The transmission of financial shocks and leverage of financial institutions: An endogenous regime switching framework

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) The transmission of financial shocks and leverage of financial institutions: An endogenous regime switching framework Kirstin Hubrich and Daniel Waggoner 2022-034 Please cite this paper as: Hubrich, Kirstin, and Daniel Waggoner (2022). “The transmission of financial shocks and leverage of financial institutions: An endogenous regime switching framework,” Finance and Economics Discussion Series 2022-034. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2022.034. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

The transmission of financial shocks and leverage of financial institutions: An endogenous regime switching framework Kirstin Hubrich Federal Reserve Board Daniel Waggoner Federal Reserve Bank of Atlanta Preliminary; This version: May 23, 2022; previous version: December 23, 2020; first version: May 2017 Abstract We conduct a novel empirical analysis of the role of leverage of financial institutions for the transmission of financial shocks to the m acroeconomy. For that purpose we develop an endogenous regime-switching structural vector autoregressive model with time-varying transition probabilities that depend on the state of the economy. We propose new identification techniques for regime switching models. Recently developed theoretical models emphasize the role of bank balance sheets for the build-up of financial instabilities and the amplification of financial shocks. We build a market-based measure of leverage of financial institutions employing institution-level data and find empirical evidence that real effects of financial shocks are amplified by the leverage of financial institutions in a financial-constraint regime. We also find evidence of heterogeneity in how depository financial institutions, global systemically important banks and selected nonbank financial institutions affect the transmission of shocks to the macroeconomy. Our results confirm the leverage ratio as a useful indicator from a policy perspective. JELClassification: C11,C32,C53,C55,E44,G21 Keywords: Regime switching models, time-varying transition probabilities, financial shocks, leverage,bankandnonbankfinancialinstitutions,heterogeneity †The views expressed in this paper are those of the authors and do not necessarily reflect those of the Federal Reserve Board or the Federal Reserve System. The authors can be contacted at kirstin.hubrich@frb.gov and daniel.f.waggoner@atl.frb.org. We thank Francesco Ferrante, Robert Tetlow and Tao Zha for valuable discussions and Jose Berrospide, Richard Clarida, Jurgen Doornik, Nathan Foley-Fisher, Jesper Linde, Helmut Luetkepohl, Sophokles Mavroides, Teodora Paligorova, Lubomir Petrasek, Fatima Pires, Steve Sharpe, Chris Sims, Gustavo Suarez, Skander van den Heuvel, Min Wei and seminar participants at Oxford University, the Federal Reserve Board, the Federal Reserve Bank of At-lanta, the Federal Reserve System and the IMF, the European Economic Association, the International Association of Applied Econometrics and NBER-NSF conferences and the conference "Innovations in Time Series Econometrics" in Berlin for useful comments. Valuable research assistance by Kevin Starnes, Rebecca John, Jack McCoy and Hongyi Wu is gratefully acknowledged. This paper was pre-viously circulated titled: "The transmission of financial shocks and leverage of banks: An endogenous regime switching framework".

1 Introduction SincetheGlobalFinancialCrisis(GFC)substantialprogresshasbeenmadeinunderstanding the interactions of financial constraints, financial market instabilities and the macroeconomy andincorporatingthoseinstandardmacreconomicmodels,butfurtherworkisneeded. Inthispaper,weaddnewempiricalevidenceontheroleofleverageoffinancialinstitutions for the transmission of financial shocks to the macroeconomy. In addition, we develop an endogenous regime switching framework with new identification techniques to conduct our novelempiricalanalysis. Wecontributetotheempiricalliteratureby(1)presentingempiricalevidencefortherole of market leverage for the amplification of the transmission of financial shocks and the implicationsfortherealeconomy; (2)providingempiricalevidenceforadifferenttransmission of financial shocks in different regimes; and (3) providing new evidence of a role of the heterogeneityoffinancialinstitutions’leverageforthedetrimentalrealeffectsofdeleveragingof financialinstitutionsandtheimplicationsfortheprobabilityofpersistenceofthefinancialconstraintregime. Inparticular,weconsiderdepositoryfinancialinstitutions,globalsystemically importantbanks(GSIBs)aswellasselectednonbankfinancialinstitutions. Themotivationforourfocusonleverageisthreefold: First,recentliteratureonstructural macroeconomicmodelsemphasizestheroleofbankbalancesheetsforthebuild-upoffinancialinstabilitiesandtheamplificationofeconomicdownturns. Second,leverageencompasses the entire balance sheet of the financial institution and therefore is a broad indicator for signalingfinancialvulnerabilities. Third,theleverageratioisaregulatorytoolcomplementaryto the(risk-weighted)capitalratio. Webuildamarket-basedmeasureofleverageoffinancialinstitutions,buildingonAdrian and Brunnermeier (2016)1. We employ financial institution level data to construct a monthly measure of leverage as book assets over market equity. Two arguments suggest a focus on market leverage: First, market leverage developments can signal a situation where financial institutionsneedtodeleveragequickly—,forinstance,ifdebtisusedtofinanceassetgrowth asforbroker-dealers(seeAdrianandShin(2014))oriffinancialinstitutionsrelyprimarilyon short-term funding (see e.g. Adrian et al. (2011) and the related literature on maturity trans- 1SeealsoPaul(2020) 1

formation). Second,marketvaluesofequityaremoreinformativeaboutfinancialinstitutions’ lossescomparedwithbookvalues. Bookequityvaluesmightnotbeatimelypredictorofbank health. Because book values incorporate information on losses with a delay, financial institutions have time to adjust their book leverage in order to avoid hitting the regulatory limit.2 Financial institutions (banks and nonbank financial institutions) might be more fragile than theirbookleveragelevelsmakethemappear. Furthermore,marketcapitalizationofafinancial institutionsisareflectionofthemarketvalueoftheequityholders’stake,andhenceanassessment by market participants of the creditworthiness of the financial institution as a borrower. Lowmarket-to-bookratiossuggestthattheassessmentofmarketparticipantsisthatfinancial institutions are more leveraged than their books suggest (see also Adrian et al. (2018)). We highlight the role of the financial fragility implied by market leverage for the transmission of financial shocks in our empirical model, which has been pointed to in a model estimated to matchfourfactsaboutbanks’leveragedynamics(seeBegenauetal.(2021)). In addition to our novel empirical analysis, we develop a regime switching vector autoregressive (RS-VAR) model with time-varying transition probabilities. We show how the Markov-switchingstructuralvectorautoregressionmodelframework—proposedinSimsand Zha(2006)andSimsetal.(2008),andemployedintheanalysisofthetransmissionoffinancialcrisesinHubrichandTetlow(2015)—isextendedinseveraldimensions. Moreprecisely, we extend previous literature on Markov-Switching models in two important dimensions: 1. Weallowfortime-varyingprobabilitiesinRS-VARmodels. Inregimeswitchingmodelswith time-varying probabilities — sometimes referred to as "endogenous switching" models — the probability of switching regime can vary over time depending on the state of the economy. 2. We propose new identification techniques for RS-VAR models, allowing a range of general, nonrecursive (over-)identification schemes of the structural shocks. Besides nonrecursive zero restrictions, these also include sign restrictions and narrative sign restrictions, thereby bringing the approaches suggested in Antolin-Diaz and Rubio-Ramirez (2018) and Arias,Rubio-RamirezandWaggoner(2018)totheclassofregimeswitchingmodels. Wealso allowfordifferentidentificationschemesindifferentregimes. Weemploytherecentlydevel- 2Sofartheinformationcontentofmarketequityaboutbooklosseshasbeenmostlyhighlightedintheaccountingliterature,indicatingthatbankshaveflexibilityinaccountingforlosses,consistentwithevidenceinBlattner etal.(2022).Thisflexibilityinaccountingforlossescanbeevenmoreprominentfornonbankfinancialinstitutions thatarepartofouranalysis. 2

opedDynamicStriatedMetropolis-Hastingssamplerforhigh-dimensionalmodelstoestimate theposteriordistributionofthemodel. Thisnewframeworkallowsustoaddresstheeconomic questionsraisedabove. The paper is structured as follows. Section 2 discusses the related theoretical and empirical literature, thereby motivating the models estimated in this paper. Section 3 presents ournewmethodologicalproposal,outlinestheestimationandevaluationofthemodelanddiscussestheidentificationissuesthatariseforRS-VARmodels. Section4containstheeconomic motivation,thedataandmodelspecificationandtheempiricalresults. Section5concludes. 2 Related economic literature and contribution of this paper We complement previous empirical studies on financial constraints and economic dynamics by providing empirical evidence on the role of leverage of financial institutions for the transmission of financial shocks to the macroeconomy, with a particular focus on market-based leverage and the differences between the role of leverage of banks and nonbank financial institutions’leverage. WedevelopanewregimeswitchingSVARmodelframeworkforourempirical analysis that is motivated in part by recent theoretical structural model developments. We discuss recent related developments in structural theoretical models and recent empirical evidence on nonlinearities in empirical models on the relation between financial constraints andthemacroeconomy. Weprovideanddiscussreferencesontheroleofleverageversusthe capital ratio as well as the motivation for the focus on market leverage in the introduction as wellasinthemotivationofourempiricalanalysisinSection4. Adiscussionoftherelationof ourmethodologicalcontributiontotheliteraturecanbefoundinSection3. Endogenousfinancialinstabilities: Theoreticalmodels Thetheoreticalliteraturehasmadeprogressrecentlyinincorporatingfinancialinstabilityand associatednonlinearitiesintomacroeconomicmodels. StructuralmodelssuchasKiyotakiand Moore(1997)aswellasBrunnermeierandSannikov(2014)illustratehowsystemicriskmight arise endogenously, determined by the choices of the model(cid:48)s decision makers. In Kiyotaki andMoore (1997)collateralconstraintsplay akeyrolefor thepropagationandamplification 3

ofshockswhileinBrunnermeierandSannikov(2014)thereductioninthevolatilityofoutput andassetpricesleadstoincreasedleverageoffinancialinstitutions.3 Bankbalancesheetsandleverageoffinancialinstitutions More recently, a number of authors introduced a more sophisticated financial sector into an otherwisestandardmacroeconomicmodel. Financialintermediaries’balancesheetshaveimplications for the institutions’ access to funds and liquidity that affect their lending activities andtherebyeconomicactivity. Afallinthevalueofabank’stradableassetsandadeclinein loan quality can adversely affect the bank(cid:48)s capital. The fall in asset prices will affect lending activity via the collateral channel. Banks have been found to limit their deposit taking in response to a decline in net worth Gertler and Kiyotaki (2010, 2015); Gertler and Karadi (2013). The importance of the bank capital channel will also depend on the extent to which nonbank financial institutions can substitute lending and liquidity provisions by banks (see e.g. DurduandZhong(2019)). Leverageoffinancialinstitutionsisanimportantcharacteristicinthepresenceoflargeand abrupt asset price movements (see e.g. Gertler and Gilchrist (2018) on the role of leverage, and Adrian and Brunnermeier (2016) and Paul (2020) for more details on the mechanisms). Theleverageratioofabankistheratiooftotalassetstoshareholderequity. Bankleverageis anindicatorforexternalfinancingopportunitiesbybanksandforrisk-takingbybanks. Itisa cyclicalindicatorandcanamplifythetransmissionofshocks. Asset prices affect the balance sheet and thereby affect leverage both in an accounting sense as well as via resulting changes in the agents’ behavior (see Paul (2020)). Financial vulnerabilities build up in boom times, when banks enlarge their balance sheets and increase their leverage, relying more on debt as opposed to equity. In the lead-up to the GFC and Great Recession there was a pronounced rise in leverage in the banking sector. During the GFCstockpricesfelldramatically,increasingthealreadyhighleverageofbanksevenfurther. At the same time, banks’ market value of assets in terms of share holdings was also shrink- 3OthercontributionsthatincorporatefinancialinstabilitiesandnonlinearitiesincludeMendoza(2010)andHe andKrishnamurthy(2019)andBoissayetal.(2016). 4

ing, reducing the access of banks to external finance (see e.g. Ferrante (2019)).4 At some point, banks had to sharply reduce the provision of loans to households and firms to obtain liquidity to avoid a bank run and insolvency (see e.g. Gertler et al. (2016) andGertler et al. (2020)). Consequently,banksweremoreconstrainedinraisingfundsandthereforewereprovidingfewer/lowervolumeloans.5 Additionally,whenbanksrealizedthatthisshockwasnot temporary,theydeleveragedgivenareducedrisktolerance,addingfurtherconstraints. 6 The discussion around Basel III and its implementation has aimed to address capital and leverage requirementsaslessonsfromtheGFC. Financialconstraintsandeconomicdynamics: empiricalevidence Only a limited number of empirical contributions allow for nonlinearities in the relation between financial constraints and the macroeconomy, while a growing empirical literature has documentedstylizedfactsontheroleoffinancialfactorsinbusinesscyclesandforthedevelopmentofafinancialcrisis. 7 HubrichandTetlow(2015)investigatewhetherafinancialcrisis isjustamanifestationofamplifiedshocksorwhetherthetransmissionofshocksdoesactually change. They find empirical support for the hypothesis of a change in the transmission of financial shocks to the US macroeconomy in episodes of high stress. Hubrich et al. (2013) analysetheeffectsoffinancialshocksonthemacroeconomyforEUandOECDcountriesand findevidencefornonlinearitiesandheterogeneityacrosscountriesinthetransmissionoffinancialshockstothemacroeconomy. Otherstudiesalsohighlightempiricalnonlinearitiesarguing that transmission channels may operate differently depending on underlying conditions, e.g. onthecredit-to-GDPgap,forinstanceAikmanetal.(2020)orfinddifferenteffectsdepending onthenatureofthefinancialshocks,i.e. whethershocksrepresenteasingoradversefinancial conditions(seeBarnichonetal.(2019)). 4Ferrante(2019),buildingontheframeworkofGertlerandKaradi(2011),extendsastandardNewKeynesian modeltoincludearichfinancialsysteminwhichfinanciallyconstrainedbankslendtofirmsandhomeownersvia defaultablelong-termloans.Inthismodelfinancialshocksaffectinglendingspreadscanbringaboutawidespread recessionthathasatitscoreadeteriorationintheequityoffinancialintermediariesandintheirleveragecapacity. 5For a recent paper suggesting a endogenous regime switching DSGE model to analyse financial crises in Mexico,seeBenignoetal.(2020). 6Notethatassetpricedrivencyclesaremorelikelyinmarket-basedbankingsystems(IMF,2009).Thebuild-up ofleverageismorelikelyinmarket-basedsystemsduetotheeffectiveuseofcollateralizationandsophisticated riskmanagementandinformation-sharingstrategies. 7Credit rises in the run-up to financial crises (Schularick and Taylor, 2012) and recessions associated with financialcrisesareusuallydeeperthannormalrecessions,especiallyiftheyareprecededwithabuild-upofcredit (Jordàetal.,2013). 5

Brunnermeieretal.(2019)employingastructuralVARidentifiedwithheteroscedasticity, also find significant output effect of financial stress shocks, measured as spread shocks, including corporate bond (GZ) spread shock that is and an interbank lending spread shock as proxiedbythe3-monthEurodollarrateoverthe3-monthTreasuries. Arecentstrandofliterature studies the distribution of future real GDP growth as a function of current financial and economic conditions or bank capital using quantile regression. They find that the estimated lowerquantilesofthedistributionoffutureGDPgrowthexhibitstrongvariationasafunction of current financial conditions (see for instance Adrian et al. (2019) and Boyarchenko et al. (2020)).8 Wecomplementthatliteraturebytakingaparametricapproach. 3 The methodology MostofthemethodologicalliteraturefocusesonmodelswithconstantprobabilitiesofMarkov switching. Following the seminal paper by Hamilton (1989), a number of contributions extendedthebasicMarkovswitchingmodelanditsestimationproceduresuggestedinthatpaper indifferentdimensions,seeforinstanceChauvet(1998),KimandNelson(1999),Frühwirth- Schnatter(2006),SimsandZha(2006)andSimsetal.(2008). Some papers propose a class of time-varying probability Markov switching regression models, including Filardo (1994), Diebold et al. (1994), Kim (2004), Kim et al. (2008) as well as Bazzi et al. (2017) and Chang et al. (2017). In these papers the probability of regime switching depends on certain variables of interest. They assume a functional form for the dependence of the probability on the state of the economy. Most papers employ a logistic function,sometimesaprobitfunctionisused(e.g. Kimetal.(2008)). Onlyafewrecentpapers allowforoccasionallybindingconstraintsinaVARcontextthatimpliessomeendogeneityof regimeswitching,andthoseincludeMavroeidis(2021),Aruobaetal.(2021)andHayashiand Koeda(2019).9 Inthispaper,weproposeaRegime-SwitchingVectorautoregressive(RS-VAR)modelwith time-varying transition probabilities, building on and extending the framework presented in Simsetal.(2008). 8SeealsoChavleishvilietal.(2021)forhowtoaddressmacroprudentialpolicyissueswitharelatedapproach. 9Two recent DSGE model proposals with endogenous switching can be found in Chang et al. (2018), and Benignoetal.(2020). 6

Weextendpreviousliteratureinseveraldimensions: 1. Weallowforatime-varyingtransitionmatrixinaregime-switchingstructuralVARmodel;2. weallowforarangeofgeneral, nonrecursiveidentificationschemes,includingsignrestrictionsandnarrativesignrestrictions that might be different in different regimes; 3. We highlight and discuss identification issues inRegime-switchingStructuralVARmodels. 3.1 TheRegime-SwitchingModelwithtime-varyingtransitionmatrix For 1 ≤ t ≤ T, let y be an n-dimensional vector of endogenous variables, let z be a kt t dimensional vector of exogenous variables, and let sc and sv be a discrete latent variables t t withsc∈{1,···,h }andsv∈{1,···,h }. Weproposeastructuralvectorautoregressionwith t c t v time-varyingtransitionmatrix(RS-SVAR) A (sc)y =A (sc)x +Ξ−1(sv)ε , (1) 0 t t + t t t t where the predetermined vector x is [y(cid:48) ,···,y(cid:48) ,z(cid:48)](cid:48) and is of dimension m = np+k.10 t t−1 t−p t The exogenous structural shocks ε are n-dimensional and assumed to be standard normal t and independent of the regime process sc and sv. The coefficient matrix A (sc) is n×n and t t 0 t invertible, A (sc) is n×m, and Ξ(sv) is n×n and diagonal, with positive diagonal elements. + t t We call sc the coefficient regime and sv the variance regime. We define the overall regime t t process to be s =h (sc−1)+sv, which can take on h=h h distinct values in {1,···,h}. t c t t c v The coefficient and variance regime processes are assumed to be independent, though this conditioncouldberelaxed.11 Wewilldenotethematrixofprobabilitiesgoverningthetransitionoftheprocessessc and t svfromtimet totimet+1byPc andPv ,respectively. ThematrixPc ish ×h andthe t t+1|t t+1|t t+1|t c c matrixPv ish ×h . Theelementinrowiandcolumn jofthesematricesistheprobability t+1|t v v oftransitingfromregime j attimet toregimeiattimet+1. Theelementsofthesematrices are all non-negative and the columns of each of these matrices must sum to one. In general, Pc and Pv can depend on the endogenous variables y ,···,y , the exogenous variables t+1|t t+1|t 1 t 10Inourempiricalapplicationtheonlyexogenousvariableisaconstant,zt=1. 11Theassumptionthatscandsvareindependentisequivalenttotheoveralltransitionmatrixfromtimettotime t t t+1beingP =Pc ⊗Pv . t+1|t t+1|t t+1|t 7

z ,···,z ,andthematricesA (·),A (·),andΞ(·). ThisimpliesthatPc andPv couldalso 1 t 0 + t+1|t t+1|t dependontheexogenousshocksε ,···,ε . Inourempiricalexamples,thetransitionmatrices 1 t willdependonlyony ,···,y ,forsomefixednon-negativevalueof(cid:96). t−(cid:96) t Inadditiontothetime-varyingtransitionmatrices,tofullyspecifytheregimeprocessesthe initialprobabilitiesmustbespecified. Wedenotetheseby pc,whichisanh -vectorwithnon- 0 c negative elements that sum to one, and pv, which is an h -vector with non-negative elements 0 v thatsumtoone. The SVAR parameters of the model will be A (·), A (·), and Ξ(·). Both the transition 0 + matrices and the initial conditions can depend on the SVAR parameters and perhaps some additional vector of parameters that we will denote by q. The transition matrices, of course, could also depend on the endogenous and exogenous data as described above. We will compactlyrepresentalltheparametersbyθ=(A (·),A (·),Ξ(·),q). Fornow,theonlyrestrictions 0 + on the SVAR parameters are that the A (·) are invertible and the Ξ(·) are diagonal matrices 0 withpositivediagonal. 3.2 IdentificationinRS-SVARmodels One of the contributions of this paper is to highlight identification issues and to propose new identification schemes for RS-SVARs. Constant parameter SVAR models with homoskedastic Gaussian shocks are not identified, but constant parameter models with heteroskedastic shocksareidentified,atleastuptotheorderingoftheequationsandsignofeachequation,see Rigobon(2003). AsimilarresultwillholdfortheRS-SVARmodelsweconsider. Forconstant parameters structural VAR models with heteroskedastic shocks, the only identification issues are determining the ordering of the equations and the sign of each equation. For RS-SVAR models these identification issues are present, but there also two additional identification issues. 3.2.1 Identificationthroughheteroskedasticity Before stating our identification result, we need a restriction to pin down the relationship between A (·), A (·), and Ξ(·). If D is any diagonal matrix with positive diagonal, then the 0 + system given by Equation (1) and the system DA (sc)y = DA (sc)x +DΞ−1(sv)ε are ob- 0 t t + t t t t 8

servationally equivalent.12 Thus a restriction is needed to force D to be the identity. In the literature, some authors have chosen the restriction Ξ(1) = I . While this certainly solves n the identification issue, it makes the first variance regime special. We will use the restriction ∑ h k= v 1 Ξ2(k)=I n ,or∑ h k= v 1 Ξ−2(k)=I n . Thisrestrictiontreatsallthevarianceregimessymmetrically and works better with the usual priors imposed on the Ξ(·). With this restriction, we havethefollowingresult. Proposition1 SupposethespanofthepredetermineddataisallofRm andtheunconditional probabilityof beingeachoverall regimeisnot zeroforeveryt. Ifh >1, then, foralmost all v parameters values, the RS-SVAR model given by Equation (1) is identified up to the ordering andsignoftheequationsandtheorderingoftheregimes. Proof. SeeAppendixA. The hypotheses of Proposition 1 are relatively mild. The first hypothesis is equivalent to the exogenous variables not being collinear and there being at least m observations The second hypothesis says that all the regimes are accessible. If all the initial probabilities are positiveorifalltheelementsofthetransitionmatricesarenon-zero,thenthishypothesiswill be satisfied. Even if neither of these is true, as long at the positions of the zeros in the initial probabilitiesandthenon-zeroelementsofthetransitionmatrixdonotexactlymatchup,then the hypothesis will be satisfied. In the next section, we discuss the precise meaning of the statement that the model is "identified up to the ordering and sign of the equations and the orderingoftheregimes." 3.2.2 OtherIdentificationIssuesinRS-VARmodels As we saw in the previous section, multiplication of the system given by Equation (1) by an invertiblematrixcanresultinanobservationallyequivalentsystem. Inthissection,wediscuss the need for three more restrictions, all arising from multiplication of the system given by Equation (1) by an invertible matrix. This will make precise what we mean by "identified up totheorderingandsignoftheequationsandtheorderingoftheregimes."(seeproposition1). 12Italsomustbethecasethatthenewparameters,DA 0 (·),DA+(·),andΞ(·)D−1,mustsatisfytherestrictions that DA 0 (·) be invertible and Ξ(·)D−1 be a diagonal matrix with positive diagonal. Since D is diagonal with positivediagonal,botharesatisfied. 9

IfoneweretopermutetherowsinEquation(1),thenonewouldobtainanobservationally equivalent system. More formally, if Q is a permutation matrix, then the system given by Equation (1) and the system Q(cid:48)A (sc)y =Q(cid:48)A (sc)x +Q(cid:48)Ξ−1(sv)QQ(cid:48)ε are observationally 0 t t + t t t t equivalent.13 So,onemusthavearestrictionthatpicksauniqueorderingoftherowsoutofthe n!possibleordering. SincetherowsinEquation(1)correspondtoequationsandeachequation containsasingleshock,orderingtherowsisreferredtoasorderingtheequationsorordering the shocks. This is also referred to as identifying, or naming, the equations or shocks. For instance, we could order the equations so that the financial shock always appears in the first equation. In this case we have identified or named the financial shock. If no such restriction is imposed, then we say the system given by Equation (1) is identified up to an ordering of the equations. The restrictions we will employ to order the equations will be discussed in Section3.3. If one were to multiply any equation in any coefficient regime in the system given by Equation(1)byminusone,thenonewouldobtainanobservationallyequivalentsystem. More formally, if D(k ), for 1≤k ≤h , is a diagonal matrix with plus or minus ones along the c c c diagonal, then the system given by Equation (1) and the system given by D(sc)A (sc)y = t 0 t t D(sc)A (sc)x +Ξ−1(sv)ε are observationally equivalent. So, one must have a restriction to t + t t t t determine the sign of each equation in each coefficient regime. Such a restriction can be thought of as giving an interpretation to a positive shock. In the constant parameter case, Waggoner and Zha (2003) suggest a class of restrictions for determining the sign of each equation. Foreachcoefficientregime, wewillusearestrictionfromthisclassofrestrictions. Inparticular,foreachcoefficientregimeandequation,wewillrestrictthesignofthatequation so that the impulse response of some variable at some horizon to a positive shock in that equationhasaparticularsign. Intheaboveexample,wherethefinancialshockisorderedfirst, one could require that the contemporaneous response of output growth to a positive financial shock to be negative. In addition, one could require that the contemporaneous response of the financial conditions index to a positive financial shock to be positive. These would be 13Ifσ(·)isapermutationof(1,···,n),thenQ=[e σ(1) ,···,e σ(n) ],whereejisthe jthcolumnofthen×nidentity matrix, isthecolumn permutationmatrixassociatedwithσ(·). PermutationmatricesareorthogonalandifAis anyn×nmatrix,thenAQpermutesthecolumnsAbyσ(·)andQ(cid:48)ApermutestherowsofAbyσ(·). So,ifQisa permutationmatrixandDisadiagonalmatrix,thenQ(cid:48)DQpermutesthediagonalelementsofD.Thus,Q(cid:48)A (·)is 0 invertiable,Q(cid:48)Ξ−1(·)Qisadiagonalmatrixwithpositivediagonal,andQ(cid:48)εt isstandardnormal. 10

consistent with a positive financial shock being detrimental to the outlook in growth. If no such conditions were imposed, then we say the system given by Equation (1) is identified up tosign. In addition to identifying (or ordering) the equations, one must also identify (or order) boththecoefficientandvarianceregimes. Ifσ (·)isapermutationof(1,···,h )andσ (·)is c c v apermutationof(1,···,h ),thenwecandefinenewdiscretelatentvariablesbys˜c =σ−1(sc) v t c t and s˜v =σ−1(sv). If Q is the h ×h column permutation matrix associated with σ (·) and t v t c c c c Q is the h ×h column permutation matrix associated with σ (·), then the transition matrix v v v v and initial probabilities for s˜c are Q(cid:48)Pc Q and Q(cid:48)pc and the transition matrix and initial t c t+1|t c c 0 probabilities for s˜v are Q(cid:48)Pv Q and Q(cid:48)pv. Furthermore, the system given by Equation (1) t v t+1|t v v o is observationally equivalent to the system given by A˜ (s˜c)y =A˜ (s˜c)x +Ξ˜−1(s˜v)ε , where 0 t t + t t t t A˜ (k )=A (σ (k )), A˜ (k )=A (σ (k )), and Ξ˜(k )=Ξ(σ (k )), for 1≤k ≤h and 1≤ 0 c 0 c c + c + c c v v v c c k ≤h . Thuswemusthavearestrictionforselectingauniqueorderingamongtheh !possible v v c orderings of the coefficient regimes and a restriction for selecting a unique ordering among the h ! possible orderings of the variance regimes. This can be thought of as identifying, or v naming,eachcoefficientregimeandeachvarianceregime. Ifnosuchrestrictionsareimposed, thenwesaythesystemgivenbyEquation(1)isidentifieduptoanorderingoftheregimes. The restrictionsthatwewillemployfortheorderingoftheregimeswillbediscussedinSection3.3. 3.3 IdentifyingShocksandRegimes Evenintheconstantparametercasewithheteroskedasticshocks,comingupwithplausiblerestrictionstoidentifytheequationsisdifficult. Thisisoneofthedisadvantagesofidentification viaheteroskedasticity. Intheregimeswitchingcase,thisdifficultyiscompoundedbecausethe coefficientandvarianceregimesmustalsobeidentified. Inthissection,wewilldiscussthree different techniques for achieving both of these goals, sign restrictions, zero restrictions, and narrativerestrictions. Combinationsofthesetechniquescanalsobeused. Last,wewilldiscuss techniquesforidentifyingthevarianceregimes. 11

3.3.1 SignRestrictions Sign restrictions on the impulse responses have long been used to identify the shocks in the case of constant parameters with homoskedastic shocks, though this identification is only set identification. In the case of regime switching parameters with heteroskedastic shocks, sign restrictions on the impulse responses can be used to identify both the equations and regimes. Even better, because there are only finitely many ways to identify the equations or regimes, in some cases sign restrictions can uniquely identify the equations or regimes, not just set identify. These ideas can best be explained by an example. For instance, suppose that one of the coefficient regimes is called the financial constraint regime and will be ordered first and that one of the shocks is the financial shock and will be ordered first. Note that the ordering of the equations must be the same across all coefficient regimes, so the financial shock would have to be ordered first in all the coefficient regimes. One could impose the restriction that the contemporaneous impulse response, conditional on being in the financial constraint regime, to a positive financial shock is positive for both financial conditions index and leverage and negative for output growth and interest rates. For some parameter values, in no regime is there a shock whose impulse responses satisfy this pattern, so that parameter wouldberejected. Inthissense,multiplesignrestrictionsontheimpulseresponsestoagiven shockinagivenregimeimplythatthemodelisoveridentified.14 Forotherparametervalues, therecouldbeauniqueregimeandshockwhoseimpulseresponsesatisfiedthispattern. Inthis case the sign restrictions uniquely determine the financial constraint regime and the financial shock and this regime and equation could be ordered first if that were not already the case. Finally,itcouldbethecasethattherearemultipleregimesorshockswhoseimpulseresponse satisfiedthispattern. Inthiscaseeitherthefinancialconstraintregimeorthefinancialshock, or both, would not be uniquely determined. As the number of sign restrictions increase, one would expect that the number of parameters that were rejected to increase and the number of parameters that did not uniquely determine both the regime and equation to decrease. If different sign restrictions on the impulse responses to the same shock across all the different 14A single sign restriction on the impulse responses to a given shock in a given coefficient regime does not imposeoveridentifyingrestrictionsbecausethesignofanyequationinanycoefficientregimecouldbechanged. Alternatively,aswesawinSection3.2.2,asinglesignrestrictionontheimpulseresponsetoagivenshockina givenregimecouldbethoughtofasarestrictiondeterminingthesignofthegivenequationinthegivenregime. 12

regimesareimposed,theneithertheparameterwillberejectedortheregimeswillbeuniquely determined. Similarly,ifdifferentsignrestrictionsontheimpulseresponsestoallthedifferent shocks,inanyregime,areimposed,theneithertheparameterwillberejectedortheequations will be uniquely determined. In this example, only one regime and one equation were being determined, though this idea could easily be extended to determining multiple regimes, i.e. partialidentification,andequationsoralloftheregimesandequations 3.3.2 ZeroRestrictions Zero restrictions on the contemporaneous and predetermined parameters have also long been used to identify the shocks in the case of constant parameters with homoskedastic shocks. Theseideascanalsobeusedtoidentifytheregimesandequations. Givenacoefficientregime, ifthepatternofzerorestrictionsonthecontemporaneousandpredeterminedparametersinthat regime is different from the pattern in all other coefficient regimes, then the given coefficient regimeisuniquelydeterminedbythezerorestrictions,foralmostallparametervalues. Given an equation, if for all other equation there exists a coefficient regime such that the patterns of zero restrictions in those two equations differ in that regime, then the given equation is uniquely determined by the zero restrictions, for almost all parameter values. In the case of heteroskedasticshocks,asinglezerorestrictionoveridentifiesthemodel. Usingdifferentpatterns of zero restrictions across different equations and coefficient regimes is a powerful way ofidentifyingtheequationsandregimes. However,suchrestrictionscanseverelyoveridentify themodelresultinginapoorfittothedata. Theyshouldbeusedjudiciouslyandincombinationwithsignandnarrativerestrictions. 3.3.3 NarrativeRestrictions Narrative restrictions were used by Antolín-Díaz and Rubio-Ramírez (2018) to set identify shocks in the case of constant parameters with homoskedastic shocks. Even earlier, Sims and Zha (2006) used similar ideas to identify the regimes in a Markov switching monetary policy model. We describe this procedure in the context of our model. In the middle of Septemberin2008,LehmanBrothersfailed. OurcontentionisthatbyOctoberof2008,there was a high probability that the economy was in what we will call the financial constraint 13

regime,whichwewillorderfirst. Wecanusethistouniquelydeterminethefinancialconstraint regime. In our case, since there are only two coefficient regimes, if we can determine the financialconstraintregime,thentheotherregime,whichwecallthenormalregime,willalso be uniquely determined. The first step is to define what we mean when we say that there is high probability that we are in coefficient regime k at time t. To do this one must choose a cutoff probability, and if the smoothed probability that we are in coefficient regime k at time t is greater than the cutoff probability, then we say we there is a high probability that we are in coefficient regime k at timet. If the cutoff probability is greater than or equal to 0.5, then thereiseitherauniquecoefficientregimethatisofhighprobabilityattimet ornocoefficient regime that is of high probability at time t. Among the parameter values for which there is a unique coefficient regime that is of high probability in October of 2008, we will order the coefficient regimes so that this regime is first and call the first regime the financial constraint regime. If one was unsure of the exact period that economy was in the financial constraint regime, then one could choose a window about October of 2008, and then say the financial constraint regime was the regime that was in high probability over most of this window. For ourmodels,wefoundthatneitherthechoiceofwindowaboutOctoberof2008northecutoff probability,withinreason,affectedthedeterminationofthefinancialconstraintregime. Similar ideas could be used to identify the equations. In our model, we wish to uniquely determinewhatwewillcallthefinancialshock. Again,inOctoberof2008,therewasamassive decline in industrial production and it is our contention that the financial shock caused mostofthisdecline. Wefirstmustdefinewhatwemeanbywhenwesaythat"shockkcaused mostofthedeclineinindustrialproductionattimet". Foreachoverallregime,onecancomputetheexpectedvalueofeachtimet shock,conditionalontheoverallregimeattimet. One couldthencomputetheexpectedcontemporaneousimpulseresponseofindustrialproduction to each timet shock, conditional on the overall regime at timet. Finally, using the smoothed probabilities,onecouldthencomputetheexpectedcontemporaneousimpulseresponseofindustrial production to each time t shock. The time t shock with largest negative expected contemporaneousimpulseresponseofindustrialproductioncausedmostofthedeclineinindustrial production at timet. Except in knife edge cases, there is a unique shock that caused mostofthedeclineinindustrialproductioninOctoberof2008andwewillordertheequations 14

sothatshockisfirstandcallthefirstshockthefinancialshock. Aswithorderingtheregimes, if one was unsure of the exact period that the financial shock was dominant, then one could chooseawindowaboutOctoberof2008, andthensaythefinancialshockwastheshockthat causedthemostcumulativedeclineinindustrialproductionoverthiswindow. 3.3.4 IdentifyingtheVarianceRegimes In some ways, identifying the variance regimes is more straightforward. The inverse of the diagonal elements of Ξ(·) directly scale the structural shocks, and thus have an economic interpretation. For instance, in our modelswe are interested in thefinancial shock, whichwe order first. So, we order the variance regimes, after we have ordered the coefficient regimes, so that the first diagonal element of the Ξ(·) are in increasing order. This would imply that the first variance regime would have the most variability, at least in terms of the effect of the financial shock. We should point out that under this ordering, in the first variance regime the impulse response to a financial shock will have the largest response for all variables at all horizonsinallcoefficientregimes. 3.4 TheTransitionMatricesandInitialProbabilities Forthevarianceregimewewilluseaconstanttransitionmatrixandforthecoefficientregime we will have time-varying transition matrices of a particular functional form. Our methodology will certainly allow for both the coefficient and variance regimes to have time-varying transition matrices of completely general functional forms, but in the interest of parsimony, werestrictthevarianceregimetohaveaconstanttransitionmatrixandthediagonalelements of the coefficient regime transition matrix to be a logistic transformation of a linear function of the endogenous variables. The off-diagonal elements of the coefficient regime transition matrix will be a constant times one minus the diagonal element from the same column. For 1≤i,j≤h ,wewilldenotetheconstantelementsinthevarianceregimetransitionmatrixby v qv i,j andnotethat∑ h i= v 1 qv i,j =1. Letq v denotethevectorcontainingalltheqv i,j . Let p (i,j)denotethetime-varyingprobabilityofswitchingfromregime j attimet to t+1|t regimeiattimet+1. Weassumethatthetime-varyingprobabilityofstayinginthe jthregime 15

attimet+1,giventhatweareinthe jth regimeattimet,isoftheform 1 p (j,j)= . (2) t+1|t 1+exp(γ¯ j −∑ (cid:96) k=1 γ j,k y t−k+1 ) The scalars γ¯ are the location parameters and the n-vectors γ are the slope parameters. In j j,k keepingwithourdesiretobeparsimonious,inmostofourexamples,(cid:96)=1andonlyafewof theelementsofγ willbeallowedtobenon-zero. Wewillgatheralloftheseparametersthat j,k arenotrestrictedtozerointoavectorthatwewilldenotebyγ. In the case of only two coefficient regimes, the diagonal elements completely determine thetransitionmatrix. Iftherearemorethantwocoefficientregimes,thenfori(cid:54)= j, p (i,j)=qc (1−p (j,j)), (3) t+1|t i,j t+1,t where the qc are non-negative constants such that ∑ hc qc = 1, under the convention that i,j i=1 i,j qc =0. Letq denotethevectorcontainingalltheqc thatarenotrestrictedtobezero. j,j c i,j We will choose the initial probabilities in both the coefficient and variance regime processessothatalltheregimeshaveequalprobability. Thischoiceismandatedbythefactthat we want the initial probabilities to be invariant to permutations of either of the coefficient or variance regime. If this was not the case, then the initial probabilities would determine, at leastpartially,theorderingofthecoefficientandvarianceregimes. Unlessonewasverysure abouttheinitialregime,thiswouldnotlikelyresultinasatisfactoryconditionfororderingthe coefficientandvarianceregimes. Inthecaseofthevarianceregime,whichhasconstanttransitionmatrix,choosingtheinitial variance probabilities to be the ergodic probabilities would also be invariant to permutations of the variance regime. This would be a permissible choice and not increase the number of parameters. However,inmostcasesthiswouldnotdeliversubstantiallydifferentresults. Therearenoparameterscontrollingtheinitialprobabilitiesandtheparameterscontrolling thetransitionmatricesare(q ,q ,γ). Asstatedbefore,wegatheralloftheseparametersintoa v c vectorthatwewilldenotebyq. 16

3.5 ThePriors Inthissectionwedescribethepriorsthatwewillemploy. For each 1 ≤ k ≤ h , we will use the same Sims-Zha prior on each (A (k ),A (k )). c c 0 c + c For the hyperparameters in this priors, we will follow the recommendations in Sims and Zha (1998)formonthlydata. Foreach1≤k ≤h ,denotethediagonalelementsofΞ(k )byξ ,for1≤ j≤n. Recall v v v kv,j that we use the restriction that either ∑ hv ξ2 =1 or ∑ hv ξ−2 =1, for each 1≤ j ≤n. kv=1 kv,j kv=1 kv,j We will use the uniform prior across the ξ . This can be easily implemented as a Dirichlet kv,j distribution over (ξ ,···,ξ ), for each 1≤ j ≤n, with all the Dirichlet hyperparameters 1,j hv,j equaltoone. For the variance regime transition matrices, the parameters are q , for 1≤i,j≤h . We i,j v willuseaDirichletprioron(qv ,···,qv ),foreach1≤ j≤h . Fortheoff-diagonalelements, 1,j hv,j v wewillchoosetheDirichlethyperparameterstoallbeequaltoone. Forthediagonalelements, we will choose the hyperparameter to match the desired duration of each variance regime, thoughwewillassumethatthedurationisthesameacrossallvarianceregimes. For the coefficient regime transition matrices, the parameters are (q ,γ). We will use a c Dirichlet prior on (qc ,···,qc ,qc ,···,qc ), for each 1≤ j≤h . We will choose the 1,j j−1,j j+1,j hv,j v Dirichlethyperparameterstoallbeequaltoone,sothedistributionwillbeuniform. Fortheγ, we will use independent normal distributions. We recommend standardizing all the variables controllingthediagonalelementsofthecoefficientregimetransitionmatrices. Inthiscasewe find that using independent normal distributions works well. Alternatively, if one had prior opinions about the means and variances of the variables controlling the diagonal elements of the coefficient regime transition matrices, then these could be used to set the means and variancesoftheelementsofγ. Because we assume that the initial probabilities are all equal, there are no parameters associatedwiththeinitialparameters. 3.6 ThePosterior,FilteredandSmoothedProbabilities In this section, we give formulas for the posterior, filtered and smoothed probabilities. For completeness,theseformulaswillbeexplicitlyderivedinAppendixB. Toderiveexpressions 17

fortheposteriorandfilteredprobabilities,weneedthefollowingassumptionabouttheexogenousvariables. Assumption1(Exogeniety) p(z |S ,Y,Z ,θ)= p(z |Y,Z ),whereS =(s ,···,s ),Y = t+1 t t t t+1 t t t 1 t t (y ,···,y ),andZ =(z ,···,z ). 1 t t 1 t The idea is that p(z |Y,Z ) is the true, but unknown, distribution of z , conditional on t+1 t t t+1 Y and Z , and knowing the path of regimes or the model parameters provides no additional t t information. Note that this assumption implies that the conditional distribution of z does t not depend on the regimes. Under this assumption, if p(θ) is the prior, then the posterior is proportionalto T h P(θ|Y ,Z )∝ p(θ)∏ ∑ p(y |s ,Y ,Z ,θ)p(s |Y ,Z ,θ). (4) T T t t t−1 t t t−1 t−1 t=1st=1 Inordertocomputetheposterior,wemustbeabletoexplicitlycomputetheconditionallikelihood, p(y |s ,Y ,Z ,θ),andthefilteredprobabilities, p(s |Y ,Z ,θ). ForRS-SVARmodt t t−1 t t t−1 t−1 els, the conditional likelihood is normal and easy to compute. Given the initial probabilities, the filtered probabilities can be recursively computed via the Hamilton filter. The recursive formulasare h p(s |Y,Z ,θ)= ∑ P (s ,s )p(s |Y,Z ,θ) (5) t+1 t t t+1|t t+1 t t t t st=1 p(y |s ,Y ,Z ,θ)p(s |Y ,Z ,θ) t t t−1 t t t−1 t−1 p(s |Y,Z ,θ)= (6) t t t ∑ h st=1 p(y t |s t ,Y t−1 ,Z t ,θ)p(s t |Y t−1 ,Z t−1 ,θ) Often, one is more interested in the smoothed probabilities, p(s |Y ,Z ,θ). This can be t T T done using backward recursion in the Hamilton smoother. The formula for the backward recursionis h p(s |s ,Y,Z ,θ)p(s |Y ,Z ,θ) p(s |Y ,Z ,θ)= p(s |Y,Z ,θ) ∑ t+1 t t t t+1 T T . (7) t T T t t t p(s |Y,Z ,θ) st+1=1 t+1 t t Since p(s |Y ,Z ,θ)canbeobtainedfromthelaststepoftheHamiltonfilter,wecanstartthe T T T backwardrecursionats andthenrecursivelycomputethesmoothedprobabilities. T 18

3.7 Estimation Having efficient and accurate samplers for simulating the posterior distribution is crucial for Bayesian analysis. There are a number of recent papers proposing new methods to compute posterior distributions, e.g. Durham et al. (2014), Herbst and Schorfheide (2014), Bognanni andHerbst(2018),andWaggoneretal.(2016). We employ the dynamic striated Metropolis-Hastings (DSMH) sampler (see Waggoner et al. (2016)) [henceforth WWZ (2016)] to our new model class to derive the whole posterior distribution. The DSMH sampler is grounded in the Metropolis-Hastings algorithm, it poolsthestrengthsfromtheequi-energyandsequentialMonteCarlosamplerswhileavoiding the weaknesses of the standard Metropolis-Hastings algorithm and those of importance sampling. The basic idea of the Dynamic Striated Metropolis-Hastings sampler is to move from atractableinitialdistributiononecansamplefromandtransformtheinitialdistributiongradually to the desired posterior distribution through a sequence of stages. Dynamic adjustment is used for sampling form the target distribution at each stage when progressing to the next stage. Thesamplerutilizesimportanceweightsonlyforinitialdrawateachstagetoavoidthe degeneracyproblemofotherSequentialMonteCarlosamplers.15 4 Empirical Analysis 4.1 Motivation Wecomplementpreviousempiricalstudiesonfinancialconstraintsandeconomicdynamicsby providingempiricalevidenceontheroleofleverageoffinancialinstitutionsforthetransmission of financial shocks to the macroeconomy using our proposed regimes switching model with new identification techniques. Our motivation to focus on leverage is threefold: First, therecenttheoreticalliteratureemphasizestheroleofbankbalancesheetsforthebuild-upof financialinstabilitiesandtheamplificationofeconomicdownturns. Second,leverageencompassestheentirebalancesheetofthefinancialinstitutionandthereforeisabroadindicatorfor signalingfinancialvulnerabilities. Third,weaimtocontributetothediscussionregardingthe 15Weusethefollowingsettings: 50stages,50striations,100Groups,2000drawswithineachgroup;overall, wehavetherefore200000draws. 19

usefulnessoftheleverageratiofromafinancialstabilitypolicyperspective.16 We empirically investigate the role of balance sheets of financial institutions for the amplification of financial shocks, differences in the transmission of financial shocks in different regimes,andtheheterogeneityoffinancialinstitutionsandimplicationsforthepersistenceof financial constraint regimes. We build on Adrian and Brunnermeier (2016) and Paul (2019) andconstructanovelmarketmeasureofleverageoffinancialinstitutions. Weemployourproposedregimeswitchingstructuralvectorautoregressive(SVAR)modelforthisinvestigation. 4.2 Data,modelspecificationandidentification 4.2.1 Data Leverage of financial institutions is an important regulatory indicator of financial sector vulnerabilities, since highly levered financial institutions are less able to absorb losses. Recent literatureonstructuralmacroeconomicmodelsemphasizestheroleofbankbalancesheetsfor the build-up of financial instabilities and the amplification of economic downturns. Furthermore, leverage encompasses the entire balance sheet of the financial institution and therefore isabroadindicatorforsignalingfinancialvulnerabilities.17 In our empirical analysis we use a market value, micro-data based measure of leverage of financial institutions, building on Adrian and Brunnermeier (2016). We employ the CRSP/Compustatmergeddatabasethatcoversabroadrangeofpubliclylisteddepositoryand nondepositoryinstitutions,bankholdingcompaniesandnonbanks. Market leverage is constructed using market value equity, not the book value, i.e. based on the expected present discounted value of future cash flows of a financial institution, its creditors and its shareholders; in contrast, book values depend on specific accounting rules. Therefore, theleveragemeasurewithmarketequitythatweareusingheretakesintoaccount that before the global financial crisis the leverage of financial institutions only rose mildly, since as debt went up also the market value of asset prices increased. During the crisis asset 16NotethatthesupplementaryleverageratiohasbeenintroducedintheUSin2014.Theaimistocounterbalance the build-up of systemic risk by limiting the risk weights compression (downweighting of seemingly low risk investments)duringboomsandthereforeaddamorecountercyclicalmeasurethanariskweightedcapitalratio. Foradiscussion,seee.g.GambacortaandKarmakar(2018). 17UnderBaselIII,thesupplementaryleverageratioisintroducedasameasurethattreatsallexposuresequally, independentofanyriskassessment. Thenon-riskweightedleverageratioisintendedtoavoidthatbanksleverup theirbalancesheetbyinvestinginassetsthatappearinlow-riskcategories. 20

prices collapsed, while financial institutions were not able to reduce their accumulated debt burdenasquicklyleadingtoasharpincreaseinleverageusingthemarketvalueofequity. Note thatourdatabaseincludesalllistedfinancialinstitutions,includingabroadrangeofdepository and non-depository credit institutions and a range of nonbank institutions, including security brokers and dealers. We use data from the Fundamentals Quarterly and Security Monthly of the CRSP/Compustat Merged database. We compute a novel monthly market leverage measure based on monthly market equity and quarterly interpolated series for book assets, or - alternatively - liabilities. Our measure builds on Adrian and Brunnermeier (2016), but goesbeyondpreviousliteraturethatuseslinearinterpolationtoconvertquarterlybookvalues. We employ monthly call reports data for interpolation of the quarterly book assets and book liabilities. The source of the data used for monthly interpolation of the quarterly book assets and liabilities is a monthly survey of a sample of the commercial banks in the call reports.18 We compute leverage as book assets over market equity as well as market equity plus book liabilitiesovermarketequity. 19 Recentliteraturehashighlightedthatbookandmarketleveragedivergesubstantiallyduring crises (see also Begenau et al. (2021). We provide evidence that market leverage is a usefulindicatorformonitoringfinancialinstitutionssinceitreflectsmarketdevelopmentsina timelyway. Theadvantageofusingfinancialinstitutionsleveldataalsoisthattheyallowusto analyzetheeconomicimplicationsofheterogeneityacrossfinancialinstitutions.20 Inparticular,wefocusoncomparingmodelspecificationswithleverageofdepositoryinstitutions,with leverage of Global Systemically Important Banks (GSIBs) and with leverage of a particular groupofnonbankfinancialinstitutions,namelysecuritiesbrokersanddealers. In addition to this novel monthly measure of market based leverage, we use US monthly data, seasonally adjusted, for a sample from 1988(12) to 2019(12). We include the following variables: Output growth is measured in terms of growth in industrial production, we 18TheH.8datafromtheFederalReservepresentsanestimateofweeklyaggregatebalancesheets(assetsand liabilities)ofcommercialbanksintheUnitedStates. Thedataarebasedonweeklyreportsfrom875commercial banks. 19Thelatterisoftenreferredtoasbeingamorereliablemeasureofmarketleveragesincebookliabilitiesarea betterproxiformarketliabilitiesthanbookassetsformarketassets,andthatiswhatweuseforourmainempirical analyses. 20NotethatthetreatmentofmergersandacquisitionsinCompustatisasfollows:Whenfirmsmerge,thefinancialbalancesheetitemsofthetargetfirmgetsabsorbedintothebalancesheetitemsoftheacquirer. Therefore, whenthetargetfirm’sdataseriesends,theacquirer’sdataseriesreflectsthetargetfinancialbalancesheetitems. ThisprovidesthebackgroundforhowstructuralchangesinthecourseoftheGFCwillbereflectedinthisdataset. 21

include core CPI inflation and the 2-year Treasury rate; market leverage of financial institutions or leverage of particular financial institutions such as Global Systemically Important Banks (GSIBs) and security brokers and dealers. Market leverage is proxied by book assets overmarketequity,asdescribedabove,andweincludeabroadfinancialconditionsindexincludingspreads,financialmarketvolatilitymeasuresandotherfinancialconditionsindicators spanning a broad range of financial markets and financial intermediaries, since we are interested in investigating heterogeneity. It is published by the Federal Reserve Bank of Chicago. We chose this broad measure of financial conditions since we are interested in investigating andcomparingtheroleofheterogeneityofleverageoffinancialinstitutionsforeconomicoutcomes. 4.2.2 Modelspecification Weemployourproposedregimeswitchingvectorautoregressive(RS-VAR)modelwithtimevarying transition probabilities. We allow for two regimes in the VAR coefficients, which we label ’financial constraint’ regime and ’normal’ times. We make the regime probability dependent on the financial variables in the model since we are primarily interested in the transmission of financial shocks. We also allow for two regimes for the variances that follow aMarkovprocessasawaytomodelheteroskedasticity. Inourregimeswitchingmodelswithtime-varyingprobabilities-thatcanalsobereferred toas"endogenousswitching"models-thetransitionprobabilityofbeinginoneregimeinthe next period, given that we are in a particular regime in this period, can vary over time. We modelthetransitionmatrixtodependonthestateoftheeconomy,namelywemakeitdependent on the financial variables in our system. To identify the regimes and structural shocks, we employ sign restrictions and, alternatively, narrative sign restrictions, thereby extending the approaches suggested in Antolín-Díaz and Rubio-Ramírez (2018) and Arias et al. (2018) to regime switching models. We also allow for different identification schemes in different regimes. Weexplainthedetailsinthenextsection. 22

4.2.3 IdentifyingRegimesandShocks Regime identification In Section 3.3, we discussed how to identify the regimes using narrativerestrictions. Inthissectionwegivethespecificdetailsofhowweassignedtheregimes to either the financial constraint regime or the normal regime. For each regime, we counted thenumberofmonthsbetween2008(9)and2009(8),inclusive,thattheprobabilityofbeingin thatregimewasgreaterthan0.70. Whicheverregimehadthelargercountwaslabeledthe’financialconstraint’regimeandorderedfirst. Theotherregimewaslabeledthe’normal’regime and ordered second. The assignment of the financial constraint regime was robust to varying thecutoffprobabilityfrom0.50through0.95andchoosingashorterperiodaroundtheGlobal financialcrisis,orevenonlyusing2008(10). Putinanotherway,thedatawasveryinformative andclearastowhichregimetheeconomywasinduringtheGlobalfinancialcrisis. Shock identification In addition to identifying the regimes, in our RS-VAR model we also need to identify the shocks. In particular, we want to identify the financial shock, which we will order first. We used sign restrictions on the impulse responses to achieve this. The contemporaneousresponsetoapositivefinancialshockinthefinancialconstraintregimewas restricted to be negative for output, inflation and short-term interest rate, but positive for the financialconditionsindexandleverage. Thecontemporaneousresponsetoapositivefinancial shock in the normal regime was restricted to be positive for the financial conditions index only. Overall, in about 20 percent of the draws these sign restrictions uniquely identified the financialshock. Alternativeshockidentification Wehavecarriedoutfurtherempiricalanalysesusingnarrativerestrictionsasanalternativeshockidentificationapproach. Thisisanewclassofnarrative restrictions developed for constant parameter structural VAR models (see Antonlin-Diaz andRubio-Ramirez, 2018)thatweextendtotheregimeswitchingstructuralVARmodelsetup. The structural parameters are constrained in such a way around key historical events that structural shocks and historical decomposition agree with the narrative. Those narrative sign restrictionscombinetheappealofnarrativeswiththeadvantagesofsignrestrictions. Asmall number of key historical events (not whole time series) are used for identification, thereby 23

avoidingmeasurementerrorinnarrativetimeseriesthathavebeenpresentedinearlierliterature. In our implementation we identify the financial shocks as the one that explains most of the variation in output growth during the Global financial crisis. In addition, a considerably smaller set of sign restrictions are imposed than were when only sign restrictions were used. We only impose the positive response of the financial conditions contemporaneously, as well as the negative initial output growth response and the initial positive response of leverage. We present the empirical results using standard sign restrictions as our baseline results in the nextfewsections,andshowsomekeyresultsfornarrativesignrestrictionsinSection4.8and AppendixC. 4.3 Regimeprobabilities In the following we use smoothed probabilities as well as the time-varying probabilities and theirassociationtohistoricaleventstointerpretthedifferentregimes. Weusethespecification withmarketleverageofGSIBsasourbaselinespecificationanddiscussthoseresultsfirst. We chooseamodelspecificationwithendogenousregimeswitchingdrivenbythethreefinancial variables in our model, namely the 2-year Treasury rate, leverage and financial conditions.21 Thischoiceismotivatedbyourinterestinthefinancialshocktransmission. Figure1presentsthesmoothedprobabilityofthefirstcoefficientregimeallowingforboth variance regimes based on our endogenous regime switching specification. We present the estimateofthemedianoftheposteriordistribution.22 Weinterpretthisregimeasa"financial constraint"regime; itcoverstheendoftheSavings&Loancrisis, the1990/91recession, the Russian debt default, the GFC and the related recession. Note that the filtered probabilities - that are particularly useful for determining the financial constraint regime in real-time - are presented in Figure 2 and are very similar to the smoothed probabilities. We will use the impulseresponsestoafinancialshockpresentedinthenextsectiontoshedlightontheeconomic dynamicsinthisregime. Figure3displaysthetime-varyingprobabilityofstayinginthefinancialconstraintregime, 21Thismodelallowsforfourregimesgiventhedifferentcombinationsofvarianceandcoefficientregimes. We focusononecoefficientregimeallowingforbothvarianceregimestooccurduringthisregimeinwhatispresented inthecharts. 22Notethattheerrorbandsforthesmoothedprobabilitiesarerathertightaroundthemedian. 24

Figure1: Smoothedprobabilitiesofcoefficientregime Figure2: Filteredprobabilitiesofcoefficientregime Figure3: Time-varyingprobabilityofstayinginlowinterestrateregime,conditionalofbeing inthatregime 25

conditionalonbeinginthatregime(thelatterprobabilityasindicatedinFigure1). Thefigure illustrates that the probability of staying in the financial constraint regime is one at the end of 2008 and at the beginning of 2009, but declines sharply in 2009/2010 down to nearly 0.6 percent, increasing again during the European sovereign debt crisis before declining again further.23 This figure illustrates the value of a model with endogenous regime switching for economic interpretation. Time-varying probabilities with error bands for the specification with GSIBs’ leverage are presented in Appendix D and illustrate that uncertainty around this probability of staying in the financial constraint regime is small during the GFC and even thoughtitincreaseswhentheprobabilityisdecliningthedecline, subsequentriseandfurther declinearesignificant. 4.4 Thetransmissionoffinancialshocks An important contribution of our paper is to shed light on the role of leverage of financial institutions for the transmission of financial shocks to the real economy. Our model specificationallowsfortwocoefficientregimesthatdependendogenouslyonthefinancialvariables in the system and for two variance regimes. We now discuss the transmission of a financial shock starting with the financial constraint regime and then compare the results to normal timesshockresponses. First, we investigate the impulse responses to a financial shock in our model when including leverage of the GSIB institutions under supervision of the Federal Reserve Board.24 Figure 4 presents the impulse responses to a one standard deviation financial shock in the financial constraint regime (for one particular variance regime) conditional on staying in the financialconstraintregime. Inotherwords,weassumethattheeconomystaysinthefinancial constraint regime for 12 months after the shock has hit. The median response of the whole posteriordistributionisdisplayedwith68%errorbands. Theimpulseresponsesshowanoutputresponsethatturnsouttobesignificantlynegative, 23Notethatthetime-varyingprobabilityofstayinginaregimecanonlybeinterpretedduringthetimewhenthere isahighprobabilityofbeinginthatregimecorrespondingtothesmoothedprobabilitydepictedintheprevious Figure. 24Capitalization of GSIBs are regularly monitored in the Financial Stability Report published by the Federal Reserve. TheyincludeBankofAmericaCorporation,theBankofNewYorkMellonCorporation,CitigroupInc., TheGoldmanSachsGroup,Inc.,JPMorganChase Co.,MorganStanley,StateStreetCorporation,WellsFargo Company. 26

Figure4: ImpulseResponsestoafinancialshockinthefinancialconstraintregime,GSIBs Note: IRFsconditionalonregime;redline:medianresponse,bluelines:lowerandupperboundofthe 68percenterrorbounds;financialshock: 1stdshocktofinancialconditionsindex;identificationwith contemporaneous sign restrictions, median and 68% error bands, output growth (IP), core inflation (CPI),2-yTreasuryyield,marketleverage,Financialconditionsindex(ChicagoFed) largeandprotracted,aswouldhavebeenexpectedinthefinancialconstraintregime. Wefind thatleverageofGSIBsinitiallyincreasesduetoasharpdeclineinassetprices,andthenstarts declining since the GSIBs deleverage in response to a financial shock in financial constraints episodes. We interpret that as evidence that deleveraging can lead to amplification effects with adverse implications for the real economy. GSIBs deleverage by liquidating assets, for instance by carrying out (fire) sales of securities and/or extending fewer loans while existing loans mature. This implies a reduction in overall credit supply. At the same time, it will be moredemandingtogetexternalfinancingduetoadeclineincollateralvalue. Next, we compare the responses of the economy in the financial constraint regime and in normal times. Figure 5 shows the responses to a financial shock in normal times. We find that output growth shows a large negative response in normal times, but that response is non-persistent in contrast to the financial constraint regime. Also, in contrast to the financial constraintregimemarketleverageremainsinsignificantovertheentirehorizon. Since these impulse responses provide an average response for the respective regime and are conditional on the regime, we shed more light on the role of leverage during the GFC 27

Figure5: ImpulseResponsestoafinancialshockinnormalregime. GSIBs Note: IRFsconditionalonregime;redline:medianresponse,bluelines:lowerandupperboundofthe 68percenterrorbounds;financialshock: 1stdshocktofinancialconditionsindex;identificationwith contemporaneous sign restrictions, median and 68% error bands, output growth (IP), core inflation (CPI),2-yTreasuryyield,marketleverage,Financialconditionsindex(ChicagoFed) usingacounterfactualanalysisinSection4.6. Beforethatweturntoinvestigatingtheroleof theheterogeneityoffinancialinstitutionsintermsofleverageforthetransmissionoffinancial shocks. Heterogeneityoffinancialinstitutions: LeverageofDepositoryInstitutions Oursecondsetofresultsisfordepositoryfinancialinstitutions(includingcommercialbanks, savings and loans, and credit unions)25 from our CRSP/COMPUSTAT database of listed institutions. Again, sign restrictions are only imposed contemporaneously on all endogenous variablesinresponsetoapositivefinancialshockinoneoftheregimes,andonlyononeshock intheotherregime. IncomparisontotheresultsfortheGSIBsourfindingsforDepositoryInstitutionsdisplay a similar median output growth response for a given financial conditions tightening (Figure 6). However, the results do not display asymmetric responses in the tails of the posterior distribution as for the GSIBs, and hence - in contrast to the model with GSIBs leverage - 25Depositoryinstitutionsarefinancialinstitutionsthatreceivemoneyfromdepositorstolendouttoborrowers suchascommercialbanksandtheotherinstitutionslistedhere. 28

negative growth outcomes appear not to be more likely than positive outcomes. We also find that, as for GSIBs (see Figure 4), market leverage initially increases significantly due to the sharp decline in asset prices and then starts declining gradually since financial institutions deleverage. Figure6: ImpulseResponsestoafinancialshockinthefinancialconstraintregime,depository institutions Note: IRFsconditionalonregime;redline:medianresponse,bluelines:lowerandupperboundofthe 68percenterrorbounds;financialshock: 1stdshocktofinancialconditionsindex;identificationwith contemporaneous sign restrictions, median and 68% error bands, output growth (IP), core inflation (CPI),2-yTreasuryyield,marketleverage,Financialconditionsindex(ChicagoFed) Theresponsestoafinancialshockinnormaltimes(Figure7)showlargeandnon-persistent outputgrowtheffectsandnosignificantleverageresponse,similartotheresponseswesawfor GSIBs(Figure5). 4.5 Heterogeneity: Securitiesbrokersanddealers Nextweexaminetheroleofheterogeneitybyincludingleverageofsecuritybrokersanddealersinthemodel. The financial constraint regime is again identified by sign restrictions only contemporaneously on all endogenous variables in one of the regimes, as for previous specifications. As pointed out in Aramonte et al. (2021) higher leverage does not need to correspond to larger balance sheets, but broker dealers clearly used debt to finance asset growth (see Adrian and 29

Figure 7: Impulse Responses to a financial shock in normal regime, depository financial institutions Note: IRFsconditionalonregime;redline:medianresponse,bluelines:lowerandupperboundofthe 68percenterrorbounds;financialshock: 1stdshocktofinancialconditionsindex;identificationwith contemporaneous sign restrictions, median and 68% error bands, output growth (IP), core inflation (CPI),2-yTreasuryyield,marketleverage,Financialconditionsindex(ChicagoFed) Figure8: ImpulseResponsestoafinancialshockinthefinancialconstraintregime,securities brokersanddealers Note: IRFsconditionalonregime;redline:medianresponse,bluelines:lowerandupperboundofthe 68percenterrorbounds;financialshock: 1stdshocktofinancialconditionsindex;identificationwith contemporaneous sign restrictions, median and 68% error bands, output growth (IP), core inflation (CPI),marketleverage=assets/marketequity,Financialconditionsindex(ChicagoFed) 30

Shin(2014)). We find similar results for the real economy implications as before, with a protracted negativeoutputgrowthinresponsetoafinancialshock,withasymmetricresponsesinthetails oftheposteriordistributionasfortheGSIBsleveragespecification. However,marketleverage increases more on impact than for other financial institutions and then immediately declines due to deleveraging. We interpret that as reflecting that The dealers’ willingness to take risk amplified the growth of the dealer balance sheets going into the crisis, causing crisis losses andasubsequentsharpcontractionofbalancesheetspost-crisis. 4.6 Counterfactual: LeverageofGISBs Figure9: Counterfactual: LeverageconstantasofOctober2008 Note: Probabilityofstayinginfinancialconstraintregime,redline: counterfactualdatawithconstant GSIBs leverage, black line: actual data; Output growth (IP), core inflation (core CPI), 2-y Treasury yield,leverage(market-based,GSIBs),financialconditionsindex(ChicagoFed) To shed further light on the role of GSIBs leverage in the financial constraint regime, we carry out a counterfactual zooming into the GFC. We simulate the data based on what would have happened if (market) leverage of GSIBs would have remained constant as of October 2008. This experiment is designed to focus on the deleveraging process and its potential adverseeffectsontherealeconomy,byavoidingthefinalsharpincreaseofmarketleverageat the onset of the GFC and subsequent fall. As illustrated in Figure 9 this would have implied 31

a less pronounced tightening of the financial conditions in the Fall of 2008 and subsequent quickerrecovery. Also,thiscounterfactualwouldhaveimpliedadramaticallysmallerdecline inoutputgrowth(about20percentagepointsintermsofgrowthinindustrialproduction)and a subsequent quicker recovery in output growth and related inflation increases despite higher short-terminterestrates. Counterfactual probability of staying in the financial constraint regime: Leverage of GISBsversusDepositoryInstitutions We also compute the counterfactual probability of staying in the financial constraint regime forthecaseofkeepingleverageofGSIBsconstant. Figure10: Counterfactualprobabilityofstayinginthefinancialconstraintregime,GSIBs Note: Blackline:mediantime-varyingprobabilityofstayinginfinancialconstraintregime,blackline: uppererrorbound,redline: lowerbound;leftpanel: actual,rightpane: counterfactual Figure 10 shows that the counterfactual probability of staying in the financial constraint regime (right panel) declines more than the actual probability (left panel). Some of the outcomesindicateamuchlowerprobabilityofstayinginthefinancialconstraintregimeunderthe counterfactual scenario of constant leverage, namely as low as 0.1. This is in line with what might be expected since the median output response might be similar than for other financial institutions,butoutcomesmightbemoredetrimentalinresponsetofinancialshocksforGSIBs thatareparticularlyhighlyleveraged. 32

Indeed, this is what is illustrated in Figure 11 for a system including depository financial institutions’ leverage where the lower bound probability of staying in the financial constraint regimedoesnotgodownbelow0.90. Figure11: Counterfactualprobabilityofstayinginthefinancialconstraintregime,depository financialinstitutions Note: Blackline:mediantime-varyingprobabilityofstayinginfinancialconstraintregime,blackline: uppererrorbound,redline: lowerbound;leftpanel: actual,rightpane: counterfactual 4.7 Marketleverageandfinancialconditions To illustrate that market leverage and financial conditions take distinct roles for the transmission of financial shocks, we have carried out a number of counterfactuals. We hold the financialconditionsindexconstantasofOctober2008(tomakeitsimilarintermsoftimingto ourotherexperimentthatfocusesonthedeleveragingprocess)andcomputethecounterfactual probabilityofstayinginthefinancialconstraintregimeinthemodelwithleverageofGSIBs. Thecounterfactualprobabilityofstayingintheconstraintregimeisnotgoingdownasin the case of holding leverage constant. We interpret this as evidence that NFCI and leverage providedifferentcharacterizationsofthefinancialconditionsoftheeconomyandhavedifferentimplicationsforthepropagationofshocksandthepersistenceoftheconstraintregime. It also illustrates that it is not the market price that is the sole driver for our results, since that wouldbebehindboththefinancialconditionsindexandleverage. 33

WehavealsoestimatedourmodelthatincludesleverageofGSIBswiththeGZspreadinsteadofthethebroadfinancialconditionsindextoillustratethatitisnottheleveragemeasures or stock market variables related to financial institutions that are behind our results. We get verysimilarresultsinthattheprobabilityofstayingintheconstraintregimeisdecliningmuch morethantheactualprobability,confirmingourresultthatithelpstopreventthedeleveraging processthathasadverseimplicationsfortherealeconomy. Figure12: Counterfactualprobabilityofstayinginthefinancialconstraintregime,GSIBs Note: Blackline:mediantime-varyingprobabilityofstayinginfinancialconstraintregime,blackline: uppererrorbound,redline: lowerbound;leftpanel: actual,rightpane: counterfactual 4.8 Sensitivity: Narrativerestrictionsforshockidentification Wehaveinvestigatedthesensitivityofourresultstoimposingnarrativerestrictionsforshock identification, extending the proposed identification approach by Rubio-Ramirez et al (2018) to our regime switching model. We combine the narrative restrictions with sign restrictions and compare the results with our findings when using standard sign restrictions presented above. We generally find that our results are rather robust when using narrative restrictions. It is noteworthy, that when using narrative restrictions we need fewer sign restrictions than withthestandardsignrestrictionapproach. TheresultsfortheRS-VARincludingleverageof GSIBsarepresentedinAppendixC. 34

5 Conclusions We conduct a novel empirical analysis on the role of leverage of financial institutions for the transmissionoffinancialshockstothemacroeconomy. Tothatendwedevelopanendogenous regime-switching structural vector autoregressive model with time-varying transition probabilities. First, we allow for the transition probabilities to be dependent on the state of the economy, and thereby to be time-varying. Second, we propose new identification schemes for RS-VAR models, extending sign and narrative restrictions to the regime switching model class. To facilitate economic interpretation, we allow the identification restrictions to differ acrossregimes. Oneofourcontributionsisalsotohighlightarangeofidentificationissuesin thecontextofregimeswitchingmodels. Employing this new modelling framework we provide a novel empirical analysis of the roleofmarketleverageoffinancialinstitutionsforthetransmissionoffinancialshockstothe macroeconomy. We construct a new monthly market-based measure of leverage of financial institutions as book assets over market equity, building on Adrian and Brunnermeier (2016) byemployingfinancialinstitutionleveldata. Wecontributetotheempiricalliteratureby(1)presentingempiricalevidencefortherole of market leverage for the amplification of the transmission of financial shocks and the implicationsfortherealeconomy; (2)providingempiricalevidenceforadifferenttransmission of financial shocks in different regimes; and (3) providing new evidence of a role of the heterogeneityoffinancialinstitutions’leverageforthedetrimentalrealeffectsofdeleveragingof financialinstitutionsandtheimplicationsfortheprobabilityofpersistenceofthefinancialconstraintregime. Inparticular,weconsiderdepositoryfinancialinstitutions,globalsystemically importantbanks(GSIBs)aswellasselectednonbankfinancialinstitutions. Themotivationforourfocusonleverageisthreefold: First,recentliteratureonstructural macroeconomicmodelsemphasizestheroleofbankbalancesheetsforthebuild-upoffinancialinstabilitiesandtheamplificationofeconomicdownturns. Second,leverageencompasses the entire balance sheet of the financial institution and therefore is a broad indicator for signalingfinancialvulnerabilities. Third,theleverageratioisaregulatorytoolcomplementaryto the(risk-weighted)capitalratio. 35

Webuildamarket-basedmeasureofleverageoffinancialinstitutions,buildingonAdrian and Brunnermeier (2016) using institution level balance sheet data 26. We employ financial institutionleveldatatoconstructamonthlymeasureofleverageasbookassetsovermarketequity. Twoargumentssuggestafocusonmarketleverage: First,marketleveragedevelopments cansignalasituationwherefinancialinstitutionsneedtodeleveragequickly—, forinstance, ifdebtisusedtofinanceassetgrowthasforbroker-dealers(seeAdrianandShin(2014))orif financialinstitutionsrelyprimarilyonshort-termfunding(seee.g. Adrianetal.(2011)andthe related literature on maturity transformation). Second, market values of equity are more informativeaboutfinancialinstitutions’lossescomparedwithbookvalues. Bookequityvalues mightnotbeatimelypredictorofbankhealth. Becausebookvaluesincorporateinformation onlosseswithadelay,financialinstitutionshavetimetoadjusttheirbookleverageinorderto avoid hitting the regulatory limit.27 Financial institutions (banks and nonbank financial institutions)mightbemorefragilethantheirbookleveragelevelsmakethemappear. Furthermore, marketcapitalizationofafinancialinstitutionsisareflectionofthemarketvalueoftheequity holders’stake, andhenceanassessmentbymarketparticipantsofthecreditworthinessofthe financial institution as a borrower. Low market-to-book ratios suggest that the assessment of market participants is that financial institutions are more leveraged than their books suggest (seealsoAdrianetal.(2018)). Ourempiricalresultshighlightthefollowingconclusionsand implications: (1) Our empirical findings support the conclusion from theoretical macroeconomicmodelsfortheimportanceofbankbalancesheetsinfinancialconstraintregimesversus normaltimesforthetransmissionofshockstothemacroeconomy. (2)Deleveragingoffinancialinstitutionscanleadtoprocyclicalfinancialamplificationeffectswithadverseimplications fortherealeconomy. Our empirical evidence indicates the importance of monitoring market-based leverage of financial institutions for financial stability and shows that market leverage provides timely informationformonitoring. Wehighlighttheroleofthefinancialfragilityimpliedbymarket leverageforthetransmissionoffinancialshocksinourempiricalmodel,thathasbeenpointed 26SeealsoPaul(2020) 27Sofartheinformationcontentofmarketequityaboutbooklosseshasbeenmostlyhighlightedintheaccountingliterature,indicatingthatbankshaveflexibilityinaccountingforlosses,consistentwithevidenceinBlattner etal.(2022).Thisflexibilityinaccountingforlossescanbeevenmoreprominentfornonbankfinancialinstitutions thatarepartofouranalysis. 36

toinanestimatedmodelframework(seeBegenauetal.(2021)). Wealsoprovideevidencefor aroleoftheheterogeneityoffinancialinstitutions’leverageforthedetrimentalrealeffectsof deleveragingoffinancialinstitutions. Weshowthedifferencesintheimplicationsofleverage of depository institutions, GSIBs and nonbank financial institutions for economic outcomes andfortheprobabilityofpersistenceofthefinancialconstraintregime. Ourresultssuggestthat deleveragingofGSIBScanhavemuchmoredetrimentaleffectsonmacroeconomicoutcomes than deleveraging of depository institutions in financial constraint regimes with implications fortheprobabilityofstayinginafinancialconstraintregime. Overall, our results confirm that leverage is a useful indicator from a financial stability perspective, and we highlight in particular the usefulness of market-leverage. It appears that sofartheinformationcontentofmarketequityaboutbooklosseshasbeenmostlyhighlighted intheaccountingliterature,indicatingthatbankshaveflexibilityinaccountingforlosses. One might argue that this flexibility in accounting for losses is even more prominent for nonbank financial institutions, highlighting the relevance of our analysis of heterogeneity of financial institutionsforregulatoryconsiderations. Ourresultsontheroleofmarket-basedleverageraisethequestionhowtolowertherelated financialvulnerability. Begenauetal.(2021),forinstance,arguethatstricteraccountingrules (i.e. faster loan loss recognition) could achieve lower financial fragility and thus mitigate the impact of shocks. Aramonte et al. (2021) develop a conceptual framework that emphasises the central role of leverage fluctuations for the propagation of systemic risk due to nonbank financial institutions. We plan to explore the role of nonbank financial institutions further in future research given the heterogeneity of nonbank financial institutions and in light of the currentdiscussionabouttheappropriatedesignofnonbankfinancialinstitutionregulations. 37

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A Proof of Proposition 1 ThefollowinglemmaiskeytoprovingProposition1. Inboththestatementofthelemmaand itsproof,itwillalwaysbeassumedthat1≤k≤rand1≤m≤s. Lemma1 Let A ,···,A be invertible n×n matrices and let D ,···,D be distinct n×n di- 1 r 1 s agonalmatricessuchthatthediagonalelementsofeachD m aredistinctand∑ s m=1 D m =I n . If A˜ 1 ,···,A˜ r aren×nmatrices,D˜ 1 ,···,D˜ s aren×ndiagonalmatricessuchthat∑ s m=1 D˜ m =I n , π(k,m)isafunctionthatisapermutationof{1,···,s}foreachk,andA(cid:48)D A =A˜(cid:48)D˜ A˜ , k m k k π(k,m) k thenπ(k,m)isindependentofk,A˜ =E Q(cid:48)A andD˜ =Q(cid:48)D Q,whereQisapermutation k k k π(k,m) m matrixandE isadiagonalmatrixwithplusorminusonesalongthediagonal. k Proof. Sinceπ(k,m)isapermutationof{1,···,s}foreachk, (cid:32) (cid:33) (cid:32) (cid:33) s s I =(A(cid:48))−1 ∑ A(cid:48)D A A−1=(A(cid:48))−1 ∑ A˜(cid:48)D˜ A˜ A−1=(A˜ A−1)(cid:48)(A˜ A−1), n k k m k k k k π(k,m) k k k k k k m=1 m=1 itfollowsthat(A˜ A−1)(cid:48)≡Qˆ isanorthogonalmatrixandA˜ =Qˆ(cid:48)A . Thus, k k k k k k A(cid:48)D A =A˜(cid:48)D˜ A˜ =A(cid:48)Qˆ D˜ Qˆ(cid:48)A , k m k k π(k,m) k k k π(k,m) k k which implies that D = Qˆ D˜ Qˆ(cid:48). Because the eigenvalues of a symmetric matrix are m k π(k,m) k unique up to an ordering, the D are distinct, and the diagonal elements of each D are dism m tinct, the permutation π(k,m) does not depend on k. If Q is the column permutation matrix associated with π, then for each k there exists a diagonal matrix E , with plus or minus ones k alongthediagonal,suchthatQˆ =QE . k Proof of Proposition 1. Throughout this proof, it will be assumed that 1 ≤ k ≤ h , c 1≤m≤h , and 1≤t ≤T. Let M(k)=A−1(k)A (k) andV(k,m)=A−1(k)Ξ−2(m)A−1(k)(cid:48). v 0 + 0 0 ForalmostallA (k),A (k)andΞ(m),theM(k)aredistinct,theV(k,m)aredistinct,theΞ(m) 0 + are distinct, and the diagonal elements of each Ξ(m) are distinct. So, we can assume that the M(k)aredistinct,theV(k,m)aredistinct,theΞ(m)aredistinct,andthediagonalelementsof eachΞ(m)aredistinct. Thedistributionofy ,conditionalonx ,isamixtureofh=h h normaldistributionswith t t c v meansM(k)x ,variancesV(k,m),andweights p((sc,sv)=(k,m)|x ). Foramixtureofdistinct t t t t A.1

normaldistributionswithpositiveweights,themeansandvariancesofthenormaldistributions in the mixture and their weights are uniquely determined by the distribution of the mixture. If some of the weights were zero, then the normal distributions corresponding to those zero weights would not be determined, though the weights themselves would be. Because the V(k,m)aredistinct,theconditionaldistributionofy isamixtureofhdistinctnormalnormal t distributions. If the conditional probabilities, p((sc,sv)=(k,m)|x ), were zero for all x , then the unt t t t conditional probabilities, p((sc,sv) = (k,m)), would also be zero. So by the hypotheses of t t Proposition 1, p((sc,sv)=(k,m)|x ) cannot be identically zero for all t. Because the weight t t t associated with each (k,m) is non-zero for some t, this implies that the normal distributions andtheirweightsareuniquelydeterminedbytheconditionaldistributionsofthey . t IfthesystemA˜ (s˜c)y =A˜ (s˜c)x +Ξ˜−1(s˜v)ε generatesobservationallyequivalentdataas 0 t t + t t t t thesystemgivenbyEquation(1),thentheremustexistfunctionsπ (k,m,t)∈{1,···,h )and c c π (k,m,t)∈{1,···,h )suchthat v v M(k)x =A˜−1(π (k,m,t))A˜ (π (k,m,t))x (8) t 0 c + c t V(k,m)= (cid:0) A˜−1(π (k,m,t))Ξ˜−2(π (k,m,t))A˜−1(π (k,m,t))(cid:48)(cid:1)−1 (9) 0 c v 0 c By the hypotheses of Proposition 1, the predetermined data span all of Rm, so Equation (8) impliesthat M(k)=A˜−1(π (k,m,t))A˜ (π (k,m,t)) (10) 0 c + c Since the left hand sides of Equations (9) and (10) do not depend on t, neither can the right handside. SincetheboththeM(k)andtheV(k,m)aredistinct,thisimpliesthatneitherπ nor c π dependont. SincethelefthandsideofEquation(10)doesnotdependonmandtheM(k) v aredistinct,π doesnotdependonm. Sowecanwriteπ (k,m,t)=π (k),π (k,m,t)=π(k,m) c c c v andEquations(9)becomes V(k,m)= (cid:0) A˜−1(π (k))Ξ˜−2(π (k,m))A˜−1(π (k))(cid:48)(cid:1)−1 (11) 0 c v 0 c BecausetheV(k,m)aredistinct,itmustbethecasethatπ (k,m)isapermutationof{1,···,h }, v v A.2

for each k. We can now apply Lemma 1 with r =h , s=h , and π(k,m)=π (k,m). When c v v the system is normalized so that ∑ h m v =1 Ξ2(m) = I n , we take A k = A 0 (k), A˜ k = A˜ 0 (π c (k)), D m =Ξ2(m), and D˜ m =Ξ˜2(m). When the system is normalized by ∑ h m v =1 Ξ−2(m)=I n , we take A =A−1(k)(cid:48), A˜ =A˜−1(π (k))(cid:48), D =Ξ−2(m), and D˜ =Ξ˜−2(m). In either case, we k 0 k 0 c m m obtain that π (k,m) does not depend on k and can be written as π (k,m)=π (m) and there v v v exists a permutation matrix Q and diagonal matrices E , with plus or minus ones along the k diagonal, such that A˜ (π (k))=E Q(cid:48)A (k) and Ξ(π (m))=Q(cid:48)Ξ(m)Q. Thus, Equation (10) 0 c k 0 v implies that A˜ (π (k))=E Q(cid:48)A (k). So, under the hypotheses of Proposition 1, for almost + c k + all A (·), A (·), and Ξ(·), the system given by Equation (1) is identified up to the sign and 0 + orderingoftheequationsandtheorderingoftheregimes. B The Likelihood, Posterior, Filtered and Smoothed Probabilities In this appendix we derive the formulas for computing the the likelihood, the posterior, and filteredandsmoothedprobabilites. Thelikelihoodis p(Y ,Z |θ) T T p(Y |Z ,θ)= (12) T T p(Z |θ) T = ∏ t T =1 p(y t ,z t |Y t−1 ,Z t−1 ,θ) (13) p(Z |θ) T = ∏ t T =1 ∑ h st=1 p(s t ,y t ,z t |Y t−1 ,Z t−1 ,θ) (14) p(Z |θ) T = ∏ t T =1 ∑ h st=1 p(y t |s t ,Y t−1 ,Z t ,θ)p(z t |s t ,Y t−1 ,Z t−1 ,θ)p(s t |Y t−1 ,Z t−1 ,θ) (15) p(Z |θ) T = ∏ t T =1 p(z t |Y t−1 ,Z t−1 ) ∏ T ∑ h p(y |s ,Y ,Z ,θ)p(s |Y ,Z ,θ), (16) t t t−1 t t t−1 t−1 p(Z |θ) T t=1st=1 where Equations (12), (13), and (15) follow from Bayes’ rule, Equation (14) is obtained by integratingouts ,andEquation(16)followsfromAssumption1andarearrangementofterms. t NotethatAssumption1doesnot, ingeneral, implythat p(Z |θ)= p(Z ). If p(θ)istheprior, t t A.3

thentheposterioris p(Y ,Z ,θ) T T P(θ|Y ,Z )= (17) T T p(Y ,Z ) T T p(Y |Z ,θ)p(Z |θ)p(θ) T T T = (18) p(Y ,Z ) T T = ∏ t T =1 p(z t |Y t−1 ,Z t−1 ) p(θ)∏ T ∑ h p(y |s ,Y ,Z ,θ)p(s |Y ,Z ,θ), (19) t t t−1 t t t−1 t−1 p(Y ,Z ) T T t=1st=1 whereEquations(17)and(18)followfromBayesruleandEquation(19)followsbysubstitutingtheexpressionforthelikelihoodandcancelingandrearrangingterms. So,theposterioris proportionalto T h p(θ)∏ ∑ p(y |s ,Y ,Z ,θ)p(s |Y ,Z ,θ). t t t−1 t t t−1 t−1 t=1st=1 TherecursiveformulasforthetheHamiltonfilterarederivednext. h p(s |Y,Z ,θ)= ∑ p(s ,s |Y,Z ,θ) (20) t+1 t t t+1 t t t st=1 h = ∑ p(s |s ,Y,Z ,θ)p(s |Y,Z ,θ), (21) t+1 t t t t t t st=1 where Equation (20) is obtained by integrating out s and Equation (21) follows from Bayes’ t rule. p(s ,y ,z |Yt−1,Zt−1,θ) p(s |Yt,Zt,θ)= t t t 1 1 (22) t 1 1 p(y ,z |Yt−1,Zt−1,θ) t t 1 1 p(s ,y ,z |Yt−1,Zt−1,θ) = t t t 1 1 (23) ∑ h st=1 p(s t ,y t ,z t |Y 1 t−1,Z 1 t−1,θ) p(y |s ,Yt−1,Zt,θ)p(z |s ,Yt−1,Zt−1,θ)p(s |Yt−1,Zt−1,θ) = t t 1 1 t t 1 1 t 1 1 (24) ∑ h st=1 p(y t |s t ,Y 1 t−1,Z 1 t,θ)p(z t |s t ,Y 1 t−1,Z 1 t−1,θ)p(s t |Y 1 t−1,Z 1 t−1,θ) p(z |Yt−1,Zt−1)p(y |s ,Yt−1,Zt,θ)p(s |Yt−1,Zt−1,θ) = t 1 1 t t 1 1 t 1 1 (25) p(z t |Y 1 t−1,Z 1 t−1)∑ h st=1 p(y t |s t ,Y 1 t−1,Z 1 t,θ)p(s t |Y 1 t−1,Z 1 t−1,θ) p(y |s ,Yt−1,Zt,θ)p(s |Yt−1,Zt−1,θ) = t t 1 1 t 1 1 , (26) ∑ h st=1 p(y t |s t ,Y 1 t−1,Z 1 t,θ)p(s t |Y 1 t−1,Z 1 t−1,θ) where Equations (22) and (24) follow from Bayes’ rule, Equation (23) is obtained by integrating out s in the denominator, Equation (25) follows from Assumption 1 and rearranging t terms,andEquation(26)followsbycancelingterms. A.4

Before deriving the formulas for the smoothed probabilities, we must first develop more notation. For 1 ≤ τ ≤ t ≤ T, let St denote (s ,s ,···,s ). We also need the following τ τ τ+1 t assumption: Assumption2 p(s |Y ,Z ,S ,θ)= p(s |Y ,Z ,s ,θ) (27) t t−1 t−1 t−1 t t−1 t−1 t−1 p(y |Y ,Z ,S ,θ)= p(y |Y ,Z ,s ,θ) (28) t t−1 t−1 t−1 t t−1 t−1 t−1 Forthemodelsdiscussedinthispaper,Assumption2holds. ItispossibletoweakenAssumption2byreplacings ontherighthandsidesofEquations(27)and(28)withSt−1,forsome t−1 t−k fixedvalueofk≥1. ItfollowsfromBayes’ruleandAssumptions1and2Equation(27)thatfor1≤τ≤t ≤T p(y ,z |St,Y ,Z ,θ)= p(y |s ,Y ,Z ,θ)p(z |Y ,Z ). (29) t t τ t−1 t−1 t t t−1 t t t−1 t−1 So,1≤τ ≤τ ≤t ≤T 0 1 p(y ,z |St ,Y ,Z ,θ)= p(y ,z |St ,Y ,Z ,θ). (30) t t τ0 t−1 t−1 t t τ1 t−1 t−1 For1≤t+1<τ≤T, p(s ,s ,y ,z |Sτ−1,Y ,Z ,θ) p(s |Sτ ,Y ,Z ,θ)= t τ τ τ t+1 τ−1 τ−1 (31) t t+1 τ τ p(s ,y ,z |Sτ−1,Y ,Z ,θ) τ τ τ t+1 τ−1 τ−1 p(y ,z |Sτ,Y ,Z ,θ)p(s |Sτ−1,Y ,Z ,θ)p(s |Sτ−1,Y ,Z ,θ) = τ τ t τ−1 τ−1 τ t τ−1 τ−1 t t+1 τ−1 τ−1 (32) p(y ,z |Sτ ,Y ,Z ,θ)p(s |Sτ−1,Y ,Z ,θ) τ τ t+1 τ−1 τ−1 τ t+1 τ−1 τ−1 p(s |s ,Y ,Z ,θ)p(s |Sτ−1,Y ,Z ,θ) = τ τ−1 τ−1 τ−1 t t+1 τ−1 τ−1 (33) p(s |s ,Y ,Z ,θ) τ τ−1 τ−1 τ−1 = p(s |Sτ−1,Y ,Z ,θ), (34) t t+1 τ−1 τ−1 where Equations (31) and (32) follow from Bayes’ rule, Equation (33) follows from Equation(30),cancellation,andAssumption2Equation(28),andEquation(34)followsfromcan- A.5

cellation. Thus,byarecursiveargument, p(s |ST ,Y ,Z ,θ)= p(s |s ,Y ,Z ,θ). (35) t t+1 T T t t+1 t+1 t+1 But p(s ,s ,y ,z |Y,Z ,θ) t t+1 t+1 t+1 t t p(s |s ,Y ,Z ,θ)= (36) t t+1 t+1 t+1 p(s ,y ,z |Y,Z ,θ) t+1 t+1 t+1 t t p(y ,z |s ,s ,Y,Z ,θ)p(s |s ,Y,Z ,θ)p(s |Y,Z ,θ) t+1 t+1 t+1 t t t t+1 t t t t t t = (37) p(y ,z |s ,Y,Z ,θ)p(s |Y,Z ,θ) t+1 t+1 t+1 t t t+1 t t p(s |s ,Y,Z ,θ)p(s |Y,Z ,θ) t+1 t t t t t t = , (38) p(s |Y,Z ,θ) t+1 t t whereEquations(36)and(37)followfromBayes’rule,andEquation(38)followsfromEquation(30)andcancellation. Thus p(s |s ,Y,Z ,θ)p(s |Y,Z ,θ) p(s |ST ,Y ,Z ,θ)= t+1 t t t t t t . (39) t t+1 T T p(s |Y,Z ,θ) t+1 t t Since p(s |Y ,Z ,θ)= ∑ p(s ,ST |Y ,Z ,θ) (40) t T T t t+1 T T ST t+1 = ∑ p(s |ST ,Y ,Z ,θ)p(ST |Y ,Z ,θ) (41) t t+1 T T t+1 T T ST t+1 (cid:32) h p(s |s ,Y,Z ,θ)p(s |Y ,Z ,θ) = p(s |Y,Z ,θ) ∑ t+1 t t t t+1 T T t t t p(s |Y,Z ,θ) st+1=1 t+1 t t (cid:33) ×∑ p(ST |s ,Y ,Z ,θ) (42) t+2 t+1 T T ST t+2 h p(s |s ,Y,Z ,θ)p(s |Y ,Z ,θ) = p(s |Y,Z ,θ) ∑ t+1 t t t t+1 T T , (43) t t t p(s |Y,Z ,θ) st+1=1 t+1 t t where Equation (40) is obtained by integrating out ST , Equation (41) follows from Bayes’ t+1 rule,Equation(42)followsfromEquation(39)andrearrangingterms,andEquation(43)followsbecause p(ST |s ,Y ,Z ,θ)isadensityandintegrates(sums)toone. t+2 t+1 T T A.6

C Impulse responses with narrative restrictions FigureA.1: ImpulseResponsestoafinancialshockinthefinancialconstraintregime,narrativesignrestrictions,GSIBs Note: Impulse Responses to financial shock, narrative sign restrictions, (conditional on staying in) the financial constraint regime; red line: median response, blue lines: lower and upperbound of the 68percenterrorbounds;financialshock: 1stdshocktofinancialconditionsindex;identificationwith contemporaneous sign restrictions, median and 68% error bands, output growth (IP), core inflation (CPI),2-yTreasuryyield,marketleverage,Financialconditionsindex(ChicagoFed) FigureA.1presentstheresultsforthefinancialconstraintregimeforamodelwithleverage of GSIBs and identification with narrative sign restrictions, where the shock is identified that explainsmostofthevariationinoutputgrowthduringGFC. Responsestoafinancialshock(median)showaprotractednegativeoutputresponse. Marketleverageinitiallyincreases,thendeclinesduetodeleveraging. WearriveatthesameconclusionasinourbaselinespecificationfortheGSIBs: Deleveragingcanleadtoamplification effects with adverse implications for the real economy. In normal times, the responses to a financialshock(median)indicatesmall,nonpersistentnegativeoutputresponseandinsignificantmarketleverage(FigureA.2)aswiththestandardsignrestrictionapproach. A.7

Figure A.2: Impulse Responses to a financial shock in normal regime, narrative sign restrictions,GSIBs Note: Impulse Responses to financial shock, narrative sign restrictions, (conditional on staying in) the financial constraint regime; red line: median response, blue lines: lower and upperbound of the 68percenterrorbounds;financialshock: 1stdshocktofinancialconditionsindex;identificationwith contemporaneous sign restrictions, median and 68% error bands, output growth (IP), core inflation (CPI),2-yTreasuryyield,marketleverage,Financialconditionsindex(ChicagoFed) D Time-varying Probabilities with error bands Figure A.3: Time-varying probability of staying in low interest rate regime with 68 % error bands,conditionalofbeinginthatregime,GSIBs A.8