissingdataandthelatentfactorconditionalontheobservationsusing thesimulationsmoother; • givenadrawofthethefulldataandthelatentfactors,drawtheparametersfromtheirposterior. Thealgorithmisinitializedbyusingtheparametersassociatedtoprincipalcomponentscomputedbyfittingmissingdata byasplinefunction. 2.2 Data We study the in-sample properties of the HDFM model and its accuracy in a real-time forecast evaluation by using a relatively small dataset for the US economy, including the most popular coincident indicators: real GDP (GDP), real disposableincome(DSPI),employment(EMP),industrialproduction(IP),andrealretailsales(RRS).13 Inaddition,we includethepurchasingmanagerindex(PMI)because,duetothetimelinessofitsrelease,itprovidesanextremelyuseful information.14 The variables are transformed in order to achieve stationarity. Real GDP enters in the model in terms of quarter-onquartergrowthrates,whilerealincome,employment,industrialproduction,andrealretailsalesenterthemodelinterms 10We leave for future research the task of conducting inference on the degree of prior tightness, which could be done followingthelinesofGiannoneetal.(2012). 11Wedonotneedtorescalethevariancesofthepriorstoadjustforthedifferentscaleofthevariables,asitiscustomary inBVARapplications,sinceweperforminferenceusingstandardizeddata. 12For a general discussion about the formulation of the state space in presence of missing data, see Ban´bura et al. (2014). 13ThesearealsothemostrelevantindicatorsconstantlymonitoredbytheNBERtodetectanddatepeaksandthroughs inthebusinesscycle. 14TheimportanceofsurveydatafornowcastinghasbeendocumentedbyGiannoneetal.(2008),Giannoneetal.(2009), Angelinietal.(2011),LahiriandMonokroussos(2013). Forasurvey,seeBan´buraetal.(2011,2013). 6

of month-on-month growth rates. The PMI is stationary by construction, therefore it enters the model without being transformed.15 This dataset is characterized by mixed frequencies because real GDP is sampled quarterly, while all the other variables aresampledmonthly. Inordertodealwiththisissue,wetreatrealGDPasobservableinthelastmonthofthequarter. Thefirsttwomonthsofthequarteraretreatedasmissingobservations. Thisapproachisconvenientsince,asexplained insection2.1,thealgorithmusedforinferencecaneasilydealwithmissingdata. The main benchmark in our forecasting evaluation is the Survey of Professional Forecasters (SPF). For the sake of comparability,weexactlyreplicatetheinformationsetavailabletotheprofessionalforecastersatthetimetheyproduced theirownforecasts.16 Specifically,theforecastsaregeneratedeveryquarterwiththeinformationavailableonthe14th of thesecondmonthinthequarter,whichisroughlyinlinewiththedeadlineforthesubmissionoftheSPFquestionnaires. Theforecastingevaluationiscarriedoutusingelevenyearsofvintages,rangingfromQ1-2003toQ4-2013. ForeachrealtimedatavintagethesamplestartsinJanuary1993. Westarttheevaluationin2003inordertohaveafirstestimation sampleoftenyears. 17 The real-time exercise introduces an additional source of missing data due to the different availabilities of the data at thetimeforecastsaregenerated. Infact,onthe14th ofeachsecondmonthofthequarter,realGDPisavailableforthe previousquarter(e.g.,inFebruaryrealGDPisavailableuptoQ4ofthepreviousyear),employmentandPMIareavailable uptothepreviousmonth(e.g.,inFebruarytheyareavailableuptoJanuary),realretailsales,andrealdisposableincome are available up to two months before (e.g., in February they are available up to December). Industrial production is usuallyreleasedatmidmonth(betweenthe13thandthe17thofeachmonth)and, hence, dependingonthevintage, it can have either the same availability of employment and PMI or the same availability of disposable income and retail sales. 3 The synthetic business cycle indicator and dynamic heterogeneity In this Section we perform in-sample inference using data from the last vintage in our dataset (February 2014). The real-timeevaluationisconductedinthenextSection. Figure1plotstherealGDPgrowthrateagainsttheothervariablesincludedinthedatabase. INSERTFIGURE1HERE Some features stand out. First, all variables tend to comove with GDP, especially during periods of downturn. Second, 15Each month, survey respondents are asked to assess their organizations’ performance based on a comparison of the currentmonthtothepreviousmonth,seehttp://www.ism.ws/files/ISMReport/ROBBroch08.pdf. 16RealtimevintagesaredownloadedfromtheFederalReserveBankofSt. Louishttp://alfred.stlouisfed.org/. 17Weusearecursiveupdatingschemeinourout-of-sampleforecastingevaluation,i.e.,foreveryvintagetheestimation samplealwaysstartsinJanuary1993. 7

real disposable income, industrial production and real retail sales display very noisy short-run fluctuations, which tend to hide the lower-frequency fluctuations. Third, the variables exhibit a different lead-lag pattern, which is particularly visible around the great recession. PMI and employment growth tend to lag GDP growth; RRS has a more coincident pattern,whereasDSPIandIPdisplayleadingdynamics,providinganearlysignaloftherecession. Figure 2 plots the six variables versus the HDFM business cycle indicator (median, 16th and 84th quantiles of the distribution),whichisexplicitlydevisedtoaccountforheterogeneouslead-lagstructureofthevariables. INSERTFIGURE2HERE In general, the HDFM indicator tracks our variables very well. This validates the strategy of estimating a dynamic factormodeltocapturethecomovementamongthevariables. Inaddition,theindicatorissmootherthantheindividual variables,suggestingthatalargepartoftheirhigh-frequencyfluctuationsareofidiosyncraticnature. Moreindetails,the indicator is roughly coincident with DSPI, IP and RRS (first three sub-plots) and it clearly leads EMP, PMI and GDP (lastthreesub-plots),henceitprovidesan“average”ofthevariableswhosedynamicheterogeneityisproperlytakeninto account. Ontheotherhand,traditionalmethodsforfactorextractionwouldassignmostoftheweightstothevariables withthemostpersistentdynamicsandlessvolatility(EMPandPMIinourcase),andtheunderlyingestimatedindicator wouldbeheavilyshapedbytheseseries. Toillustratethispoint,Figure3comparestheindicatorextractedbyemploying the HDFM in Section 2.1 with four of the most common methods for factor extraction: the average of the monthly variables included in the panel, the first principal component (PC) of the monthly variables, the Chicago Fed National Activity Index (CFNAI) and the factor extracted from a model that imposes dynamic homogeneity. The latter is the posterior mode of the common factor in our model, estimated under the restriction of complete dynamic homogeneity (DFM).18 INSERTFIGURE3HERE TheHDFMindicator,whichisdesignedtoexploitthedynamicheterogeneityofthevariables,leadsalltheotherindicators whichdonottakeintoaccountthisimportantfeatureofthedata. Itisworthstressingthatthisisapuremodelingissue, notrelatedatallwiththedimensionoftheinformationset;indeed,theCFNAIindex,whichisextractedfromapanelof 85monthlyseries,alsolagsthedynamicsofourindicator.19 These results have also non trivial implications on the traditional simulation exercises, which are performed to assess thesystemdynamicsaftersomeexogenousshock. Toclarifythispoint,Figure4reportstheimpulseresponsefunctions (IRFs) of the (log-)levels of the six variables to a common shock, that is an exogenous shock to the synthetic business cycle index. The red line refers to the median IRF estimated by means of the DFM model, which imposes dynamic 18Notice that with flat prior the posterior mode of the model parameters corresponds to the Maximum Likelihood estimates. FollowingDozetal.(2012)MaximumLikelihoodestimationisperformedbyusingtheEMalgorithminitialized by principal components. The algorithm is modified to account for arbitrary patterns of missing data following the procedureofBan´buraandModugno(2014). Thealgorithmhasbeenshowntobecomputationallyefficientandfeasible evenwithhigh-dimensionaldata. RecentresultsbyJungbackeretal.(2011)andJungbackerandKoopman(2015)show howcomputationalefficiencycanbefurtherimproved. 19Seehttps://www.chicagofed.org/publications/cfnai/index 8

homogeneity in the effects of the exogenous shock to the synthetic business cycle indicator. The blue lines refer to the IRFs(median,16thand84thquantilesofthedistribution)oftheHDFM.Forallvariables,theIRFsofthelog-levelsare obtainedbycumulatingtheIRFsofthegrowthrates,withtheexceptionofPMI,whichisnottransformed,andforwhich themodelproducesdirectlytheIRFsofthelevels. INSERTFIGURE4HERE Themostimportantdifferencebetweenthetwoapproachesisthat,whendynamicheterogeneityisexcludedbyassumption, theIRFshaveessentiallythesamedynamics,uptoare-scalingfactorgivenbythefactorloadings. Indeed,thecumulated IRFsforamodelwithdynamichomogeneityis: ∂x i,t+h =λi,0 (cid:88) h bj (3) ∂ut j=0 where bj are the coefficients of the polynomial b(L) = a(L)−1, and where a(L) is the polynomial that captures the dynamics of the common factor as described in Section 2.1. As noticed above, the only difference among the IRFs of differentvariablesistheirloadingsλi,0. Instead,accountingforthedynamicheterogeneity,theIRFsareallowedtodiffer: ∂x i,t+h = (cid:88) h ci,j (4) ∂ut j=0 whereci,j arethecoefficientsofthepolynomialc(L)=θi(L)φi(L)−1a(L)−1. Inthiscase,theIRFsforaspecificvariable willdiffernotonlybyare-scalingfactor, butalsobythepotentiallydifferentimportancethatlagsofthevariableitself and of the factors have in explaining the fluctuations. This is evident in Figure 4, where the dynamics captured by the redlinesarealikeamongvariables. Instead,whenadifferentlead-lagstructureamongthevariablesisallowed,theIRFs may have different dynamics. In fact, the blue lines in the figures show that the variables have heterogeneous patterns aftertheshock. ThemoststrikingexampleisPMI,thatdisplaysaclearhump-shapedreactiontoashocktothecommon componentintheHDFM,whilethisisnotthecasefortheDFM. 4 Evaluation of the density nowcasts Dynamic factor models are known to perform very well as forecasting tools (see Stock and Watson, 2011, for a survey). However, the specification we advocate in this paper is richly parameterized, hence parameters estimation uncertainty andoverfittingareanimportantconcern. Forthisreason,weevaluatetheout-of-sample(real-time)predictiveabilityof themodel. Bayesian estimation methods allow us to rigorously account for all sources of uncertainty and, hence, we put particular emphasis on density forecast evaluation. We test the density nowcast accuracy of our HDFM against two popular benchmarks: theGDPnowcastsfromtheSurveyofProfessionalForecastersandthosefromana¨ıveautoregressivemodel. 9

Toourknowledge,thisisthefirstpaperthatcomparesprobabilisticforecastsofmodelsandinstitutionsinafullyreal-time perspective. For the sake of comparability, our out-of-sample exercise is designed to replicate the features of the SPF. Specifically, we ask what the model would predict if used in “real time” to answer the SPF questionnaire for GDP growth. More indetails, wecollected44vintagesofdatawhichwereavailable, inreal time, totheforecastersin thequartersbetween 2003Q1 and 2013Q4 and, at each point in time, we use the model to derive a nowcast of the GDP growth rate in the current calendar year. For example, by using the data vintage available in the first quarter of 2003, we nowcast GDP growthinthatquarterandweforecastGDPgrowthinthesubsequentquartersof2003toderivetheannualgrowthrate for2003. Theannualgrowthrategt forGDPinthecalendaryearty isdefinedasthegrowthrateintheaveragelevelofGDPover thefourquartersinyearty,comparedtotheaverageannualleveloverthefourquartersinyearty−1: (cid:32) (cid:33) gty =100∗ GDP G Q1 D ,t P y Q − 1 1 ,t + y G + D G P D Q P 2 Q ,ty 2, − ty 1+ + G G D D P P Q Q 3 3 , , t t y y − + 1 G + D G P D Q P 4 Q ,ty 4,ty−1 −1 whereGDPQj,ty isthelevelofGDPinthejth quarterofyearty. SinceGDPentersintermsofquarterlygrowthintheHDFM,thecalendaryeargrowthneedtobederivedstartingfrom the quarterly growth profiles. This is achieved in two steps: first we approximate the year-over-year (yoy) growth rates as a four quarters moving average of annualized quarter-over-quarter (qoq) growth rates; second, we approximate the calendaryeargrowthastheaverageoftheyoygrowthrateswithinthecalendaryear.20 Onceagain,ourchoiceisdrivenbythefactthattheSPFdensityforecastsareonlypubliclyavailableforthisdefinitionof thegrowthrate.21 Thena¨ıvebenchmarkisanautoregressivemodelofordertwo. Whenlookingatdatainrealtime,one issue to address is which data vintage is used to compute the outcome of the target variable. Our choice is to consider thefirstvintageinthesampleinwhichthedataforthefullcalendaryeararemadeavailable. Figure5reportsthenowcasts(median,16thand84thquantiles,dashedlines)ofannualGDPgrowthfortheHDFMand theautoregressivemodel(AR).22 ThebluesolidlineinthechartsreferstotherealizationsoftheannualGDP.Forboth models,wereportthenowcastscomputedineachquarteroftheyear.23 INSERTFIGURE5HERE Figure 5 shows that the density nowcasts of the HDFM are generally centered around the outcome, already in the first quarteroftheyear,differentlyfromtheARnowcasts. TheuncertaintyonthegrowthrateofGDPinthecalendaryear decreasesand,byconsequence,thenowcastdensitiesbecomenarrower. 20Formally, defining tq as the last quarter of the calendar year of interest, the computation is equivalent to (1+L+ L2+L3)(1−L4)logGDPtq ×400. ForarecentdiscussionseeRichardCrumpandMoench(2014). 21TheSPFalsotargetstheGDPgrowthrateinthecurrentquarter,butthedensityforecastsforthisdefinitionarenot available. 22The order of the autoregressive model is set equal to two, as suggested by the Akaike criteria computed in first estimation sample (1993-2003). Results are similar when using only one lag, when the selection is updated recursively andwhenthecoefficientsofthebenchmarkarerestrictedtothoseoftherandomwalk. 23ThedensitynowcastsoftheSPFarenotincludedinthefigurebecausetheyareavailableonlyintermsofhistograms. 10

Next,weevaluatemoreformallywhethertheHDFMdensitynowcastsareagoodapproximationofthetruedatadensities, by testing the uniformity of the probability integral transforms (PITs).24 The PITs are the value of the predictive cumulativedistributionevaluatedatthetruerealizedvaluesofthevariablesandarewidelyusedtoassessthecalibration of density forecasts (most recent works include Aastveit et al., 2011; Mitchell and Wallis, 2011; Geweke and Amisano, 2010;Clark,2011). Infact,Dieboldetal.(1998)showthat,ifthedensityforecastsapproximatewellthetruedensity(i.e., are“wellcalibrated”),thenthePITsshouldbeuniformlydistributedintheinterval[0−1]. Assessingtheuniformityof thePITsisequivalenttocheckingwhethertheinversenormaltransformationofthePITsisstandardnormal. Wecompare thefirstfourthsamplemomentsofthePITsinversenormaltransformationaredifferentfromthefirstfourmomentsofthe standardnormaldistribution(zero,one,zeroandthreerespectively). Table2reportsthefoursamplemoments(columns twotofive)foreachofthenowcastscomputedinthefourquartersoftheyear(rowstwotofive). FollowingBaiandNg (2005)wereporttheheteroskedasticityandautocorrelationconsistent(HAC)standarddeviationestimatortoprovidea roughideaofthestatisticalsignificance.25 Table 2: Tests of normality, HDFM nowcasts. Quarter First moment Second moment Third moment Fourth moment Q1 -0.47 0.52 -0.56 0.72 (0.58) (0.70) (1.02) (1.42) Q2 -0.20 0.83 -0.09 1.60 (0.94) (0.99) (1.97) (2.50) Q3 0.35 0.92 0.70 2.20 (0.93) (1.22) (2.78) (5.11) Q4 0.52 0.77 0.89 1.24 (0.74) (0.84) (1.24) (1.68) Note: Samplemomentsinthefourquarters. Standarddeviationinparentheses. Table 2 shows that all the sample moments are close to the theoretical values for the standard normal distribution, indicating that the density nowcasts of the HDFM model are well calibrated. We now turn to the analysis of the “relative”accuracyoftheHDFMdensitynowcasts,comparingtheirlog-scorestothoseofthealternatives. In the SPF the forecasters are asked to report, among other things, a density forecast by allocating probabilities to ranges of possible future outcomes of the annual growth of GDP. The lower bottom interval and the upper interval of the range are open bins, and the interior bins have equal lengths of 1 percentage points.26 Individual responses are aggregatedbycomputingaverageprobabilities. Forthesakeofcomparability,weorganizetheoutputofthemodel-based nowcasts(HDFMandAR)alongthesamelinesoftheSPFquestionnaire. Inotherwords,forallmodels,wecomputethe percentage(frequency)oftheoutcomesthatfallinthedifferentbinsidentifiedintheSurveyofProfessionalForecasters. 24Notice that our evaluation is more demanding that the traditional residual based diagnostics since the predictive densitiesarecomputedinrealtime,henceaccountingforparameterestimationuncertaintyandoverfitting. 25The implied t-statistics should be taken with caution since the asymptotic distribution is non standard due to the recursiveestimationoftheparameters(seeMcCrackenandClark,2013). 26Untilthefirstquarterof2009theupperboundofthelowerbottomintervalis2%. Startinginthesecondquarterof 2009itis3%. 11

Then,wecomputethelog-scores,foreachmodelandperiod,definedasthelogarithmofthefrequencyofthebinincluding the observed annual GDP growth rate. The higher the mean of the log-scores, the higher the accuracy of the density nowcasts. In Table 3, column one indicates the quarter in which the nowcast for the calendar year is produced. In the second column, we report the average HDFM log-scores. For the AR (column three) and SPF (column four), instead, wereportthedifferenceoftheaveragelog-scoreswiththeHDFMcounterpart: positivevaluesindicatethattheaverage log-scoreofthespecificmodelishigherthantheaveragelog-scoreoftheHDFMforthatspecificquarter,andviceversa. TheHACestimateofitsstandarddeviationarereportedinparentheses.27 Table 3: Evaluation of density nowcasts for the calendar year, average log-scores. Quarter HDFM AR minus HDFM SPF minus HDFM Q1 -1.27 -0.16 0.15 (0.24) (0.11) Q2 -1.17 0.22 0.12 (0.13) (0.10) Q3 -0.68 -0.10 -0.03 (0.12) (0.18) Q4 -0.18 -0.01 -0.22 (0.03) (0.07) Note: HDFM (column two), average log-scores. AR (column three) and SPF (column four), average log-scores minus average HDFM log-scores. Standarddeviationinparentheses. ResultsinthefirstcolumnofTable3showthat,asexpected,theaccuracyimprovesasmoreinformationbecomesavailable duringtheyear. TheresultsinthesecondcolumnindicatethattheHDFMisgenerallymoreaccuratethantheARmodel. Thethirdcolumnindicatesthat,whiletheSPFdensitynowcastsaremoreaccuratethanthoseoftheHDFMinthefirst twoquartersoftheyear,theoppositeistrueinthethirdandfourthquarteroftheyear. However,thestandarddeviations ofthesamplemeanofthedifferenceinlog-scores(inparentheses)arequitelargecomparedtotheaveragedifferencesin log-scores,sothedifferencesareunlikelytobestatisticallysignificant. Sincetheevaluationsampleisshort,theforecasting evaluationshouldnotbeseenasahorserace,butratherasanassessmentofthevalidityofthemodel,aimingtoascertain that the accuracy of the density nowcasts is preserved, in spite of the proliferation of parameters resulting from taking intoaccountgeneralpatternsofdynamicheterogeneity.28 Figures6to8zoomonthreespecificcalendaryears,the2008,2009,and2010. InthethreeFigureswereporttheevolution overthefourquartersof2008,2009,and2010ofthedensitynowcasts,informofhistograms,oftheHDFM,theAR,and theSPF. INSERTFIGURE6to8HERE 27Seefootnote25. 28Results not reported here show that the HDFM model does not significantly outperform the homogenous model in terms ofreal-time out-of-sampleforecasting accuracy. Thisresult indicates thatdynamic heterogeneity, although itis a featureofthedata,isnotsoprominenttosignificantlyimprovetheforecastingperformanceofthemodel,atleastnotin theshortevaluationsampleconsideredhere. 12

Figures 6 to 8 show that, particularly in the most acute part of the recession, the HDFM outperforms the AR model andprovidesverysimilaroutcomestotheSPF.Thisresultshowsthataccountingfordifferentsourcesofinformationis important(forexample,surveyswereanimportantsourceofinformationtotimelycapturethegreatrecession). Moreover, ithighlightshowtheHDFM,inspiteofitsmechanicalnature,isabletoreplicatetheoutcomesofthesurveyofprofessional forecastersthat,presumably,incorporateshumanjudgement. Theseabilityofmechanicalmodelstoreplicatebestpractice innowcastingGDPgrowthhasbeenextensivelydocumentedinthecontextofpointforecasts. Thefindingaboveindicate thatthisstylizedfactalsoholdsfordensityforecasts. 5 Conclusions A synthetic indicator of economic activity should condense, in a timely and reliable manner, the information of several alternativeobservablemeasures. Theindicatorproposedinthispaperisbasedonadynamicfactormodelthatexplicitly allowsfordynamicheterogeneityintheeffectsofthecommonfactoronthevariables. Sincethemodelisrichlyparameterized,wecontrolforoverfittingbycombiningsampleinformationwithapriorbeliefthattheeffectsoflaggedfactorson theobservedeconomicindicatorsaremoreimportanttheshorterthelag. Empiricalresultssupportourmodelingstrategy andindicatethatitisimportantandfeasibletoaccountforgeneralpatternsofdynamicheterogeneityinthecontextof dynamicfactormodels. Indeed,inferencebasedonourframeworkprovidesatimelyaccountofthebusinesscyclepeaks andthroughsand,inrealtime,itprovidesaccurateandwell-calibratedpredictivedensities. In this paper, we have focused on a relatively small set of indicators that have been pre-classified as coincident on the basis of a long-established tradition in business cycle analysis. However, the general framework can be used to analyze more general dataset. Indeed, because of the high level of generality, the dynamic heterogenous factor model allows to analyzesimultaneouslyavarietyofindicators,withouttheneedofpre-testingorexpertjudgementfortheclassification basedonlead-lagpatterns. Thiscanbeparticularlyimportantwhendealingwithdatasetscharacterizedbyblurredlines of separation between coincident, leading and lagging indicators, as it tends to be the case when considering additional indicators and other countries. Evidence in this direction has been provided by Luciani and Ricci (2013) who have successfullyusedourmethodologytonowcastNorway. References Aastveit, K. A., K. R. Gerdrup, A. S. Jore, and L. A. Thorsrud (2011, September). Nowcasting GDP in Real-Time: A DensityCombinationApproach. WorkingPapers0003,CentreforAppliedMacro-andPetroleumeconomics(CAMP), BINorwegianBusinessSchool. Angelini, E., G. Camba-M´endez, D. Giannone, L. Reichlin, and G. Ru¨nstler (2011, February). Short-term forecasts of euroareaGDPgrowth. Econometrics Journal 14(1),C25–C44. Aruoba,S.B.,F.X.Diebold,andC.Scotti(2009). Real-TimeMeasurementofBusinessConditions. JournalofBusiness & Economic Statistics 27(4),417–427. 13

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Figure 1: GDP and other variables DSPI IP 2 2 0 0 −2 −2 −4 −4 Jan00 Jan10 Jan00 Jan10 RRS EMP 2 2 0 0 −2 −2 −4 −4 Jan00 Jan10 Jan00 Jan10 PMI 2 Variable 0 GDP −2 −4 Jan00 Jan10 Note: redline: quarter-on-quarterrealGDPgrowthrate;blueline: month-on-monthrealdisposableincomegrowthrate(DSPI), month-on-monthindustrialproductiongrowthrate(IP),month-on-monthrealretailsalesgrowthrate(RRS),month-on-monthemployment growthrate(EMP),andlevelofthepurchasingmanagerindex(PMI).Duetothedifferentsamplingfrequency,GDPgrowthisreportedas constantinthethreemonthsofeachquarter. Sources: U.S.BureauofEconomicAnalysis(BEA),GrossDomesticProduct(GDP),Disposable IncomeGrowthRate(DSPI);BoardofGovernorsoftheFederalReserveSystem,IndustrialProductionIndex(IP);FederalReserveBankofSt. Louis,RealRetailSalesGrowthRate(RRS);U.S.BureauofLaborStatistics,Employmentgrowthrate(EMP);InstituteforSupply Management,PurchasingManagersIndex(PMI). 17

Figure 2: Common factor and variables DSPI IP 4 2 2 0 0 −2 −2 −4 −4 −6 −6 Jan00 Jan10 Jan00 Jan10 RRS EMP 2 5 0 0 −2 −4 −5 −6 Jan00 Jan10 Jan00 Jan10 PMI GDP 2 2 0 0 −2 −2 −4 −4 −6 −6 Jan00 Jan10 Jan00 Jan10 Factor Variable 68% c.i. Note: Bluelines: HDFMbusinesscycleindicator(mediansolidline,16thand84thquantilesdashedlines);Redline: month-on-monthreal disposableincomegrowthrate(DSPI),month-on-monthindustrialproductiongrowthrate(IP),month-on-monthrealretailsalesgrowthrate (RRS),month-on-monthemploymentgrowthrate(EMP),thelevelofthepurchasingmanagerindex(PMI),andthequarter-on-quarterreal GDPgrowthrate,reportedasaconstantinthethreemonthsofeachquarter. Sources: U.S.BureauofEconomicAnalysis(BEA),Gross DomesticProduct(GDP),DisposableIncomeGrowthRate(DSPI);BoardofGovernorsoftheFederalReserveSystem,IndustrialProduction Index(IP);FederalReserveBankofSt. Louis,RealRetailSalesGrowthRate(RRS);U.S.BureauofLaborStatistics,Employmentgrowthrate (EMP);InstituteforSupplyManagement,PurchasingManagersIndex(PMI). 18

Figure 3: HDFM and other indicators Mean PC 2 2 0 0 −2 −2 −4 −4 −6 −6 Jan95 Jan00 Jan05 Jan10 Jan95 Jan00 Jan05 Jan10 CFNAI DFM 2 2 0 0 −2 −2 −4 −4 −6 −6 Jan95 Jan00 Jan05 Jan10 Jan95 Jan00 Jan05 Jan10 Indicator HDFM 68% c.i. Note: Bluelines: HDFMbusinesscycleindicator(mediansolidline,16thand84thquantilesdashedlines);Redline: simpleaverageofthe variables(mean),firstprincipalcomponentofthevariables(PC),theChicagoFednationalactivityindex(CFNAI)andthefactorextracted fromthehomogeneousdynamicfactormodel(DFM). 19

Figure 4: IRFs of all variables to a common shock DSPI IP 2 4 1 3 0 2 1 −1 0 5 10 15 20 0 5 10 15 20 months after the shock months after the shock RRS EMP 3 2 1 1 0.5 0 −1 0 0 5 10 15 20 0 5 10 15 20 months after the shock months after the shock PMI GDP 2 5 4 1 3 2 0 1 0 5 10 15 20 0 1 2 3 4 5 6 7 months after the shock quarters after the shock HDFM DFM 68% c.i. Note: Bluelines: HDFMIRFofthelog-levelsofthevariables(exceptforPMI,forwhichwereportlevels)toacommonshock(mediansolid line,16thand84thquantilesdashedlines);Redline: DFMIRFofthelog-levelsofthevariables(exceptforPMI,forwhichwereportlevels)of thevariablestoacommonshock(median). 20

Figure 5: Nowcasts for the calendar year - 2003 to 2013 6 4 2 0 −2 −4 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 HDFM retrauQ 1 6 4 2 0 −2 −4 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 AR 6 4 2 0 −2 −4 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 HDFM retrauQ 2 6 4 2 0 −2 −4 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 AR 6 4 2 0 −2 −4 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 HDFM retrauQ 3 6 4 2 0 −2 −4 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 AR 6 4 2 0 −2 −4 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 HDFM retrauQ 4 6 4 2 0 −2 −4 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 AR Note: Leftpanels: HDFMnowcasts(dashedgreen)andout-turns(solidblue). Rightpanels: ARnowcasts(dashedred)andout-turns(solid blue). Allpanelsreportmedian,16thand84thquantilesofthedensitynowcasts. Fromtoptobottom,nowcastsproducedinquarter1,2,3and 4. 21

Figure 6: Case study - calendar year 2008 Q1−08 Q2−08 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −2 0 2 4 6 −2 0 2 4 6 Q3−08 Q4−08 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −2 0 2 4 6 −2 0 2 4 6 Actual HDFM SPF AR Note: Reddashedline: observedcalendaryearrealGDPgrowthrate;x-axis: SPFbins;Bluebars: probabilitiesassignedbytheHDFMtothe bins;Purplebars: probabilitiesassignedbytheARtothebins;Greenbars: probabilitiesassignedbytheSPFtothebins. 22

Figure 7: Case study - calendar year 2009 Q1−09 Q2−09 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −2 0 2 4 6 −4 −2 0 2 4 6 Q3−09 Q4−09 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −4 −2 0 2 4 6 −4 −2 0 2 4 6 Actual HDFM SPF AR Note: Reddashedline: observedcalendaryearrealGDPgrowthrate;x-axis: SPFbins;Bluebars: probabilitiesassignedbytheHDFMtothe bins;Purplebars: probabilitiesassignedbytheARtothebins;Greenbars: probabilitiesassignedbytheSPFtothebins. 23

Figure 8: Case study - calendar year 2010 Q1−10 Q2−10 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −4 −2 0 2 4 6 −4 −2 0 2 4 6 Q3−10 Q4−10 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −4 −2 0 2 4 6 −4 −2 0 2 4 6 Actual HDFM SPF AR Note: Reddashedline: observedcalendaryearrealGDPgrowthrate;x-axis: SPFbins;Bluebars: probabilitiesassignedbytheHDFMtothe bins;Purplebars: probabilitiesassignedbytheARtothebins;Greenbars: probabilitiesassignedbytheSPFtothebins. 24