P aper No. 907 Banks, shadow banks, and business c

DEstimation

Thissectioncontainsdetailsaboutthemodel’sestimation.Wediscusstheparameteri- zationandpresenttwomeasuresofmodelfit.

D.1CalibratedParameters

Anumberofmodelparametershaveaclearcounterpartinthedata,sowecalibratethem tomatchthemeaninoursample.Theseincludetheannualgrowthrateoftheeconomy µ_z^∗=1.01%;theannualinflationrateπ=1.50%;theannualrateofinvestment-specific technologicalchangeΥ=0.78%;theshareofgovernmentspendinginGDPη^g=0.24; thecapitalshareinproductionα=0.37;andthetaxratesonconsumptionτ^c=0.19, laborincomeτ^l=0.12,andcapitalincomeτ^k=0.33.

Wefixafewotherparametersasfollows.Thediscountfactorofpatienthouseholds _βp_{equals0.9998,whichpinsdowntheannualizednominalinterestrateRto1.87%.}

Thediscountfactorofimpatienthouseholdsβⁱmustbelowerthanβ^pandissetto 0.97.Capitaldepreciationδandlaborsupplyelasticityσ_Larefixedat0.025and1, respectively.Wesetthesteady-statepriceandwagemarkupsλ_pandλ_wto1.20and 1.05,followingtheliterature.AllcalibratedparametersappearatthetopofTable4.

D.2EstimatedParameters

Weestimatetheremaining41parameterswithBayesiantechniques.Thebottompanel ofTable4reportstheirpriorandposteriordensities.Mostofthestructuralparameters arecommonintheliteratureandareassignedstandardpriors.13Ourresultsfallinline withpreviousstudies.Forinstance,thepolicyresponsetoinflationα_πis2.19while theinterestratesmoothingcoefficientρ_pis0.66,consistentwithexistingestimates. PosteriorvaluesfortheCalvopriceξ_pandwageξ_wstickiness,of0.83and0.77respec- tively,aresensiblecomparedto0.91and0.74foundbySmetsandWouters(2003). Consumptionhabitofpatientb_c^pandimpatienthouseholdsbⁱ_c,at0.82and0.75,look reasonable.Oneexceptionisthepriceindexationparameterι_p.Overtheperiodfrom 1999Q1to2019Q4coveredbyoursample,inflationintheeurozonewasremarkably lowandstable.Theestimationprocedureaccommodatesthisbyascribingahighvalue of0.93tothiscoefficient.

Wenowdiscussthelesshabitualparameters.ThecostofadjustinghousingS^h′′is

essentialtosmooththedynamicsofimpatienthouseholdhousing,andhencehousehold debt,avariableweobserve.Becauseitiscostlytodisposeofhousingimmediately, impatienthouseholdsreactgraduallytoshocks.TheposteriormodeofS^h′′,at52,is

13ThemainreferencesareSmetsandWouters(2003,2007),Justiniano,Primiceri,andTambalotti (2010),andChristiano,Motto,andRostagno(2014).

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whatittakestodisciplinethedynamicsofhouseholddebt.Anotherimportantparameter istheshareκofpatienthouseholdsintheeconomy.Wesetitspriorto0.5basedon theobservationthatatleasthalfofhouseholdsinEuropeholdsomeformofdebt. Ourposteriorestimateis0.74,suggestingfinancially-constrainedconsumersrepresent roughlyaquarterofthepopulation,inaccordwithmicroevidenceinKaplan,Violante, andWeidner(2014).Next,wesetthepriormeanofthesteady-statedefaultprobability ofhouseholdsFⁱ(ω¯ⁱ)andentrepreneursF^e(ω¯^e)toanannualpercentagerateof3.We findahigherestimateforFⁱ(ω¯ⁱ),consistentwiththefactthathouseholdsdefaultmore thanfirms.14Finally,twoparametersµ^handµ^karespecifictothefinancialsector.

Theygovernthemarkupofshadowbanksinthesecuritizedmortgageandbusinessloan markets,respectively.Wecentertheirprioraroundameanof40percent,intheupper rangeofdataonpre-taxreturnonequity.Posteriorestimatesarelower,at0.19forµ^h and0.22forµ^k,indicatingthemodeldoesnotneedahighdegreeoffinancialfrictions

toperformwellquantitatively.
Weturntotheexogenousprocesses.EachrowinthelowerpartofTable4corre-

spondstoashockandreportsapair(ρ,σ).Theonlyexceptionisthemonetarypolicy shockwhoseautocorrelationwesettozero,giventhatthereisalreadyasmoothing parameterintheTaylorrule.Themarketsentimentshockstandsout,withthehighest standarddeviationofallshocks,0.07,andapersistenceof0.939.Thisautocorrelation islowerthanwhattheliteratureusuallyfindsforfinancialanduncertaintyshocks,typi- cally0.95–0.98,andisonereasonwhytheresponsesofourmodeltoamarketsentiment shockareclosetothoseofalessstructuralVAR.