Financial and Macroeconomic Data Through the Lens of a Nonlinear Dynamic Factor Model

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Financial and Macroeconomic Data Through the Lens of a Nonlinear Dynamic Factor Model Pablo A. Guerro´n Quintana, Alexey Khazanov, Molin Zhong 2023-027 Please cite this paper as: Guerr´on Quintana, Pablo A., Alexey Khazanov, and Molin Zhong (2023). “Financial and Macroeconomic Data Through the Lens of a Nonlinear Dynamic Factor Model,” Finance and Economics Discussion Series 2023-027. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2023.027. 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.

Financial and Macroeconomic Data Through the Lens of a Nonlinear Dynamic Factor Model∗ Pablo A. Guerrón Quintana† Alexey Khazanov‡ Molin Zhong§ April 26, 2023 Abstract Throughthelensofanonlineardynamicfactormodel,westudytheroleofexogenousshocksandinternalpropagationforcesindrivingthefluctuationsofmacroeconomicandfinancialdata. Theproposedmodel1)allowsfornonlineardynamicsinthe state and measurement equations; 2) can generate asymmetric, state-dependent, and size-dependentresponses ofobservablestoshocks; and3)can producetime-varying volatility and asymmetric tail risks in predictive distributions. We find evidence in favorofnonlineardynamicsintwoimportantU.S.applications. Thefirstusesinterest rate data to extract a factor allowing for an effective lower bound and nonlinear dynamics. Our estimated factor coheres well with the historical narrative of monetary policy. We find that allowing for an effective lower bound constraint is crucial. The second recovers a credit cycle. The nonlinear component of the factor boosts credit growthinboomtimeswhilehindersitsrecoverypost-crisis. Shocksinacreditcrunch periodaremoreamplifiedandpersistforlongercomparedwithshocksduringacredit boom. Keywords— Interest rates, effective lower bound, credit cycle, asymmetric dynamics, predictivedistributions,tailrisk ∗WethankToddClarkandTaeyoungDohforinsightfulcommentsandsuggestions. Wealsobenefited fromcommentsbyBoraganAruoba,MarkBognanni,ThorstenDrautzburg,JesusFernandez-Villaverde,AndrewFoerster,ManuelGonzales-Astudillo,LucaGuerrieri,ArthurLewbel,MicheleModugno,JuanRubio- Ramirez,FrankSchorfheide,DonghoSong,andseminarparticipantsattheBoardofGovernors,European CentralBank,BankofEngland,CEF2021,IAAE2021,NBER-NSFSBIES2021,NBER-NSFTimeSeries2021, SystemEconometricsMeetings2021,forusefulcomments. Allerrorsareours. ThispaperdoesnotnecessarilyreflecttheviewsoftheFederalReserveSystemoritsBoardofGovernors. †BostonCollege,email: pguerron@gmail.com. ‡TheHebrewUniversityofJerusalem,email: alexey.khazanov@mail.huji.ac.il. §BoardofGovernors,email: Molin.Zhong@frb.gov. 1

1 Introduction What are the drivers of financial and macroeconomic variables - external shocks or endogenous propagation? This is a timeless question in economics. Understanding the origins of economic fluctuationsbecomesevermoreimportantduringdownturnsbecauseoftheirimplicationsforthe scope of economic policies. To address this question, dynamic factor models (DFM) have played a crucial role (Stock and Watson, 2016). Since their introduction in the 1970s, DFMs have relied mostlyonavectorautoregressiverepresentationthatimposesalinearrelationbetweenthefactor todayanditspastaswellasbetweenthefactorandtheobservables.1 However,thereissubstantial evidence showing that economic variables possess significant nonlinearities (Baker et al., 2016, Fernández-Villaverde et al., 2015b, Justiniano and Primiceri, 2008). Furthermore, financial data aremorepronetosuddenchanges,particularlysoduringtimesofcrises(GilchristandZakrajsek, 2012, Ludvigson et al., 2021). In this paper, we study macroeconomic and financial data through thelensofadynamicfactormodelthatincorporatesnonlinearitiesinthemeasurementandstate equations. The nonlinearities in the DFM allow us to exploit many features emphasized in the recent macroeconomicandfinancialliteraturethatpreviousworkonfactormodelshavelargelyleftunexplored. For example, we can use our new nonlinear dynamic factor model (NLDF) to examine the importance of nonlinear dynamics in the state equation during moments of high volatility in theeconomysuchastheGlobalFinancialCrises(GFC);toconstructpointandpredictivedensity estimators in the presence of nonlinearities; and to study the truncated relation between factors andobservablesasintheshadowinterestrateliterature(WuandXia,2016). Importantly,werely onacoherentframeworktosimultaneouslystudyalloftheseforcestogether. Ournonlinearfactormodelisinspiredbytheprunedsecondorderstate-space(2ndSS)model discussed in Kim et al. (2008) and Andreasen et al. (2017) as the approximate solution of a nonlineardynamicstochasticgeneralequilibriummodel. AsdiscussedinFernandez-Villaverdeetal. (2016)andthereferencestherein,animportantfeatureoftheprunedsolutionisthatitcancapture several types of nonlinearities with reasonable accuracy, a feature that we aim to exploit in this paper. Were-interpretthe2ndSSframeworkinthecontextofadynamicfactormodelwhosefactor evolvesaccordingtothestate-spacemodel’sstateequation. Weallowthemeasurementequation to be potentially nonlinear if economic theory suggests it. This accommodates situations where theobservablesareboundedandallowsforthepresenceofnon-additivemeasurementerrors. Thenonlinearstatedynamicsgeneratenovelimplicationsbothintermsoftheimpulseresponse 1Thestandardrepresentationfollows: Y =ΛF +e ; F =Ψ(L)F +v . t t t t t−1 t Here, the first and second expressions correspond to the measurement and state equations, respectively. Theshockse andv areassumednormallydistributedandindependentovertimeandcross-sectionally. L t t isthelagoperator. 2

functions(IRFs)andthepredictivedensities. Althoughthenonlinearitiesarespecifiedatthefactor level,theyarepassedthroughtotheobservablevariablesviathemeasurementequation. TheIRFs producedbythemodelhavethreeinterestingproperties. First,theyareasymmetric,meaningthat positive shocks produce differently-shaped IRFs compared to same-sized negative shocks. Second, theyarestate-dependent, meaningthattheshapesandmagnitudesoftheIRFsaredifferent dependingoninitialconditions. Finally,theyaresize-dependent,meaningthatonestandarddeviationshocksgeneratedifferentshapesintheIRFscomparedtotwostandarddeviationshocks. The modelhasimplicationsforhigher-ordermomentsaswell,leadingtorichdistributionaldynamics. Despite homoskedastic and normally distributed innovations, the unconditional distribution of thestatesisnon-normalduetotheasymmetriesimpliedbythemodel. Moreover,themodelgenerates predictive densities that have time-varying volatility and asymmetric tail risk movements. ThelatterfactisakeypropertydocumentedbyAdrianetal.(2019)inthemacrodata. WiththeNLDFmodelinhand,weanalyzetheroleofexternalforcesandinternalpropagation in two macroeconomic andfinancial cases. In our first case, we estimatethe shadow interest rate model along the lines of Wu and Xia (2016). Our exercise extracts a common factor from a series ofU.S.forwardrateswhilerespectingtheeffectivelowerboundintheshort-maturityrates. There are two key differences from Wu and Xia (2016). First, we model the yields in first differences, followingtherecommendationsofOnatskiandWang(2021)andCrumpandGospodinov(2022). Second,weinvestigatethepossibilityofnonlinearfactordynamicsintheinterestrates. Thisexerciseismotivatedbytheliteraturedebatingwhethertherewerestructuralchangesinthebehavior of longer-term yields brought about by the effective lower bound (ELB) constraining monetary policy(SwansonandWilliams,2014). Ournonlinearfactormodelprovidesoneavenuetotestthis questionempiricallybecauseitallowsforstate-dependence. We find that allowing for the effective lower bound constraint is crucial, both in terms of estimating a yield curve factor that coheres with the historical narrative of monetary policy and in modelfit. ThisresultisinlinewithresultsfoundinWuandXia(2016). Therearetwotimeperiods wherethepresenceoftheELBaffectstheestimationresults. Thefirstisinthelate2003andearly 2004period,whenthefedfundsratedeclinedto1percent. Thesecondisthelongspellinthezero lowerboundthatbeganintheGFC.AfterallowingfortheELBconstraint,however,wedonotfind muchevidenceofnonlinearfactordynamics,whichsuggeststhatthedynamicsoftheentireyield curvedidnotappreciablychangeuponenteringtheELBperiod. Thekeynonlinearitytoaccount foristhelowerboundconstraintfortheshorter-maturityyields. Our second case estimates a nonlinear credit cycle as the common component of U.S. credit growthdynamicsacrossfoursectors: nonfinancialbusiness,household,financial,andpublicsectors. Theinvestigationismotivatedbytheextensivetheoreticalandempiricalliteraturedocumentingtheimportanceofcreditgrowthandleverageinunderstandingthecreditcycleandespecially nonlinearamplificationofshocks(Bernankeetal.,1999,BrunnermeierandSannikov,2014,SchularickandTaylor,2012). Weinvestigatethepresenceofcommonnonlineardynamicsacrosssectors 3

in a time series context. Our nonlinear factor captures the slow rise and rapid declines common across credit sectors. The results highlight the importance of a slow-moving second order factor that boosts credit growth in boom times – notably beginning in the mid 1990s until the onset of theGFC.ItscollapseintheGFCplaysakeyroleinthesluggishpost-crisisrecoveryofcredit. The effects of the same-sized shock to the credit cycle factor is different depending on the state of the economy. Shocks in a credit crunch period are more amplified and persist for longer compared withshocksduringacreditboom. Negativeshocksleadtoincreasesinthestandarddeviationof the credit factor predictive distribution. The combination of the decline in mean and increase in volatilitygenerateslargermovementsindownsidetailriskcomparedtoupsiderisk. RelatedLiterature Theliteraturehasconsideredsomeformsofnonlinearitiesinfactormodels like Markov switching, time-varying parameters, and stochastic volatility. However, to the best of our knowledge, there is no work on models that allow for second order dynamics in the state equation or general nonlinearities in the measurement equation. The classic example these days ofanonlinearityinthedataisthezerolowerboundimposedonshort-terminterestrates. Ingeneral,wefindlowerandupperboundswhenwedealwithpercentageslikelabormarkettightness, transition probabilities, and job finding and separation rates, so investigating a factor model that candealwiththesesituationsisimportant. Ourworkisparticularlyclosetotworecentpapersintheliterature. First,Aruobaetal.(2017a) introduce the quadratic autoregressive process (QAR), which allows quadratic terms in lagged regressorsaswellasGARCHfeatures. Likeourapproach,theyrelyontheprunedrepresentation to generate a stable model but their study concentrates on univariate models and posits that the observables follow the QAR. In contrast, our factor model framework can be modified to admit differentclassesofnonlinearitiesliketheoneintroducedbytheeffectivelowerbound. Second,GorodnichenkoandNg(2017)usetheinsightsfromthesecondordersolutionofDSGE models to obtain restrictions on the dynamics of observables and its squared values. From this, theauthorsextractafactorthatmimicsthedynamicsofalevelstatevariableandanotheronethat displays stochastic volatility features. There are important difference between our papers. While GorodnichenkoandNg(2017)’sapproachisbasedontheapproximatedsolutionproposedbyBenigno et al. (2013), our representation arises from the perturbed solution of a nonlinear dynamic stochastic general equilibrium model (Andreasen et al., 2017). They extract factors based on singularvaluedecomposition. Incontrast, weestimatethenonlinearsystemusinglikelihood-based methods,whichallowsus,amongotherthings,tobuildpredictivedensities,filterthemostlikely stateoftheeconomy,orreportIRFsconditionalonthestateoftheeconomy. Finally,Gorodnichenko andNg(2017)’smethodologyrequiresthatobservablesbemeasuredwithouterrorandrulesout thepossibilityofkinksinthedata. More broadly, we contribute to the large literature on factor models. An in-depth review of linearfactormodelsisgivenbyStockandWatson(2016). Amongtherecentadvances,Banburaand 4

Modugno(2014)allowformissingdatawitharbitrarypatternsinestimatinglinearfactormodels byusinganexpectationsmaximizationalgorithm. Chauvet(1998)usesalinearfactormodelwith regimeswitchestoestimatebusinesscycles. AruobaandDiebold(2010)leavesnonlinearfactorsas ato-dotask,althoughwithafocusonMarkovswitchingregimesratherthanthetypewepropose. Shintani (2005) estimates a nonparametric diffusion model for forecasting Japanese data. Chen et al. (2021) analyzes a semiparametric panel data model where latent factors are modeled in a nonparametricfashion. Chengetal.(2016)proposesalinearDFMthatallowsforbreaksinloadings and/orthenumberoffactors,whichisanalternativeviewoftheworld. TheyfindthattheGreat Recessionledtoachangeinthefactorloadingsandtheemergenceofanewfactor. Carrascoand Rossi(2016)considersforecastingwithmisspecificedfactormodels. Finally,ourstudyisalsorelatedtoworkthatdepartsfromGaussianshocks(Gourierouxetal., 2019). Aruoba et al. (2021) estimate a structural VAR model that allows coefficients to switch dependingonwhethertheeconomyisattheELB.Theresultsonthedistributionalimplicationsofthe NLDFalsoconnectustoagrowingliteratureontailrisksanddistributionalasymmetries(Adrian etal.,2019). Therestofthepaperisorganizedasfollows. Thenextsectiondiscussesthenonlineardynamic factormodelusingasimpleexamplewithtwoobservables. WemotivatethefactormodelbyconnectingittotheprunedsolutionofanonlinearDSGEmodel. InSection3,wehighlightthenovel implicationsofthenonlinearDFMformoments,IRFs,andpredictivedensities. Section4discusses ourtwoempiricalapplications. Someconcludingremarksareinthefinalsection. 2 The Nonlinear Dynamic Factor Model In this section, we introduce the nonlinear dynamic factor model and then discuss the motivation for using our factor dynamics. Next, we discuss possible specifications of the measurement equation. Finally,weclosethesectionbypresentingourestimationalgorithms. 2.1 Model Specification Weconsiderthefollowingnonlineardynamicfactormodel: Measurement: y = G(f )+η(cid:15) (1) t t t Factordynamics: f = H(f )+σν . (2) t t−1 t Here,(cid:15) isanN×1vectorofiid N(0,I )innovations,ν isaK×1vectorofiid N(0,I )innovations, t N t K y isanN ×1vectorofobservedvariables,andf istheK×1underlyingfactor. G(·)andH(·)are t t general,possiblynonlinearfunctions. Inaddition,weassumethattheH(·)functionisatleasttwice differentiable. η isanN ×N diagonalmatrixofstandarddeviationsandσ isaK ×K matrixthat isthesquarerootofavariancecovariancematrix. Theadditivemeasurementerrorassumptionis 5

foreaseofexposition. OurframeworkcaneasilyhandlemultiplicativeerrorslikeinHwang(1986) ornonadditiveerrors. For empirical applications involving data such as unemployment, GDP growth, or inflation rates, a typical desirable feature of the factor process is stationarity. A generic H(·) function, estimated using limited data range, may imply explosive dynamics of the factor and, thus, of the observables. To avoid this problem, we use the pruned motion equation (Kim et al., 2008, Andreasenetal.,2017)thathaseasily-verifiablestationarityconditions. Tomaketheanalysismoreconcrete,consideraNLDFmodelinwhichthemeasurementequationislinearinasingleunderlyingfactorandwetakeaprunedsecondorderapproximationtothe function H(·). We adopt the single factor specification for the rest of the paper. Let f denote the t underlyingfactorandff andfs itsfirstandsecondordertermssuchthatf = c+ff +fs. Then t t t t t theprunedsystemis y = Gf +η(cid:15) t t t     f t = c+f t f +f t s –1stand2ndorderfactors  ff = h ff +σν (3) t x t−1 t   (cid:16) (cid:17)2  fs = h fs + 1h ff . t x t−1 2 xx t−1 Thefirstordertermfollowsthesameprocessasalineardynamicfactormodelwithpersistence parameter governed by h . The exogenous shocks ν perturb the first order term on impact. The x t secondordertermdependsonthesquareofthelaggedfirstorderterm,withh modulatingthe xx importance this relationship. The second order term also has persistence determined by h . The x exact structure of this process is discussed in detail as the second order solution to a dynamic equilibriummodelthatpreventsexplosivepaths(Andreasenetal.,2017). Therearetwomainattractivepropertiestothisstructure. First,themodelallowsforrichnonlinearitiesduetothepresenceofthesecondorderterm. Specifically,themodelcangenerateasymmetric, state-dependent, and size-dependent impulse response functions (IRFs). Moreover, the modelcangeneratetime-varyingvolatilitythroughthestate-dependence. Wewillillustratethese propertiesindetailinSection3. Second,themodelhaseasilyverifiablestationarityconditions. As longas|h | < 1,themodelisstationary. x Thepresenceofthequadratictermh introducesanadditionaldistinctionrelativetothelinear xx factor model. Even if the shocks have zero mean, the factor has a mean different from zero. This canbeverifiedbyapplyingtheexpectationoperatoronthesecondorderterminequation3. Inthe dynamicequilibriummodelsliterature,h makesthemodel’sdeterministicsteadystatedifferent xx fromitsstochasticsteadystate. Weincludeaconstantcinthefactor’slawofmotiontoadjustthe overallfactortohavezeromeaninourapplications,althoughthisparametermayalternativelybe estimated. 6

2.2 Motivating the NLDF Model OurtimeseriesmodeliscloselyconnectedtothenonlinearsolutionofaDSGEmodel, whichwe viewasanimportantstrengthofourframework. NonlinearDSGEmodelshaverapidlygrownin popularity,spurredonbyanamplebodyofempiricalresearchwhichdocumentsthattheU.S.economyhasnonlinearfeaturessuchasstochasticvolatility(JustinianoandPrimiceri,2008,Fernández- VillaverdeandRubio-Ramírez,2007,Bloom,2009,Fernández-Villaverdeetal.,2015a),time-varying monetarypolicy(Fernández-Villaverdeetal.,2015b),andthezerolowerboundonshort-terminterestrates(Fernandez-Villaverdeetal.,2015,Gustetal.,2017,WuandXia,2016). Our NLDF model is the direct time series analogue of the pruned second order perturbation DSGEsolution. Supposethereisonlyonestatevariable(denotedf )intheDSGEmodel. Then,its t dynamicequationisapproximatedby Stateequation: f = ff +fs (4) t t t ff = h ff +σν t 1 t−1 t 1 (cid:16) (cid:17)2 fs = h fs + h ff . t 1 t−1 2 2 t−1 Here,h andh arecoefficientsandν isanormallydistributedinnovation. ComparingEquations 1 2 3 and 4, we see that their structures are the same. The key difference is that in the DSGE model solution,h andh areknowngiventhedeepparametersofthemodel. Inourtimeseriesmodel, 1 2 thecorrespondingparametersareestimatedfromthedata. If the researcher believes that the fundamental driver of the data is the one in Equation 4, it seems natural to advocate for the extraction of factors based on an approach that departs from linearity. Incorrectlyassumingalinearfactormodel, andtherebyignoringthefs term, resultsin t anestimatedfactorthatisdrivenbycounterfactuallyvolatileshocks. Thatis,theresearcherwould concludethat fluctuationsarein alargepart duetoexogenous eventsasopposed toendogenous propagation. 2.3 Specification of the Measurement Equation Giventhenonlinearitiesmodeledinthelatentfactor,anaturalbenchmarkcaseisforthemeasurementequationtobelinear,asinEquation3. Thelinearmeasurementequationallowsustoprove that the latent factor is not identified, and an additional normalization is needed. We close the sectionbydiscussingsomenonlinearmeasurementequationextensions. IdentificationwithLinearMeasurementEquation Webeginbydiscussinganidentification issue with the linear measurement equation case. To show this, it is convenient to use a twoobservableversionofourmodelwithasinglefactor: 7

    y G 1,t 1   =  f t +η(cid:15) t , y G 2,t 2 f = c+ff +fs, (5) t t t ff = h ff +σν , t x t−1 t 1 (cid:16) (cid:17)2 fs = h fs + h ff . t x t−1 2 xx t−1 Inthissystem,theunknownparametersaretheloadingcomponentsG andG ,thefactor’slinear 1 2 andquadratictermsh andh ,thestandarddeviationoftheshockσ,andthestandarddeviations x xx ofthemeasurementerrorsη. 2.1 ThesignandscaleofthefactorinthenonlinearfactormodelinEquation5arenot Proposition . identified. Proof. Considerthefollowingconstantn (cid:54)= 0andscalethesysteminequation5asfollows:     y 1,t G 1 (cid:18) 1 (cid:19)   =   nf t +η(cid:15) t n y G 2,t 2 nf = nc+nff +nfs t t t nff = h nff +nσv t x t−1 t 1 (cid:18) 1 (cid:19)2(cid:16) (cid:17)2 nfs = h nfs +n h nff . t x t−1 2 xx n t−1 Next, define f(cid:101)t = nf t , G(cid:101)1 = (cid:0) n 1(cid:1) G 1 , G(cid:101)2 = (cid:0) n 1(cid:1) G 2 , σ (cid:101) = nσ, (cid:101)h xx = (cid:0) n 1(cid:1) h xx , (cid:101) c = nc and rewrite the systemasfollows:     y 1,t G(cid:101)1   =  f(cid:101)t +(cid:15) t y 2,t G(cid:101)2 f(cid:101)t = (cid:101) c+f(cid:101) t f +f(cid:101) t s (6) f(cid:101) t f = h x f(cid:101) t f −1 +σ (cid:101) ν t 1 (cid:16) (cid:17)2 f(cid:101) t s = h x f(cid:101) t s −1 + 2 (cid:101)h xx f(cid:101) t f −1 This shows that the “tilde” model produces exactly the same observables as the baseline model. Therefore,notallparametersinthemodelareidentified. Moreover,sincencanbenegative,wedo nothavesignormagnitudeidentification. Inourapplications,wefixG = 1. ThiscorrespondstothenamedfactorapproachintheDFM 1 8

literature(StockandWatson(2016)). Nonlinear Measurement Equation Although the linear measurement equation is a leading case, in some instances, economic theory may suggest specifying nonlinearities in the measurement equation. Our model can accommodate these more complex dynamics. For instance, one couldallowforafullynonlinearmeasurementequation:     y f ,η (cid:15) 1,t t 1 1,t   = G , y f ,η (cid:15) 2,t t 2 2,t f = c+ff +fs, t t t ff = h ff +σν , t x t−1 t 1 (cid:16) (cid:17)2 fs = h fs + h ff . t x t−1 2 xx t−1 Here,G isthenonlinearfunctionmappingfrommeasurementerrorsandfactorstoobservables. Inourshadowinterestratesapplication,wespecifyaconditionthatrestrictsthelevelofinterest ratesfromgoingbelowalowerboundviaanonlinearmeasurementequation,consistentwiththe zerolowerboundrestrictiononnominalrates. 2.4 Estimation Algorithms We use Bayesian methods to estimate the model. As our model is nonlinear, we rely on particle filtering methods to approximate the likelihood (Särkkä, 2013). We use two algorithms in our empirical illustrations: a Metropolis Hastings combined with the bootstrap particle filter and a Gibbs sampling combined with the particle smoother. We believe each has their strengths. The particle filtering algorithm readily delivers the filtered factor, which may be useful in situations wheremaintainingtheinformationstructureofthefilteredvariableisimportant,suchaswhenthe filteredvariableisincludedinavectorautoregression(Fernández-Villaverdeetal.,2015a). Acaveat ofthebootstrapparticlefilteristhatitdemandshundredsofthousandsofparticlestocharacterize accurately the likelihood function. The particle Gibbs sampling algorithm delivers the smoothed estimate, which is the most accurate estimate of the factor given all of the data. Also, through its exploitationofancestorsampling(Lindstenetal.,2014),thealgorithmhasgoodmixingproperties evenwithrelativelyfewparticles–intheorderofhundreds. Thedisadvantageisthatthesampler is only approximate for our model, although Lindsten et al. (2014) shows that its performance is stillgood.2 Furtherdetailsaboutbothalgorithmsandtheircomputationalimplementationcanbe foundinAppendixSectionA. 2SeetheassociateddiscussioninSection7.2ofthatpaper. 9

Monte Carlo In the Appendix Section B, we conduct a Monte Carlo exercise to study the estimation performance. First, we show that if the data generating process is the nonlinear factor model itself, our estimation strategy can recover the true parameter values. Second, we find that the likelihood implied by a linear factor model is below the likelihood from the nonlinear model ifthedataweregeneratedfromourNLDF.Byignoringthenonlineardynamics,thelinearmodel tendstoestimateaexcessivelypersistentlinearfactor. 3 Properties of the Nonlinear Dynamic Factor Model WenowmoveontosomekeypropertiesgeneratedbytheNLDFmodel. Ourfocusisonthelatent factorwiththeunderstandingthatthesepropertiespropagatethroughtotheobservablesviathe measurementequation.3 WefocusonthreenovelfeaturesthatourNLDFmodelbringstothetable: asymmetric responses to positive versus negative shocks, state-dependent responses, and sizedependentresponses. ThesepropertiesofIRFswerepreviouslydiscussedinthecontextofsolution methodstoDSGEmodelsinAndreasenetal.(2017),butwefinditinstructivetoreviewthemhere. Alineardynamicfactormodelcannotdeliverthesetypesofimpulseresponsefunctions. 3.1 Analytical Properties of the Model Webeginbydiscussingtheanalyticalpropertiesofthemodel. Wecanwritethefactordynamicsin ausefulstatespaceform,firstpresentedinAndreasenetal.(2017),tomakeanalyticalprogresson themodel’simplicationsforthemomentsofthefactors. Tofacilitatetheexposition,letuscontinue toassumeaone-dimensionalfactor. Thenonecanwritethefactordynamicsasfollows: f = c+ff +fs (7) t t t         ff h 0 0 ff σ 0 0 (cid:15) t x t−1 t          fs  =  0 h 1h  fs +0 0 0  (cid:15)2  (8)  t   x 2 xx t−1    t  (cid:16) (cid:17)2  (cid:16) (cid:17)2    ff 0 0 h2 ff 0 σ2 2σh ff (cid:15) t x t−1 x t−1 t Foreaseofnotation,wewritethestatespaceasfollows: z = Az +Bζ (9) t t−1 t 3Therelationshipbetweenthepropertiesofthefactorandtheobservablesismoststraightforwardinthe benchmarkcaseofalinearmeasurementequation. Withanonlinearmeasurementequation,therewillin generalbeanonlineartransformationofthelatentfactortotheobservablevariables. 10

    ff (cid:15) t t     where z t =   f t s   and ζ t =   (cid:15)2 t   . The matrices A and B contain the corresponding (cid:16) (cid:17)2   ff ff (cid:15) t t−1 t parametersinthestateequation. Itispossibletoshowthattheinnovationsζ areintertemporally t uncorrelated,inotherwordsE[ζ ζ ] = 0fori (cid:54)= j. Inthissection,whendiscussingconditional t+i t+j momentsattimet,theconditioningsetisthepasthistoryoffactors{ff,fs,ff ,fs ,...}. t t t−1 t−1 FirstMomentDynamics Webeginbydiscussingfirstmomentdynamics. Weemphasizefour parts: thepersistenceofthefactor,asymmetryintheimpulseresponsetoapositiveversusnegative shock,state-dependencyintheresponse,andsize-dependencyintheresponse. Theoverallfactorhaspersistencefromboththefirstandsecondorderfactors. Thefirstorder factor,ff,hasapersistenceequaltoh . Thesecondorderfactor,fs,hasapersistencegreaterthan t x t h . Thisisbecausewhilefs followsanautoregressiveprocesswithparameterh ,its"innovation" x t x (cid:16) (cid:17)2 depends on ff , which itself is persistent with parameter h2. As long as h > 0, which is t−1 x x usuallythecaseformacroeconomicandfinancialdata,thesecondorderfactorismorepersistent thanthefirstorderone. The(cid:15)2componentoftheinnovationgeneratesanasymmetricresponsetoapositiveversusnegt ativeshock. Thisisbecausewhileapositiveshockincreases(cid:15) –andthereforeff –andanegative t t shockdecreasesit,bothapositiveandnegativeshockincreases(cid:15)2. Theimportantparametergovt (cid:16) (cid:17)2 erning the direction of the asymmetry is h . This parameter governs how ff relates to fs, xx t−1 t andthereforehowtheeffectsof(cid:15)2 passthroughtotheoverallfactor. Ifthesignofh ispositive, t xx thenapositiveshockincreasesff andtheresponseof(cid:15)2 increasesfs. Anegativeshockdecreases t t t ff buttheresponseof(cid:15)2 stillincreasesfs. Theeffectisreversedifthesignofh isnegative. t t t xx Theimpulseresponseisstate-dependent,whichcomesfromtheff (cid:15) termintheinnovation. t−1 t (cid:16) (cid:17)2 Thesignofff determinestheeffectofashockto(cid:15) on ff . Additionally,themagnitudeofff t−1 t t t−1 determinestheamountoftime-varyingvolatility,whichwediscussmoreinthenextsubsection. The (cid:15)2 term also creates size-dependencies in the response to a shock. This means that a two t standard deviation shock does not generate double the responses of a one standard deviation shock. Thiseffectfollowsstraightforwardlyfromthequadratictransformation. Thisisanimportantfeatureofourmodelbecauseitcangeneratestrongamplificationtoshocksduringdownturn episodes. ThegeneralformulaforconditionalmeandynamicsathorizonhisshowninEquation10. On topofthepreviousdiscussion,therearetwoadditionalpointstomentionfromthisequation. First, the h term determines the persistence property of the entire system, as is expected. If |h | < 1, x x theconditionalmeanresponsesconverge. Second,the(cid:15)2innovationproducesanon-zerolong-run t (cid:16) (cid:17)2 meanfor ff . t 11

  0 h−1 (cid:88)   E t z t+h = Ahz t + Ai  0  (10)   i=0 σ2 VolatilityDynamics Themodelgeneratestime-varyingvolatilityviathestate-dependencein- (cid:16) (cid:17)2 herent in the second order factor. The magnitude of ff modulates the effect of (cid:15) on ff . A t−1 t t (cid:16) (cid:17)2 larger value of ff means that the same-sized shock generates a larger response in ff . Int−1 t tuitively, the quadratic component of the model responds by more the further ff is away from t−1 0. h−1 V (z ) = (cid:88) AiBV (ζ )B(cid:48)(cid:0) A(cid:48)(cid:1)i (11) t t+h t t+h−i i=0   1 0 hh−1ff x t   V t (ζ t+h ) =   0 2 0   (12)  (cid:18) (cid:19)  hh−1ff 0 E (cid:16) ff (cid:17)2  x t t t+h−1 Theformulafortheh-stepaheadconditionalvarianceofthesystemisshowninEquations11 and 12. The latter equation shows that it is the ff (cid:15) term that generates time-varying volatility t−1 t (cid:16) (cid:17)2 in the system. The conditional variance from this term depends on ff and therefore has a t−1 persistenceofh2. AsEquation11shows,theconditionalvarianceofz atvarioushorizonsisthen x t adiscountedsumoftheconditionalvarianceofζ fromt+1upthrought+h. t Relationship Between First and Second Moments Ourmodelgeneratesanon-zerocorre- (cid:16) (cid:17)2 lation between conditional first and second moments. This can be seen by noticing that ff t−1 simultaneously determines the conditional mean of fs and the conditional volatility of shocks to t (cid:16) (cid:17)2 ff . Simultaneousmovementsinmeanandvolatilityisanimportantmechanismidentifiedin t thegrowth-at-riskliteraturetogenerateasymmetrictailriskbehavior(Adrianetal.,2019). (cid:16) (cid:17)2 Moreformally,wecanexaminetheconditionalcovariancebetweenff and ff andfs.4 We t t t beginbydiscussingshort-runconditionalcorrelationsandthenmoveontounconditionalcorrelations. TheseexpressionscanbederivedfromthegeneralformulasinEquations11and12. Atone stepahead,theconditionalcovariancebetweenthefirstorderfactoranditssquareis: (cid:18) (cid:19) (cid:16) (cid:17)2 Cov ff , ff = 2σ2h ff (13) t t+1 t+1 x t (cid:16) (cid:17)2 4Whilewediscussthecorrelationsbetweenallofthesevariablesattimet,keepinmindthatitis ff t−1 thataffectstheoverallfactorf . t 12

Assuggestedbytheearlierdiscussiononconditionalmeandynamics,thesignofff isimport tant. Specifically,ifff ispositive,theconditionalcorrelationbetweenthetwotermsispositiveas t well,andviceversa.5 Tounderstandwhythisisthecase,notethattheshock(cid:15) determinesff t+1 t+1 (cid:16) (cid:17)2 while(cid:15)2 andff(cid:15) determines ff conditionalonknowingff. Theshocks(cid:15) and(cid:15)2 are t+1 t t+1 t+1 t t+1 t+1 uncorrelated,soanynonzerocovariancemustcomefromff(cid:15) . Whenff ispositive,thenfurther t t+1 t (cid:16) (cid:17)2 increases in the first order factor increase ff and the correlation between the first order fact+1 toranditssquareispositive,whilewhenff isnegative,furtherdecreasesinthefirstorderfactor t (cid:16) (cid:17)2 increase ff andthecorrelationisnegative. t+1 Attwostepsahead,theconditionalcovarianceis: (cid:18) (cid:19) Cov ff , (cid:16) ff (cid:17)2 = 2σ2h2(cid:0) 1+h2(cid:1) ff (14) t t+2 t+2 x x t Qualitatively, thesame mechanisms areat play asin the onestep ahead case. Thecorrelation ispositiveifff > 0andisnegativeotherwise. Thisisthecaseasboththefirstorderfactorandits t squarehavepersistentdynamics. Forexample,ifff ispositive,ff isexpectedtoremainpositive t t+1 and therefore continue producing a positive comovement between the first order factor and its square. Theoverallcovarianceisasumoftwotermsbecauseittakesintoaccountshocksatt+1 andt+2. Theconditionalcovariancebetweenthesecondorderfactorandthefirstorderfactorsquaredis 0atonestepaheadasthesecondorderfactorattimet+1ispredeterminedgiventimetinformation. Theconditionalcovariancebecomesnonzeroattwostepsaheadanditsvalueequals: (cid:18) (cid:19) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) (cid:16) (cid:17)2 1 (cid:16) (cid:17)2(cid:16) (cid:17)2 (cid:16) (cid:17)2 (cid:16) (cid:17)2 Cov fs , ff = h E ff ff −E ff E ff (15) t t+2 t+2 2 xx t t+1 t+2 t t+1 t t+2 (cid:124) (cid:123)(cid:122) (cid:125) Conditionalone-stepaheadautocovarianceof (cid:16) ff (cid:17)2 t+1 Thetwotermsaretightlyrelatedbecausethesecondorderfactordirectlyloadsontopastvalues ofthefirstorderfactorsquared. AscanbeseenbyexaminingEquation15,theconditionalcovarianceisdeterminedbytwocomponents: h andtheconditionalonestepaheadautocovarianceof xx (cid:16) (cid:17)2 ff . Ifh > 0,thenincreasesinthefirstorderfactorsquaredgenerateincreasesinthesecond t+2 xx order factor, and the conditional covariance is positive. The effects are reversed if h < 0. The xx autocovariancetermappearsbecausethetimet+2valueofthesecondorderfactorloadsontothe timet+1valueofthesquaredfirstorderfactor,sointertemporaldynamicsplayarole. We contrast the short-run comovement behavior to unconditional comovements. The unconditionalvariancecovariancematrixofthesystemisgivenbyEquations16-18: 5Thisresultisassumingh >0,whichistheempiricallyrealisticrange. x 13

V (z) = AV (z)A(cid:48)+BV (ζ)B(cid:48) (16)   1 0 0   V(ζ) = 0 2 0  (17)     0 0 E (cid:0) ff(cid:1)2   0   E(z) = (I −A)−10. (18)     σ2 ThegeneralformofV(z)isasfollows:   X 0 0   V(z) = 0 X X, (19)     0 X X where the X denotes non-zero values. Unconditionally, there is no correlation between the first orderfactoranditssquare,orthefirstorderfactorandthesecondorderfactor. Althoughthefirst orderfactordeterminesthetime-varyingvolatilityin thesystem, itunconditionallyhaszerocorrelationwithvolatility. Thisisbecausethetime-varyingvolatilitydependsonlyonthemagnitude of ff and not its sign. As the unconditional distribution of ff is symmetric around zero, this t−1 t correlationisalsozerounconditionally. Thereisdependence,however,betweenthesecondorderfactorandthevolatilityinthesystem, whichinducesarelationshipbetweenlevelandvolatilityevenunconditionally. Thisisbecausethe firstorderfactorsquaredentersasthedrivingforceofthesecondorderfactor. Unsurprisingly,the sign of h is important in governing this relationship, with a positive h generating a positive xx xx dependencebetweenlevelandvolatilityandanegativeh generatinganegativedependence. xx 3.2 Simulations We use a parameterized version of the nonlinear model and simulation methods to further illustrateitsproperties. Tothisend,letusconsiderthebaselinemodelwithtwoobservables: 14

    y G 1,t 1   =  f t +η(cid:15) t y G 2,t 2 f = c+ff +fs (20) t t t ff = h ff +σv t x t−1 t 1 (cid:16) (cid:17)2 fs = h fs + h ff , t x t−1 2 xx t−1 The baseline parameterization is c = −1 hxxσ2 , G = 1, G = 2, h = 0.45, h = 0.5, and 2(1−hx)(1−h2 x ) 1 2 x xx σ = 1. Thesettingofcguaranteesthattheoverallfactorf haszeromean. Westartourdiscussion withtheimpulseresponsefunctionsimpliedbythemodel. Then,wemoveontothedistributional implicationsoftheshocks. Akeyparameterintheanalysisish . Itssigndeterminesthedirection xx oftheasymmetry. Weseth > 0,butifh weretobenegative,thentheasymmetrieswouldbe xx xx flipped. Impulse Response Functions We focus on the three novel properties that our factor model candeliver: asymmetric,state-dependent,andsize-dependentshocks. As our model is nonlinear and state-dependent, there are two main issues when computing impulse response functions. First, as discussed in Koop et al. (1996) and Goncalves et al. (2021), inthepresenceofnonlinearities,thedifferentnotionsofimpulseresponsefunctionsdonotnecessarilycoincide. WeusethedefinitionofimpulseresponsefunctionssuggestedbyGoncalvesetal. (2021). IRF [f ] = E[f (δ)−f |Ω ] (21) δ,t−1 t+h t+h t+h t−1 whereΩ = {ff ,ff ,...,fs ,fs ,...},δisthesizeoftheinnovation,f isthebaselinevalue t−1 t−1 t−2 t−1 t−2 t+h conditional on a path of shocks {ν ,ν ,...}, and f (δ) is the counterfactual value conditional t t+1 t+h onthesamepathofshocksexceptwiththeadditionofδ attimet{ν +δ,ν ,...}. t t+1 ThisdefinitionofIRFshasseveralproperties. First,itintegratesouttheeffectsoffutureshock uncertainty, similarly to the generalized impulse response function. Second, as the time t shock is added ontoa baseline path, the exercise is bestthought of as showing theeffects of perturbing the time t innovation ν by δ. This is a slightly different thought experiment when compared to t thegeneralizedimpulseresponsefunctionone,inwhichthetimetshockisfixedatδ inthecounterfactual case (Koop et al., 1996). We prefer the definition suggested by Goncalves et al. (2021) because it maintains randomness in the period of the shock, which has important implications when thinking about higher-order moments and the distribution. Finally, our impulse response functions are state-dependent. We are explicit in our conditioning set to make this dependence cleartothereader. 15

Webeginbydiscussingtheasymmetricresponsestoshocks. ThefirstrowinFigure1showsthe IRFsofthefactorfollowingaone-standarddeviationpositiveinnovation(leftpanel)andanegative one(rightpanel),initializingthefirstorderfactorff at0.56andthesecondorderfactorfs atits −1 −1 unconditionalmeanvalue. Thiscalibrationisillustrative,andwechooseanonzerolagofthefirst order factor to showcase the state-dependency in the IRFs. The figures plots the dynamics of the factor(blueline)anditsfirst(blackdashedline)andsecondorder(blackline)components. These IRFsillustratetheasymmetrythatthemodelcangenerate. Specifically,withthisparameterization and initial condition, a positive shock persists for longer than a negative shock. This can be seen by comparing the impulse response with its first order component. The first order component is linearandthereforesymmetric. Itproducesimpulseresponsesthatarerepresentativeofthosethat comefromastandarddynamicfactormodel. Thebluelineisformedbyaddingupthefirstorder andsecondordercomponentstogether. Giventheinitialconditionsandthefactthath > 0,the xx secondordertermisalwayspositiveinthisexample. Thismeansthatthebluelineisalwaysabove thefirstorderresponse,nomatterwhethertheshockispositiveornegative. The second row of the figure illustrates the next important property that our model can produce: state-dependence. The solid blue line is the same response as in the top row. The dashed blacklinenowshowstheresponsestothesamesizedshock,butstartingataninitialconditionof ff = 3.33 and fs at its unconditional mean. Although the initial impulse is the same, as can −1 −1 be seen by the identical response at time 0, the effects of the state-dependence kick in with a one period lag. Starting from the different initial condition with an elevated first order factor, both positive and negative shocks generate larger magnitude of responses in the factor. This can understood by examining Equation 8. The lag of the first order factor determines the volatility of (cid:15) t (cid:16) (cid:17)2 inthethirdequationgoverning ff ,withaff thatislargerinmagnitudeleadingtoahigher t t−1 varianceof(cid:15) . t Theresponsesshowninthesecondrowofthefigureleadustoanotherrelatedfact,whichwe illustrateinmoredetailinthesectionondistributionalresponsestoshocksbutisworthmentioning here. The different amplification of shocks is indicative of time-varying volatility. This example illustratesthatwhenthefirstordercomponentstartsoutatalargervalueinmagnitude,theoverall nonlinearfactoralsobecomesmorevolatile. Wecontrastthisstate-dependencewithfactormodels of exogenous stochastic volatility, such as Del Negro and Otrok (2008), where movements in the volatilityofthefactorareduetoseparateshocks. Finally,thethirdrowofFigure1showsthesize-dependenceoftheimpulseresponsefunctions. What we mean by this is that the shape of the impulse response function changes depending on the size of the shock. The blue line is the same one that we have carried over from the previous rows. Thedashedblacklineistheresponseatthesameinitialconditions,buttoashockthatistwo standard deviation in size instead of one standard deviation. On impact, the response is double that of the one standard deviation shock. In the next period, however, the shapes of the impulse responsefunction changefor positiveand negativeshocks. Thisfact isespecially clearin thisex- 16

Positive Shock Negative Shock 1 First order 0 0.8 Second order -0.2 0.6 Sum -0.4 0.4 -0.6 0.2 -0.8 0 -1 0 4 8 12 16 20 24 0 4 8 12 16 20 24 1.5 0 f f = 0.56 -1 -0.2 1 f f = 3.33 -0.4 -1 -0.6 0.5 -0.8 0 -1 0 4 8 12 16 20 24 0 4 8 12 16 20 24 2.5 0.5 1 std 2 0 2 std 1.5 -0.5 1 -1 0.5 -1.5 0 -2 0 4 8 12 16 20 24 0 4 8 12 16 20 24 Figure1: Row1: Asymmetricresponsestothesame-sizedshockinacalibratedmodel. Thisfigureshows theeffectsofapositiveshockontheleftpanelinblue. Thedashedblacklineisthefirstorderresponsewhile thesolidblacklineisthesecondorderresponse. Row2: State-dependentresponsesinacalibratedmodel. TheresponsestoapositiveshockareintheleftpanelwiththebluelinebeingthesameasinRow1andthe dashedblacklinetheresponsetothesameshockbutatadifferentinitialcondition. Row3: Size-dependent responsesinacalibratedmodel. Theresponsestoapositiveshockareintheleftpanelwiththeblueline beingthesameasinRow1andthedashedblacklinetheresponsetoashocktwicethesizeastheonethat generatestheblueline. Inallrows,therightpanelshowstheresponsestoanegativeshock. ample after a negative shock, in which the impulse response function turns positive two periods afterimpactfollowingatwostandarddeviationshock,whileitstaysnegativefollowingaonestandarddeviationshock. Thisisbecauseatwostandarddeviationchangetothefirstordercomponent changesthesecondordercomponentbymorethandoublethatofthefirstordercomponent. This greater-than-proportionateresponseofthesecondordercomponentgeneratesthedifferentshape oftheoverallimpulseresponsefunction. 17

Distributional implications Despitebeingdrivenbynormal,homoskedasticshocks,theNLDFmodelproducesrichnon-normalities inthedistributionofthefactor,whichthenfeedsintothedistributionoftheobservables. Webegin by discussing the unconditional distribution of the factor. Next, we move on to the time-varying volatilityandtailriskthatourmodelcanproduce. Figure12inAppendixSectionCshowsthedistributionofthefactorproducedbyalongsimulationofthemodel. ThedistributionproducedbytheNLDFmodelisnotnormal,asisevidenced byitspositiveskew. ItsKelleyskewness, whichmeasurestheshareofthedistancefromthe90th percentiletothe10thpercentilethatisabovethemedianversusbelowthemedian,is0.07.6 Bycontrast,thedistributionproducedbythefirstordercomponentisnormallydistributed,andtherefore hasaKelleyskewnessof0. Focusing only on the unconditional distribution, however, masks important dynamics of the distributionsfollowingdisturbances. InFigure2,weillustratethisfeatureofourmodelbysimulatingthepredictivedistributionsatvarioushorizonsfollowingthesameone-standarddeviation shock that we began discussing in the top row of Figure 1.7 The blue distribution is the baseline density – the one that characterizes the possible outcomes if we simulate the model from the initialconditions. Thedashedblacklineisthedensitythatrealizesifwehadapositiveonestandard deviation shock at period 0. Our model implies that such a shock leads to a positive shift in the distribution on impact. Crucially, in period 1, the distribution widens out . The differences in the distributionspersistthroughperiod3andbyperiod10,theeffectsoftheshockarelargelygone. Figure 3 shows the impulse response functions of the first and higher-order moments to the shock. Forthestandarddeviationandtailriskresponses,wecomputetheimpulseresponsefunctionsasdifferencesinthestandarddeviationandshortfallandlongriseofthe+1ShockandBaselinedistributions. Thestandarddeviationofthedistributionincreases,peakinginthefirstperiod after the shock. These effects lead to a large rise in the 5% longrise, as the combination of a rise in the mean and increase in the standard deviation work together to greatly increase the upside tailrisk. The5%shortfallalsoincreases,butbymuchless,asthemeanincreaseiscounteractedby theincreasedstandarddeviation. Therefore,throughmovementsinthehigher-ordermomentsof thedistribution,themodelcangeneratedistinctasymmetriesinthemovementsoftheupperand lower tails of the distribution, in line with the stylized facts documented by Adrian et al. (2019). TheseIRFsalsoreinforcethefactthatourmodelcangeneratetime-varyingvolatilitythroughthe nonlineardynamics. Figures 13 and 14 in Appendix Section C show the corresponding distributional responses to a one standard deviation negative shock. A negative shock lowers the mean and decreases the 6TheformulaforKelleyskewnessis Q90+Q10−2∗Q50 whereQisthequantileofthedistribution. Q90−Q10 7Wegeneratethesedistributionsbysimulating100,000pathsfromtheinitialcondition. Inthebaseline case, we take draws from the DGP. In the "+1 Shock" case, we add a one standard deviation shock to the impactperiod’sinnovationsfromthebaseline. Aftertheimpactperiod,weusetheexactsamedrawsofthe innovationsinbothscenarios. 18

Period 0 Period 1 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -5 0 5 -5 0 5 10 15 Period 3 Period 10 0.4 0.4 Baseline +1 Shock 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -5 0 5 10 15 -5 0 5 10 Figure 2: Dynamicsofthedistributionofthedemeanedoverallfactoratvariousperiodsafterashockin acalibratedmodel. Thebluelineisthedistributionafterapositiveshockandthedashedblacklineisthe distributionwithoutashock. Period0istheperiodoftheshock. standard deviation of the distribution. These factors together again generate a more persistent negativemovementinthelongrise,butlesspersistenteffectsontheshortfall. Comparingthepositiveandnegativeresponses,onefeatureweseeisthattheasymmetrieswedocumentforthefirst momentalsocarryovertohighermomentsaswell. Thedeclinesinthestandarddeviationandtail riskaresmallerinmagnitudewhencomparedtotheincreasesinthosefeaturesofthedistribution followingapositiveshock. 3.3 Variants to NLDF Before we move to the estimation section, we briefly discuss potential ways in which the model couldbeextendedtostudydatathatdemandsaricherfactorstructure. 19

Mean response 1 0.5 0 0 4 8 12 16 20 24 Std response 0.3 0.2 0.1 0 0 5 10 15 20 25 Tail risk response 2 Shortfall Longrise 1 0 0 4 8 12 16 20 24 Figure 3: IRFs of the mean, standard deviation, and 5% shortfall and longrise of the demeaned overall factorinresponsetoapositiveshockatperiod0inacalibratedmodel. Inthethirdpanel,thesolidblueline showstheresponseoftheshortfallandthesolidredlineshowstheresponseofthelongrise. Beyond a 2nd-order Representation Our choice of the 2nd-order pruned representation for the factor dynamics is based on its being parsimonious and the familiarity that macroeconomists have on perturbation methods. But our exposition is general enough that one can use, for example, projection methods to approximate the functions G and H. This alternative can capture richer nonlinearities that monomials cannot model. LetΨ (·)denotetheChebyshevpolynomialofdegreei. Thenthenonlinearstateequation i canbeapproximatedby n (cid:88) f t = θ i Ψ i (f(cid:101)t−1 )+σν t . i=0 Here,θ i areparameterstobeestimatedandf(cid:101)t−1 isatransformationoftheoriginalt−1factorsuch that is bounded between -1 and 1.8 However, this option comes at the cost of potentially more complexlikelihoodwhenestimatingthemodel. 8ThistransformationisnecessarybecauseChebyshevpolynominalsaredefinedintheinterval[−1,1]. 20

Multidimensional state One can in theory easily expand our model to accommodate more factors. Below we still use 2 observablesbutweaddanadditionalfactor.     y x 1,t 1,t   = G  + η (cid:15) t , y 2×2 x [2×2][2×1] 2,t 2,t     x x 1,t 1,t−1   = H + Σ ν t . x x 2×2 2,t 2,t−1 Here,thefunctionH(·)isthenonlinearmapbetweenthefactorsyesterdayandthefactorstoday. If oneextendstheprunedrepresentationfromabovetothetwo-factorcase,theresultsfromProposition 2.1 carry over. Specifically, two factors must be named; i.e., their loadings in two of the observableequationsmustbesetto1. 4 Macro & Finance Applications Revisited Now we use the nonlinear factor model to study the role of exogenous and internal forces in the dynamics of macro and financial series. We have two empirical applications. First, we estimate a shadowratemodelmotivatedbytheworkofWuandXia(2016). Second,weestimateanonlinear creditcyclefactor. 4.1 Shadow rate Short-termyieldshaverecentlyhittheirlowerboundconstraints,promptingmodificationstoexistingyieldcurvemodelstoaccountforthisbehavior. Animportantadvancementcamewithmodels thatconsideredanELBconstraintonshort-termyieldsandtherebyallowedthelatentyieldcurve factortoturnnegative. Notably, WuandXia(2016)showedthattheseshadowratemodelsdeliveredameaningfulmeasureofmonetarypolicyconditionsevenwhentheshortratewasstucknear zero. Theusualassumptionmadeinshadowratemodelsisthattheyieldfactordynamicsarelinear anddonotchangeuponenteringtheELB.Implicitly,thisassumptionpresumesthattheeconomy does not undergo structural changes due to constrained monetary policy, which is at odds with someofthetheoreticalliterature(Fernandez-Villaverdeetal.,2015,Aruobaetal.,2017b)andsupportedbyothers(WuandZhang,2019,Bernanke,2020). Someempiricalstudieshaveinvestigated whether structural changes in the economy occurred as a result of the ELB constraining policy (Swanson and Williams, 2014, Debortoli et al., 2019, Wu and Zhang, 2019, Aruoba et al., 2021). Amongtheseworks,oneofparticularrelevanceforourpurposesisSwansonandWilliams(2014), 21

who investigate if the behavior of yields at longer-term maturities in response to news changed duringtheELB. Ourworkcontributestothisdebatebyinvestigatingwhetherthereisevidenceofnonlinearities in the factor dynamics of a shadow rate model. We simultaneously model ELB restrictions on the yields in the measurement equation and nonlinear factor dynamics. If there were structural changes in the yield curve movements due to the ELB, our model would be able to capture them throughthesecondorderfactor. Anothercontributionofourworkisthatwemodeltheyieldsin firstdifferencesinsteadoflevelsandweshowhowtodosowhileaccountingfortheELB(Onatski andWang,2021,CrumpandGospodinov,2022). EmpiricalSetup Weestimateourmodelonthefirstdifferencesofonemonthforwardratedata at 3 months, 6 months, 1, 2, 3, 5, and 10 year maturities from February 1990 to September 2019, following Wu and Xia (2016) to construct the data.9 We impose a 0.3% effective lower bound on theseries. Weconsiderthefollowingmodelofdifferencesinforwardrates:  ∆forwardh = m + G h (c+f t f +f t s)+ηhεh t ifS(cid:98) t h >= 0.3 (22) t h −m +ηhεh otherwise h t (cid:16) (cid:16) (cid:17)(cid:17) where c = −1 2(1−h h x x ) x ( σ 1 2 −h2 x ) , S(cid:98) t h = (cid:80)t τ=2 m h +G h c+f τ f +f τ s + forwardh 1 , ∆forwardh t = forwardh−forwardh ,andindexhstandsforthematurity. Wemodelthelatentfactoraccording t t−1 tooursecondorderdynamics: ff = h ff +σν (23) t x t−1 t 1 (cid:16) (cid:17)2 fs = h fs + h ff t x t−1 2 xx t−1 WenormalizethefactorloadingG onthe3−monthratetobe1. Wealsoallowforyield-specific 1 constantsm thatcaptureanydifferencesintheaverageforwardratechangesacrossmaturities. h Therearetwokeynonlinearitiesinthemodel. First,thereisanELBconstraintinthemeasurement equation, which removes any influence of the factor if it predicts a rate lower than 0.3% in levels. Instead, themeasurementequationissetto∆forwardh = ηh(cid:15)h. AttheELB,theobserved t t changeintheforwardrateshouldbe0%asthelevelisstuckat0.3%. Themeasurementerrorpicks uparesidualdifference. Second,thelatentfactordynamicsareallowedtobenonlinear. WeestimatethenonlinearmodelusingMetropolisHastingscombinedwiththebootstrapparticlefilter.10 Weuse500,000particlesintheparticlefilter. Wetake510,000drawsfromtheposterior distributionwithaburninof210,000. Weconstructtheposteriordistributionsforourresultsby 9FurtherdetailsaboutthedataconstructioncanbefoundinAppendixSectionD. 10TheestimationdetailsandpriordistributionscanbefoundinAppendixSectionsAandD. 22

takingevery100thparameterdrawfromtheremaining300,000draws. Asacomparison,wealsoestimatealinearversionofthemodelwherewedonotmodeltheELB in the measurement equation and impose linear factor dynamics.11 Insofar as the nonlinearities we consider are empirically important, the nonlinear model should fit the data better than the linearmodeldoes. Iftheh terminthenonlinearmodelisestimatedtobeinsignificant,thenwe xx interpretthatasevidenceagainststate-dependentdynamicsattheELB. Historical Estimates of the Forward Rate Factor We first discuss the historical filtered estimates of the forward rate factor implied by our nonlinear model and compare it with a linear modelthatdoesnotaccountfortheELBrestrictionandimposesonlylinearfactordynamics. The toppanelinFigure4showsthefilteredestimatesofthefactorproducedbyournonlinearmodel. The factor captures the contours of historical yield curve movements. For example, we catch the rapiddropinshort-andlong-termratesintheearly1990s,whichwasthenfollowedbyatightening cycle in monetary policy during the middle of the decade. In 1995-1996, the 10 year forward ratedroppedovertwopercentagepoints,whichourfactorcaptures. Movingontothe2000s,our estimated factor reflects the easing cycle in the early 2000s followed by tightening beginning in 2004. The factor then rapidly drops entering into the GFC. In the middle of the GFC, short-term interestrateshittheirELB,whichishighlightedbythegrayareainthefigure. Becauseourmodel can account for an ELB in the measurement equation, however, it still estimates variability in the factor,chieflythecontinueddeclineinlong-termrates. Thelackofvariationofshorter-termmaturitiesleadstoawideningoftheuncertaintyintheestimatedfactorasmoreforwardrateshittheir ELB.Finally,asweexittheELB,thefactorcapturestheriseinshort-termforwardrates. It is worth emphasizing that our factor matches some important events in recent monetary policy(redlinesinFigure4). Forexample,thereisasharpdeclineinthefactorintheaftermathof 9/11. AsimilarsuddendropisobservedaroundtheannouncementofQE1. Incontrast,theTaper Tantrum of 2013 coincides with a rise in the factor. Interestingly, our estimated factor shows that QE3didnotresultinachangeofthestanceofmonetarypolicy. Thebottompanelshowsthecorrespondingfilteredestimatesfromalinearmodel. Outsideof the highlighted ELB period in gray, the two models estimate similar factors, foreshadowing the limited role played by the second order factor dynamics. During the ELB period, however, the factorestimatesdiverge. Thelinearmodelisconstrainedbythefactthatshort-termforwardrates have no variation, so they are stuck at 0, while long-term rates continued to vary. Indeed, these fluctuationsinthelongermaturityratesinformthedynamicsofourfactorduringthezerolower bound episodes. On balance, the linear model estimates little variation during the ELB period, therebysacrificingfittothelong-termyields. Figure5showsthemodelimplicationsforthefilteredleveloftheshadow3monthand10year 11WeestimatethismodelwiththeMetropolisHastingsalgorithmandtheKalmanFilter. Furtherdetails canbefoundinAppendixSectionsD. 23

Filtered Forward Rate Factor - Nonlinear Model 1 1 1 /9 1 E Q 2 E Q ts iw T 3 E Q m u rtn 0.5 n o ita re p O a T re p a T 0 -0.5 1990 1995 2000 2005 2010 2015 2020 Filtered Forward Rate Factor - Linear Model 1 0.5 0 -0.5 1990 1995 2000 2005 2010 2015 2020 Figure 4: Filteredestimatesoftheforwardratefactorestimatedbythenonlinearmodel(toppanel)and linear model (bottom panel) and 80% credible sets (blue shaded area). The gray shaded areas denote the timeperiodsinwhichthe3monthforwardrateisattheELB. rates in red. We calculate this by fo (cid:92) rward h,sh = (cid:80)t m + G (cid:16) c+ff +fs (cid:17) + forwardh. We t τ=2 h h t t 1 ignoretheELBrestrictionsplacedontheforwardrate,andsotheseestimatesarebestinterpreted as shadow rates (hence the sh in the superscript). In the top row, we also show the Wu and Xia (2016)shadowfederalfundsrateinblueforcomparison. Asournonlinearmodelallowstheshadowratetogonegative,wecaptureremarkablysimilar dynamics to Wu and Xia (2016). Namely, our shadow 3 month rate continues to trend down into 2014and2015,beforeliftingoffinearly2016. WuandXia(2016)interpretthedeclineoftheshadow rates during this period as evidence of the effectiveness of unconventional monetary policy, the effects of which can be seen in the decline of longer-term unconstrained rates. The influence of thelonger-maturityratesontheshadowratebetween2009and2015canbeseenintheinsetatthe bottom left panel in Figure 5. Both the 10-year and the shadow rates feature a downward trend andhavebroadlysimilardynamics. Incontrast,thelinearmodel’sestimatesdonotcaptureanyof thesemovementsintheELBperiod,andweseealargelyflatpredictionofthe3monthratefrom thefactor. Importance of the Nonlinear Components How important are the nonlinear additions to the model? From the filtered estimates, we see clear evidence that the ELB restriction tangibly changesthefactorestimates. Moreover,amarginallikelihoodcomparisonbetweenthetwomodels 24

Nonlinear Model Linear Model Shadow 3 Month Rate Shadow 3 Month Rate 10 10 8 8 6 6 4 4 2 2 0 0 -2 -2 -4 -4 Model -6 Data -6 Wu-Xia -8 -8 1990 1995 2000 2005 2010 2015 2020 1990 1995 2000 2005 2010 2015 2020 10 Year Rate 10 Year Rate 10 10 8 8 6 6 4 4 2 2 6 0 0 5 -2 -2 4 3 -4 -4 2 -6 -6 1 -8 -8 1990 1995 2000 2005 2010 2015 2020 1990 1995 2000 2005 2010 2015 2020 0 2009 2010 2011 2012 2013 2014 2015 Figure 5: Filteredestimatesoftheshadow3monthforwardrate(toprow)andthe10yearrate(bottom row) by the nonlinear model (first column) and linear model (second column). The red lines denote the filtered estimates with 80% credible sets (red shaded area). The dot dashed magenta line is the observed data. ThedashedbluelineistheshadowfederalfundsrateestimatedbyWuandXia(2016). showsthatthenonlinearmodelisheavilyfavoredbythedataat640versus575logpoints.12 Taken together, these results suggest that allowing for nonlinearities is important to understand yield curvedynamics. The central question in our investigation is whether the ELB produced structural changes in yield curve dynamics, and therefore changes in the behavior of longer-term forward rates that werenotconstrainedbytheELB.ThisquestioncanbeansweredbyexaminingwhetherthestatisticalgainsfromthenonlinearmodelareprimarilyduetotheELBconstraintonthemeasurement equation,thesecondorderfactordynamics,orboth. Table2inSectionDoftheAppendixshows the80%crediblesetsofparameterestimates. There,wecanseethatthecrediblesetsforh –the xx key parameter that governs the second order factor – ranges from −0.01 to 0.47. These estimates 12We compute the marginal likelihood using the modified harmonic mean (Geweke, 1999). We use a truncationparameterof0.95. Theresultsaresimilarfortruncationparametersof0.5and0.75. 25

contain0,andatbestcanbecharacterizedasmarginallysignificant,suggestingthathigher-order factordynamicsplayalimitedrole. Second,wecanalsoestimateaversionofthemodelwherewe maintaintheELBrestrictioninEquation22,butweimposelinearfactordynamics. Thisversionof the model produces filtered factors similar to the fully nonlinear model and fits the data slightly betterinamarginallikelihoodsense(642logpointsversus640). Our empirical evidence then is in favor of the idea that the ELB mainly was a restriction on thebehaviorofshort-termyields. Thereislittleevidenceofnonlinearitiesinthefactordynamics, at least using our model. Therefore, dynamics of the factor continued to propagate linearly as in unconstrainedtimes. 4.2 Nonlinear Credit Cycle Since the GFC, economists have once again taken a close look at the importance of credit growth formacroeconomicfluctuations(SchularickandTaylor,2012). Excessivecreditbuildupsoftenprecedefinancialcrisesandleveragecanfurtheramplifyshocks. Moreover, itisnotenoughtofocus on one credit sector, but instead a broad monitoring framework is needed (Adrian et al., 2015). For instance, Mian et al. (2017) emphasize the importance of household debt to GDP as a predictoroflowerGDPgrowthandhigherunemploymentworldwide. Corporateleveragemayleadto distorted investment decisions due to debt overhang effects (Gomes et al., 2016). Financial sector leverage can amplify shocks via the financial accelerator and binding borrowing constraints (Bernanke et al., 1999, Gertler and Karadi, 2011). Finally, as discussed in Jorda et al. (2016), high levels of public debt tend to prolong the pain of private sector deleveraging. Taking center stage inthesestudiesistheimportanceofcreditgrowth. Our second application investigates the importance of a common component in real credit growth in the United States across the nonfinancial business, household, financial, and public sectors from 1952:Q1 to 2021:Q4. Credit growth across different sectors may move together due to common factors such as changes in risk appetite, financial technology, or structural reforms. Moreover,economictheorysuggeststhepotentialimportanceofnonlinearitiesindeterminingthe dynamicsofcreditgrowth. Minsky(1977)describesaneconomythatmayexperiencearapidcontractionincreditafteralongboomwithspeculativelendingasexpectationsrapidlychange. Bordalo et al. (2021) formalizes these dynamics in a model with diagnostic expectations. Several papers highlight the role of occasionally binding borrowing constraints in modeling U.S. business andcreditcycles(BrunnermeierandSannikov,2014,GuerrieriandIacoviello,2017). Weviewour modelasoneavenuetocheckhowimportantthenonlineardynamicsareinthedatawithoutneedingtoresorttoafully-specifiedstructuralmodel. Weestimateaonefactorversionofournonlinearfactormodel(Equation3). Weuseaparticle Gibbssamplingalgorithmwith100particlesandtake1.5milliondrawsfromtheposteriordistribution,burninginthefirst600,000. Toformourposteriordistribution,wetakeevery300thdraw for a total of 3000 draws. For posterior distributions of impulse response functions and distribu- 26

tionalmoments,whichrequireheaviercomputation,weuse1000drawsoftheparameters. Further detailsabouttheestimation,includingthepriorspecification,canbefoundinSectionsAandEof theAppendix. CreditGrowthDynamics Figure15inAppendixSsectionDshowsthedatathatweusetoestimatethemodel,whichisnormalizedU.S.realcreditgrowthinthenonfinancialbusiness,household, financial, and government sectors. The data come from the Z.1 Financial Accounts data providedbytheFederalReserveBoard.13 Private credit growth generally increases during expansions and declines during recessions, althoughthetroughsofnonfinancialbusinessandfinancialcreditgrowthlagthetroughsofrecessions. Thethreeprivatecreditgrowthseriesarefairlypositivelycorrelated,rangingfrom0.4to0.5. Inthenonfinancialbusinessandhouseholdsectors,creditgrowthexhibitsanimportantasymmetry,withexpansionsmarkedbysteadystronggrowthandrecessionsassociatedwithsharpviolent declines. These dynamics have implications for higher-order moments, with the Kelly skewness of nonfinancial business credit growth at −0.23 and household credit growth at −0.15. Financial creditgrowthexperiencedrapiddeclinesintheGFC,butoverallhasaskewnesscloseto0. Ontheotherhand,governmentcreditgrowthismildlynegativelycorrelatedtothethreeother series,owingtothefactthatithasincreasedinrecentrecessions. Theserieshasadistinctpositive skewduetoseverallargespikesinpublicdebt. Historical Credit Cycle Estimates Ourestimatesprovideevidenceofanonlinearfactorthat wecallthecreditcycle. Foridentificationpurposes,wefixthefactorloadingfornonfinancialbusinesscreditgrowthat1. Thefactorpositivelyloadsontothehouseholdandfinancialsectors,with 100%ofdrawsbeingabove0inbothcases. Indeed,theposteriormedianofthefactorloadingon household credit is 1.3, with nearly all draws above 1, while the posterior median of the loading on financial credit is around 1. The factor therefore is heavily informed by the common cyclical comovementofthethreeprivatecreditgrowthseries. Thefactor,however,alsoplaysaroleinunderstandingthepubliccreditgrowthdynamics. Ithasafactorloadingof−0.2onthegovernment credit growth series, with nearly all draws less than 0. Therefore, the factor broadly captures the correlationdynamicswedocumentedinthedata. Figure 6 shows the smoothed factor estimates. In the top panel, the red line is the posterior median of the nonlinear factor estimates along with the 68% credible bands.14 The credit cycle factor was strong throughout the 1960s before the recession in 1969. It then rebounded before collapsing again during the mid 1970s recession, with similar dynamics repeating again in the late 1970s to early 1980s. The frequency of the credit cycle lengthened afterwards, with a robust expansion in the 1980s before declining again in the late 1980s and early 1990s with the savings 13ThedetailsofthedataconstructioncanalsobefoundinSectionEoftheAppendix. 14Weshow68%bandsinsteadof80%inthepreviousapplicationbecauseweareusingquarterlyasopposedtomonthlydata(e.g. StockandWatson(2016)). 27

Factor Estimates 2 First-order factor 1.5 Nonlinear factor 1 0.5 0 -0.5 -1 -1.5 -2 1960 1970 1980 1990 2000 2010 2020 Second order factor 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 1960 1970 1980 1990 2000 2010 2020 Figure 6: SmoothedfactorestimatesproducedbytheNLDFmodel. Thetoppanelshowsthedemeaned factorestimates,withtheredlinebeingtheestimateofthenonlinearfactorandthebluelinetheestimate ofthefirstorderfactor. Theshadedareasdenote68%crediblesets. Thebottompanelshowsthedemeaned secondorderfactorwith68%crediblesets. and loan crises. Following that episode was a prolonged expansion through the 1990s and 2000s before the collapse in the GFC. The recovery from the GFC was especially slow, with the credit factor still below its mean even over 10 years after the recession. This finding points to a secular stagnation in financial markets. That is, the GFC resulted in a significant and permanent change infinancialmarkets. Indeed,wecanseethatthestagnationcapturedbythefactorarisesfromthe dynamicsofcreditinthehouseholdandfinancialsectors. Thebluelineandshadedareasarethecorrespondingmovementsofthefirstorderfactoronly forcomparisonpurposes. Thesearethecounterfactualestimatesofthefactorifwehadseth to0 xx acrossallofthedraws,holdingallelseequal. Thebottompanelshowstheestimatesofthesecond order factor adjusted to have 0 mean. The nonlinear component of the model was significantly positive starting in the 1970s, providing a boost to credit growth. It then declined to negative territory in the late 1980s during the savings and loan crisis. The factor again turned positive for a15yearstretchbeginningintheearly1990suntiltheGFC,whenthesecondorderfactorswung heavilynegative. Thisnegativeswingcontributedtothesluggishrecoveryofcreditgrowthpostcrisis. Importance of the Nonlinear Factor How important is the nonlinear factor when modeling the credit cycle? We answer this question in three ways. First, we look at the unconditional distributionofthenonlinearmodelcomparedtoacounterfactualonewithonlythefirstorderfactor 28

Table1: UnconditionalMomentsImpliedbytheNonlinearDFMandtheLinearFactorOnly (cid:16) (cid:17) Skewness 5%Shortfall 95%Longrise Corr fs, (cid:0) ff(cid:1)2 VarianceDecomp Nonlinear −0.18 −1.91 1.22 −0.53 9.69 [−0.27,−0.07] [−2.59,−1.38] [1.08,1.37] [−0.54,−0.51] [1.41,25.78] LinearOnly −0.00 −1.34 1.33 − − [−0.00,0.00] [−1.49,−1.20] [−1.48,−1.19] − − ThetableshowstheKellyskewness,5th/95thshortfallandlongrise,andthevariancedecompositionshowingthepercentageofunconditionalvariationimpliedbythesecondorderfactor. "Nonlinear"referstothe full model while "Linear Only" refers to a counterfactual in which h = 0 for all of the draws, keeping xx everythingelsethesame. Theheadlinenumberistheposteriormedianwhilethenumbersinbracketsare the16/84crediblesets. active. Second,weinvestigatethestate-dependenteffectsofshocksconditioningonthreetimeperiods: thecreditboominthemid2000s,thebustinthelate2000sandearly2010s,andamixedcase inthelate1980s. Wefindthelasttimeperiodlistedparticularlyinterestingasithadapositivefirst orderandoverallcreditfactor,butanegativesecondorderfactor. Thisisincontrasttothefirsttwo time periods, in which the first and second order factors had the same signs. Finally, we look at thestandarddeviationandtailriskeffectsofshocks. Itisimportanttoreemphasizethatinalinear dynamicfactormodel,shocksdonothavestate-dependentnorhigher-ordermomenteffects. UnconditionalDistribution A key implication of the nonlinear model is that the unconditional distribution of the factor isnotnormallydistributed,eventhoughtheexogenousinnovationstothesystemare. Thisisnot the case if we ignore the second order component. In examining the credit growth data, we saw someevidence ofasymmetries. Thesefeaturesofthe datainformtheestimation ofthenonlinear model. Table 1 shows that the nonlinear model generates a negative Kelly skewness, with mass belowthemedianofthedistributioncoveringnearly60%ofthetotaldistancefromthe10thtothe 90th percentiles. The credible sets of the skewness estimates are wide, reflecting the difficulty in pinning down the magnitude of the higher order moments. However, there is evidence that the skewnessisnegativeatthe68thpercentilecrediblesets,asseeninthetable. Thiscontinuestobe the case at the 80th percentile sets as well. The second row of the table shows the corresponding estimatesforthelinearonlymodel. WithalinearprocessandGaussianshocks,themodelcannot generateanyskewness. ThenexttwocolumnsinTable1showtheestimatesoftheloweranduppertailsofthedistribution. Asareference,themeanofthefactorbyassumptionis0. Thenonlinearmodelgenerates a distribution that has higher probability on large declines in the credit cycle as opposed to large increases. Thisasymmetrictailbehaviorisconsistentwiththenegativeskewnesspreviouslydiscussed. Incomparison,thelinearmodelgeneratessymmetrictailbehavior. 29

Underlying the skewness and tail risk behavior of the model is a strong correlation between the level and volatility components of the nonlinear factor. The second to last column of Table 1 showsthemodel-impliedcorrelationbetweenthesecondorderfactor,whichentersintothelevel of the nonlinear factor, and the square of the first order factor, which determines the conditional volatility of the innovations to the second order factor. This correlation is −0.53, which suggests thattheconditionalvolatilityofthecreditcycleincreasesasthecreditcycledeclines. Thisisagain consistent with the idea that credit expansions are smoother than credit contractions. Moreover, itcanalsogeneratethenegativeskewnessandlonglowertailscoupledwithshortuppertailswe see. In addition to examining deviations from normality, we can also compute the unconditional variancedecompositionoftheoverallfactorintoitslinearandnonlinearcomponents. Ifthesecond orderfactorshareofoverallfluctuationsishigh,thenitisfurtherevidencethatnonlinearitiesplay an important role in the credit cycle. The last column shows this variance decomposition for the second order factor. Its median estimate is around 10%, indicating a secondary, although still quantitativelyrelevantrole. Similartotheresultsbefore,itscrediblesetiswide. State-DependentandAsymmetricEffectsofShocks Twokeyaspectsofthenonlinearcreditcyclearestate-dependentandasymmetricresponsesto shocks. Figure7showstheresponsestoonestandarddeviationpositiveandnegativeshockstothe creditcyclefactor. Theredlineandshadedareasaretheresponsesfromthenonlinearmodelwhile the blue line and shaded areas are the responses from the linear model. The first row shows the effectsofapositiveshockwhilethesecondrowshowstheeffectsofanegativeshock. Thecolumns condition on the smoothed state estimates of three different time periods: a boom period in the mid2000s,thebustaftertheGFCin2010,andamixedcaseleadingintotheearly1990srecession. Duringthecreditboomperiod,whereboththefirstorderandsecondorderfactorswerepositive,theexpectedpathofthecreditfactoractuallybehavessimilarlytothelinearonlymodel. The persistence of the first order factor, governed by the h parameter has a posterior mean of 0.92, x with the nonlinear factor showing similar intertemporal dynamics. Moving to the credit crunch period in 2010, the first and second order factors both were negative. This generates a response totheshockthatismorepersistentandwithaslighthumpshapeintheinitialquarters. Thereis a change in the conditional volatility of the shock when compared to the credit boom period as well,withthemagnitudesoftheresponsestothesamesizedshocklargerinthequartersafterits realization. Thesefindingsareconsistentwiththeunconditionaldistributionresults,whichfound anegativerelationshipbetweenthelevelofthecreditfactoranditsconditionalvolatility. Thefinal column of the figure shows a mixed period before the early 1990s recession. The smoothed first orderfactorwaspositivebutdecliningwhilethesecondorderfactorbecamenegative. Thiscombinationofstatesleadstoaresponsetothecreditfactorshockthatdiesoutmorequicklycompared toboththecreditboomandcrunchstates. Thisistruebothforapositiveandnegativeshock. Insummary, thenonlinearmodelexhibitsevidenceofstate-dependenceintheresponsestoa 30

Credit Boom Credit Crunch Credit Mix 0.4 0.4 0.4 Nonlinear k Linear c o h 0.2 0.2 0.2 S e v itis 0 0 0 o P -0.2 -0.2 -0.2 0 4 8 12 16 20 24 0 4 8 12 16 20 24 0 4 8 12 16 20 24 0.2 0.2 0.2 k c o h 0 0 0 S e v ita g-0.2 -0.2 -0.2 e N -0.4 -0.4 -0.4 0 4 8 12 16 20 24 0 4 8 12 16 20 24 0 4 8 12 16 20 24 Figure 7: StatedependentIRFsproducedbytheNLDFmodel. Theredlinesdenotetheresponsesofthe overall factor following a positive shock (top row) and negative shock (bottom row) while the blue lines denotetheresponsesofthelinearcomponentofthemodel. Thefirstcolumnconditionsonacreditboom periodinthemid2000s, thesecondcolumnconditionsonacreditbusttimeperiodin2010, andthefinal columnconditionsonamixedcasebeforetheearly1990srecession. Theshadedareasdenote68%credible sets. shock. Theresultsareinlinewiththeoreticalpredictionsaswell. Whenthecreditcycleisstrong,a creditfactorshockbehavesapproximatelylinearly. Thesetimescorrespondtoperiodsofslackborrowingconstraintsandeasycredit(GuerrieriandIacoviello,2017). Timesimmediatelyaftercredit crunches generate amplification and persistence as borrowing constraints tighten. Our empirical resultssuggestthatdynamicsinthedataareconsistentwiththesetheories. Wealsocommentbrieflyontheasymmetryintheresponsestopositiveversusnegativeshocks. Acrossallofthetimeperiods,thereisevidencethatanegativeshockgeneratesalargerandmore persistent response when compared to a positive shock. Negative shocks lead to a response approximately10%largerinmagnitudewhencomparedtopositiveones.15 Finally, we search for evidence of size-dependencies in the response to a shock. We identify twohistoricalepisodesinwhichthemodelestimateslargeshocks: 1980:Q2and2008:Q2. Then,we askwhetheratwostandarddeviationshockgeneratesadifferentresponsewhencomparedtotwo timesaonestandarddeviationshockduringthesetimes. Wefindlittleevidenceofthismechanism atplayforeitherapositiveornegativeshock. 15In Section E of the Appendix, we present results on the difference between the magnitudes between positiveandnegativeshocks. 31

Higher-OrderMomentEffectsofShocks Figure 8shows thehigher-order momenteffects ofshocks. Forthese results, we conditionon the credit crunch state, although many of the qualitative features we discuss apply to the other timesaswell. Mean response Std response Tail risk response 0.4 0.6 0 SF k c0.3 LR o h -0.05 0.4 S e0.2 v itis -0.1 0.2 o0.1 P 0 -0.15 0 0 4 8 12 16 20 24 0 4 8 12 16 20 24 0 4 8 12 16 20 24 0 0.15 0 k c o h -0.1 0.1 -0.2 S e v-0.2 ita g 0.05 -0.4 e-0.3 N 0 -0.4 -0.6 0 4 8 12 16 20 24 0 4 8 12 16 20 24 0 4 8 12 16 20 24 Figure 8: IRFs of the mean, standard deviation, and 5% shortfall and longrise of the demeaned overall factor produced by the NLDF model during the credit crunch time period in 2010. The responses to a positive shock are shown in the top row and the responses to a negative shock are shown in the bottom row. Inthethirdcolumn,thebluelinesdenotetheshortfallresponsewhiletheredlinesdenotethelongrise response. Theshadedareasare68%crediblesets. A positive shock leads to an increase in the mean and a decline in the volatility of the credit factor predictive distribution, as seen on the first two columns of the figure. Moving to the last column, we see the effects that these shocks have on the tail risk of the predictive distribution. Theshortfallincreasesmorethanthelongrisedoesbecausetheincreaseinmeananddecreasein volatilitybothleadtothelowertailofthedistributionshiftingleftward. Ontheotherhand,these effects partially cancel each other out on the upper end of the distribution, generating the more mutedresponse. The bottom row shows the response to a negative shock. The responses flip in sign, with the shock generating an increase in the volatility. Both the shortfall and longrise decline, with the decline in the shortfall still greater than the decline in the longrise. Taken together, these results suggestthatacreditcycleshockproduceslargermovesindownsideriskrelativetoupsiderisk. An adversecreditcycleshocklowersthefactoronaverage,anditalsoincreasestheriskofparticularly large declines due to an increase in volatility. In contrast, a positive shock increases the factor on average,andalsofurtherdecreasestheriskoflargedeclinesduetoadeclineinvolatility. 32

5 Conclusion Weproposeaparsimoniousnonlineardynamicfactormodel,thatisbuiltarounda second pruned order factor equation. In this model, the propagation of shocks is asymmetric, state-dependent andsize-dependent,andstationarityisguaranteedbyconstruction. Theapplicationoftheparticle filter to evaluate the likelihood and extract the factor allows us to augment the nonlinear factor motion with nonlinearities in the measurement equation, which makes the model applicable to macroeconomicenvironmentsinwhichvariablescanbeconstrained. Weinvestigatethepropertiesofthemodelandillustratethenonlinearmeasurementequation, estimatingthe modelàlaWuandXia(2016)withameasurementequationthatspecishadowrate fiesaneffectivelowerboundonU.S.data. Weshowhowtheextracted factorandthus shadowrate conclusionsregardingthemonetaryconditionsdifferbetweenmodels: onewithanonlinearmeasurementequationandsecondorderfactordynamicsandanotherthatisastandardlinearfactor model. The former predicts an easing of monetary conditions during the ELB period, while the latterdoesnotprovideevidenceofsuch. Our credit cycle application emphasizes the importance of a second order component when measuringthecreditcycle. Thisnonlinearityleadstostate-dependentimpulseresponsefunctions andchangesinthehigher-ordermomentsinresponsetoshocks. There are several directions in which our work can be expanded. We mentioned already the multidimensionalfactorinthemaintext. Anotherfruitfulavenueistousethenonlineardynamic factormodelwithaVARinthesamefashionastheFAVARmodel(StockandWatson,2016). 33

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Appendices A Estimation Algorithms In this section of the Appendix, we provide further details about the two estimation algorithms thatweuse. ThefirstoneisaMetropolisHastingsalgorithmusingtheparticlefilter. Thesecond isaparticleGibbssamplingalgorithm. WerefertoourbenchmarkNLDFmodel,shownagainin Equation24forconvenience. y = Gf +η(cid:15) t t t     f t = c+f t f +f t s  ff = h ff +σν (24) t x t−1 t   (cid:16) (cid:17)2  fs = h fs + 1h ff . t x t−1 2 xx t−1 Here,weassumethatc = −1 hxxσ2 . 2(1−hx)(1−h2 x ) A.1 Metropolis Hastings with Bootstrap Particle Filter OurMetropolisHastingsalgorithmisasfollows: 1. ProposenewsetofparametersΘprop = (cid:8) Gprop,ηprop,hprop,hprop,σ2,prop(cid:9) x xx • In practice, we break up the proposals into three blocks: Block 1 (factor equation) Θprop = (cid:8) hprop,hprop,σ2,prop(cid:9);Block2(measurementequationloadings)Θprop = {Gprop}; 1 x xx 2 Block3(measurementequationvariances)Θprop = {ηprop}. Foreachblock,wetake50 3 draws,holdingtheparametersintheotherblocksattheirpreviouslyacceptedvalues. Θprop = Θcurr +0.95S ζ +0.05S ζ , ζ ∼ N(0,I) i = 1,2,3,4 i i i,1 1 i,2 2 i • We tune the variance covariance matrix of the proposals S and S in an adaptive i,1 i,2 fashion over the first 30,000 draws of the algorithm. S is calculated using variance i,1 covariancematrixfromallofthepreviousdrawsmultipliedbyascalingparameterthat decreasesiftheprevious250drawswithintheblockhadanacceptanceratebelow10%. S isadiagonalmatrixthatismeanttointroducesomeindependentnoisewithinthe i,2 proposal. Itismultipliedbyaseparatescalingparameterthatdecreasesiftheprevious 250drawswithintheblockhadanacceptanceratebelow10%. 2. Evaluatethelikelihoodoftheproposedparametersusingthebootstrapparticlefilter(Särkkä, 2013). 39

• Initialize particle filter: For particles j = 1,...,N. To take a draw from the unconditionaldistribution,wesimulatethemodelfor500periodsandusethefinalperiodofthe simulationtodetermine: f f,(j) ,f s,(j) ,f s,(j). Notethatf s,(j) isafunctionoff f,(j) ,f s,(j), 0 0 1 1 0 0 soitisknown. Wesetw (j) = 1forallparticles. t Fort = 1,...,T: • Predictionstep: (cid:110) (cid:111) Givenparticlesandweightsatt−1: f f,(j) ,f s,(j) ,w (j) t−1 t t−1 (cid:110) (cid:111) (a) Forparticlesj = 1,...,N. Drawanewparticle f f,(j) ,f s,(j) from t t+1 f,(j) f,(j) f = h f +σν t x t−1 t 1 (cid:16) (cid:17)2 s,(j) s,(j) f,(j) f = h f + h f t+1 x t 2 xx t (b) Calculateweights: (j) f,(j) s,(j) ω = p(y |f ,f ), j = 1,...,N t t t t • Updatestep: (a) Definenormalizedweights: w (j) = ω t (j)w t ( − j) 1 . (cid:101)t 1 (cid:80)ω(j)w(j) N t t−1 (cid:110) (cid:111) (b) Resamplefrommultinomialdistribution ω (j) ,w (j) andsetw (j) = 1. t (cid:101)t t • Computeconditionallikelihood N p(y |Y ) ≈ 1 (cid:88) ω (j) w (j) . (25) t 1:t−1 N t t−1 i=1 Theoveralllikelihoodisthenp(y|Θprop,Θcurr) = (cid:81)T p(y |Y ). i −i t=1 t 1:t−1 3. Weaccepttheproposalwithprobability (cid:26)p(y|Θprop,Θcurr)g(Θprop,Θcurr) (cid:27) prob = max i −i i −i ,1 (26) p(y|Θcurr,Θcurr)g(Θcurr,Θcurr) i −i i −i whereg(.)isthepriordistribution. A.2 Gibbs Sampling with Particle Smoother OurGibbssamplingalgorithmisasfollows: 1. DrawG,η givenff,fs,andy . Thisisastandardlinearregressionmodel. t t t 40

2. Drawh ,h givenσ,G,η,ff,fs,andy . x xx t t t WeusearandomwalkMetropolissteptodrawh andh . Giventhecurrentaccepteddraw x xx ofh andh ,ourproposalisasfollows: x xx     hprop h  x  =  x +Shζ, ζ ∼ N(0,I) hprop h xx xx Wethrowawaydrawsthatviolatethestationarityconditionshprop > 1. x Given proposed hprop and hprop, we calculate its likelihood. The new parameters change c x xx andfs. t Weupdate: 1 hpropσ2 cprop = − xx (cid:16) (cid:17) 2 (1−hprop) 1−(hprop) 2 x x and 1 (cid:16) (cid:17)2 fs,prop = hpropfs,prop+ hprop ff t x t−1 2 xx t−1 Weinitializefs,prop = fs. 0 0 Wethenformthelikelihoodoftheproposal,whichcanbecalculatedintwoparts. Thefirst isbasedonthemeasurementequationandthesecondisfromthetransitionequationofthe firstorderfactor: (cid:16) (cid:17) y −G cprop+ff +fs,prop = η(cid:15) t t t t (27) ff −hpropff = σν t x t−1 t Weaccepttheproposalwithprobability: (cid:81)T p (cid:16) y |cprop,G,η,ff,fs,prop (cid:17) p (cid:16) ff|hprop,σ,ff (cid:17) g(hprop,hprop)   t=1 t t t trans t x t−1 x xx  prob = max ,1 (cid:16) (cid:17) (cid:16) (cid:17)  (cid:81)T p y |ccurr,G,η,ff,fs p ff|hcurr,σ,ff g(hcurr,hcurr)  t=1 t t t trans t x t−1 x xx (28) where p(y |.) denotes the likelihood from the measurement equation, p (ff|.) denotes t trans t thelikelihoodfromthetransitionequation,andg(.)isthepriordistribution. 3. Drawσ2 givenG,η,ff,fs,andy . t t t 41

We draw σ2 using a random walk Metropolis step. Given the current accepted draw of σ2, ourproposalisasfollows: σ2,prop = σ2+Sσι, ι ∼ N(0,I) Wethrowoutdrawsthatarenegative. Giventheproposedσ2,prop,wecalculateitslikelihood. Thenewparameterschangec. Weupdate: 1 h σ2,prop cprop = − xx (29) 2(1−h )(1−h2) x x Wethenformthelikelihoodoftheproposal,whichcanbecalculatedintwoparts. Thefirst isbasedonthemeasurementequationandthesecondisfromthetransitionequationofthe firstorderfactor: (cid:16) (cid:17) y −G cprop+ff +fs = η(cid:15) t t t t (30) ff −h ff = σpropν t x t−1 t Weaccepttheproposalwithprobability: (cid:81)T p (cid:16) y |cprop,G,η,ff,fs (cid:17) p (cid:16) ff|h ,σ2,prop,ff (cid:17) g (cid:0) σ2,prop(cid:1)   t=1 t t t trans t x t−1  prob = max ,1 (31) (cid:16) (cid:17) (cid:16) (cid:17)  (cid:81)T p y |ccurr,G,η,ff,fs p ff|h ,σ2,curr,ff g(σ2,curr)  t=1 t t t trans t x t−1 where p(y |.) denotes the likelihood from the measurement equation, p (ff|.) denotes t trans t thelikelihoodfromthetransitionequation,andg(.)isthepriordistribution. 4. Draw ff,fs given σ, G, η, h , h , and y using the particle Gibbs sampler with ancestor t t x xx t sampling. We discuss our implementation of the sampler here, but further details of the algorithmcanbefoundinLindstenetal.(2014). • Initialize particle smoother: For particles j = 1,...,N − 1. To take a draw from the unconditional distribution, we simulate the model for 500 periods and use the final period of the simulation to determine: f f,(j) ,f s,(j) ,f s,(j). Note that f s,(j) is a function 0 0 1 1 off f,(j) ,f s,(j),soitisknown. 0 0 • Drawfirstperiod: Forparticlesj = 1,...,N −1. Wedeterminef f,(j) ,f s,(j) bysimula- 1 2 tion. 42

• Fixfinalparticle: Fixf f,(N) ,f s,(N) ,f f,(N) ,f s,(N) ,f s,(N)equaltoff,∗,fs,∗,ff,∗,fs,∗,fs,∗, 0 0 1 1 2 0 0 1 1 2 where∗denotestheacceptedpreviousdraw. • Setweights: Computew (j) = p(y1|f 1 f,(j),f 1 s,(j)) forj = 1,...,N. 1 (cid:80)N jj=1 p(y1|f 1 f,(jj),f 1 s,(jj)) Fort = 2,...,T : • Sampleindicestosetancestorsforeachparticle: Forparticlesj = 1,...,N −1. Draw a (j) fromthedistributionw . Simulatethefollowing: t t−1 f,(j) f,(a(j)) f = h f t +σν t x t−1 t (32) s,(j) s,a(j) 1 (cid:16) f,(j) (cid:17)2 f = h f t + h f t+1 x t 2 xx t • Fixfinalparticle: Fixf f,(N) equaltoff,∗. t t • Computeauxiliaryweightsforfixedparticle: Forj = 1,...,N. Wecomputetheauxiliaryweightsforthefixedparticleasfollows: aux,(j) (j) f,(N) s,(j) f,(N) f,(j) f,(N) s,(N(cid:48)) f,(N) f,(N) w = w p(y |f ,f )g(f |f )p(y |f ,f )g(f |f ) t t−1 t t t t t−1 t+1 t+1 t+1 t+1 t (33) Whencalculatingf s,(N(cid:48)),wehavetotakeintoaccountthefactthatf s,(N(cid:48)) dependson t+1 t+1 f s,(j). Therefore,f s,(N(cid:48)) doesnotequalf s,(N). Theformulais: t t+1 t+1 f s,(N(cid:48)) = h f s,(j) + 1 h (cid:16) f f,(N) (cid:17)2 (34) t+1 x t 2 xx t Note that this formula comes from Equation 23 in the Lindsten, Jordan, Schon (2014), "ParticleGibbswithAncestorSampling",JournalofMachineLearningResearchpaper withlag= 2. OurmodelisadegeneratestatespacemodeldiscussedinSection7.2of thatpaper. Wecanviewourmodelalternativelyasanon-Markovianmodelwithone factorff. Seeassociateddiscussionthere. t • SampletheassociatedancestorindexforparticleN: Wesamplea (N) fromthedistrit butionwaux. Notethatwehavetoupdatef s,(N) tomakeitconsistentwiththeselected t t+1 ancestor: f s,(N) = h f s,(a( t N)) + 1 h (cid:16) f f,(N) (cid:17)2 (35) t+1 x t 2 xx t • Setweights: Computew (j) = p(yt|f t f,(j),f t s,(j)) forj = 1,...,N. t (cid:80)N jj=1 p(yt|f t f,(jj),f t s,(jj)) Note that for the t = T, we do not have to update fs because it is the end of the t+1 43

sample. When computing the auxiliary weights for the fixed particle, we also do not considertheT +1likelihood. • Sample selected states: Sample ∗ according to w . Set ff,∗,fs,∗ equal to the sampled T t t state. 44

B Monte Carlo Results To better understand the estimation of our model, we turn to a Monte Carlo experiment. Here, we show that if the true data generating process is the nonlinear dynamic factor model, our estimationstrategysuccessfullyrecoversallparameters. Weassumethattheunderlyingmodelisour benchmark NLDF model with the following parameters: c = 0,h = 0.85, h = 2.15 , σ = 0.18, x xx diag(η) = [0.54,0.06,0.79,1.08,0.39], G = [1,0.17,1.5,2.21,0.56]. We generate 50 series of length T = 1000,startingfromff = 0,fs = 0. 0 0 Withthesyntheticdatainhand,wethenestimatethelinearfactormodelandourbenchmark NLDFmodelwithalinearmeasurementequation. ThemodelsareestimatedusingtheMetropolis Hastings and particle filter procedure detailed in Section 2.4 with 200,000 MCMC draws. We assumeflatpriorsforalloftheparameters. The parameter estimates converge to the true values under correct specification. As seen in the left panel in Figure 9, the log likelihood is higher for the nonlinear model (vertical axis) than itisforthelinearone(horizontalaxis)acrossallsimulations. Theaveragedifferencebetweenthe loglikelihoodsinthenonlinearandlinearmodelsis80points–thedifferencecanbeaslowas48 pointsandashighas127points. Correspondingly,themeansquareerrors(MSE)ofthefactorsare smallerinthenonlinearfactorversion(rightpanelinFigure9).16 We report the estimands of the state equation’s parameters in Figures 10 and 11. Whereas the nonlinear factor model’s estimate for h (y-axis) is clustered around its true value, the linear x estimate (x-axis) is about 14% more persistent. This over-persistence is compensated for with a downward bias estimate of the factor innovation volatility. This is needed so the factor delivers secondmomentsconsistentwiththedata. Incontrast,thevolatilityestimatefromtheNLDFmodel isaroundthetruevalue. Furthermore,thesecondordercomponent(h )isestimatedclosetoits xx truevalue. -3000 -3050 -3100 -3150 -3200 -3250 -3300 -3350 -3350 -3300 -3250 -3200 -3150 -3100 -3050 -3000 Log likelihood (Linear model) )ledom raenilnoN( doohilekil goL 0.038 0.036 0.034 0.032 0.03 0.028 0.026 0.026 0.028 0.03 0.032 0.034 0.036 0.038 MSE (Linear model) )ledom raenilnoN( ESM Figure9: Performanceofestimatedlinearandnonlinearmodelsonsimulateddata 16Themeansquareerrorisdefinedas (cid:80)T t= = 1 1000(fˆ t|t−ft)2,wherefˆ isthefactorfilteredfromtheestimated T t|t model(linearornonlinear),andf isthetruesimulatedfactor. t 45

1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.8 0.8 0.85 0.9 0.95 1 Linear model ledom raenilnoN Estimates of h x -1 True value -1.2 -1.4 -1.6 -1.8 -2 -2.2 -2.4 -2.6 -2.8 -3 -3 -2.5 -2 -1.5 -1 Linear model ledom raenilnoN Estimates of log( ) Figure10: Estimationbiasoflinearandnonlinearmodelsonsimulateddata Figure11: Performanceofnonlinearmodelinestimationofh onsimulateddata xx 46

C Further Simulation Results 0.4 NLDF First-order 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -6 -4 -2 0 2 4 6 8 10 12 Figure 12: Unconditional distribution of the overall factor in a calibrated model. The blue line denotes the unconditional distribution of the demeaned overall factor in the NLDF model. The dashed black line denotestheunconditionaldistributionofthefirstorderfactor. 47

Period 0 Period 1 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -10 -5 0 5 -5 0 5 10 Period 3 Period 10 0.4 0.4 Baseline -1 Shock 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -5 0 5 10 -5 0 5 10 Figure13: Dynamicsofthedistributionofthedemeanedoverallfactoratvariousperiodsafterashockin acalibratedmodel. Thebluelineisthedistributionafteranegativeshockandthedashedblacklineisthe distributionwithoutashock. Period0istheperiodoftheshock. 48

Mean response 0 -0.5 -1 0 4 8 12 16 20 24 Std response 0 -0.05 -0.1 -0.15 0 5 10 15 20 25 Tail risk response 0 Shortfall Longrise -0.5 -1 0 4 8 12 16 20 24 Figure14: IRFsofthemean,standarddeviation,andtailriskofthedemeanedoverallfactorinresponseto anegativeshockatperiod0inacalibratedmodel. Inthethirdpanel,thesolidbluelineshowstheresponse oftheshortfallandthesolidredlineshowstheresponseofthelongrise. 49

D Shadow Rate: Additional Results D.1 Details on Data Construction We followed the approach of Wu and Xia (2016) in constructing the 1-month forward rates for 7 maturities based on the nominal yield curve data from Gurkaynak et al. (2007). We use the code providedbyWuandXia(2016). Theend-of-monthmonthlydataspanstheperiodfromJan,1990 toSep,2019,andthematuritiesusedarethesameasintheoriginalpaper: 3and6months,1,2,5, 7,and10years. WedownloadtheWuandXia(2016)shadowratefromCynthiaWu’swebsite. U.S. Credit Growth by Sector 8 Nonfinancial Business Household Financial 6 Government 4 2 0 -2 -4 -6 1960 1970 1980 1990 2000 2010 2020 Figure 15: Normalized real credit growth by sector in the United States: 1952:Q1-2021:Q4 with NBER recession shading. The solid blue line is nonfinancial business credit growth, the dot dashed red line is household credit growth, the dotted magenta line is financial sector credit growth, and the dashed green lineisgovernmentcreditgrowth. D.2 Tailoring the Estimation to the Shadow Rate Model WediscussherehowwemodifytheestimationpresentedinAppendixSectionA.1toaccountfor thenonlinearmeasurementequationinEquation22,reproducedbelowforconvenience.  ∆forwardh = m + G h (c+f t f +f t s)+ηhεh t ifS(cid:98) t h >= 0.3 (36) t h −m +ηhεh otherwise h t 50

(cid:16) (cid:16) (cid:17)(cid:17) where c = − 2 1 (1−h h x x ) x ( σ 1 2 −h2 x ) , S(cid:98) t h = (cid:80)t τ=2 m h +G h c+f τ f +f τ s + forwardh 1 , ∆forwardh t = forwardh−forwardh ,andindexhstandsforthematurity. Wemodelthelatentfactoraccording t t−1 tooursecondorderdynamics: ff = h ff +σν (37) t x t−1 t 1 (cid:16) (cid:17)2 fs = h fs + h ff t x t−1 2 xx t−1 RelativetothebenchmarkNLDFwithlinearmeasurementequation,therearethreemaindifferences. First,wehaveforwardratespecificconstantsm thatcapturethelong-runmeanofeach h series. ThisisastraightforwardadditiontotheMetropolisHastingsalgorithm,andweaddablock totheestimationprocedure. Second,wehavetokeeptrackofS(cid:98)t intheparticlefilter,whichisthe sumoftheentirepathoftheparticle. Toaccountforthis,weaddanadditionalcomponenttothe particlecalled (cid:16) (cid:17) (j) f,(j) s,(j) (j) S(cid:98) = c+f +f +S(cid:98) . t t t t−1 TheconversionfromS(cid:98) (j) toS(cid:98) h,(j) foreachmaturityhisstraightforwardfromtheirrespective t t formulas. Finally, the measurement equation that we use to evaluate the weight of the particle in the particlefilterchangesdependingonwhetherS(cid:98) h,(j)isaboveorbelow0.3. Thisaffectstheprediction t stepinouralgorithm. D.3 Estimation of the Linear Model Wealsoestimatealinearversionofthemodelwhichremovesthenonlinearityinthemeasurement equation and only allows for first order factor dynamics. We use the same Metropolis Hastings schemeaslaidoutinAppendixSectionA.1,withtwodifferences. First,wehavetheextraparametersm thatweestimateasanadditionalblock. Second,weusetheKalmanFilterinsteadofthe h particlefiltertoestimatethemodel,asitisnowalinearGaussianstatespacemodel. 51

D.4 Parameter Estimates Table2: ParameterEstimates Prior NLDFMandELB LinearfactorandELB Linear h N(0.5,1) 0.192 0.183 0.188 x (0.132,0.249) (0.118,0.242) (0.126,0.261) h N(0,5) 0.272 0.000 0.000 xx (-0.010,0.474) (0.000,0.000) (0.000,0.000) σ2 IW(v =4,η =1) 0.050 0.055 0.031 (0.045,0.056) (0.048,0.062) (0.026,0.035) G N(0,5) 1.000 1.000 1.000 1 (1.000,1.000) (1.000,1.000) (1.000,1.000) G N(0,5) 1.171 1.110 1.290 2 (1.121,1.218) (1.059,1.174) (1.209,1.383) G N(0,5) 1.410 1.354 1.623 3 (1.356,1.464) (1.306,1.422) (1.542,1.735) G N(0,5) 1.466 1.385 1.704 4 (1.418,1.513) (1.300,1.483) (1.607,1.827) G N(0,5) 1.022 0.947 1.123 5 (0.944,1.099) (0.860,1.034) (1.011,1.248) G N(0,5) 0.779 0.775 0.809 6 (0.708,0.871) (0.682,0.878) (0.686,0.920) G N(0,5) 0.635 0.616 0.575 7 (0.580,0.706) (0.537,0.712) (0.451,0.685) η2 IW (cid:0) v =4,η = 1Std (cid:0) ∆forward1(cid:1)(cid:1) 0.015 0.016 0.019 1 5 (0.013,0.016) (0.014,0.017) (0.018,0.021) η2 IW (cid:0) v =4,η = 1Std (cid:0) ∆forward2(cid:1)(cid:1) 0.008 0.008 0.010 2 5 (0.007,0.009) (0.008,0.009) (0.009,0.011) η2 IW (cid:0) v =4,η = 1Std (cid:0) ∆forward3(cid:1)(cid:1) 0.002 0.001 0.002 3 5 (0.001,0.002) (0.001,0.002) (0.002,0.003) η2 IW (cid:0) v =4,η = 1Std (cid:0) ∆forward4(cid:1)(cid:1) 0.013 0.013 0.015 4 5 (0.012,0.015) (0.012,0.015) (0.014,0.017) η2 IW (cid:0) v =4,η = 1Std (cid:0) ∆forward5(cid:1)(cid:1) 0.043 0.045 0.055 5 5 (0.039,0.047) (0.041,0.051) (0.050,0.061) η2 IW (cid:0) v =4,η = 1Std (cid:0) ∆forward6(cid:1)(cid:1) 0.057 0.058 0.070 6 5 (0.052,0.064) (0.052,0.064) (0.065,0.078) η2 IW (cid:0) v =4,η = 1Std (cid:0) ∆forward7(cid:1)(cid:1) 0.063 0.064 0.071 7 5 (0.058,0.068) (0.059,0.072) (0.065,0.079) Medianvaluesoftheposteriorarereported. 10%and90%inbrackets. Loglikelihoodsarereportedatthe mode. 52

Prior NLDFMandELB LinearfactorandELB Linear m N(−0.019,1) -0.019 -0.020 -0.019 1 (-0.021,-0.017) (-0.022,-0.017) (-0.024,-0.014) m N(−0.019,1) -0.021 -0.020 -0.019 2 (-0.023,-0.019) (-0.022,-0.018) (-0.024,-0.015) m N(−0.019,1) -0.022 -0.021 -0.019 3 (-0.025,-0.020) (-0.024,-0.019) (-0.024,-0.015) m N(−0.019,1) -0.022 -0.020 -0.019 4 (-0.025,-0.019) (-0.024,-0.018) (-0.024,-0.015) m N(−0.019,1) -0.010 -0.008 -0.019 5 (-0.012,-0.008) (-0.011,-0.006) (-0.024,-0.014) m N(−0.019,1) -0.007 -0.007 -0.019 6 (-0.009,-0.005) (-0.009,-0.005) (-0.025,-0.014) m N(−0.019,1) -0.005 -0.005 -0.019 7 (-0.007,-0.004) (-0.007,-0.004) (-0.024,-0.014) LL 723.731 723.742 636.693 Medianvaluesoftheposteriorarereported. 10%and90%inbrackets. Loglikelihoodsarereportedatthe mode. 53

E Nonlinear Credit Cycle: Additional Results E.1 Details on Data Construction Ourdatarunsfrom1952:Q1to2021:Q4ataquarterlyfrequency. OurdataarefromtheZ.1Financial AccountsanddownloadedfromFRED.Thesedataarenotseasonallyadjusted,andweseasonally adjustthemusingtheCensusX-13SeasonalAdjustmentprocedureimplementedinEviews12. We deflatetheseasonallyadjusteddatabytheseasonallyadjustedGDPdeflatortoturnthemintoreal values. Fornonfinancialbusinessdebt,weusethecategoryNonfinancialBusiness,DebtSecuritiesand Loans, Liability, Level (BOGZ1FL144104005Q). For household debt, we use the category Households and Nonprofit Organizations, Debt Securities and Loans, Liability, Level (TCMILBSHNO). For financial sector debt, we use the category Domestic Financial Sectors, Debt Securities and Loans, Liability, Level (TCMDODFS). Finally, for government debt, we sum the categories FederalGovernment,DebtSecuritiesandLoans,Liability,Level(FGTCMDODNS)andStateandLocal Governments,DebtSecuritiesandLoans,Liability,Level(SLGTCMDODNS).Weseasonallyadjust thefederalandstateandlocalgovernmentdebtseparatelybeforesummingthemup. E.2 Difference Between Positive and Negative Shocks Mean response 0 -0.02 -0.04 0 4 8 12 16 20 24 10-3 Std response 0 -5 -10 0 4 8 12 16 20 24 Tail risk response 0 SF -0.02 LR -0.04 -0.06 0 4 8 12 16 20 24 Figure16: CreditBoomState(mid2000s):Draw-by-drawdifferencesbetweenpositiveandnegativeshocks onthemean,standarddeviation,andtailriskresponses. Shadedareasdenote68%crediblesets. 54

Mean response 0 -0.02 -0.04 0 4 8 12 16 20 24 10-3 Std response 0 -2 -4 0 4 8 12 16 20 24 Tail risk response 0 SF -0.02 LR -0.04 0 4 8 12 16 20 24 Figure 17: Credit Crunch State(2010): Draw-by-draw differencesbetween positive and negative shocks onthemean,standarddeviation,andtailriskresponses. Shadedareasdenote68%crediblesets. Mean response 0.05 0 -0.05 0 4 8 12 16 20 24 Std response 0.01 0 -0.01 0 4 8 12 16 20 24 Tail risk response 0.1 SF 0.05 LR 0 -0.05 0 4 8 12 16 20 24 Figure18: CreditMixState(Beforetheearly1990srecession): Draw-by-drawdifferencesbetweenpositive and negative shocks on the mean, standard deviation, and tail risk responses. Shaded areas denote 68% crediblesets. 55

E.3 Parameter Estimates Table3: ParameterEstimates Prior NLDFM h N(0.5,1) 0.922 x (0.901,0.938) h N(0,5) -0.130 xx (-0.223,-0.053) σ2 IW(v =4,η =1) 0.062 (0.050,0.077) G N(0,5) 1.000 1 (1.000,1.000) G N(0,5) 1.318 2 (1.162,1.470) G N(0,5) 0.983 3 (0.881,1.100) G N(0,5) -0.220 4 (-0.327,-0.116) η2 IW (v =4,η =1) 0.626 1 (0.551,0.698) η2 IW (v =4,η =1) 0.282 2 (0.234,0.347) η2 IW (v =4,η =1) 0.592 3 (0.537,0.651) η2 IW (v =4,η =1) 0.968 4 (0.888,1.05) Medianvaluesoftheposteriorarereported. 16%and84%inbrackets. 56