Approximately Right?: Global v. Local Methods for Open-Economy Models with Incomplete Markets

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Approximately Right?: Global v. Local Methods for Open-Economy Models with Incomplete Markets Oliver de Groot, C. Bora Durdu and Enrique G. Mendoza 2020-006 Please cite this paper as: de Groot, Oliver, C. Bora Durdu and Enrique G. Mendoza (2020). “Approximately Right?: Global v. Local Methods for Open-Economy Models with Incomplete Markets,” Finance and Economics Discussion Series 2020-006. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2020.006. 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.

Approximately Right?: Global v. Local Methods for ∗ Open-Economy Models with Incomplete Markets OliverdeGroot C.BoraDurdu EnriqueG.Mendoza UniversityofLiverpool FederalReserveBoard UniversityofPennsylvania, NBER&PIER January2020 Abstract Globalandlocalmethodsarewidelyusedininternationalmacroeconomicstoanalyzeincompletemarketsmodels. Westudysolutionsforanendowmenteconomy,anRBCmodelandaSudden Stops model with an occasionally binding credit constraint. First-order, second-order, risky steady state and DynareOBC solutions are compared v. fixed-point-iteration global solutions in the time and frequency domains. The solutions differ in key respects, including measures ofprecautionarysavings,cyclicalmoments,impulseresponsefunctions,financialpremiaand macro responses to credit constraints, and periodograms of consumption, foreign assets and netexports. Theglobalmethodiseasytoimplementandfasterthanlocalmethodsfortheendowmentmodel. LocalmethodsarefasterfortheRBCmodelandtheglobalandDynareOBC solutionsareofcomparablespeed. Thesefindingsfavorglobalmethodsexceptwhenprevented bythecurseofdimensionalityandurgecautionwhenusinglocalmethods. Ofthelatter,firstordersolutionsarepreferablebecauseresultsareverysimilartosecond-ordermethods. JELClassification: F41,E44,D82 Keywords: Solutionmethods;SuddenStops;IncompleteMarkets;Precautionarysavings;Occasionallybindingconstraints ∗We thank Tom Holden, Matteo Iacoviello, and Rob Vigfusson for helpful discussions and comments, and Holt Dwyer, Alex Martin and Sergio Villalvazo for excellent research assistance. We are also grateful for comments by conference and seminar participants at the 2017 Dynare Conference, the 2018 SED Meeting, the 2018 North American Meetings of the Econometric Society, the Bank of Finland, George Washington University, and the 2018 CentralBankMacroeconomicModelingWorkshop. Correspondence: oliverdegroot@gmail.com, bora.durdu@frb.gov, egme@sas.upenn.edu. Theviewsexpressedinthispaperarethoseoftheauthorsandshouldnotbeattributedtothe BoardofGovernorsoftheFederalReserveSystemoritsstaff.

1 Introduction All major strands of the International Macroeconomics literature study topics in which incompleteassetmarketsplayakeyrole(e.g.,businesscyclesinemergingeconomies,sovereigndefault, SuddenStops,globalimbalances,nominalrigidities,macroprudentialregulation,unconventional monetary policies, currency carry trade, etc). Since the dynamics of external wealth or net foreignassets(NFA)generallylackanalyticsolutionsunderincompletemarkets,researchersrelyon numericalmethodstostudytheimplicationsoftheoreticalmodels. Choosingtheappropriatenumericalmethodsisdifficult,however,becausedeterministicmodelsyieldstationaryequilibriadependentoninitialconditionsand,underuncertainty,marketincompletenessmakestheevolution of wealth state-contingent and influenced by precautionary savings. Certainty equivalence fails andtheNFApositiongrowsinfinitelylargeiftherateofinterestequalstherateoftimepreference. Theliteraturefollowstwoapproachestoaddresstheseissues. Thefirstapproach,basedonthe influentialworkbySchmitt-GrohéandUribe(2003),modifiesthemodelsbyintroducing“stationarityinducing”assumptionsthatyieldawell-defineddeterministicsteadystateforNFA,independent of initial conditions, and implements log-linear or first-order (1OA) approximations around that steady state, recovering certainty equivalence. Schmitt-Grohé and Uribe proposed introducing one of three assumptions: a debt-elastic interest rate (DEIR) function by which the real interest rate rises as NFA falls below its steady state, quadratic costs that make NFA costly to deviate from steady state, or a subjective discount factor that varies with individual or aggregate consumption.1 Important innovations were made subsequently to improve local methods, including second-(2OA)andhigher-orderapproximationmethods(e.g.,DevereuxandSutherland(2010), Fernandez-Villaverdeetal. (2011)),theriskysteadystate(RSS)methodproposedbyCoeurdacier et al. (2011), and the algorithms for solving models with occasionally binding constraints (e.g., OccBinbyGuerrieriandIacoviello(2015),DynareOBCbyHolden(2016a)). Thesecondapproachusesglobalmethodstosolvedirectlyforthenonlineardecisionrulesand long-run distribution of wealth of the models in their original form. These methods are analogous to those used in other fields in Macroeconomics, particularly to solve models of heterogeneous agents with incomplete markets. Open-economy applications of these methods date back to Mendoza’s (1991) RBC model of a small open economy, and there are now many applications in quantitative studies of sovereign default (e.g., Aguiar and Gopinath (2006), Arellano (2008)), 1Schmitt-GrohéandUribeshowedthatthebusinesscyclemomentsofanRBCsmall-open-economymodelsolved usinganyoftheseassumptionsaresimilar,andimpulseresponsefunctionstoaTFPshockarevirtuallyidentical. 1

emerging markets business cycles (e.g., Neumeyer and Perri (2005), Mendoza (1995), Uribe and Yue (2006)), global imbalances (e.g., Mendoza et al. (2009)), Sudden Stops (e.g., Mendoza and Smith(2006),Durduet. al(2009),Mendoza(2010)),andmacro-financialregulation(e.g.,Bianchi (2011),Benignoetal. (2016),Bianchietal. (2012),Schmitt-GrohéandUribe(2017)). Tables1and2documenttheusageoflocalandglobalmethodsforsolvingopen-economymodelswithincompletemarketsinresearcharticlesandpolicyinstitutions. Table1includes61papers. Thelistismeanttobeillustrative,becauseconstructingacompletelistofthearticlesintheliteraturethatusethesemethodsisbeyondthescopeofthispaper. Itincludesthe50mostcitedpapers inGoogleScholarthatciteSchmitt-GrohéandUribe(2003),excludingtextbooksandreviewarticles. Italsoincludesallquantitativepapersinthereferencesofthispaperthatarenotinthattop-50list, andseveralwell-knownpapersgoingbacktotheearly1990swhenthefirstnumericalsolutionsof open-economy models with incomplete markets were produced. Table 2 lists the models used in eightpolicyinstitutions,usinginformationobtainedfrompubliclyavailabledocuments. Table 1 shows that both local and global methods are widely used in research. 1OA is the most common local method, and from the assumptions to induce stationarity, DEIR is the most common. Table 2 shows that all eight policy models use the 1OA method. Five of these models useDEIR.OfallsolutionsthatuseDEIR,themajoritysetthevalueofthedebtelasticityparameter ψ toanarbitrarysmallnumber,withtheaimofpreventingtheDEIRfunctionfromplayingarole otherthaninducingstationarity,sinceideallythisfunctionshouldbeanendogenousobject.2 The ψ values range from 0.00001 to 0.01, and the most common is 0.001, which is the value proposed by Schmitt-Grohé and Uribe.3 In other cases, the value of ψ is set by calibration or obtained via estimation,althoughforsomepolicymodelsthevalueofψand/orthemethodfollowedtosetitare notreported. Incalibratedcases(threeresearchpapersandthreepolicymodels),ψ isinthe0.01- 0.1range,andinestimatedcases(fourresearchpapersandonepolicymodel),thepointestimates orthemediansofposteriordistributionsinBayesianestimationareinthe0.00014-2.8range. Whileglobalmethodshavetheadvantagethatthemodelsaresolvedintheiroriginalformand capturethedynamicsofwealthaccurately,theysufferfromthetraditionalcurse-of-dimensionality problem: Theyareimpracticalinlargemodelsbecausethemethodsbecomeexponentiallyineffi- 2Garcia-Ciccoetal. (2010)explainthat,followingSchmitt-GrohéandUribe(2003),thestandardpracticeistosetψ toasmallvaluebecausetheDEIRfunctionaimstoobtainindependenceofthedeterministicsteadystatefrominitial conditionswithoutaffectingcyclicaldynamics.Theyalsostudiedamodelinwhichψrepresentsfinancialfrictions,and inthiscasetheyestimatedψusingBayesianmethods. 3Note,however,thattheDEIRfunctionalformsarenotalwaysthesame,soψ valuesarenotdirectlycomparable. Whenrelevantforourquantitativeanalysis,wecontrolforthisbymakingcomparisonsintermsoftheelasticityofthe interestratewithrespecttopercentdeviationsofNFAfromsteadystate. 2

cientasthenumberofendogenousstatevariablesrises. Thelocalmethodshavetheadvantagethat theycanbeappliedinlarge-scalemodels,buthavetheshortcomingthattheyneedthestationarityinducingassumptionsthatarenotpartoftheoriginalmodels,andmoreimportantly,asweshowin thispaper,theseextraassumptionsmaynotbeinnocuousforthenumericalresults. Hence,these tradeoffsposetwokeyquestions: Underwhatconditionsarelocalsolutionsbetterapproximations tothe“exact”solutionsobtainedwithglobalmethods? Whenthoseconditionsaresatisfied,which featuresoftheglobalsolutionsareapproximatedaccuratelyandwhicharenot? This paper provides answers to these questions by comparing local and global solutions. For the local methods, we use 1OA, 2OA, RSS, and DynareOBC. For the global methods, we use the fixed-pointiteration(FiPIt)algorithmproposedbyMendozaandVillalvazo(2019),whichapplies fixed-pointiterationtosolveforNFA(assetprices)intheEulerequationofbonds(capital).4 Withtheglobalmethod,theexistenceofawell-definedstochasticsteadystatefollowsfromthe sameconditionasintheBewley-Aiyagari-Huggetclassofheterogeneous-agentsmodels: therate of timepreference mustbe lowerthan theinterest rate.5 In multicountrymodels thisis ageneral equilibrium result, because if the rate of interest equals the rate of time preference, all countries would desire an infinitely large stock of NFA for self-insurance, which is inconsistent with world generalequilibrium(seeMendozaetal. (2009)). Hence,assuminganinterestratelowerthanthe rateoftimepreferenceinsmall-open-economymodelsisanimplicationoftheassumptionthatthe interestrateisaworld-determinedprice. Withthelocalmethods,theDEIRfunctionisconstructed sothatatapre-determinedsteadystatetherateofinterestequalstherateoftimepreference. We compare local v. global solutions for three widely-used small open economy models: An endowment economy model, a real business cycle (RBC) model, and a model of Sudden Stops (SS)withanoccasionally-bindingcreditconstraintlinkedtomarketprices. Ineachcase,westart with“baselinecalibrations”inwhichthelocalmethodsuseDEIRwithψ = 0.001andthecenterof approximation of the 1OA, 2OA and DynareOBC methods is the deterministic steady state, and RSS is centered at its risky steady state. Then we consider “targeted calibrations,” for which ψ is calibrated to match the first-order autocorrelation of NFA in the global solutions.6 For RSS and 4WealsosolvedthemodelproposedinSection2usingvaluefunctioniterationandobtainedverysimilarresults (seeAppendixD.1). Thismethodisslowinmodelswithmorethanoneendogenousstatevariableandoftenrequires efficientequilibria,butdoesnotrequiredifferentiabilityorconvexityoftheoptimizationproblems,whichFiPItrequires. Moreover,FiPItdoesnotusethecontractionmappingpropertysoitsconvergenceisnotguaranteed. 5Preferences with endogenous discounting proposed by Uzawa (1968) and Epstein (1983) also support a welldefinedlimitingdistributionofNFA.InAppendixSectionD.1, wereportresultsoftheglobalsolutionofthemodel ofSection2obtainedwiththesepreferences. 6WealsostudiedanalternativeinwhichthecenterofapproximationistheaverageNFAoftheglobalsolutions,but 3

DynareOBC, we also solve variants without DEIR in which the rate of interest is lower than the rateoftimepreference,sothatcreditconstraintsbindatthedeterministicsteadystate. Weconduct both time-domain comparisons of statistical moments and dynamics in response to shocks, and frequency-domain comparisons based on spectral density analysis. In addition, for the SS model wecomparemultipliers,financialpremiaandmacroresponseswhenthecreditconstraintbinds. The results show important differences between local and global solutions, which are due to differencesinthedecisionruledrivingNFA,thekeyendogenousstatevariableinopeneconomy models. In the global solutions, the NFA decision rule is solved for “exactly,” and the model’s equilibriumstochasticprocessesarejointlydeterminedbythisdecisionrule(andtheoneforcapital intheRBCandSSmodels)andthestochasticprocessofexogenousshocks. Inthelocalsolutions (otherthanDynareOBC),theNFAdecisionruleisafirst-orsecond-orderapproximationaround the deterministic steady state or the risky steady state for the RSS method. Hence, differences across the solution methods can be summarized by comparing NFA decision rules, particularly their first-order autocorrelations and their implications for two key moments: (a) the mean of NFA, which is an indicator of precautionary savings and the critical variable in research topics likeglobalimbalancesorreservesaccumulation;and(b)theautocorrelationofnetexports,since netexportsarealsoakeyvariableintheanalysisofopen-economymodels(seeMendoza(1991), AguiarandGopinath(2006),andGarciaCiccoetal. (2010)). Globalmethodspindownthe“true”NFAautocorrelation,whilelocalsolutionsyieldapproximationsthatdifferdependingonthesolutionmethodandthecriteriausedtosetψandthecenter of approximation. Both global and local methods produce a near-unit-root NFA process. In all calibrated exercises, the NFA autocorrelations exceed 0.95. The autocorrelations differ slightly, however,andbecausetheyarenear-unit-rootsthesesmalldifferenceshavemajorimplications. The largest difference between global and local solutions is in the long-run average of NFA. Small differences in the autocorrelation and intercepts of the NFA decision rules cause large differences in mean NFA, and the local methods can yield NFA averages above or below the global solution. The local methods also yield very different results for the effects of parameter changes thatalterprecautionarysavings. Forinstance,theglobalsolutionpredictslargeincreasesinmean NFAwithhighervariabilityofshocks,lowerrateoftimepreferenceorhighercoefficientofrelative risk aversion (CRRA). In contrast, the 1OA method preserves certainty equivalence, and hence keepsmeanNFAequaltothedeterministicstationaryequilibriumimpliedbytheDEIRfunction. targetingtheautocorrelationproducesaclosermatchtotheglobalsolutions. 4

The2OAandRSSmethodsproducelong-runaveragesofNFAthataremuchlargerorsmallerthan themeanoftheglobalmethods,dependingonthemodel,theparameterchangeconsidered,and whether we use baseline or targeted calibrations. For example, changing income volatility in the endowmenteconomy,the2OAmethodoverestimatesaverageNFAwhiletheRSSunderestimates itusingbaselinecalibrations,whilefortargetedcalibrationsbothmethodsunderestimateaverage NFAsignificantly. Similarly,DynareOBCcalibratedtoasteadystateinwhichthecreditconstraint binds(doesnotbind)yieldsmeanNFAmarkedlybelow(above)theglobalsolution. Thus,amajor drawbackoflocalmethodsisthattheydopoorlyatquantifyingprecautionarysavings. SmalldifferencesintheautocorrelationofNFAalsoyieldlargedifferencesintheautocorrelationsofthenetexports-GDPratio(NX),becauseNXisaquasifirst-differenceofanear-unit-root NFAprocess,asshowninSection2. Forinstance,intheendowmentmodelwiththebaselinecalibration,theglobalsolutionpredictsthatraisingthepersistenceofincomefromnear0to0.8makes theautocorrelationofNFArisefrom0.83to0.99butthatofNXvariesfrom-0.09to0.77. Incontrast, the20AandRSSsolutionspredictthattheautocorrelationofNFAalwaysexceeds0.99whilethat of NX varies from 0.24 to 0.95. For a given autocorrelation of income in the 0-0.8 range, the local solutionsalwaysoverestimatetheautocorrelationsofNFAandNX. Thelocalmethodsdobetteratmatchingthemomentsoftheglobalsolutionwiththetargeted calibrations(i.e. withψ settomatchtheautocorrelationofNFAintheglobalsolution). Averages of NFA get closer to the global solution. The autocorrelations of NFA and NX are similar when the persistence of income is such that the autocorrelation of NFA in the global solution is greater orequaltothatobtainedwiththebaselinecalibration(0.977),butforlesspersistentincome,and hencelowerautocorrelationsofNFAintheglobalsolution,againthelocalsolutionsoverestimate theautocorrelationsofNFAandNX.Moreover,thetargetedcalibrationapproachhasthedrawback thatitrequiresobtainingfirsttheglobalsolutionsoastofindthefirst-orderautocorrelationofNFA tocalibrateψ,anddoingthisagainforanyparametricchangethatalterstheNFAautocorrelation. Comparing alternative local methods also yields important findings. 1OA, 2OA and RSS solutions yield similar second- and higher-order moments and similar impulse response functions forallendogenousvariablesintheendowmentandRBCmodels. Weuseanalyticsolutionsofthe endowmentmodelNFAdecisionrulestoshowthatthisisduetotwofeaturesoftheresults: First, the coefficient on lagged NFA is nearly the same in the RSS and 2OA solutions when ψ is small (lessthan0.1),unlessthedeterministicandriskysteadystatesofNFAdifferbyaverylargemargin (at least 40 percentage points of GDP). Second, the coefficients in the square and interaction 5

termsofthe2OAdecisionrulesareverysmall. Asaresult,thethreemethodsyielddifferentfirst momentsbecauseoftheirdifferentcentersofapproximation(anddifferentinterceptfor1OA),but thesecond-andhigher-ordermomentsandimpulseresponsefunctionsareaboutthesame. Targetedcalibrationsrequireincreasingψ fromthecommoncalibrationsettingof0.001tovaluesof0.0469(0.0063)and0.0469(0.0045)forthe2OAandRSSmethodsappliedtotheendowment (RBC) economy respectively. These variations imply large increases in the elasticity of the DEIR function, by factors of 4.5 to 25, which make this function play a more significant role than when ψ = 0.001. In particular, the required ψ values effectively make deviations of NFA from steady state too costly, and as a result even the first moments of 2OA and RSS are similar and they are both also close to the 1OA solution (i.e., certainty equivalence approximately holds). In these cases,1OAshouldbethepreferablelocalmethod,butitalsomeansthatprecautionarysavingsare disregarded entirely. On the other hand, keeping ψ = 0.001 implies that predictions about some keymoments,liketheautocorrelationofNX,deviatesignificantlyfromthe“true”solutions. The global and local solutions also show important differences in the moments that measure thevariability,persistenceandoutput-correlationofconsumptioninthethreemodels. Incontrast, and in line with the findings of Schmitt-Grohé and Uribe (2003), the moments for output and investment in the RBC and Sudden Stops models are similar. This is an implication of the small excess return on capital typical of RBC models, which keeps the capital decision rule close to the oneconsistentwithFisherianseparationofinvestmentfromsavingduetoarbitrageofassetreturns. Thefrequency-domainanalysisshowsthatlocalandglobalmethodsyieldequilibriumstochasticprocessesthatdisplaydifferentbehavioratmostfrequencies,andnotjustinbusinesscyclemoments and long-run averages. Non-parametric periodograms for the endowment economy with the baseline calibrations show that local methods overestimate significantly the contribution of low-frequency movements to the variability of NFA and NX, which is consistent with the result indicating that the local methods overestimate the autocorrelation of NFA. As a result, the local methods also overestimate (underestimate) the contribution of low-frequency (high-frequency) movements to consumption fluctuations. The targeted calibrations perform much better at approximatingthespectraldensityofNFAandNX,butforconsumptiontheystillunderestimatethe contributionofhigh-frequencyfluctuationsbyalargemargin. SimilarresultsareobtainedforNFA, NX and consumption in the RBC model, but in this case the local and global methods yield similarperiodogramsforcapital, investmentandoutput, inlinewiththefindingthatthetime-series momentsforthesevariablesaresimilarwithbothmethods. 6

Standard 2OA and RSS methods cannot be used for solving the SS model, because they cannot handle the occasionally binding constraint. We used instead Holden’s (2016a) DynareOBC toolkit. DynareOBC uses also a local approximation but introduces news shocks which hit every timetheconstraintisviolatedandpushtherelevantvariablesbacktotheconstraint. Toensurethe solutionisconsistentwithrationalexpectations,thesenewsshocksareconstructedasiftheywere expected by agents along a perfect-foresight path and so are akin to “endogenous news shocks.” This method, when solved in first order and without integrating over future uncertainty, ignores precautionarysavingsandtherisk ofalternativefuturepathsinwhichtheconstraintmayormay notbind,andhenceitalsoignorestheequityriskpremium. WeexaminedDynareOBCsolutions withandwithouttheconstraintbindingatthedeterministicsteadystate. TheresultsshowthatourfindingsfromtheendowmentandRBCmodelextendtotheSSmodel. Localandglobalsolutionsyieldlargedifferencesintheamountofprecautionarysavingsinduced bythecreditconstraint,businesscyclemoments,theprobabilityofhittingtheconstraint,impulse responses, and spectral densities. Moreover, the near-unit-root nature of NFA increases Dynare- OBCexecutiontimeconsiderably,becauseitrequiresmultiple,longperfect-foresightpathstoform the news shocks realizations needed to implement the constraint, and long time-series simulations to attain convergence of long-run moments. DynareOBC also underestimates significantly the tightness of the credit constraint and its effects on financial premia and macro responses. In particular,ityieldsmeanshadowinterestpremiawhentheconstraintbindsof0.13to0.8percent v. 2.6 percent in the global solution and average equity premia of 0.1 to 0.64 percent v. 2.2 percent. Inturn,lowerequityreturnsimplyhigherequitypricesandinvestmentwhentheconstraint binds,andhencehigherborrowingcapacity. Asaresult,DynareOBCwiththeconstraintbinding atsteadystateyieldsweakerSuddenStopmacroresponseswhenthecreditconstraintbinds,and DynareOBCwiththeconstraintnotbindingatsteadystatecannotcaptureSuddenStopresponses. In terms of computational performance, the FiPIt method is easy to implement in Matlab. It is faster than the local methods for solving the endowment model and of comparable speed to DynareOBC for solving the Sudden Stops model, but it is slower than local methods for the RBC model. Forallthreemodels,however,thelocalmethodsyieldmuchlargerEulerequationerrors. Thispaperisrelatedtothebroaderliteraturecomparingsolutionmethodsofnonlinearrational expectations models. The comprehensive study by Taylor and Uhlig (1990) showed that several methods that approximate global solutions of a canonical RBC model yield different results for simulated sample paths, density functions and cyclical moments. Dou et al. (2019) compared 7

1OA,2OAandOccBinlocalsolutionsv. globalsolutionsofaDSGEmodelwithfinancialfrictions andfoundthatthelocalsolutionsapproximatepoorlythemodel’snonlineardynamicsandyield biased impulse responses. Rabitsch et al. (2015) compared the DS local method proposed by Devereux and Sutherland (2010) for solving portfolio allocations in a two-country, incompletemarketsmodelv. aglobalmethod. IntheDSmethod,non-stationarityoftheNFApositionremains anissue, butgivenNFAityieldsanaccurateportfoliostructure. TheyfoundthattheDSmethod isaccurateonlywithparticularcalibrationsandwithsymmetriccountrieswithlong-runNFAset to 0. With asymmetric countries, and using endogenous discounting to induce stationarity, the DSmethod performspoorly unlessthe centerof approximationmatches theglobalsolution, and more so if NFA decision rules are nonlinear. Our work differs from these studies in three key respects: Westudy1OA,2OA,RSSandDynareOBCmethodsusingthedominantDEIRapproachto inducestationarity;wecompareresultsinboththetimeandfrequencydomains;andweconsider endowment,RBCandSSmodels,andforthelatterwecompareglobalv. DynareOBCsolutions. It is also worth noting that Holden (2016b) compared DynareOBC v. global solutions of a small open economy model with endowment income (i.e., NFA as the single endogenous state) and subject to a constant debt limit and two non-negativity constraints as occasionally binding constraints. Our results differ in that we solve an SS model with two endogenous states (capital and NFA) and a credit constraint that depends on both states and on asset prices. Holden found thatDynareOBCisnotfarfromtheglobalsolution,whilewefoundthatthesolutionsdiffersharply. Therestofthepaperisorganizedasfollows. Section2comparestheendowmentmodelsolutions, providing analytic and quantitative results. Section 3 compares RBC model solutions. Section4comparesglobalv. DynareOBCsolutionsoftheSSmodel. Section5providesconclusions. 2 Endowment economy 2.1 Modelstructureandequilibrium Considerfirstasmallopeneconomymodelwithstochasticendowmentincome. Weusethissetup toderiveanalyticresultsandcharacterizeNFAdynamicsunderincompletemarkets. Theeconomy isinhabitedbyarepresentativeagentwithpreferencesgivenby: (cid:40) (cid:88) ∞ (cid:41) c1−σ E βtu(c ) , u(c ) = t . (1) 0 t t 1−σ t=0 8

whereβ ∈ (0,1)isthesubjectivediscountfactor,c isconsumptionandσ istheCRRAcoefficient. t Theeconomy’sresourceconstraintis: c = ezty¯−A+b −qb , (2) t t t+1 where ezty¯is stochastic income or GDP that fluctuates around a mean y¯with shocks z t of exponential term ezt, b t denotes the NFA position in one-period, non-state-contingent discount bonds tradedinafrictionlessglobalcreditmarketataanexogenouspriceq = 1 ,wherer istheworld 1+r real interest rate, and A is a constant that represents investment and government expenditures (which is introduced so that the model can be calibrated to observed average consumption-GDP ratios).7 IncomeshocksfollowanAR(1)process: z = ρ z +σ εz whereεz isi.i.d. t z t−1 z t t The agent chooses the optimal sequences of bonds and consumption so as to maximize (1) subjectto(2). Thisoptimizationproblemisanalogoustotheonesolvedbyasingleindividualin heterogeneous-agent models of precautionary savings (e.g., Bewley (1977), Aiyagari (1994) and Hugget(1993)). Asinthosemodels,theInadaconditionofCRRAutilityimpliesthatthemarginal utilityofconsumptiongoestoinfinityasconsumptiongoestozerofromabove. Thisimpliesthat the small open economy faces Aiyagari’s Natural Debt Limit (NDL), by which the NFA position never exceeds the annuity value of the worst realization of net income b t+1 ≥ −min(ezty¯−A)/r, otherwise agents would be exposed to the possibility of nonpositive consumption with positive probability. Wecanalsoimposeatighterad-hocdebtlimitϕ,followingAiyagari(1994),suchthat b t+1 ≥ ϕ ≥ −min(ezty¯−A)/r. Asweshowlater,thisisusefulformodelcalibration. Usingtheresourceconstraint,wecanexpresstheEulerequationforbondsasfollows: u (ezty¯−A+b −qb ) = (1+r)βE [u (ezt+1y¯−A+b −qb )]+µ , (3) c t t+1 t c t+1 t+2 t whereu (t)isthemarginalutilityofc andµ istheLagrangemultiplierofthedebtlimit. c t t Acompetitiveequilibriumforthiseconomyisdefinedbystochasticsequences[c ,b ]∞ that t t+1 t=0 satisfyequation(3)foralltwithb ≥ ϕ. Aswediscussnext,theincompletenessofassetmarkets t+1 playsacriticalroleindeterminingkeyfeaturesofthisequilibrium. Considerfirsttheequilibriumundercompletemarketsofcontingentclaims. Ifincomeshocks are idiosyncratic to the small open economy, the economy diversifies away all the risk of its endowment fluctuations. As a result, the equilibrium features a constant consumption stream and 7Weassumethatrisconstant,butstudylatertheimplicationsofallowingittobestochastic. 9

theeconomy’swealthpositionistime-andstate-invariant. Thesolutionwouldbethesameasina perfect-foresightmodelwithβ(1+r) = 1andwealth(thepresentvalueofincomeplustheinitial NFAholdings)scaledtorepresentthesamewealthasinthecompletemarketseconomy. Underincompletemarkets,theequilibriumdifferssharply,becausewealthbecomesstate-contingent andconsumptioncannotattainaperfectlysmoothpath. AsshowninChapter18ofLjungqvistand Sargent(2012),theEulerequation(3)impliesthatM ≡ (1+r)tβtu (t)formsasupermartingale, t c whichconvergesalmostsurelytoanon-negativerandomvariablebecauseoftheSupermartingale Convergence Theorem. If β(1+r) ≥ 1, this convergence implies that consumption and NFA divergetoinfinitybecausemarginalutilityconvergestozeroalmostsurely,whichisthecauseofthe non-stationarity problem that led to the use of the DEIR function in local solution methods. The economybuildsaninfinitelylargestockofprecautionarysavingssothatself-insurancecansustain aconsumptionpaththatsatisfiesu (t) ≥ β(1+r)E [u (t+1)]andstillleadstoconvergenceinM . c t c t Hence, in this case the model lacks a “nice” stochastic stationary state with a well-defined limitingdistributionofNFA.Incontrast,ifβ(1+r) < 1,theeconomyattainsawell-definedstochastic steadystatewithfinitelong-runaveragesofassetsandconsumption,andtherestofthemoments of the model’s endogenous variables are also well-defined. A crude intuition is that the opposing forces of the incentive to save for self-insurance and the pro-borrowing incentive implied by β(1+r) < 1makeNFAfluctuatewithinanergodicset. WhenNFAfallstoomuchtheprecautionarysavingsforceprevailsandwhenNFArisestoomuchthepro-borrowingforceprevails. Itisimportanttonotethatβ(1+r) < 1isalsoageneralequilibriumoutcomeinmulti-country modelswithincompletemarkets,becauseotherwiseallcountrieswouldwantaninfiniteamount of NFA, which is inconsistent with world market clearing in the market of risk-free assets (see Mendoza et al. (2009)). Thus, in small-open-economy models β(1+r) < 1 is not an arbitrary assumptionbutanimplicationoftheassumptionthattherealinterestrateisaworld-determined price. Thisalsoimpliesthattheissuesraisedherearenotparticulartosmallopeneconomymodels, theyarerelevantformulti-countrymodelsandclosed-economymodelswithincompletemarkets. 2.2 Globalmethods Global methods solve for the competitive equilibrium in recursive form. Since the equilibrium is efficientinthiseconomy,wecanrepresentitasthesolutiontothisdynamicprogrammingproblem: (cid:40) (cid:41) V(b,z) = max c1−σ +β (cid:88) π(z(cid:48),z)V (cid:0) b(cid:48),z(cid:48)(cid:1) , (4) c,b(cid:48) 1−σ z(cid:48) 10

s.t. c = ezy¯−A+b−qb(cid:48), b(cid:48) ≥ ϕ. TheAR(1)processofincomeisapproximatedasadiscreteMarkovchainwithtransitionprobabilitymatrixπ(z(cid:48),z). ThesolutiontotheBellmanequationischaracterizedbyadecisionruleb(cid:48)(b,z) andtheassociatedvaluefunctionV(b,z). ThisdecisionruleandtheMarkovprocessoftheshocks induceajointergodic(unconditional)distributionofNFAandincomeλ(b,z). We solve for b(cid:48)(b,z) over a discrete state space of (b,z) pairs using the FiPIt method.8 This method solves the recursive equilibrium conditions using a fixed-point iteration algorithm (see Mendoza and Villalvazo (2019) for details). For this simple model, the FiPIt method iterates on thefollowingrepresentationoftheEulerequation: (cid:40) (cid:88) (cid:20) (cid:16) (cid:17)−σ (cid:21) (cid:41)− σ 1 c (b,z) = βR π(z(cid:48),z) c (ˆb(cid:48)(b,z),z(cid:48)) (5) j+1 j j z(cid:48) Givenaconjectureofthedecisionrule ˆb(cid:48)(b,z)initerationj,theassociatedconsumptionfunctionis j c (b,z) = ezy¯−A+b−qˆb(cid:48)(b,z). Thisconsumptionfunctionisinterpolatedoveritsfirstargumentin j j ordertodeterminec (ˆb(cid:48)(b,z),z(cid:48)),sothattheaboveequationsolvesdirectlyforanewconsumption j j function c (b,z). Using the resource constraint, the new consumption function yields a new j+1 decision rule for bonds b(cid:48) (b,z), which is re-set to b(cid:48) (b,z) = ϕ if b(cid:48) (b,z) ≤ ϕ. Then the j+1 j+1 j+1 decisionruleconjectureisupdatedto ˆb(cid:48) (b,z)asaconvexcombinationof ˆb(cid:48)(b,z)andb(cid:48) (b,z), j+1 j j+1 andtheprocessisrepeateduntilb(cid:48) (b,z) =ˆb(cid:48)(b,z)uptoaconvergencecriterion. j+1 j FiPIt uses a standard fixed-point iteration approach to solve transcendental equations so that the Euler equation solves directly instead of using a non-linear solver. Mendoza and Villalvazo (2019) provide a detailed comparison of this algorithm with other Euler equation methods, including time iteration and endogenous grids methods. They show that this method is fast and easytoimplementandoptimizeinMatlabformodelswithoneortwoendogenousstatevariables, includingproblemswithoccasionallybindingconstraints. Incontrast,time-iterationwithtwoendogenousstatesrequiressolvingtwononlinearsimultaneousequationsandtheendogenousgrids methodrequirescomplexinterpolationtechniquesbecausetheendogenousgridsareirregular. The global method solves the model in its “original form” without additional assumptions to impose stationarity. If β(1 + r) = 1, the true solution is that NFA diverges to infinity, which is unpleasant but is the equilibrium outcome. β(1+r) < 1 is, however, the relevant case, because 8WeshowinAppendixD.1thatsolvingbyvaluefunctioniterationyieldsnearlyidenticalresults. 11

as noted above it is implied by world general equilibrium. Note also that with β(1+r) < 1 the deterministicstationarystateconvergestothenaturalorad-hocdebtlimit,withconsumptionfalling atagrossrateof(β(1+r))1/σ. Hence,withoutquantitativeanalysis,wecanpredictalreadythat the long-run average of NFA in the stochastic, incomplete-markets model will differ significantly fromthedeterministicsteadystateandthatthedifferenceisduetoprecautionarysavings. 2.3 Localmethods The local methods solve for the competitive equilibrium using a local approximation of the optimality conditions (equations (2) and (3)) either around the deterministic steady state (bdss) for the 1OA and 2OA methods or the risky steady state (brss) for the RSS solutions. Also, since the modelinitsoriginalformyieldsadeterministicstationarystatethatiseitherdependentoninitial conditions(ifβ(1+r) = 1)orthedebtlimit(ifβ(1+r) < 1),1OAand2OAmethodsrequireone of the stationarity-inducing assumptions. As noted in the Introduction, the most widely used of theseassumptionsistointroducetheDEIRfunction,whichgenerallyadoptsthefollowingform: (cid:104) (cid:105) 1+r = 1+r+ψ ebdss−bt+1 −1 , (6) t whereψdeterminestheelasticityofr withrespecttoNFA.ForsmalldeviationsofNFAfrombdss, t theelasticityofr withrespecttopercentdeviationsofb frombdss isgivenbyηr ≡ −ψbdss. t t+1 We implement the 1OA and 2OA methods following Scmitt-Grohe and Uribe (2004) and the RSS method following Coeurdacier et al. (2011).9 1OA and 2OA solve for local approximations aroundbdss instandardfashion,bysolvingafirst-orsecond-orderapproximationofthedecision rulesjointlywithapproximationsofthesameorderofthemodel’soptimalityconditions. Incontrast, RSS uses brss as center of approximation and assumes β(1+r) < 1 (see the Appendix for details). Theriskysteadystateaimstotakeintoaccountfuturerisk,sothatthecenterofapproximation may approximate better the precautionary savings motive. The value of brss is obtained fromasecond-orderapproximationtotheconditionalexpectationofthesteady-stateEulerequation,whichissolvedjointlywiththecoefficientsofafirst-orderapproximationofthedecisionrules usingaconditionalsecond-orderapproximationofthefullequilibriumconditions’Jacobian,which requiresthethirdderivativesoftheequilibriumconditions. AsexplainedbydeGroot(2014),this second part of the solution is crucial to obtain stationary NFA dynamics in the RSS solution. We 9UsingAndreasenetal. (2018)tosolveusing2OAand3OAyieldssimilarfindings. KliemandUhlig(2016)also proposedaRSSmethod,inthecontextofanassetpricingmodel. 12

also consider a variant of the RSS method in which brss is computed in the same way, but is then usedtogetherwiththeDEIRfunctionandstandardfirst-orderapproximationsofthedecisionrules andequilibriumconditionstoobtainstationarydynamics. WerefertotheoriginalRSSmethodas “fullRSS”andthealternativewiththeDEIRfunctionas“partialRSS.”PartialRSSisfasterand,for otherthanfirst-ordermoments,ityieldssimilarresultsasfullRSSinourquantitativeapplications. 2.4 Calibration WeusethesamebaselinecalibrationasinDurduetal. (2009),whichwasbasedonannualdatafor Mexico. Table3liststhecalibrationparameters. Thetableshowsasubsetofparameterscommon tolocalandglobalsolutions,aswellastheparametersthatareparticulartoeach,includingforthe localmethodsbothbaselineandtargetedcalibrationsofψ. The common baseline calibration parameters are set as follows. The CRRA coefficient is set to σ = 2, which is a standard value. Mean income is normalized to y¯ = 1. The steady state real interest rate is set to 5.9 percent, which is the average of Uribe and Yue’s (2006) real interest rate includingtheEMBIspreadforMexico. ThetargetaverageratioofnetforeignassetstoGDPisset to−44percent,whichistheaverageofMexico’sNFA-GDPratioovertheperiod1985–2004inthe databaseconstructedbyLaneandMilesi-Ferretti(2006),andthetargetconsumption-GDPratiois 0.692,whichistheaverageratioinMexicandataforthe1965–2005period. Theseparametervalues andthesteady-stateresourceconstraintimplythatA = y¯+rb−c = 0.282. Thestandarddeviation and first-order autocorrelation of the endowment process are set to estimates obtained using the HP-detrendedcyclicalcomponentofGDPcomputedalsofromtheMexicandata. Thisyieldsσ = z 0.0327 and ρ = 0.597. When using local methods, these parameters define the AR(1) process of z income shocks that is part of the equilibrium conditions. In the global solution, we approximate thisprocessasafive-pointMarkovchainusingTauchenandHussey(1991)’squadraturemethod. Thebaselinecalibrationfortheglobalsolutionsrequiresalsocalibratingthevaluesofϕandβ. AsexplainedinDurduetal. (2009),theseparametersaresettovaluessuchthatthemodelmatches the−0.44longrunaverageoftheNFA-GDPratioobservedinMexicandatatogetherwithMexico’s cyclicalvariabilityofprivateconsumptionof3.28percentoverthe1965–2005period. Thisimplies ϕ = −0.51andβ = 0.94. Twoparametersarerequiredtoidentifythecalibration,becausewhilethe averageNFA-GDPratiocanbematchedbysimplyadjustingϕ,thiscanresultinstochasticsteady statesinwhichthedistributionofbondholdingsisclusterednearthedebtlimitandconsumption fluctuatestoomuch,orhasahighvarianceandconsumptionfluctuatestoolittle. 13

In the baseline calibration for the local methods, we set the deterministic steady state of NFA to the same value of ϕ in the calibration of the global solutions, hence bdss = −0.51. This is done so that both solutions have the same bdss (recall that the deterministic steady state of the global solution is ϕ because β(1+r) < 1). The value of β for the local methods (except in the full RSS solution) is set to 1 , because the functional form of the DEIR function implies that the rate of 1+r interestequalstherateoftimepreferenceinthedeterministicsteadystate. Note,however,thatthe discountfactorsoftheglobalandlocalcalibrationsdifferonlyslightly(0.944v. 0.94). Thebaseline value of ψ is the commonly-used value of 0.001, taken from Schmitt-Grohé and Uribe (2003). In the targeted calibrations, we set ψ to values so that the solution for a given local method matches theautocorrelationofNFAobtainedwiththeglobalsolution. Thisyieldsψ = 0.0469forboththe 2OA and RSS. We do this because, as we show below, approximating closely the autocorrelation of the equilibrium law of motion of NFA gives the local methods the best chance to match the globalsolution. Recallalsothatforagivenvalueofψ theelasticityofdebtwithrespecttopercent deviationsofNFAfromsteadystatevarieswithbdss,sinceηr ≡ −ψbdss forsmalldeviations. 2.5 Comparisonofquantitativeresults We compare global and local solutions along five dimensions. First, the NFA decision rule and its implications for net exports. Second, the full set of long-run moments. Third, impulse response functions for income shocks. Fourth, spectral density functions. Fifth, an environment withinterest-rateshocks,inadditiontoincomeshocks. a)DecisionrulesofNFAandNetExports We start by examining differences in the first-order autocorrelations of NFA and net exports (nx). Given that the trade balance is a quasi first-difference of NFA, since nx = qb −b , and t t+1 t assumingthatbfollowsanAR(1)processwithautocorrelationcoefficientρ ,weshowinAppendix b SectionAthatthefirst-orderautocorrelationofnetexports(ρ )is:10 nx q2ρ +ρ −q−qρ2 ρ = b b b. (7) nx 1+q2−2qρ b Hence, ifρ iscloseto1, asistypicallythecaseinmodelswithnon-state-contingentassets, small b differencesinρ inducelargedifferencesinρ . Thus,smallerrorsinthelocalsolutionforρ can b nx b resultinlargeerrorsintheirsolutionforρ . Weshowbelowthatthisisindeedthecasehere. nx 10NFAisanAR(1)processinthe1OAandRSSsolutions. Inthe2OAsolutionitincludessquaredandinteraction termsinb t−1andy t,butthesetermsarequantitativelynegligibleinallofourexperiments. 14

Theglobalmethodsproduce“exactsolutions”forρ andρ thatarecomputedusingλ(b,z), b nx π(z(cid:48),z),b(cid:48)(b,z)andthedefinitionofnetexports. Thelocalmethodsproducesolutionsimpliedby thelocaldecisionrules. The2OAdecisionruleofNFAcanbeexpressedas: ˜b = h ˜b +h y˜ + 1 (cid:16) h ˜b2+h y˜2 (cid:17) +h ˜b y˜ + 1 (cid:0) h σ2(cid:1) (8) t+1 b t y t 2 bb t yy t by t t 2 σzσz z where ˜b ≡ b −bdss andy˜ ≡ y −y¯. The1OAandRSSmethodshavesimilarexpressions,except t t t t that they only have the first two terms in the right-hand side, because they both use a first-order approximationofthedecisionrules. TheRSSmethodalsodiffersinthatitusesbrssinsteadofbdss. Thekeycoefficienttoanalyzeish ,becauseitisthemaindeterminantofρ . Thisisthecaseeven b b forthe2OAsolutionsbecauseinallofourquantitativeapplicationsh ,h andh arenegligibly bb yy by small.11 The term h σ2 is also important, because it isolates the effect of income variability on σzσz z meanNFA.Itisanestimateoftheamountofprecautionarysavingsthatthe2OAsolutioncaptures. Moreover, since this is the only term that is quantitatively relevant of those that distinguish 2OA from 1OA solutions, and both of these solutions have the same h term (as shown by Schmittb GrohéandUribe(2004)),theseresultsalsoimplythatthe2OAand1OAsolutionsshouldbevery similar,exceptintheirfirstmoments. FortheRSSmethod,deGroot(2014)showedthatincomevolatilitymattersfordeterminingbrss because the coefficient of variation of consumption (relative to its risky steady state) is constant, atalevelthatdependsonβ,r andσ.12 Intuitively,thiscapturesprecautionarysavingsbecause,if incomevariabilityrisesandthesharesofincomeallocatedtosavingsv. consumptionremainunchanged,thevolatilityofconsumptionwouldincrease. ButbyincreasingNFArelativetoincome, more of the disposable income comes from interest income, so that the coefficient of variation of consumption can remain constant. Since the RSS decision rule follows from a first-order approximation, however, the ρ value will differ from that implied by the 1OA and 2OA solutions only b to the extent that bdss and brss differ, and as we document below, this requires larger differences than those obtained under the baseline and targeted calibrations. Hence, 1OA, 2OA and partial RSSsolutionsarelikelytobeverysimilar,exceptfordifferencesinfirstmoments. Wenowstudyhowψ andthecenterapproximationaffectthedeterminationofh inthelocal b decision rules. For tractability, assume log utility and i.i.d income. Under these assumptions, we 11Forinstance,aswedocumentinSectionB.3oftheAppendix,forthe2OAsolutionwithbaselinecalibration: h bb = 0.004,h by =0.005,andh yy =0.00005. 12Corollary5indeGroot(2014)showsthat var(c) = 2 1−βR. (crss)2 σ(1+σ) βR 15

showinAppendixC.3thath inallthelocalmethodsisgivenbythefollowingexpression:13 b (cid:113) R+eb∗ψ(1−b∗ψ+ψ)− R2+2eb∗ψ(b∗ψ+ψ−1)R+e2b∗ψ(1−b∗ψ+ψ)2 h (ψ,b∗) = , (9) b 2eb∗ψ whereR ≡ 1+randb∗ = bdssfor1OAand2OAorbrssforRSS.Sincethetermsh ,h andh in(8) bb yy by are quantitatively irrelevant, it follows that ρ (ψ,b∗) ≈ h (ψ,b∗) for 1OA, 2OA and RSS methods. b b Hence, eq. (9) can be used to analyze how ψ and b∗ affect the autocorrelation of the equilibrium stochastic process of NFA produced by the local methods. Moreover, the above expression also impliesthatthevalueofh obtainedwith1OAand2OAdiffersfromtheRSSsolutiononlytothe b extentthatbdss andbrss differ. Equation (9) formalizes the argument that setting the value of ψ imposes implicitly the equilibriumautocorrelationofNFA,andinparticularchoosingverylowvaluesofψ impliesvaluesof ρ close to 1. For given R and b∗, ψ determines ρ and in fact, as the numerical results reported b b below show, ρ is decreasing (increasing) in ψ for relatively low (high) ψ. For the 1OA and 2OA b methods, eq. (9) describes fully the relationship between ψ and ρ , because b∗ = bdss and bdss is b exogenous, but for the RSS method we need to consider that b∗ = brss and brss is solved together withthecoefficientsofthedecisionrulesfor ˜b andc˜,whichalsodependonψ. t+1 t Equation(9)canalsobeusedtoillustratethenon-stationarityofthelocalsolutionsifastationarityinducing transformation is not used. If ψ = 0, the solution of ρ (ψ,b∗) has two roots, R or 1, so b NFAisnon-stationary. Incontrast(andassumingb∗ = 0forsimplicity),ifψ > 0thesmallerofthe tworootsthatsolveρ (ψ,0)islessthanunitary,andthusyieldsastablesolution.14 b Westudynumericallyhowvariationsinψ andb∗ alterρ (ψ,b∗). Tothisend,wesetR = 1.059 b as in the baseline calibration and solve for ρ (ψ,b∗) or a set of values of ψ and b∗. The results are b summarizedinFigure1. TheFigureplotsρ (ψ,b∗)forψ intheinterval[0,0.9]andthreevaluesof b b∗: 0,-0.41(brss inthebaselinecalibration)and-0.51(bdss inthebaselinecalibration). Figure 1 yields a key finding: The value of ρ is nearly identical across 2OA and RSS for any b 0 ≤ ψ ≤ 0.1, which is an interval that includes the baseline and targeted calibration values and also all the values of ηr implied by the ψ values used in the papers listed in Table 1.15 Hence, for 13ThisresultappliesforbothfullandpartialRSS. 14Schmitt-GrohéandUribe(2003)obtainedsimilarresultsbyderivingtheanalyticsolutionofthedecisionruleof NFAforanendowmenteconomywithlogutilityandusingthestationarity-inducingassumptionbywhichtherateof timepreferencedependsonaggregateconsumption. 15Thehighestψintheliteratureisψ=2.8inGarcia-Ciccoetal. (2010),andwiththeirvalueofbdss =−0.007yields ηr =−0.0196.Fortheinterval0≤ψ≤0.1withourbdss =−0.51weobtainanintervalofelasticities0≥ηr ≥−0.051. Inourbaselineandtargetedcalibrations,ψ=0.001,0.0468whichimpliesηr =−0.0051,−0.0239. 16

thevaluesofψusedintheliterature,thechoiceofapproximatingaroundbdss v. brss andsolvingwith1OA, 2OA or partial RSS does not make a difference. The two steady-state estimates would have to differ much more than what the baseline calibration and small variations around it would predict. We starttonoticeanon-negligibledifferenceonlyifbdss ismorethanfortypercentagepointsofGDP below brss. Moreover, since in the baseline and targeted calibrations it is also the case that the termsrelatedtothequadraticandinteractiontermsofthe2OAdecisionruleofbarenearlyzero,it followsthatwecanexpectthe2OAandRSSsolutionstoproducesimilarsecondandhigher-order momentsforallendogenousvariables,astheresultsreportedbelowwillconfirm.16 Figure 1 also shows that ρ switches from decreasing to increasing in ψ at a sufficiently high b valueofψ(i.e.,ψ ≥ 0.5). However,ψ ≥ 0.5wouldimplymuchlargervaluesofηr thanthoseused intheliterature,includingestimatedmodels. The above findings indicate that the implications of ρ for ρ derived in condition (7) by asb nx suming that NFA follows an AR(1) process actually apply to the equilibrium stochastic processes producedbythelocalmethods. TheDEIRfunctionwithverysmallψimposesvaluesofρ near1, b andsmalldifferencesbetweentheseimposedvaluesandtheexactsolutionsoftheglobalsolution resultinlargedifferencesinthepredictedvaluesofρ ,aswedocumentnext. nx Table4illustratestherelationshipbetweenρ andρ intheglobal(GLB)andlocalsolutions. b nx Toinducechangesinρ ,wevaryρ from0to0.8. Paneli)showstheGLBresultsusingthebaseline b z calibration. Panel ii) shows the 2OA and partial RSS results for their baseline calibration with ψ = 0.001. Paneliii)showslocalsolutionsfortargetedcalibrationswithψ settomatchρ = 0.996 b (thevalueintheGLBsolutionshowninTable5)whichimpliedψ = 0.0469forthe2OAandpartial RSSsolutions. Paneliv)showsanadditionalscenarioinwhich,foreachvalueofρ obtainedwith b theGLBsolutionasρ changes,were-calibratethevalueofψ inthelocalsolutionssoastomatch z thatvalueofρ (thecorrespondingvaluesofψ arealsoshowinthispanel). b The first result evident in Table 4 is that 2OA and partial RSS results are always very similar, indicating that the gap between bdss and brss and the quadratic and interaction terms in the 2OA decisionrulesarenotlargeenoughtoaltersignificantlythevaluesofρ andρ ofthetwomethods b nx underanyofthecombinationsofρ andψ considered. Recallalsothatforthesamereasons1OA z and2OAsolutionsarenearlythesame. Paneli)showsthatasρ risesfrom0to0.8,theGLBsolutionindicatesthatthetruevalueofρ z b 16Theanalyticsolutionforh b (ψ,b∗)isstrictlyvalidonlyforlogutilityandi.i.d. shocks,buttheseimplicationsofthe analysisstillholdquantitativelyinthesolutionswithAR(1)shocks. 17

rises from 0.82 to 0.99 and ρ rises from almost -0.1 to 0.85. Thus, as predicted by eq. (7), small nx variationsinρ near1causelargechangesinρ . Incontrast,Panelii)showsthatwithψ = 0.001, b nx the local solutions yield values of ρ that always exceed 0.99, which in turn yield values of ρ b nx rangingfrom0.3to0.95. Theerrorsrelativetotheglobalsolutionsarelarge. Forρ = 0, thetrue z valuesofρ andρ are0.83and-0.1respectively,whilethe2OAandRSSmethodsyieldρ = 0.99 b nx b and values of ρ of 0.27 and 0.24 respectively. For the calibrated value of ρ = 0.597 estimated nx z from the Mexican data, the global solution yields ρ = 0.977 and ρ = 0.54, while the 2OA and b nx RSS methods yield ρ = 0.999 and ρ = 0.82 (see Table 5).17 Thus, these results show that in b nx orderforthelocalmethodstoyieldanaccurateapproximationtothemodel’struevalueofρ to nx contrastwiththedata,theyfirstneedtoproduceavalueofρ veryclosetothetruevalue. b Panel iii) shows that the local solutions perform better using the targeted calibrations with ψ = 0.0469. In this case, GLB and local solutions yield the same ρ for the calibrated value of ρ b z (0.597)byconstruction. Forlowervaluesofρ ,thelocalsolutionsoverestimateslightlythevalues z ofρ andρ relativetotheGLBsolutions. Paneliv)showsthat,ifwere-calibrateψ aswechange b nx ρ sothatthelocalsolutionsmatchtheρ oftheGLBsolutionsineachcolumnoftheTable,thelocal z b methodsdoagoodjobatmatchingtheglobalsolutions. Thisistruebyconstructionforρ ,butthe b solutions of ρ are also close. In these results, however, the value of ψ had to be increased as ρ nx z falls, up to a value of 0.185 for ρ = 0. The required values of ψ range from 0.027 to 0.185. These z aresignificantlylargerthantheidealvalueof0.001thatkeepstheDEIRinessential,andeffectively theymakedeviationsofNFAfromitssteadystateverycostly. Moreover, knowingthetruevalue ofρ thatψ needstotargetrequiresfirstsolvingthemodelwithglobalmethods. b b)Long-runmomentsandprecautionarysavings Table5showslong-runmomentsproducedbytheglobalandlocalsolutions,includingforthe latterresultsfrombaselineandtargetedcalibrationsandforpartialandfullRSS.18 ThelocalsolutionsunderthebaselinecalibrationdopoorlyatmatchingtheGLBmoments. Theglobalsolution yieldsanaverageNFApositionof-41.3percentofGDP,nearly10percentagepointshigherthanthe -51percentatthedeterministicsteadystate(whichistheaverageforthe1OAsolutionbecauseof certaintyequivalence). The2OAandpartialRSSmethodsyieldlong-runaveragesof-28.2and-45.1 percentofGDPrespectively. Theformer(latter)overestimates(underesestimates)precautionary savingsbyabout13(4)percentagepoints. ThefullRSSsolutionyieldsamuchloweraverageNFA 17The1OAand2OAsolutionsarenearlyidentical,andhenceweomittheformerfromtheTable. 181OAsolutionsarenotshownbecausetheyarenearlythesameasthe2OAsolutions,exceptfortheaverages. 18

position of nearly -1120 percent of GDP. This is because this method has the same βR < 1 of the globalmethod,butitlackstheoccasionallybindingdebtlimitϕthatallowstheglobalmethodto simultaneouslymatchthemeanofNFAandthevariabilityofconsumptionobservedinthedata.19 Hence, the full RSS generates much higher debt and yields a consumption process with a much lowermeanandhighervariabilitythaninboththeGLBsolutionandtheMexicandata. Local solutions with baseline calibrations also do poorly at matching the rest of the moments shownintheTable. Thethreelocalmethodsoverestimatesignificantlythevariabilityandpersistenceofc,nxandb. ThecorrelationswithGDPshowtheoppositepattern. Forthesehigher-order moments, full and partial RSS generate similar values, so the large differences in means do not translate into large differences in higher-order moments. Since partial RSS is much faster and is muchclosertothetruemeanofNFA,weconcludethatpartialRSSisbetterthanfullRSS. Thelocalmethodsagainperformbetteratapproximatingtheglobalresultsifweswitchtotargetedcalibrations. Themajorexceptionisthattheydoworseatcapturingtheeffectofprecautionary savings, with both RSS and 2OA solutions yielding an average of b/y of nearly -0.51. These averagesareveryclosetothedeterministicsteadystate,andthisoccursbecause(asexplainedbelow) higherψ isakintoahighercostofmovingbawayfromitssteadystate.20 Thelocalsolutionswith thetargetedcalibrationsalsohavethedrawbackthattheystilloverestimatetheautocorrelationof consumption,astheydidunderthebaselinecalibration. Fortherestofthemoments,thetargeted calibrationsdeliverabetterapproximationtotheGLBsolutionthanthebaselinecalibrations,but again they require knowledge of the global solution to determine the target value of ψ and the impliedψ valuesaremuchlargerthan0.001. TheresultsinTable4showingthatthe2OAandpartialRSSsolutionsyieldsimilarρ andρ b nx extendstonearlyallthemomentsshowninTable5underbothbaselineandtargetedcalibrations. The only exception is the mean of b/y under the baseline calibration, which is -0.28 with 2OA v. -0.45withRSS,butunderthetargetedcalibrationeventhismomentisnearlythesameacross2OA and RSS. This is further evidence indicating that the different centers of approximation in these solutionsandtheextratermsinthe2OAdecisionruleshavenegligiblequantitativeeffects. Figure 2 provides further evidence of the inaccuracy of the local methods at accounting for precautionarysavings. Panels(a)and(b)showhowaverageNFAchangeswithσ andρ under z z 19FullRSSalsodoesnotfacetheNDL,sinceityieldsstationarydynamicsaroundbrss,whichmakestheeffectsofthe Inadaconditiononu(cid:48)(c)irrelevantandthemultiplierontheNDLisignored. 20Wecouldtargetψ tomatchthemeanofNFAintheGLBsolution(-0.41)instead,butthenthelocalmethodsdo poorly at matching the GLB value of ρ nx. Using this approach, ρ nx = 0.74 and 0.88 for the 2OA and RSS solutions respectively,whereasρ nx =0.543intheglobalsolution. 19

thebaselinecalibration. Panels(c)and(d)dothesamebutforthetargetedcalibrations. Recallthat for1OAsolutions,certaintyequivalenceimpliesthatmeanofNFAisalways-0.51(thedeterministic steadystate),regardlessofthevariabilityandpersistenceofincomeandthevalueofψ. These plots yield two important findings. First, the local methods cannot approximate accurately the long-run averages of NFA produced by the GLB solutions in general, and hence they yield incorrect measures of precautionary savings. The continuous, blue curves for the GLB solutionsshowthatthemodelembodiesastrongprecautionarysavingsmotive. Panels(a)and(c) ((b) and (d)) indicate that increasing the standard deviation (autocorrelation) of income from 1 to8percent(0to0.8)increasesthelong-runaverageofNFAfrom-0.5tonearzero(-0.47to-0.34). In contrast, the local solutions with the baseline calibrations show that the 2OA method overestimatestheincreaseinprecautionarysavingssignificantly,withagapthatwidensasthevariability or persistence of income increase, while the partial RSS method mostly underestimates average NFA, although with a smaller error in absolute value than the 2OA solution. Note also that for highenoughρ inPanel(b),partialRSSactuallypredictsslightlyhighermeanNFAthanGLB. z Thesecondfindingisthatthelocalmethodswithtargetedcalibrationsareworsethanthebaselinecalibrationsatcapturingprecautionarysavings. Panelsc)andd)showthattheaverageofNFA in the targeted calibrations increases very slightly above bdss as the variability and persistence of incomerise. Usingψ =0.1and0.2,theresultisevenstronger,withmeanNFAbecominginvariant toincomevariabilityandpersistence. Moreover,sincethetermsdependingonthequadraticand interactiontermsofthe2OAsolutionscontinuetobequantitativelyirrelevant,thisresultsuggests thatunlessψiskeptverylow,the1OAsolutionsarenearlythesameasthe2OAandRSSsolutions in all dimensions, even long-run averages. In addition, the 2OA and RSS solutions also become nearly identical, since brss becomes very similar to bdss. Thus, while calibrating ψ to match the persistenceofNFAintheGLBsolutionimprovesthelocalmethods’abilitytomatchsecond-and higher-ordermoments,italsoremovesprecautionarysavingsalmostentirelyandrendersthe2OA andRSSsolutionsquantitativelyconsistentwithcertaintyequivalence! Thefactthatprecautionarysavingsnearlyvanishfromthe2OAandpartialRSSsolutionsasψ risesimpliesthatthetermsdrivingthedeviationsoftheunconditionalaveragesofbfrombdss are vanishing too. To shed light on why this happens, we use again the decision rules for log utility andi.i.d. shocks(togetherwiththequantitativeresultthatthequadraticandinteractiontermsof 20

the2OAsolutionsarenegligible)toexpresstheunconditionalmeansofthelocalsolutionsas: h g2 E[b]2OA = bdss+σ2 σzσz , E[b]RSS = bdss+σ2 y , (10) z2(1−h ) z2ψ b where g is the coefficient of the linearized consumption decision rule on income. We showed y earlierthath isdecreasinginψforψ < 0.5. Hence,thedenominatorintheright-hand-sideofboth b of the above expressions rises with ψ, which brings the unconditional average closer to bdss. The coefficients h and g also depend on ψ, so there are also effects of ψ on mean NFA operating σzσz y through these coefficients, but the quantitative evidence indicating that for ψ ≥ 0.047 the mean NFAisaboutthesameasbdss indicatesthattheseeffectsareeitherworkinginthesamedirection asthoseoperatingviathedenominatorsoftheexpressions,ortheyaremorethanoffsetbythem. TheeconomicintuitionfortheaboveresultfollowsfromtheargumentbySchmitt-Grohéand Uribe(2003)showingthattheDEIRfunctionissimilartoasetupwithoutDEIRbutwhereagents incuraquadraticcost(ψ˜/2)(b −bdss)2 fordeviatingfromsteady-statebondholdings. Thelogt+1 linearized Euler equations of the two formulations become equivalent if we set ψ˜ = ψ/(1 + r). Hence, the model with DEIR can be re-interpreted as a model in which agents are penalized for deviatingfrombdss,andthecostincreaseswithψ.21 Moreover,thecosthasvariableandfixedcomponents,sinceitcanbedecomposedintothesetwoterms: (ψ˜/2)(b −2bdss)b and(ψ˜/2)bdss. If t+1 t+1 thefixedcostislargerthanthebenefitderivedfromprecautionarysavings,itwouldbesuboptimal to allow the long-run average of bonds to deviate from bdss. Thus, local solutions using targeted calibrationshavetheshortcomingthatwhiletheycanapproximatebetterthetruevaluesofsecondandhigher-ordermomentsbysettingψ highenoughtomatchthetruevalueofρ ,itonlytakesa b modestincreaseinψtomakeprecautionarysavingsnearlyvanishandtomake1OA,2OAandRSS solutionsaboutthesameinalldimensions. ThelastpanelinTable5showsperformancemetricsofthealgorithms. FiPItyieldstheglobal solutionin5.9seconds,whilethelocalmethodstake44to68percentlonger(seetheTablefootnote fordetailsonthesoftwareandhardwareusedinallthecomputations). TheGLBsolutionisalso significantlymoreaccurate,asindicatedbythemuchsmallermaximumerrorintheEulerequation andthemeanandmaximumpercentdifferencesindecisionrulesforNFAandconsumptionofthe localv. GLBsolutions. TheNFAlocaldecisionrulesshowaverage(maximum)differencesranging from7.5to12(12.3to37.8)percent. 21WithDEIR,forb t+1 <bdss(b t+1 >bdss)agentspaymore(getless)forborrowing(saving)more. 21

c)Impulseresponsefunctions Figure 3 provides impulse response functions to a one-standard-deviation income shock. IncomedynamicsareshowninPanelg. Forthelocalsolutions,Panelsa,c,ande(b,d,andf)show resultsforthebaseline(targeted)calibration. Ineachplot,theGLBsolutionisshowninblue. Consumption and output are in percent deviations from their long-run means, while NFA/GDP and nx/GDPareindifferencesrelativetotheirlong-runmeans(sinceratiosarealreadyinpercent). The plots show nearly identical impulse responses for 1OA, 2OA and RSS in each calibration case, inlinewiththeresultsthattheh coefficientsofthebonddecisionrulesaresimilarandthe b quadraticandinteractiontermsofthe2OAsolutionsareverysmall. Ontheotherhand,thelocal impulse responses with the baseline calibration differ sharply from the GLB ones. GLB predicts a weaker decline in NFA (i.e., less borrowing) and much faster mean reversion than the local solutions. Accordingly, consumptionfallsnearlytwiceasmuchonimpactintheGLBsolution, and continues to decline before recovering, displaying also faster mean reversion. These differences in consumption responses also reflect smaller trade deficits on impact and in the first periods of transition and a faster recovery into trade surpluses with the GLB solution. Local solutions with targeted calibrations yield impulse responses that approximate better the GLB solutions, but still show noticeable discrepancies. In particular, the local solutions now overestimate the fall in consumptiononimpact. d)Spectraldensityanalysis Wecomparenextnonparametricsampleperiodogramsofthevarioussolutions. Thegoalisto determine whether in addition to the differences in time-series properties, the methods differ in theirpredictionsabouttherelevanceoffluctuationsatdifferentfrequenciesforoverallvariability. Figure4showsperiodogramsforsimulateddataofb,candnxcorrespondingtoamultivariate spectrumusingaBartlettwindowwiththesmoothingparametersetto100.22 Theseperiodograms arecomputedbasedonlongtime-seriessimulationsincluding4500periods. They-axisshowsthe population spectrum, the x-axis shows the frequency in years, and the vertical lines isolate the businesscyclefrequency. Thepanelsontheleft(right)areforthebaseline(targeted)calibrations. Asinthepreviouscharts,theplotsfortheGLBsolutionareidenticalinbothsetsofplots,because theglobalsolutionhasasinglecalibration. Inaddition,aswiththetime-seriesresults,thespectral density functions are nearly identical for 2OA and RSS methods, because the local decision rules 22WefollowHamilton(1994)insettingthevalueofthesmoothingparameter.Theresultsforparametricestimatesof thespectraldensitiesaregenerallysmootherbutshowsimilarpatternsasthoseofthenonparametricestimates. 22

havesimilarh termsandthequadraticandinteractiontermsofthe2OAsolutionsareirrelevant. b AlltheperiodogramsaregenerallydownwardslopingbecausetheequilibriumstochasticprocessesaresimilartoAR(1)processes. Hence,thecontributionoflowerfrequenciestothevariances of the variables exceeds that of business cycle and lower frequencies. The results show, however, that the local methods under the baseline calibration overestimate the contribution of low frequency movements to the total variance of all three series, which is consistent with their slower mean-reversionandhighervaluesofρ relativetotheGLBsolution. Moreover,whilethecontribub tionoffluctuationsatthebusinesscyclefrequencyorhigherforthevariabilityofbisslightlyhigher withthelocalsolutionsthanintheGLBsolution,fornxthelocalmethodsoverestimateitandfor ctheyunderestimateit. Inparticular,thelocalmethodsunderestimatesignificantlythefractionof consumption fluctuations explained by business cycle and higher frequencies and under-predict significantlythecontributionoflowfrequencies. Fortargetedcalibrations,theperiodogramsofbarenearlythesameintheGLBandlocalsolutionsalmostbyconstruction,becausethetargetedcalibrationsarebuilttomatchtheAR(1)coefficientoftheGLBsolution. However,theperiodogramsofcandnxforthelocalsolutionsstilldiffer sharplyfromtheGLBones. Theystillunderestimatesignificantlythecontributionofconsumption fluctuationsatbusinesscycleandhigherfrequenciestooverallconsumptionvariance. e)Interest-rateshocks We examine next the effects of adding interest-rate shocks. We do this for two reasons. First, because Courdacier et al. (2011) and de Groot (2013) showed that interest-rate shocks play an important role in the quantitative performance of the RSS method, and second to facilitate comparisonsoftheendowmentmodelresultswiththoseoftheRBCmodel,whichalsohasinterest-rate shocks. Thegrossrealinterestrateisnowdefinedas: R t = ez t RR¯ ,wherez t R isashockwithexponentialsupportandR¯ isthemeaninterestrate. ThetwoshocksfollowadiagonalVARrepresentation:           z ρ 0 z εz σ2 σ t z t−1 t εz εz,εR   =  · + , Σ =  . (11) zR 0 ρ zR εR σ σ2 t zR t−1 t εz,εR εR whereΣisthevariance-covariancematrixoftheinnovations. Notethatifthevarianceofinterestrateshocksiszero,theendowmentshockprocessisidenticaltotheoneusedearlierinthisSection. The value of ρ is the same 0.597 of the original calibration. We minimize the size of the state z spaceintheGLBsolutionbyusingaMarkovchainfortwoshocksdefinedbytheSimplePersistence 23

Rule, which imposes the same autocorrelation on both shocks. Hence, we set ρ = 0.597.23 The zR value of σ2 is set at 0.00069, so that given ρ we obtain a standard deviation of the endowment εz z (cid:112) incomeofσ = σ2 /(1−ρ2) = 0.0327,whichisthevaluefromtheMexicandataintheoriginal z εz z calibration. Forthetermsthatinvolvetheinterest-rateprocess,wesolvethemodelwithvaluesof σ2 andσ suchthatσ takesvaluesrangingfrom0to2.5percentandthecorrelationbetween εR εz,εR zR endowment income and the interest rate is ρ = −0.669, which matches the correlation of the z,zR interest rate with TFP in Mendoza (2010), and is also the value we use in the calibration of the RBC model in the next Section. The values of σ and σ2 change as we change σ , and they εz,εR εR zR aregivenby: σ = (1−ρ ρ )ρ σ σ andσ2 = σ2 /(1−ρ2 ). εz,εR z zR z,zR z zR εR zR zR The Simple Persistence Markov chain is defined by a set of pairs of realizations of the shocks (z,zR)andamatrixπ oftransitionprobabilitiesofmovingfromanyrealizationpairtoanyother pair in one period. Each shock has two realizations equal to plus/minus one-standard deviation ofeachshock(z = −z = 0.0327,zR = −zR = σ ,withσ rangingfrom0to2.5percent). The 1 2 1 2 zR zR SimplePersistenceruleproducesa4x4matrixπ withelementsdefinedbyaformulasuchthatthe standarddeviationsoftheshocksmatchtherealizationvalues,andtheunconditionalcorrelation andautocorrelationsoftheshocksmatchthevaluessetinthecalibration. With interest-rate shocks, a well-defined limiting distribution of NFA requires βR¯ < 1, otherwise βtΠt R diverges to infinity (see Chamberlain and Wilson (2000)). In addition, there are j=1 j realizations, andhencehistoriesofrealizations, withR lower(higher)thanR¯ formanyperiods, t whichimplymuchweaker(stronger)precautionarysavingsincentivesthanwithaconstantinterestratesetatR¯ . ThisiseasiertoseeforhistoriesofrealizationswithβR > 1,becausetheyproduce t equilibrium subsequences where b can grow very large, since there is no pro-borrowing effect t+1 duetoβR < 1offsettingtheprecautionarysavingsincentive.24 Atsomepoint,eachofthesehistot riesshiftstohistoriesofinterest-raterealizationswithsufficientlylowR toinducemean-reversion t inNFA.NotealsothattheNDLisnowcomputedwiththehighestrealizationofR −1,soitwill t betighterthanwhencomputedwithR¯−1. TheseeffectsareatworkonlyintheGLBsolution,not in the local ones, because they result from expectations of histories of future shocks that take the economyfarfromitslong-runaverageandthedeterministicsteadystate. WhenusingtheDEIRfunction,theinterestrateismodifiedsothatinterest-rateshockshitonly 23ThisisreasonablebecauseintheMexicandataρ zandρ zR are0.537and0.572,respectively(seeMendoza(2010)). 24ReducingR¯ keepingσ zR constantaccentuatestheseeffects,becausehistorieswithevenlargergapsbetweenβand R tarepossibleandwithhigherprobability. 24

theworldinterestratecomponent,notthepremiumdrivenbysteadystatedeviationsofb : t+1 (cid:104) (cid:105) 1+r t = ez t r R¯+ψ ebdss−bt+1 −1 . (12) Table 6 shows some of the key moments produced by the different solution methods under baseline and targeted calibrations as σ rises from 0 to 2.5 percent. The baseline and targeted zR calibrationsshowninTable3areleftunchanged. FortheGLBsolution,weshowresultswithboth the calibrated ad-hoc debt limit (ϕ = −0.51) and the NDL, with the aim of comparing the roles that the ad-hoc debt limit and interest-rate shocks play in inducing higher mean NFA outcomes, andwiththesimilareffectofinterest-rateshocksinlocalsolutions. Comparing the GLB solution with the solutions under the baseline calibration for partial RSS and 2OA, we find that increasing σ from 0 to 2.5 percent has a much stronger effect on mean zR NFAinthelocalsolutionsthanintheGLBsolution. Intheformer, E(b/y)increasesby140(109) percentagepointsofGDPforthepartialRSS(2OA)solution,andactuallyturnsfromnegativeto positive, while in GLB it increases by just about 3 percentage points of GDP. Notice also that the ability of the partial RSS v. the 2OA solutions to generate precautionary savings changes as the variabilityoftheinterestraterises. Withlowornointerest-ratevariability,2OAgeneratessignificantlymoreprecautionarysavings(E(b/y) = −0.285v. −0.451),butforinterest-ratevariabilityof 2.5percenttheoppositeistrue(E(b/y) = 0.806v. 0.942). Largerinterest-rateshocksalsoalterthe resultthatthebaselineRSSand2OAsolutionshavesimilarsecond-andhigher-ordermoments. Theabovefindingssuggestthatinterest-rateshocksinthepartialRSSsolutionswithbaseline ψcouldbehelpfulformatchingmeanNFA,playingtheroleofNDLintheglobalsolutioncalibration. Thisstrategyfails,however,becauseconsumptionfluctuatestoomuchinallthescenariosfor partialRSSand20Asolutions. AllthelocalsolutionsshowninTable6overestimatethevariability ofconsumptionintheGLBsolutionsbyratiosrangingfrom1.04(forpartialRSSwithtargetedψ andσ = 0.5%)to4.01(forpartialRSSwithbaselineψ andσ = 2.5%). zR zR Comparing now GLB solutions v. local solutions with targeted calibrations, we find that the adjustment-cost-like effect of higher ψ bringing mean NFA close to the deterministic steady state stilldominates. ThelocalsolutionsyieldsmallincreasesinE(b/y)ofabout1.8percentagepoints (withσ < 1.5%thereisalmostnochange)andsecond-andhigher-ordermomentsforRSSand zR 2OAareagainverysimilar. Hence,theresultthathigherψvaluesneutralizeprecautionarysavings andyieldverysimilar1OA,2OAandRSSsolutionsisrobusttoaddinginterest-rateshocks. 25

Table6alsoshowsthatfullandpartial(baseline)RSSsolutionsdonotyieldsimilarsecond-and higher-ordermomentsonceinterest-rateshocksareadded. ThefullRSSsolutiongeneratessharply higher variability in consumption and NFA, higher autocorrelations in net exports, and very low valuesofE(b/y). Infact,fullRSSisclosertotheGLBsolutionthatreplacesthead-hocdebtlimit withtheNDLthantothebaselineortargetedpartialRSSsolutions. ThefullRSSsolutionandthe GLBsolutionwiththeNDLhave,however,themajorshortcomingthattheyproduceunreasonably largenetdebtpositionsof3to11timestheincomeoftheeconomy! Moreover,fornetinterestrates justanotchbelowtherateoftimepreference(0.0638),thefullRSSmethodalwaysgeneratesmuch lowervaluesofE(b/y)thantheglobalsolutionswitheitherad-hocornaturaldebtlimits. Atlow interestrates,theRSSsolutionviolatestheNDLveryoften(e.g.,forR¯ = 1.01,NDLis−68.44while theaverage NFAofthefullRSSsolutionis−69.62). Hence,fullRSSperformspoorlyingeneralat approximating accurately the long-run average of NFA, even if we remove the ad-hoc debt limit fromtheGLBsolution. Summingup,theendowmentmodelanalysisyieldsfourkeyfindings: First,localsolutionsfor baseline calibrations perform poorly in several key dimensions, such as the mean of net foreign assets, the cyclical moments of consumption and net exports, the impulse response functions to incomeshocks,andthecharacteristicsofspectraldensityfunctions. Second,targetedcalibrations perform better, but they are still unable to match some key features of the global solutions, and inordertoconstructthemoneneedstheexactvalueofρ fromtheglobalsolutiontocalibrateψ. b Moreover, the implied values of ψ (0.0469 for both the 2OA and RSS solutions) are much higher than the 0.001 baseline value used to keep the DEIR function inessential and imply much higher interest-rateelasticities. ThesehighψvaluesalsomakemovingNFAfromitssteadystatetoocostly, thereby causing precautionary savings to vanish. Third, the targeted calibrations also underperformsignificantlyatcapturingtheeffectsofparametervariations,particularlythosethataffectthe precautionarysavingsmotive. Inordertokeeptargetedcalibrationsclosetotheglobalsolutions,ψ needstoberecalibratedtomatchtheautocorrelationofNFAofeachglobalsolution,whichmakes targetedcalibrationsimpractical. Fourth, allthelocalmethods(1OA,2OAandpartialRSS)yield similarsolutions, becausesecond-ordertermsofthedecisionrules, otherthanthevarianceterm, arequantitativelysmallandthedeterministicandriskysteadystatesarenotsufficientlydifferent. Forψ = 0.001onlythefirstmomentsdiffer,whileforψ = 0.047orhighereventhefirstmoments aresimilar. Theseresultsarelargelyrobusttotheadditionofinterest-rateshocks,exceptthatlocal solutions with ψ = 0.001 can generate sizable precautionary savings effects as interest-rate vari- 26

abilityrises. Moreover,wealsofoundthatforthebaselineψvaluethefullandpartialRSSsolutions yieldsimilarsecond-andhigher-ordermomentswithoutinterest-rateshocks,butnotwhenthese shocksarepresent,andthefullRSSalwaysyieldsexcessivelylargeaveragenetdebt-incomeratios (largerthanGLBsolutionsconstrainedonlybythenaturaldebtlimit). 3 Real business cycle model WecomparenextlocalandglobalsolutionsforaworkhorseRBCmodelbasedonthoseproposed byNeumeyerandPerri(2005),UribeandYue(2006)andMendoza(2010). 3.1 Modelstructureandequilibrium As in Mendoza (2010), we characterize the model’s competitive equilibrium as the solution to a representative firm-household problem that is akin to a planner’s problem, except that the wage ratew enteringinthecalculationofworkingcapitalistakenasgivenbytherepresentativeagent t and set to satisfy the equilibrium condition that equates w with the marginal disutility of labor. t The economy produces gross output using a Cobb-Douglas technology that requires capital, k , t labor,L ,andimportedinputs,υ : t t exp(εA)F(k ,L ,υ ) = exp(εA)kγLαυη, 0 ≤ α,γ,η ≤ 1, α+γ +η = 1, εA > 0. (13) t t t t t t t t t Gross output is a tradable good sold at a world-determined price which is the numeraire and is assumed to be constant and equal to 1. The price of imported inputs is also determined in world markets,withtherelativepriceoftheseinputsintermsoftradablegoodsgivenbyp = pexp(εP), t t where p is the mean price and εP is a terms-of-trade shock. There are also TFP shocks, εA, and t t interestrateshocksεR. t AstandardworkingcapitalconstraintrequiresafractionφofthecostofL andυ tobepaidin t t advanceofsales. Workingcapitalloansareobtainedfromforeignlendersatthebeginningofeach periodandrepaidattheend,sothatthefinancingcostoftheseloansisthenetinterestrateR −1. t Capital is costly to adjust, with adjustment costs per unit of net investment (k − k ) det+1 t (cid:16) (cid:17) termined by the function Ψ(kt+1−kt) = a kt+1−kt , with a ≥ 0. This functional form satisfies kt 2 kt Hayashi’s conditions so that the average and marginal values of Tobin’s Q are equal at equilibrium. 27

Therepresentativefirm-householdchooses[c ,L ,i ,υ ,b ,k ]∞ soastomaximize: t t t t t+1 t+1 t=0  (cid:16) Lω (cid:17)1−σ  (cid:88) ∞ c t − ω t   E βt , (14) 0 1−σ   t=0  subjecttothisbudgetconstraint: c (1+τ)+i = exp(εA)F(k ,L ,υ )−p υ −φ(R −1)(w L +p υ )−q b +b , (15) t t t t t t t t t t t t t t t+1 t with w set to satisfy the labor supply condition w = L¯ω−1 , and L¯ denoting the aggregate labor t t t t allocationtakenasgivenbytheagent. Theonlyfinancialassetavailableisanon-state-contingent discount bond traded in world markets at price q . The left-hand-side of the resource constraint t is the sum of consumption, inclusive of an ad-valorem tax τ which will be used to calibrate the ratio of government expenditures to GDP, plus gross investment, i , where i = δk + (k − t t t t+1 (cid:104) (cid:16) (cid:17)(cid:105) k ) 1+Ψ kt+1−kt andδ denotesthedepreciationrateofphysicalcapital. Theright-hand-side t kt equals total supply, which consists of GDP (gross output minus the cost of intermediate goods, exp(εAF(k ,L − t,υ ) − p υ ) net of foreign interest payments on working capital loans (φ(R − t t t t t t 1)(w L +p υ ))minus(plus)netresourceslent(borrowed)abroad(q b −b ). Thetradebalance t t t t t t+1 t isthereforeq b −b +φ(R −1)(w L +p υ ) = GDP −c (1+τ)−i t t+1 t t t t t t t t t Thecompetitiveequilibriumisdefinedbystochasticsequencesofallocations[c ,L ,k ,b ,υ ,i ]∞ t t t+1 t+1 t t 0 andprices[w ]∞ suchthat(a)therepresentativefirm-householdsolvesitsoptimizationproblem t 0 takingasgiventhewagerateandtheinitialconditions(k ,b ), (b)wagessatisfyw = L¯ω−1 , and 0 0 t t (c)thelabormarketclears: L¯ = L . t t We solve this model using the same methods as in the previous section, extended to include the capital stock as a second endogenous state. For the global solution, we use FiPIt with a state space consisting of grids of k and b with 30 and 80 nodes respectively. The algorithm iterates to convergence on the decision rule for bonds and the pricing function for capital. Mendoza and Villalvazo(2019)providefulldetails,MatlabcodesandanAppendixdescribinghowtousethem. Forthelocalsolutions,theDEIRfunctionisnowdefinedas: 1+r t = exp(εR t )R¯+ψ (cid:20) ey bd d s s s s − b y t d + ss 1 −1 (cid:21) . (16) Notethatr dependsonthegapbetweenb /ydss andbdss/ydss. Theelasticityoftheinterestrate t t+1 28

with respect to percent deviations of b from bdss is ηr ≈ ψbdss/ydss for small deviations. This t+1 facilitates comparisons across calibrations of GLB and local solutions, since output is no longer equalto1atsteadystateandNFAiscalibratedtomatchtheNFA-GDPratiointhedata. 3.2 Calibration Table7showsthecalibrationparametersfortheRBCmodel,mostofwhichweretakenfromMendoza’s (2010) calibration to Mexican data. The main difference is that we calibrate ϕ and β in the GLB solution following the same strategy as in the endowment model, by targeting those twoparameterstoapproximatethemeanNFAandthestandarddeviationofconsumptioninthe data. Note, however, that since output is endogenous, steady state GDP cannot be normalized to 1. Hence, we searched over values of β and the lower bound of the NFA grid that yield model solutionsclosetothetargetdatamoments,andthenexpressedϕasaratioofthatlowerboundto ydss. Settingβ = 0.92andϕ = −0.758(impliedbyalowerboundofNFAof-300andydss = 396) we obtained a mean NFA-GDP ratio of -0.372 (v. -0.44 in the data) and a variability ratio of consumptiontoGDPof1.29(v. 1.25inthedata). Forthelocalsolutions,wealsoproceedasinthecase oftheendowmentmodel,bysettingbdss/ydss tobethesameasintheGLBsolutionandstudying baseline and targeted calibrations of ψ. The baseline value is again ψ = 0.001 and the targeted valuesareψ = 0.0109forthe2OAsolutionandψ = 0.008fortheRSSsolution. The stochastic process of the shocks is also taken from Mendoza (2010). We characterize the jointprocessofthethreeshocksasthefollowingdiagonalVARsystem:           εA ρA 0 0 εA uA σ2 σ 0 t t−1 t uA uA,uR            εR  =  0 ρR 0 · εR + uR , Σ =  σ σ2 0 . (17)  t     t−1   t   uA,uR uR            εp 0 0 ρp εp up 0 0 σ2 t t−1 t up InlinewiththeestimatesinMendoza(2010),theabovespecificationimposestheconditionsthat the co-movement between TFP and interest-rate shocks is driven only by the covariance of their innovations, and the price shocks are independent of the other two shocks. Mendoza reports estimates of the standard deviations of the shocks of σ εA = 0.013, σ εR = 0.0196 and σ εp = 0.0335 respectively. Thefirst-orderautocorrelationcoefficientsareρA = 0.537,ρR = 0.572andρp = 0.737, andthecorrelationbetweenTFPandRisρ = −0.669. Since,asweexplainbelow,thediscrete εA,εR approximation to this VAR system in the global solution requires ρA = ρR, we set the common 29

autocorrelationtotheaverageofthetwodataestimates, andhenceρA = ρR = 0.555. Weimpose thesameconditionontheVARrepresentationoftheshocksusedforthelocalsolutions. Givenall theseestimates,theelementsoftheΣmatrixaregivenby: σ2 = 1.0273e−04,σ = −0.0047, uA uA,uR σ2 = 2.4387e−04,andσ2 = 5.1097e−04. uR up In the global solution, the shocks are approximated using symmetric two-point Markov pro- E cesses defined with the Simple Persistence Rule. These processes consist of a set of all combinationsofrealizationsoftheshocksε = (εA,εR,εP),andamatrixπ oftransitionprobabilitiesof t t t t moving from ε to ε . Each shock has two realizations equal to +/- one-standard-deviation of t t+1 theircorrespondingdatacounterparts: εA = −εA = 0.0134,εR = εR = 0.0196,εP = −εP = 0.0335, 1 2 1 2 1 2 so E contains 8 triples. The Simple Persistence Rule produces an 8x8 matrix π which yields variances,correlationsandautocorrelationsforalltheshocksthatmatchthoseinthedata,exceptthat theprocedurerequiresshocksthatarecorrelated(i.e.,εAandεR)tohavethesameautocorrelation. As noted above, we set ρA = ρR = 0.555. This restriction is immaterial, because the two shocks haveverysimilarautocorrelationcoefficientsinthedata(ρA = 0.537,ρR = 0.572). 3.3 Comparisonofquantitativeresults a)Long-runmomentsandperformancemetrics Table8presentsunconditionalmomentsofGLB,2OAandRSSsolutions. 1OAresultsareomittedbecause,aswiththeendowmentmodel,second-andhigher-ordermomentsarenearlyidentical tothoseobtainedwith2OA.25 First,wehighlightbrieflythedifferencesbetweentheRBCandendowmentresultsintheGLBsolutions: TheRBCmodelpredictshighervariabilityofconsumption relative to GDP and countercyclical net exports, both of which bring the model closer the data. Thesechangesareduetothepresenceoftheworkingcapitalconstraintandcapitalaccumulation. The former amplifies the effects of TFP and input price shocks, and induces higher imports of inputsduringexpansionsinresponsetothecountercyclicalinterest-rateshocks. Capitalaccumulationalsogeneratesanincentivetoincreaseimportsandrunexternaldeficitsduringexpansions, becauseofthepositiveautocorrelationofthethreeshockspresentinthemodel: “Goodtimes,”in which TFP is high and both input prices and the interest rate are low, have positive persistence, which makes it optimal to borrow from abroad to finance investment due to the expectation that favorablerealizationsoftheseshockswillcontinuetobeobservedinthenearfuture. Thecounter- 25AswedocumentinAppendixC.4.1,thefirst-ordercoefficientsofthedecisionrulesfor1OAand2OAsolutionsare identical,andthoseforhigher-ordertermsinthe2OAsolution,exceptthevariance,arenegligible. 30

cyclicalnetexportsduetotheseeffectscontributestotheexcessvariabilityinconsumptionrelative toGDP. Compare next the RSS and 2OA solutions under the baseline calibration. The moments for consumption,netexportsandNFAdifferslightlybetweenthesetwosolutionsintheendowment model,butintheRBCmodelthedifferencesarelarger. Thisis,however,consistentwiththeargumentspresentedearlier,becauseintheRBCmodelbrss andbdss differsharply(36v. -76inpercent of GDP), while in the endowment model the difference was too small to matter. The first-order coefficientsofthedecisionrules(reportedinAppendixC.4.1)areagainsimilarfor2OAandRSS, and the second-order coefficients of the 2OA solution yield again negligible effects, but the large differencebetweenbrssandbdssyieldslargerdifferencesinlong-runmoments. Thisisparticularly thecaseforthemeansoftheratiosofnetexportsandNFAtoGDP,whichare-4.2and73.2percent respectivelyinthe2OAsolutionv. -18.5and255.9percentintheRSSsolution. Comparinglocalv. globalsolutionsunderthebaselinecalibration,theperformanceoftheformer at approximating the GLB solution for the average NFA-GDP ratio worsens markedly in the RBC model v. the endowment model. In particular, while for the endowment model the 2OA and RSS methods produced mean ratios of -0.28 and -0.45, relative to -0.41 in the GLB solution, in the RBC model they produce positive ratios of 0.73 and 2.56 respectively (i.e., the economy is a net lender) relative to -0.38 in the GLB solution. Hence, the precautionary savings motive is sharplyoverstatedbythelocalsolutions. ThisispartlybecausetheRBCmodelincludesinterestrate shocks, and we documented earlier that when these shocks are included 2OA and RSS solutions overestimate significantly the mean NFA position even in the endowment model.26 These findings are also in line with results reported by de Groot (2014), showing large, positive mean NFA-GDPratiosof3.6and41inthetwostableequilibriaproducedbythefullRSSmethodforan endowmenteconomy.27 For second- and higher-order moments, the results are largely in line with what we observed in the endowment model. In particular, the local solutions overestimate again the persistence of the balance of trade. The GLB solution generates an autocorrelation of net exports around 0.71 whereasbothlocalmethodsgeneratevaluesaround0.85. ThisoccursagainbecauseNFAisanearunit root process and small differences in the autocorrelation of NFA (0.996 in GLB v. 0.999 in 26In the endowment model with σ zR = 2.5%, the 2OA (RSS) method produced a mean NFA-GDP ratio of 0.806 (0.942),v.-0.38intheGLBsolution. 27Interestingly,deGroot’sanalysisshowingspuriousmultiplicityofthefullRSSsolutionshowsanadditionalweaknessofthismethod,namelythatitcanproducetwostablesolutionswhereastheexactglobalsolutionisunique. 31

2OAand0.998inRSS)implylargedifferencesintheautocorrelationofnetexports. Moreover,and also in line with the endowment model results, the local solutions overestimate significantly the variabilityofconsumption,netexportsandNFArelativetoGDP. Despitethedifferencesinthemomentsforconsumption,netexportsandNFA,thecyclicalmoments for investment, capital, imported inputs, labor and output are similar across the solutions. For investment and the capital stock, this occurs because, as shown in Mendoza (1991), the Fisherian separation of investment from savings and consumption decisions that holds strictly under perfectforesight,holdsapproximatelyintheRBCmodel. Intuitively,theRBCmodelisinthewide classofmodelsconsistentwithnegligibleequitypremia,andinthelimitwithzeropremiumFisherianseparationholdsexactly. Inaddition,theGHHstructureofpreferencespreventsconsumption andsavingsfromaffectinglaborsupply,andhenceoutputandallfactorsofproduction. Thenear- Fisherianseparationpropertyisverifiedinthenegligiblecoefficientsofthecapitaldecisionruleson laggedNFAinthe2OAandRSSsolutionsandthenear-zeronumericalderivativesofthedecision rulefork(cid:48)(b,k,ε)withrespecttobintheglobalsolutions(thelargestofwhichwas0.0064). Considernextthelocalsolutionswithtargetedcalibrations,forwhichmatchingtheGLBvalue of ρ required ψ = 0.0109 and ψ = 0.008 in the 2OA and RSS solutions, respectively (see Table b 7). These are considerably smaller than the value needed for the targeted calibrations of the endowmenteconomy(0.0469forboth2OAandRSS).Thesedifferences, togetherwiththedifferent NFA-GDPratiosinthedeterministicsteadystatesoftheendowmentandRBCmodels,implyvaluesofηr of0.0083and0.0061forthe2OAandRSSsolutionsoftheRBCmodelrespectively,lower byafactorof3thanthe0.0239fortheendowmentmodelsolutions. Thisisthecasemainlybecause theGLBsolutionofρ ishigherintheRBCthanintheendowmentmodel(0.996v. 0.977). b ThelowerψvaluesunderthetargetedcalibrationsoftheRBCmodelv. theendowmentmodel also imply that the mean of NFA can now rise above the deterministic steady state by non-trivial margins,becausetheimplicitcostofdeviatingfrombdss issmaller. Thisisevenmorethecasefor the targeted RSS solution, which has a lower ψ than the 2OA solution and thus allows E(b/y) to risebymore(−0.397v. −0.62inthe2OAsolutionand−0.758inthedeterministicsteadystate). ThegapbetweenbrssandbdssintheRSSand20Asolutionsnarrowsinthetargetedcalibrations relative to the baseline calibrations: brss is now −0.591, compared with −0.758 for bdss. With this smaller difference, we recover the result that the decision rules for RSS and 2OA yield similar second-andhigher-ordermomentsandsimilarimpulseresponsefunctions,asshownbelow. Asintheendowmentmodel, targetedcalibrationsgenerallyyieldmomentsclosertotheGLB 32

solutionthanbaselinecalibrations. Itisstillthecase,however,thatinordertotargetthecalibration ofthelocalsolutionsweneedtoknowtheGLBsolutionforρ . ThetargetedRSSsolutionperforms b markedlybetterthanthe2OAsolutioninthatityieldsameanNFA-GDPratiomuchclosertothe GLB solution. 2OA yields a ratio of −0.62, nearly 24 percentage points of GDP lower than the true value (−0.38), whereas the RSS solution yields about −0.4, just 1.3 percentage points below the true value. The targeted RBC calibrations, however, do not get as close to the GLB solution moments as in the case of the endowment model, even with the RSS solution: The variability of theNFA-GDPratioisroughlyhalfofwhattheGLBsolutionyieldsanditscorrelationwithGDPis 3.5timesbigger. Theleverageratioisalsomuchlessvariableandhasamuchlowercorrelationwith GDP. Fisherian separation continues to approximately hold, so moments for output, investment, andfactorsofproductionaresimilarinthetargetedcalibrationsandtheGLBsolution. In terms of execution times, the local solutions are still faster than the GLB solution but with a much smaller margin than in the endowment model solutions. The local solutions take about 2/3rdsofthetimetakenupbytheGLBsolution(whichtakes61seconds),insteadof1/4thforthe endowmentmodel. The1OAsolutionhasasimilarexecutiontime,andasexplainedearlieryields similarsecond-andhigher-ordermoments. Theaccuracyoflocalmethodresultsshowsimilarlimitations as in the endowment model: the local solutions yield significantly larger Euler equation errors (for capital and NFA with RSS and for capital with 2OA), the average (maximum) differences in the decision rules for k and c are in the 1.7–1.9 (5.1–6.6) percent range, and those for NFAaremuchlargeratabove8(50)percentfortheaverage(maximum)respectively. Theselarge differences occur at the debt limit ϕ, because the local methods do not handle it as occasionally binding. Weconductedarobustnessanalysisoftheresultsreportedherebyalteringthevaluesofsome ofthemodel’skeyparameters. ThedetailsareprovidedinSectionDoftheAppendix. WeexaminedscenariosincreasingthevariabilityofTFP,inputpriceandinterest-rateshocksoneatatime, as well as increasing the coefficient of relative risk aversion, the correlation between interest rate andTFPshocks,andthesubjectivediscountfactor. Asinthecaseoftheendowmentmodel,local solutionswithafixedvalueofψcalibratedtomatchρ inthebaselineGLBsolutionarenotuseful b foranalyzingtheeffectsofanyoftheseparameterchanges,becausetheyyieldsolutionsthatdiffer sharplyfromtheGLBsolutionsforthesameparametervariations. Inparticular,thelocalsolutions continuetoperformpoorlyatcapturingprecautionarysavingseffects(i.e.,thetruesolutionforthe 33

meanoftheNFA-GDPratiodifferssharplyfromwhatthelocalsolutionsyield).28 Inaddition,the localsolutionsunderestimatethevariabilityofNFAandnetexports,overestimate(underestimate) thecorrelationsofNFAandconsumption(netexports)withGDP,andunderestimatetheautocorrelation of net exports. The local solutions are closer to the GLB solutions if we re-calibrate ψ to target the new value of ρ from the GLB solution for each new parameterization, but this implies b having obtained the GLB solution first and in addition the long-run moments are not as close to thoseoftheGLBsolutionasinthecaseoftheendowmentmodel. b)Impulseresponsefunctions Figures 5 and 6 show impulse response functions to a negative one-standard-deviation TFP shockfortheGLBandlocalsolutions(baselineandtargeted).29 Weplotonlythefirst100periods tohighlightthedifferencesacrossthesolutions. Allimpulseresponsesreturntozeroinabout500 periods. As in the endowment model, impulse responses for 1OA and 2OA are nearly identical under baseline and targeted calibrations. This occurs because again the first-order coefficients of decisionrulesareidentical,andthesecond-orderterms(otherthanthevarianceterms)arequantitativelyirrelevant. Henceweomittheplotsforthe1OAresults. Forthebaselinecalibration,RSSyieldsmarkedlydifferentresponsesforNFA-GDPratio(panel a.),consumption(panelb.) andthenetexports-GDPratio(panelc.) than2OA.Thisisbecause,as notedearlier,thegapbetweenbrssandbdssislargeenoughtoaffecttheresults. However,sincenear- Fisherianseparationstillholds,theothervariables(capital,investment,labor,importedinputsand GDP)displaysimilarresponsesinthetwolocalsolutions. 2OA and RSS baseline impulse responses differ sharply form those of the GLB solution. In particular, RSS overestimates the initial rise in the NFA-GDP ratio while 2OA underestimates it (see panel a.). In fact, RSS yields above-average NFA-GDP ratios for the first 17 periods, while in both GLB and 2OA the NFA position is always below average. After the 15th period, the two localsolutionspredictlargermeandeviationsoftheNFA-GDPratiothanGLB.Thelocalsolutions remain uniformly above the GLB solution until mean reversion is attained. These differences in NFAarereflectedindifferencesinconsumptionandnetexports(seepanelsb. andc.). Initially,the meandeviationofconsumptionundertheGLBsolutionislower(higher)thanintheRSS(2OA) solution. After the 30th period, the GLB solution yields significantly smaller mean deviations of 28Theonlyexceptionwastheexperimentdoublingthevariabilityofinputpriceshocks.Thishasminoreffectsbecause imported inputsare only 10percent of grossoutput, so thattheir share inGDP net ofworking capital is11 percent (0.1/(1−0.1)=0.11).Thus,risingσ (cid:15)p from3.4to6.8percentincreasesnetincomevariabilityjustanotch. 29Wealsocomputedimpulseresponsefunctionsforinterest-rateandinputpriceshocksinAppendixD. 34

consumption than the two local solutions and the opposite is observed for the net exports-GDP ratio. Differencesininvestment,outputandfactorsofproductionarelessnoticeablebecausenear- Fisherianseparationholds,butstillcapitalfallsslightlymoreinitiallyintheGLBthaninthelocal solutions,andthenbetweenperiods15and80theGLBsolutionrisesslightlyabovethelocalones. Sincethereisnowealtheffectonlaborsupply,thesedifferencesincapitalstockdynamicstranslate intoqualitativelysimilarbutquantitativelysmallerdifferencesinlabor,importedinputsandGDP. Under the targeted calibrations (Figure 6), the gap between brss and bdss becomes again too small to make a difference for the 2OA and RSS impulse responses. Hence, our findings for the endowment and RBC models indicate that, if the choice is limited to local methods, a 1OA solution is simpler and nearly identical to 2OA and RSS solutions. Relative to the GLB solution, the targetedlocalsolutionsstillfailtomatchimportantfeaturesoftheGLBimpulseresponses. Initial differencesaresmallerthanwiththebaselinesolutions,andnowtheNFA-GDPratioalwaysshows negativedeviationsfromitsmeaninallthreesolutions. Beyondthe15thto20thperiod,however, NFA-GDP,consumptionandthenetexports-GDPratiointhetargetedsolutionsdiffersharplyfrom the GLB solution, with similar qualitative features as with respect to the baseline solutions, and insomecaseswithevenlargerquantitativedifferences. Thereasonforthisisthat,eventoughthe targetedcalibrationsforcethesameρ acrossGLBandlocalsolutions,therequiredhighervalues b ofψ implythatNFAhasmuchlessvariabilitythanintheGLBsolution(seeTable8). Thehigher volatilitywithsimilarpersistenceintheGLBsolutionyieldanimpulseresponseforNFA-GDPthat rises more initially and then drops more before returning to zero in the long run. In contrast, in thelocalsolutionsthehighψ valuesmakelargedeviationsofNFA-GDPfromitsmeantoocostly, and hence NFA-GDP never falls more than about 2 percentage points below its mean (v. about 7 percentagepointsintheGLBsolution). c)Spectraldensityfunctions Figures 7 and 8 show nonparametric periodograms for the key macroeconomic aggregates of theRBCmodelproducedbytheglobalandlocalmethods. Asinthecaseoftheendowmentmodel, alltheperiodogramsaredownwardsloping,indicatingthatlowerfrequenciescontributemoreto the variability of the simulated data than business cycle and higher frequencies. In contrast with theendowmenteconomy,however,theperiodogramsforNFA,consumptionandnetexportsproducedbythe20AandRSSsolutionsunderthebaselinecalibrationaredifferent,becauseintheRBC baseline calibration the gap between brss and bdss is large enough for 2OA and RSS results to differ. Theotherperiodogramsfor2OAandRSSaresimilarbecauseofthenear-Fisherian-separation 35

propertynotedearlier. RelativetotheGLBsolution,2OAandRSSperiodogramsshowdifferences thatarelessstarkthanfortheendowmentmodel,butRSSstilloverstatesthecontributionofbusiness cycle and higher frequencies to the variability of NFA, and 2OA and RSS still overstate the contribution of very low frequencies to the variability of NFA, consumption and net exports, as wellasthecontributionofbusinesscycleandhigherfrequenciestothevariabilityofGDP. Forthetargetedcalibrations,the2OAandRSSperiodogramsarenearlyidentical,reflectingthe result that in this case the gap between brss and bdss is too small to affect the results. Relative to theGLBsolution,bothRSSand2OAyieldperiodogramsthatapproximatetheirGLBcounterparts betterthanunderthebaselinecalibration,inlinewithwhatwefoundfortheendowmentmodel. ThelocalmethodsunderestimateslightlyoverallNFAandGDPvariability. Theperiodogramsfor investment and factors of production are very similar to those under both the GLB solution and thelocalbaselinecalibrations,againbecauseofthenear-Fisherian-separationproperty. Insummary,theanalysisoftheRBCmodelyieldsseveralkeyresultsinlinewiththoseobtained fortheendowmentmodel: Localmethodsdopoorlyatquantifyingtheeffectsofprecautionarysavings. Localmethodswithbaselinecalibrationsyieldverydifferentresultsthantheglobalsolution forconsumption,netexportsandNFA.Targetedcalibrationsperformbetterbutinordertotarget thevalueofψ itisnecessarytosolvethemodelgloballytofindtheexactsolutionforρ ,andthis b needs to be re-done for any parameter variation. 1OA and 2OA solutions yield nearly identical results (other than first moments), because they have identical first-order terms and the secondordertermsofthe2OAsolution(otherthanthevarianceterm)arequantitativelyirrelevant. TheRBCresultsdifferfromtheendowmentmodelresultsinthat2OAandRSSsolutionswith baselinecalibrationsdiffersignificantly,becausebrssandbdssdifferenoughtoyieldnon-negligible differencesinfirst-ordercoefficientsofthedecisionrules. Inthetargetedcalibrations,however,brss and bdss are close again, and hence 2OA and RSS solutions are very similar. Thus, with targeted calibrations, 1OA, 2OA and RSS solutions differ only in their first moments, while higher-order moments,impulseresponsesandspectraldensityfunctionsarenearlyidentical. Thismakes1OA thepreferablelocalmethodiffirstmomentsarenotbeingstudied. Asecondimportantdifference relative to the endowment model results is that the targeted local solutions are less accurate at approximatingtheGLBsolutionresultsforNFA,consumptionandnetexports. Thisisbecausethe requiredψ valuesmakefluctuationsinNFAcostlyandthisreducesNFAvariabilitytoabouthalf ofthatintheGLBsolution. Resultsforinvestment,outputandfactorallocationsaresimilaracross localandGLBsolutionsbecauseFisherianseparationofsavingsandinvestmentnearlyholds. 36

4 Sudden Stops Model This Section compares local v. global solutions for the SS model proposed by Mendoza (2010), whichaddstotheRBCmodelanoccasionallybindingcreditconstraint. Debtandworkingcapital financingcannotexceedafractionκofthemarketvalueofcapital: qbb −φR (w L +p υ ) ≥ −κq k . (18) t t+1 t t t t t t t+1 4.1 SolutionMethods For the GLB solution, we use again the FiPIt algorithm. In each iteration, the algorithm assumes firstthattheconstraintdoesnotbind,solvesforallocationsandpricesusingthemodel’soptimality conditionsandthenevaluatestheconstraint. Ifitbindstheresultsarediscardedandnewallocations and prices are solved for with the constraint holding with equality. The algorithm iterates over three recursive functions of the state variables: the NFA decision rule, the price of capital, andtheLagrangemultiplierontheborrowingconstraint(seeMendozaandVillalvazo(2019)for details). Forthelocalsolution,AppendixSectionsC.4andC.5describethemodelformulationandthe DynareOBCmethod. Thismethodtreatstheoccasionallybidingconstraintasasourceofendogenous news about the future along perfect-foresight paths. If the constraint is (is not) binding at thedeterministicsteadystate,itusesnewsshockstosolveforunconstrained(constrained)periods along those paths by representing the solution as a mixed integer linear programming problem. For instance, if the constraint does not bind at steady state, when agents anticipate that the constraint will bind at some date t+j conditional on the date-t state variables and the deterministic evolution of the exogenous shocks, this provides “news” that bond holdings will follow a path in which they are higher than otherwise. This approach is akin to assuming that agents live in a world without the constraint, but whenever they are on a path that would lead them to borrow abovewhattheconstraintallows,aseriesofnewsshockshitthatmakesthemborrowonlywhatis allowedandmoderatetheirborrowingbeforethathappens.30 ThemainoutputofDynareOBCisatime-seriessimulationconstructedbystitchingtogetherthe date-t values of perfect-foresight equilibrium paths conditional on (k ,b ) and the date-t realizat t 30Themodelwiththeconstraintisapproximatelyequivalenttothesamemodelwithouttheconstraintbutwithsequencesofnewsshockschosentoyieldthesameequilibriumasthemodelwiththeconstraint. Thisequivalenceholds exactlyifthemodelislinearandshockvariancesarezero,suchthatanyshocksthatoccuraretruly“unexpected.” 37

tionsoftheexogenousshocks. Eachpathisobtainedusinganextendedpathalgorithmthattraces equilibrium dynamics up to T periods ahead of t, with the shocks following their deterministic VAR dynamics. The extended path can be obtained using first- or higher-order approximations, butwereporthereresultsbasedontheformer.31 Thepathcomputedforagivenstartingdatetdeterminesthevaluesoftheendogenousstates(k ,b )andthesetogetherwiththerealizations t+1 t+1 oftheshocksattandtheoptimalityconditionsdeterminethedate-tvaluesofalltheendogenous variables. The rest of the path is discarded and the process is repeated at t + 1 to generate the valuesofthetime-seriessimulationforthatperiod. Theefficiencyofthismethodhingesonthree factors: (a)T needstobelargeenoughsothatfort > T nofurthernewsshocksareneeded(ifthe constraint does not bind at the deterministic steady state, T needs to be large enough so that the constraintneverbindsagain,andifitbindsatsteadystateT needstobelargeenoughsothatitalwaysbinds),(b)foreachpathrequiringnewsshocks,thealgorithmneedstofindthesequenceof newsshocksthatsupportsthecorrectequilibriumpath,and(c)thetime-seriessimulationneeds tobelongenoughforlong-runmomentsoftheendogenousvariablestoconverge. Thealgorithm islessefficientinmodelswithverypersistentdynamics,whichrequirealargeT andalongsimulationlength,andmodelsinwhichthenewsshocksareneededfrequently. Figure 9 illustrates the DynareOBC method in a simple example based on the endowment model of Section 2 with its ad-hoc debt limit and using the DEIR function so that this limit does notbindatthedeterministicsteadystate(seeAppendixC.5.1fordetails). Panels(a)and(b)show thesolutionsofconsumptionandbondholdingsfort=100to250(black,solidcurves)andeleven of the perfect-foresight paths (red, dashed curves) that generated them, with the corresponding date-t solution marked with a red circle. In panels (b) and (d), the green, dashed lines indicate thead-hocdebtlimit. Theconstraintneverbindsinsevenoftheperfect-foresightpathsshownin Panel(b)andinfouritdoes. Hence,endogenousnewsshocksareneededonlyinthelatter. Panels (c) and (d) isolate periods t=140 to 180 and show the extended path that generates the equilibrium values at t=141 (red, dashed curve). DynareOBC computes a sequence of news shocks that sustains this path as an equilibrium. The comparable path of NFA in the solution without credit constraint is also provided in Panel (d) (black, dashed curve). Panel (d) shows 31DynareOBCisequivalenttotheGuerrieri-IacovielloOccBinmethodwhenusingafirst-orderapproximation,with thedifferencethatitisguaranteedtoconvergeinfinitetime.Holden(2016b)showedthatasecond-orderapproximation integratingoverfutureuncertaintycanbeusedtoapproximateprecautionarysavingsinmodelswithsimpleconstraints, butthismethodissignificantlyslowerandfortheSSmodelproducedresultsthatdeviatesharplyfromtheGLBandfirstorderDynareOBCsolutions.Inparticular,investmentandthenetexports-GDPratiohadnegativeserialautocorrelation andNFAhadnear-zeroautocorrelation. 38

that the constraint first becomes binding along the perfect-foresight path at t=144. It also shows that, relative to the model without a credit constraint, agents choose higher bond holdings (less debt) earlier, in anticipation of the constraint becoming binding with perfect foresight (i.e., the red, dashed curve is above the black, dashed curve at t=142,143). Income is rising gradually on its path back to its deterministic steady state, so that the constraint continues to bind for several periods, until income is sufficiently high for NFA to also start rising back towards its steady state (aftert=170). Forcomparingv. theGLBsolution,itiscriticaltonotethatfirst-orderDynareOBCignoresthe risk of hitting the constraint and fluctuating across states where it binds or not. At each date t, it doesnotconsiderthehistoriesoffutureshocksandassociatedequilibriumallocationsandprices thatcanoccur,itonlyconsidersthenews-shock-adjustedperfect-foresightpathcomputedfortand the date-t shock realizations. Hence, wealth and precautionary-saving effects of the occasionally bindingconstraintareignored,andforward-lookingobjectslikeassetpricesandexcessreturnsalso abstractfromthem. ThisisimportantinSSmodels,becauseafinancialcrisiswithadeeprecession and collapsing prices occurs when the constraint binds, and the risk of these events strengthens precautionarysavingsandaltersassetpricesevenin“goodtimes”(seeMendoza(2010),Durduet al. (2009)). Incontrast,ineachDynareOBCperfect-foresightpath,iftheconstraintbinds(doesnot bind)atthedeterministicsteadystate,agentsanticipatereachingalong-runequilibriuminwhich the constraint binds (does not bind) regardless of whether it binds or not at t. They anticipate deterministicallyhitting(escaping)theconstraintatsomedatet+jifunconstrained(constrained) attandadjusttheiroptimalplansbeforet+jaccordingly,buttheseplansdonotfactorintheriskof theconstraintbecomingbindingornon-binding,andhowthisriskaffectsincentivestoself-insure, assetpricesandtheequitypremium. 4.2 Calibration ThecalibrationisverysimilartothatoftheRBCmodel(seeTable7). Thevalueofκissetto0.2,as inMendoza(2010). FortheGLBsolution, weincreasethelowerboundofthebondsgridto-200 (i.e.,atighterad-hocdebtlimit),whichisaboutabouthalfofsteadystateGDP.Wedothisbecause strongprecautionarysavingsduetothecreditconstraintimplythatNFA-GDPratiosbelow−0.192 areneverobservedintheergodicdistributionoftheSSmodel,andhencewecansetϕtoahigher valuesoastousefewernodesinthebondsgridtomakethealgorithmmoreefficient. BeforediscussingtheDynareOBCcalibration,itisimportanttonotethatiftheconstraintbinds 39

atthedeterministicsteadystate,thesteady-stateequilibriumiswelldefinedandthereisnoneed toinducestationarity. Thisfollowsfromthesteady-statebondsEulerequation: µ ss 1 = βR+ , (19) u(cid:48)(c ) ss whereµistheLagrangemultiplierontheborrowingconstraint. NotethatβR < 1 ⇐⇒ µ > 0, ss hencehavingtheconstraintbindatsteadystaterequiresβR < 1andviceversa. ThisEulerequation issolvedtogetherwiththeothersteady-stateequilibriumconditions(particularlythesteady-state resourceandcreditconstraints)tosolveforthesteady-stateequilibrium,includingc andµ . ss ss WestudyDynareOBCsolutionswithbothµ > 0andµ = 0. Thefirstislabeled“DynareOBCss ss βR < 1”andthesecond“DynareOBC-DEIR,”becauseinthelattercasetheDEIRfunctioninduces stationarity. FortheDynareOBC-βR < 1case,β isthesameasintheGLBsolution,whichensures thatthedeterministicsteadystateisthesame. Hence,inthiscaseDynareOBCandGLBuseidenticalcalibrations. FortheDynareOBC-DEIRcase,thetargetNFApositionintheDEIRfunctionisset sothatthedeterministicsteadystateoftheNFA-GDPratiomatchesthelong-runaverageobtained in the GLB solution, with β = 1/R (which is required for µ = 0) and ψ = 0.001 (which is the ss inessential value). The rationale for looking at this scenario is that in the GLB solution the constraintbindsrarelyandmeanNFAismuchhigherthanatthedeterministicsteadystate. Hence,a localapproximationaroundanunconstrainedsteadystatewouldbemoreinlinewiththeunconstrainednatureoftheGLBlong-runequilibriumsolution. 4.3 Comparisonofquantitativeresults a)Long-runmoments,impulseresponsefunctions&performancemetrics Table 9 shows that several moments of the DynareOBC solutions differ from their GLB counterparts, with smaller differences for supply-side variables as was the case for the RBC model. Long-runaveragesshowthelargestdifferences,particularlyforc,nx/y,b/yandtheleverageratio. ThelargedifferencesinE[b/y]indicatethatthelocalsolutionscontinuetoperformpoorlyinthis dimension. E[b/y] =1.5percentintheGLBsolution,relativeto−37.2percentintheRBCmodel,indicatingthatthecreditconstraintstrengthensprecautionarysavingssharply. DynareOBC-βR < 1 (DynareOBC-DEIR) underestimates (overestimates) this result significantly, yielding E[b/y] = −10(20.6)percent. Differencesthislargehaveimportantimplicationsforkeyresearchquestions. Forexample,quantifyingtheoptimalforeignreservestomanagetheriskofSuddenStopsrequires determiningfirsthowtheeconomyrespondstothisriskwithoutpolicyintervention(e.g.,Durdu 40

etal. (2009)). DynareOBCwouldyieldfindingsthatdeviatesharplyfromthecorrectresult. Several of the moments that are underestimated in DynareOBC-βR < 1 v. the GLB solution tend to be overestimated in the DynareOBC-DEIR solution. Relative to the GLB solution, DynareOBC-βR < 1 (DynareOBC-DEIR) yields markedly lower (higher) variability in NFA, net exports, consumption, and leverage, lower (higher) correlations of GDP with investment, net exports, and intermediate goods, and lower (higher) persistence in consumption and net exports. Interestingly, DynareOBC-DEIR, which was calibrated so that bdss/ydss equals E[b/y] in the GLB solutionalsoyieldsavalueofρ(b)closetotheGLBsolution(0.995v. 0.99). Hence,itisinlinewith thenotionofthe“targetedcalibrations”ofthepreviousSectionseventoughwekeptψ = 0.001. Local and global solutions also differ sharply in that the constraint binds much more often in theformerthaninthelatter (51.8and71.1percentinthetwolocalsolutionsv. 2.6percentinthe GLBsolution). Thisisdueinparttothefactthatagentsdonotrespondtotheriskoftheconstraint becomingbindinginthefirst-orderDynareOBCsolution. However,recallthatagentsdoanticipate theconstraintbecomingbindingintheperfect-foresightpathsforwhichthishappensandborrow less before hitting the constraint. Hence, although the constraint binds nearly half the time with DynareOBC, it does so with very small multipliers. Later in this Section we study in more detail theextenttowhichthelocalandglobalsolutionsdifferwhentheconstraintbinds. The long-run averages of NFA in the DynareOBC solutions are higher than the corresponding deterministic steady states, indicating that certainty equivalence does not hold, even though the perfect-foresight paths are first-order approximations: bdss/ydss is set at −0.192 (0.015) in DynareOBC-βR < 1(DynareOBC-DEIR)whileE[b/y] =−0.1(0.206). Thisisnot,however,dueto precautionarysavings,sinceDynareOBCdoesnottakeintoaccounttheriskoffutureshocksand the constraint becoming binding. Instead, this occurs because the constraint induces asymmetric (nonlinear) responses to shocks even without risk. This feature of DynareOBC can be illustrated usingagaintheendowmentmodelexampleofFigure9(seeAppendixSectionC.5.1fordetails). A negativeendowmentshockthatcausestheconstrainttobindalongtheperfect-foresightpathdeterminingthedate-tvalueofthesolutionreducesbondholdingsbylessthantheincreaseinbond holdings in response to a positive shock of the same size in absolute value. As a result, upward movements in b when positive shocks hit are larger than downward movements when negative shockshitiftheeconomyisnearoratapointwheretheconstraintbinds. Moreover,bcannotmove belowthelowerboundsetbytheconstraintbutitcanwanderofftohighvaluesaftersequencesof positiveshocks(seeagainFigure9). Hence,theoutcomeisaDynareOBCtime-serieswithobser- 41

vations“biased”abovethedeterministicsteadystate,whichimpliesameanabovebdss/ydss.32 TheGLBsolutionfeaturesasimilarasymmetrybutinadditionittakesintoaccountprecuationarysavingseffectsduetotheriskoffutureshocksandtheconstraintbecomingbinding. Itdoesnot follow,however,thattheDynareOBCsolutionsmustalwaysyieldmeanbondpositionslowerthan the GLB solution. DynareOBC-βR < 1 (DynareOBC-DEIR) yields a significantly lower (higher) mean of NFA. Both of these results abstract from precautionary savings, but in the DynareOBC- DEIR solution bdss/ydss is set equal to the value of E[b/y] in the GLB solution (0.015) and the constraint does not bind at steady state. Hence, the DynareOBC solution is “biased” above 0.015 andthereforeitmustyieldalong-runaveragehigherthanthatvalue. IfonehadtochoosebetweenDynare-βR < 1andDynare-DEIR,theformerispreferable. Both yield long-run moments that differ from the GLB solution, as shown in Table 9, but as we show laterinthisSection,Dynare-βR < 1doesbetteratapproximatingtheeffectsofthecollateralconstraint. Moreover,Dynare-βR < 1usestheexactsamecalibrationastheGLBsolutionanddoesnot requireastationarity-inducingtransformation. Incontrast, Dynare-DEIRrequiressettingavalue for bdss/ydss. We used the average of the GLB solution, but this resulted in a much higher mean fortheNFA-GDPratioandrequirespriorknowledgeofthemeanintheGLBsolution. We compare next performance metrics.33 The main result is that the speed advantage of the local methods shrinks considerably, particularly for DynareOBC-βR < 1. Speed ratios relative to the GLB solution rise to 0.901 for DynareOBC-βR < 1 and 0.70 for DynareOBC-DEIR (the GLB solution takes 268 seconds). Relative to the local RBC solutions using standard 1OA and 20A algorithms, the DynareOBC execution times are considerable higher at 244 and 188 seconds for DynareOBC-βR < 1andDynareOBC-DEIR,respectively. Thisisduetothethreedeterminantsof theefficiencyofDynareOBCnotedearliertogetherwiththefactthatNFAfollowsanear-unit-root process. Each extended path required at least 60 periods and the time-series simulations needed 100,000periodstoconvergetoinvariantmoments,particularlythoseforb/y,nx/yandthestandard deviation of consumption.34 DynareOBC-βR < 1 is slower than DynareOBC-DEIR because it requiredmoresearchestoconstructthenewsshockssequencesthatimplementtheconstraint. The speed comparisons of DynareOBC v. FiPIt need to be pondered carefully. On one hand, FiPIt suffers from the standard curse of dimensionality of GLB methods related to the number 32RecallthattheconstraintinthisexampleisafixeddebtlimitwhileintheSSmodelitdependsonq t k t+1. 33SeefootnotetoTable5forthehardwareandsoftwareusedtorecordtheperformancemetrics. 34Intuitively, consider that the estimators of the mean and autocorrelation of an AR(1) process are consistent but biasedinfinitesamples. Thebiasishigherthehigherthetrueautocorrelationbutitfallsasthesamplesizerises. Fora near-unit-rootprocess,thesampleneedstobequitelargetomaketheestimationbiasnegligible. 42

of state variables, and more so if the model specification requires using a root-finder when the constraint binds.35 But once the decision rules are solved for, generating stochastic time-series simulationsisveryfast. Ontheotherhand,thenumberofstatevariablesismuchlessofanissue for DynareOBC, but execution time rises with the required length of perfect-foresight paths, the length of the time-series simulation needed for convergence of long-run moments, and the iterationsrequiredtocomputethenews-shockssequencesthatimplementtheconstraint. Asshownin Table4ofAppendixD,DynareOBC-βR < 1ismuchslowerthanFiPItifthesimulationlengthrises to150,000periods(350secondsv. 268seconds)orwithfewerexogenousshockssothatthecurse of dimensionality is less severe (with TFP shocks only, FiPIt solves in 42 seconds v. 230 seconds with DynareOBC). Relaxing the credit constraint by setting κ to 0.3 also results in FiPIt solving muchfasterthanDynareOBC(137v. 228seconds).36 Moreover,solvinginsecond-orhigherorder and/or adding the option to integrate over future uncertainty also slows down DynareOBC considerably. Holden(2016b)notedthatsolvingamodelsimilartotheendowmentmodelofSection 2 using first-order DynareOBCintegrating over 45 periods of future uncertaintywith the highest accuracyrequired2,855seconds(withdifferenthardwarethanweused). Intermsofaccuracy,FiPItproducesagainveryaccurateGLBresults,asindicatedbythesmall maximumerrorsofthebondsandcapitalEulerequations. TheaccuracyoftheDynareOBCsolutionscannotbeassessedthiswaybecausethesolutionmethodproducesatime-seriessimulation, instead of decision rules. Hence, we follow Holden (2016b) and evaluate their accuracy by constructingconsumptiontime-seriessimulationsoftheGLBsolutionforthesameinitialconditions andsequenceofshocksasineachofthetwoDynareOBCsolutions,andcomputingthemaximum absolutevaluesofthedifferencesacrossthem. Themaximumdifferencesinlogbase10are1.292 percent with DynareOBC-βR < 1, and 1.342 percent with DynareOBC-DEIR. These are significantlylargerthanHolden’sestimateswithfortheendowmentmodel. Figure 10 shows impulse responses to a one-standard deviation negative TFP shock. The impulseresponsesfortheGLBsolutionareconditionalonstartingatthelong-runaveragesofcapital andNFA,andthoseforDynareOBCsolutionsonstartingatthedeterministicsteadystates(which for DynareOBC-DEIR are the same as the GLB averages). The GLB impulse responses are very 35AsMendozaandVillalvazo(2019)explain,thisisnotneededforseveralspecificationsofcreditconstraints.Thisis thecaseintheSSmodelwithoutworkingcapitalintheconstraint,whichreducestheFiPItruntimeby57percent. 36UsingDynareOBCalsoposeslogisticalhurdles. SinceitreliesonDynare,updatestoDynarecanmakeolderversionsofDynareOBCinoperable,andsomeversionsofDynareoperateonlyincertainoperatingsystemsandsoftware environments. Forinstance,theDynareOBCtoolboxweusedoperateswithDynare4.4.3andonlywithMatlab2016a. Dynare4.4.3operateswiththeUbuntu14.04LinuxoperatingsystembutnotwithUbuntu18.04. 43

similartothoseoftheRBCmodel,becausetheconstraintbindsveryinfrequentlyinthelefttailof the ergodic distribution. Hence, in the “correct” SS model solution, the responses of macro variablestriggeredbyshocksofstandardmagnitudesstartingofffromlong-runmeansarenearlyunaffectedbythecreditfriction. Themeansofvariablesotherthanthesupply-sidevariableschange acrosstheRBCandSSGLBsolutions,butthedeviationsfrommeanstriggeredbyidenticalshocks arenearlyunchanged. Incontrast,theDynareOBCimpulseresponsesareverydifferentfromthe local-methodsimpulseresponsesoftheRBCmodel,eventhoseforthe1OAsolution. DynareOBCimpulseresponsesofseveralvariablesdiffersharplyfromtheirGLBcounterparts. For DynareOBC-βR < 1, Panel (a) shows that NFA/GDP hardly moves, and Panel (b) shows thatnx/y movesintoasurplusonimpact,becauseofreduceddemandforimportedinputs. This occursbecausetheconstraintbindsatdate0andtheTFPshocktightenstheconstraintmore. For DynareOBC-DEIR, NFA/GDP declines, offsetting the fall in imported inputs to yield an almost unchangedtradebalance. Incontrast,intheGLBsolutionnetexportsjumponimpactnearlytwice as much as under DynareOBC-βR < 1 and NFA/GDP rises gradually to peak roughly 150 basis pointsaboveitsmean,andafterthatitfallsslowlytoatrough400basispointsbelowitsmeanbefore graduallyrevertingtoitsmean. Theresponsesofcapitalarealsoverydifferent. BothDynareOBC solutions yield a decline on impact, while in the GLB solution capital is nearly unchanged. Then capitaldeclinesslightlyandstartsrecoveringinthetwolocalsolutions,whileintheGLBsolution itfallsbynearlythreetimesasmuchreachingnearly1.5percentbelowaveragebeforestartingto recover. Qualitatively,theresponsesofconsumption,investment,labor,inputsandGDParesimilar inallthreesolutions,butthedeclinesonimpactaresignificantlylargerintheGLBsolution. b)Spectraldensityfunctions Figure11showsthenonparametricperiodogramsfortheDynareOBCandGLBsolutions. As in the endowment and RBC models, since all of the variables follow AR(1)-like processes, the periodogramsaregenerallydownwardsloping,indicatingthatlowfrequenciesaccountforalarger fractionofthevarianceofthevariablesthanbusinesscycleandhigherfrequencies. Moreover,the periodogramsfortheGLBsolutionareverysimilartothosefortheRBCGLBsolution,inlinewith thefindingthattheGLBimpulseresponsesoftheRBCandSSmodelsaresimilarbecausethecredit constraint binds infrequently. The periodograms of the DynareOBC solution differ from those of the 2OA and RSS solutions of the RBC model, so the local methods fail to match the property of theGLBsolutionsthatspectraldensitiesoftheRBCandSSmodelsaresimilar. TheDynareOBCperiodogramsforNFA,consumption,netexports,investmentandlabordiffer 44

sharplyfromtheGLBresults. ConsumptionhasthehighestvarianceintheGLBsolution(121.4), followedbyDynareOBC-DEIR(119.2)andDynareOBC-βR < 1withamuchlowervariance(65.8). Incontrast,theautocorrelationofconsumptionishighestinDynareOBC-DEIR(0.91)andaboutthe sameinGLBandDynareOBC-βR < 1(0.83). Asaresult,theconsumptionperiodogramsforthe lattertwohavethesameinterceptbuttheoneforGLBisuniformlyhigherotherwise,whiletheperiodogramforDynareOBC-DEIRhasthehighestinterceptbutisgenerallybelowtheperiodogram for GLB. The DynareOBC solutions assign significantly less consumption variability to business cycle and lower frequencies than the GLB solution. Net exports also show higher persistence in theDynareOBC-DEIRsolutionwhileDynareOBC-βR < 1andGLBhavesimilarpersistence,and opposite from what we observe for consumption, the GLB solution has less overall variance and lessvariabilityatallfrequencies. InvestmenthashighervarianceandpersistenceintheGLBthan inthelocalsolutions,andithasuniformlyhighervariabilityatallfrequencies. c)SuddenStopsandRiskEffects We compare next the results that DynareOBC and GLB yield for the effects of the credit constraint, particularly for financial premia and for sudden stop responses of macro variables when the constraint binds. The financial premia include the shadow interest rate premium (SIP), the equitypremium(EP),itscomponentsduetounpledgeablecapital((1−κ)SIP)andriskpremium (RP),andtheSharperatio(S). Asweexplainbelow,thesepremiadependonboththetightness oftheconstraintanditsriskeffects. Formacroaggregates,wecomparedeviationsfromlong-run averagesinconsumption,thenetexports-GDPratio,investment,GDP,laborandimportedinputs. SIP is the amount by which the subjective rate at which agents are willing to trade current forfutureconsumption(i.e.,theintertemporalmarginalrateofsubstitutionu(cid:48)(t)/[βE (u(cid:48)(t+1))]) t exceeds R . The bonds’ Euler equation implies that SIP = µ (1+τ)/[βE (u(cid:48)(t+1))]. Using the t t t t sameEulerequation,thedenominatorcanberewrittenwithouttheconditionalexpectation: R µ (1+τ) t t SIP = . (20) t u(cid:48)(t)−µ (1+τ) t SinceSIP = 0ifµ = 0,thisshadowinterestpremiumisrelevantonlywhentheconstraintbinds. t t We can also infer that SIP rises as the constraint becomes more binding, because µ rises and t t E (u(cid:48)(t+1))falls,sincetheconstraintforcesagentstodeferconsumption. t The equity premium is defined as EP ≡ E [Rq ]−R , where Rq ≡ (d +q )/q is the t t t+1 t t+1 t+1 t+1 t return on equity and d is the dividend payment, where d ≡ exp((cid:15)A)F (t) − δ + a(kt+1−kt)2 . t+1 t t k 2 k2 t 45

UsingtheEulerequationsforbondsandcapitalitfollowsthat: COV [u(cid:48)(t+1),Rq ] EP = (1−κ)SIP +RP , RP ≡ − t t+1 . (21) t t t t E [u(cid:48)(t+1)] t EP hastwocomponents: thestandardriskpremium(RP )drivenbytheconditionalcovariance t t ofmarginalutilityandequityreturnsandthefractionofSIP pertainingtotheshareofk that t t+1 cannotbepledgedascollateral((1−κ)SIP ). EP riseswhenµ > 0fortworeasons: First,SIP t t t t rises,asexplainedabove. Second,RP rises,becauseCOV [u(cid:48)(t+1),Rq ]becomesmorenegative t t t+1 asconsumptionishardertosmoothandE [u(cid:48)(t+1)]fallsasthecreditconstraintforcesconsumpt tion into the future. Thus, EP responds both to the tightness of the constraint itself via SIP t and to the larger risk premium that the constraint induces. The Sharpe ratio measures the compensation for risk-taking, defined as the excess returns obtained per unit of variability in returns (S = E[EP]/σ(Rq)). Followingstandardpractice,wecomputeS usinguconditionalmoments. t t For the GLB solution, the financial premia are computed for each triple (b,k,ε) in the state space by applying the above formulae using the Markov process of ε and the recursive optimal decisionrules(seeAppendixSectionD.4fordetails). Conditionalandunconditionalaveragesare thencomputedusingconditionalandunconditionaldistributionsof(b,k,ε)oftheGLBsolution. FortheDynareOBCsolution,SIP iscomputedusingthetime-seriessimulationproducedby t Holden’salgorithm. TheequitypremiumisthengeneratedasEP = (1−κ)SIP becauseRP = 0 t t t by construction, since each date-t solution is determined by a perfect-foresight path along which agentsbasedecisionsonexpectationssuchthatCOV [u(cid:48)(t+1),Rq ] = 0. Computingtheuncont t+1 ditionalcovarianceforthesimulateddatasetalsoproducesverysmallvaluesforCOV[u(cid:48)(·),Rq]. Table10reportsquintiledistributionsofµconditionalonµ > 0,theassociatedwithin-quintile averages of the financial and macro variables, their overall means and medians, and the Sharpe ratios.37 TheTableshowskeydifferencesacrossGLBandlocalsolutions. Themagnitudesofµare verysmallinallfivequintilesofbothsolutions,butthisisbecauseµisinunitsofmarginalutility withCRRApreferencesandσ = 2. Forinstance,evaluatedattheunconditionalmeansofcandL, marginalutilityisabout2.05E-05(-4.688inlogbase10). Hence,smallµvaluesdonotimplythat theconstraintisnegligiblybindingorthatmodelsolutionsaresimilar,aswedocumentbelow. Table10showsthatthelocalsolutionsyieldsignificantlysmallerresultsthantheGLBsolution for the credit constraint multipliers, the financial premia and the macro responses when the con- 37Variablesareassignedintoquintilesaccordingtothequintiledistributionofµ: ifagivenµ ibelongstoaparticular quintileofµ,thenthecorrespondingvaluesofthefinancialandmacrovariablesareassignedtothatsamequintile. 46

straintbinds. Thedifferencesgrowlargerforhigherµ(i.e.,inthefourthandfifthquintiles),and they are larger compared with the local solution that has the unconstrained deterministic steady state(DynareOBC-DEIR)v. theonewiththeconstrainedsteadystate(DynareOBC-βR < 1). For financial premia, GLB yields overall means of 2.6, 2.2, 2.1 and 0.1 percent for SIP, EP, (1−κ)SIP andRP,respectively,whileDynareOBC-βR < 1(DynareOBC-DEIR)yields0.8,0.6,0.6 and0(0.13,0.10,0.10and0)percent,respectively. IntheGLBsolution,RP isabout0.1percenton averageineachofthefivequintilesofµ,butEP stillincreasessharplywithµbecause(1−κ)SIP rises sharply. In the fifth quintile, GLB yields averages of 6.6, 5.4, and 5.3 percent for SIP, EP, and (1−κ)SIP, respectively, while DynareOBC-βR < 1 (DynareOBC-DEIR) yields 3.3, 2.7 and 2.7 (0.64, 0.51 and 0.51) percent, respectively. The local solutions underestimate SIP, and hence EP, significantly. They also miss the risk premium, but this accounts for a small fraction of the differencesinEP. GLByieldsaSharperatioof1.16,nearly5and30timestheDynareOBC-βR < 1 andDynareOBC-DEIRresults,respectively. SinceRP issmallintheGLBsolutionandzerointhe localsolutions,thesedifferencesinS aremostlyexplainedbythelargegapinSIP. LargedifferencesinSIP andEP areimportantbecausethemodelpredictsthattheycausevery differentresponsesofmacrovariableswhentheconstraintbinds. Toexplainhowfinancialpremia affect macro responses, wefollow Mendoza and Smith(2005) in using the definitionof expected returnsE[Rq ] ≡ E [(d +q )/q ]toexpresstheforwardsolutionforthepriceofcapitalas: t+1 t t+1 t+1 t     ∞ i (cid:88) (cid:89) 1 q t = E t  E (Rq ) d t+1+i (22) i=1 j=0 t t+1+j Sinceeq. (21)impliesthatE [Rq ] = (1−κ)SIP +RP +R ,lowerSIP inDynareOBCimplies t t+1 t t t t higher equity prices when µ > 0, which in turn imply weaker Fisherian deflation effects of the t bindingcreditconstraint. Moreover,sinceequitypricesandinvestmentaremonotonicfunctionsof eachotherduetotheTobinQnatureoftheinvestmentsetup,k shouldbehigher,andsoshould t+1 beborrowingcapacity(κq k ),whichiskeyfordeterminingallocationswhenµ > 0. Thisalso t t+1 t affectsfuturedividends,causingfurtherfeedbackeffectsintoequitypricesandborrowingcapacity. The differences in macro responses across the GLB and local solutions when µ > 0 reported in Table 10 reflect the above arguments. In the GLB solution, the responses are in line with the standard features of Sudden Stops (i.e., large recessions and sharp reversals in the external accounts). The mean percent declines (relative to long-run averages) are -3.6 in c, -4.1 in i, -1.0 in GDP,-0.7inL,and-1.8inv whileNX/GDP rises2.6percentagepointsaboveitslong-runaver- 47

age. Theresponsesaregenerallylargerwhentheconstraintbindsmore,reachingmeansof-4.9for cand-13.5foriwithatradebalancereversalof5.1percentagepointsforthetopquintileofµ. The DynareOBC-βR < 1solutionunderestimatesthemeanresponsesofconsumptionandnetexports (-1.9 v. -3.6 for c and 1.2 v. 2.6 for NX/GDP) and overestimates those for L, v and GDP. It also fails to match the property that the responses should be larger when the constraint binds more, asityieldsthelargestresponsesinthethirdquintileofµ. DynareOBC-DEIRperformsevenmore poorly, producing positive mean responses for c and i and a negative mean response for NX/Y (i.e., on average, when the constraint binds consumption and investment rise above their longrun means and the trade balance falls below its long-run mean). Moreover, these counterfactual responsesgrowlargerwhentheconstraintbindsmore,inthefourthandfifthquintilesofµ. DynareOBCusingβR < 1v. DEIRyieldverydifferentresultsbecauseintheformer(latter)the constraintbinds(doesnotbind)atthedeterministicsteadystate,andusingDEIRthesteadystate wassettothemeanNFAoftheGLBsolution. TheconstraintbindsofteninDynareOBC-DEIR(71 percentfrequency)butinallbutthefirstquantileofµityieldsbelowaveragenetexportsandabove average consumption and investment. This failure to produce Sudden Stops when the constraint bindsisamajorshortcomingofthissolutionvis-a-visbothGLBandDynare-βR < 1solutions. Insummary,thisSection’sresultsareconsistentwithourpreviousfindingsindicatingthatlocal solutions fail to capture important features of the global solution in terms of long-run moments, impulse-responsefunctionsandspectraldensityfunctions. Inaddition,localsolutionsfailtomatch keypropertiesoftheGLBsolutionwhentheconstraintbinds,becausefirst-orderDynareOBCabstractsfromriskeffectsanddoesnotapproximateaccuratelythefrequency,magnitudeandeffects of the occasionally binding credit constraint. DynareOBC yields large differences in the amount of precautionary savings caused by the constraint, the probability of hitting the constraint, the tightnessoftheconstraintwhenitbindsandtheassociatedfinancialpremiaandmacroresponses. 5 Conclusions Wecomparedglobalandlocalsolutionsofopen-economymodelswithincompletemarketsinthe timeandfrequencydomainsandfoundmajordifferences. Weexaminedanendowmenteconomy, an RBC model and a Sudden Stops model with an occasionally binding credit constraint. Local solutionswereproducedusing1OA,2OA,RSSandDynareOBCmethodsandtheglobalsolutions weregeneratedusingtheFiPItmethod. Mostlocalmethodsneedastationarity-inducingtransformation,forwhichwechosethewidely-usedDEIRfunctionthatmakestheworldrealinterestratea 48

decreasingfunctionoftheNFAposition. Weconsideredthestandard“inessential”approachtoset theDEIRelasticitysothattheinterestrateremainsclosetotheworldinterestrateandavariation inwhichtheelasticityiscalibratedtomatchtheautocorrelationofNFAintheglobalsolution. Themainlimitationofthelocalmethodsistheirinabilitytoapproximateaccuratelytheeffects ofprecautionarysavingsonNFA,netexportsandconsumption,evenusinghigher-ordermethods such as 2OA and RSS. For the Sudden Stops model, first-order DynareOBC has two additional disadvantages: it underestimates the tightness of the credit constraint and its effects on financial premia and macro responses to a binding constraint, and it does not capture risk effects of the credit constraint and their implications for precautionary savings and the determination of forward-looking variables like asset prices. In terms of speed, the local methods do not have a clearadvantage. TheyarefasterforsolvingtheRBCmodel,buttheFiPItalgorithmcomputesthe globalsolutionoftheendowmentmodelfasterandfortheSuddenStopsmodelFiPItandDynare- OBCareofcomparablespeed. TheglobalsolutionsalsoyieldsignificantlysmallerEulerequation errorsinallcases. ButthecurseofdimensionalityremainsalimitationoftheFiPItglobalmethod. NFAisanear-unit-rootprocessinallthreemodels. Hence,smallerrorsincalculatingits“true” autocorrelation coefficient cause local methods to produce non-trivial errors in other key items, including long-run averages of NFA, consumption and net exports, as well as various features of business cycle moments, impulse responses and spectral densities. Local solutions with the DEIR elasticity targeted to match the NFA autocorrelation of the global solution perform better, butimplyelasticitiesakintoimposinglargecostsinmovingNFAfromitssteadystateandrequire knowingtheglobalsolution. Interestingly,1OA,2OA,andRSSlocalmethodsproduceverysimilar second- and higher-order moments, impulse responses and periodograms. This is because they yield decision rules that differ mainly in their intercepts, but otherwise have similar first-order terms and negligible higher-order terms. Hence, if one is restricted to local methods only, and if firstmomentsarenotcentraltothequestionunderstudy,the1OAmethodisthebestalternative. Overall, these findings argue in favor of using global methods unless the curse of dimensionalitymakesthemunfeasible. Theresultsarerobusttoseveralparametermodifications,including settingtheDEIRelasticitytoaninessentiallowvaluev. targetingtheglobalsolution,replacingthe DEIRfunctionwiththeassumptionthattherateofinterestislowerthantherateoftimepreference, introducingdifferentshocks,andchangingthevariabilityandpersistenceofshocks. 49

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Table1: MethodsUsedtoSolveOpen-EconomyIncompleteMarketsModels Authors Year Publication Typeof Solution Stationarity ψ model method assumption Adolfsonetal. 2007 JIE SOE 1OA DEIR .145(e) AguiarandGopinath 2007 JPE SOE 1OA DEIR .001(s) AngeloniandEhrmann 2007 BEJMacro N =12 1OA DEIR .1(c) Arellano 2008 AER SOE GLB ArellanoandMendoza 2002 NBER SOE GLB BaxterandCrucini 1995 IER N =2 1OA AHC Benguietal. 2012 JME N =2 GLB BenignoandThoenissen 2008 JIMF N =2 1OA AHC Benignoetal. 2016 JME SOE GLB Bergin 2006 JIMF N =2 1OA DEIR .00384(e) Bianchi 2011 AER SOE GLB BianchiandMendoza 2018 JPE SOE GLB Bianchietal. 2012 IMFER SOE GLB Bianchietal. 2016 JIE SOE GLB Bodenstein 2011 JIE N =2 1OA ED Bodensteinetal. 2011 JIE N =2 1OA DEIR 0.0001(s) Bozetal. 2011 JME SOE 1OA DEIR .001(s) BozandMendoza 2014 JME SOE GLB Buchetal. 2005 JIMF N =2 1OA AHC CavalloandGhironi 2002 JME N =2 1OA OLG Coeurdacieretal. 2011 AERP&P SOE RSS Correiaetal. 1995 EER SOE 1OA AHC Corsettietal. 2008 RESTUD N =2 1OA ED CuadraandSapriza 2008 JIE SOE GLB Devereuxetal. 2006 EJ SOE 1OA AHC DevereuxandSutherland 2010 JMCB N =2 1OA ED DevereuxandSutherland 2011 JEEA N =2 1OA ED Durduetal. 2009 JDE SOE GLB DurduandMendoza 2006 JIE SOE GLB Endersetal. 2011 JIE N =2 1OA ED EngelandWang 2011 JIE N =2 1OA AHC Continuedonnextpage 53

Table1–continuedfrompreviouspage Authors Year Journal Typeof Solution Stationarity ψ model method assumption FernandezandChang 2013 IER SOE 1OA DEIR .001(s) Fernandez-Villaverdeetal. 2011 AER SOE 3OA AHC FogliandPerri 2006 NBER N =2 GLB Garcia-Ciccoetal. 2010 AER SOE 1OA DEIR .001(s),2.8(e) Gertleretal. 2007 JMCB SOE 1OA DEIR 0.0001(s) Ghironi 2006 JIE N =2 1OA OLG GhironiandMelitz 2005 QJE N =2 1OA AHC HatchondoandMartinez 2009 JIE SOE GLB HeathcoteandPerri 2002 JME N =2 1OA AHC HeathcoteandPerri 2013 JPE N =2 2OA,3OA JaimovichandRebelo 2008 JMCB SOE 1OA DEIR .00001(s) JustinianoandPreston 2010 JIE SOE 1OA DEIR .01(c) LubikandSchorfheide 2005 NBERMacro N =2 1OA CM Mendoza 1991 AER SOE GLB Mendoza 1992 IMFSP SOE GLB Mendoza 1995 IER SOE GLB Mendoza 2010 AER SOE GLB MendozaandSmith 2006 JIE SOE GLB Mendozaetal. 2009 JPE N =2,3 GLB MendozaandYue 2012 QJE SOE GLB Monacelli 2005 JMCB SOE 1OA CM NasonandRogers 2006 JIE SOE 1OA DEIR .00014,.007(e) NeumeyerandPerri 2005 JME SOE 1OA AHC RabanalandTuesta 2010 JEDC N =2 1OA AHC Raffo 2008 JIE N =2 1OA AHC RebeloandVegh 1995 RESTUD SOE 1OA AHC SmetsandWouters 2002 JME SOE 1OA OLG UribeandYue 2006 JIE SOE 1OA AHC Note:SOEdenotesasmallopeneconomymodel.N =denotesamulticountrymodelwithN countries.1OA,20Aand 3OAarethefirst-,second-andthird-orderapproximationmethodsrespectively,RSSistheriskysteadystatemethod, and GLB indicates models solved with global methods (including models with standard preferences and βR < 1, endogenousdiscounting, oroverlappinggenerations). Theapproachesusedtoinducestationaritywhenusinglocal methods are the debt-elastic interest rate (DEIR), asset holding costs (AHC), endogenous discounting (ED), overlappinggenerations(OLG)andcompletemarkets(CM).ForcasesusingDEIR,(s), (c)and(e)denotewhetherthe debt-elasticityparameterψwaschosentobesmall,estimated,orcalibratedrespectively. 54

Table2: SolutionMethodsUsedinPolicyModels Institution Model Typeof Solution Stationarity ψ name model method assumption BankofCanada GEM N = 5 1OA DEIR n.d. BankofEngland COMPASS SOE 1OA ED ECB NAWM N = 2andSOE 1OA DEIR .01(s) EuropeanCommission QUEST SOE 1OA DEIR .02(e) FederalReserveBoard SIGMA N = 2 1OA PAC IMF GIMF N ≥ 2 1OA OLG NorgesBank NEMO SOE 1OA DEIR n.d. Riksbank RAMSES SOE 1OA DEIR .01(c) Note: SeenotetoTable1fordetailsonabbreviations. n.d. denotesthatthereisnopublicdocumentdisclosingwhat valuewasused. Table3: CalibrationoftheEndowmentEconomyModel Notation Parameter/Variable Value 1. Commonparameters σ Coefficientofrelativeriskaversion 2.0 y Meanendowmentincome 1.00 A Absorptionconstant 0.28 R Grossworldinterestrate 1.059 σ Standarddeviationofincome(percent) 3.27 z ρ Autocorrelationofincome 0.597 z 2. Globalsolutionparameters β Discountfactor 0.940 ϕ Ad-hocdebtlimit −0.51 3. Localsolutionparameters Commonparameters β Discountfactor 0.944 ¯b DeterministicsteadystatevalueofNFA −0.51 Baselinecalibration ψ InessentialDEIRcoefficient 0.001 Targetedcalibration ψ DEIRcoefficeintfor2OA 0.0469 ψ DEIRcoefficientforRSS 0.0469 Note:2OAandRSSdenotethesecond-orderandrisky-steadystatesolutions,respectively. 55

Table4: AutocorrelationsofNetExports,NFA,andIncome: EndowmentEconomyModel ρ(cid:15) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 i)GLB ρ 0.827 0.866 0.899 0.926 0.947 0.964 0.977 0.987 0.993 NFA ρ -0.088 0.010 0.11 0.213 0.321 0.432 0.547 0.661 0.768 NX ii)Baseline(inessentialψ) 2OA ρ 0.995 0.996 0.996 0.997 0.998 0.998 0.999 0.999 0.999 NFA ρ 0.238 0.348 0.455 0.556 0.652 0.741 0.821 0.89 0.946 NX RSS ρ 0.995 0.996 0.997 0.997 0.998 0.998 0.999 0.999 1.000 NFA ρ 0.239 0.35 0.457 0.559 0.655 0.745 0.826 0.896 0.952 NX iii)Targeted(calibratedψ tomatchbaselineGLBρb) 2OA ρ 0.912 0.928 0.941 0.952 0.961 0.97 0.977 0.984 0.990 NFA ρ -0.01 0.089 0.188 0.287 0.387 0.486 0.586 0.685 0.786 NX RSS ρ 0.912 0.928 0.941 0.952 0.961 0.97 0.977 0.984 0.99 NFA ρ -0.01 0.089 0.188 0.287 0.386 0.485 0.585 0.684 0.784 NX iv)Targetedforallρ(cid:15) (calibratedψ tomatcheachGLBρb) 2OA ψ 0.185 0.158 0.13 0.106 0.083 0.064 0.046 0.034 0.027 ρ 0.827 0.866 0.899 0.926 0.947 0.964 0.977 0.987 0.993 NFA ρ -0.029 0.068 0.166 0.267 0.37 0.476 0.586 0.698 0.807 NX RSS ψ 0.185 0.158 0.13 0.106 0.083 0.064 0.046 0.034 0.027 ρ 0.827 0.866 0.899 0.926 0.947 0.964 0.977 0.987 0.993 NFA ρ -0.030 0.067 0.166 0.266 0.369 0.475 0.585 0.696 0.804 NX Note:GLB,2OAandRSSdenotetheglobal,second-orderandrisky-steadystatesolutions,respectively. 56

Table5: Long-runMoments: EndowmentEconomyModel BaselineCalibration TargetedCalibration GLB 2OA RSS 2OA RSS DEIR βR<1 DEIR DEIR DEIR ψ = na 0.001 na 0.001 0.0469 0.0469 Averages E(c) 0.694 0.701 0.093 0.692 0.689 0.689 E(nx/y) 0.022 0.015 0.625 0.025 0.028 0.028 E(b/y) -0.413 -0.282 -11.210 -0.451 -0.502 -0.506 Standarddeviationsrelativetostandarddeviationofincome σ(c)/σ(y) 0.992 1.594 1.161 1.617 1.001 0.997 σ(nx)/σ(y) 0.660 1.327 1.202 1.346 0.730 0.730 σ(nx/y)/σ(y) 0.643 1.311 1.161 1.331 0.709 0.709 σ(b)/σ(y) 7.461 62.327 1.706 40.078 6.647 6.576 σ(b/y)/σ(y) 7.735 61.989 1.892 40.213 7.174 7.118 Incomecorrelations ρ(y,c) 0.755 0.202 0.188 0.197 0.684 0.684 ρ(y,nx) 0.729 0.572 0.312 0.567 0.705 0.708 ρ(y,nx/y) 0.704 0.572 0.006 0.567 0.705 0.708 ρ(y,b) 0.449 0.128 0.070 0.124 0.489 0.488 ρ(y,b/y) 0.549 0.156 0.445 0.149 5.593 0.592 First-orderautocorrelations ρ 0.840 0.995 0.996 0.995 0.929 0.929 c ρ 0.543 0.819 0.934 0.823 0.583 0.582 nx ρ 0.551 0.826 0.995 0.830 0.591 0.590 nx/y ρ 0.977 0.999 0.999 0.999 0.977 0.977 b ρ 0.961 0.998 0.953 0.998 0.958 0.959 b/y Performancemetrics Executiontime(secs.) 5.9 8.5 n.a. 9.9 8.5 9.8 ratiorel. toGLB 1.0 1.441 n.a. 1.678 1.441 1.661 Max. Abs. Eulereq. errors 9.60E-05 1.10E-03 4.45E-03 1.00E-03 2.60E-03 2.50E-03 Decisionrulediffb 0.120(0.248) n.a. 0.099(0.378) 0.086(0.127) 0.075(0.123) Decisionrulediffc 0.025(0.049) n.a. 0.028(0.055) 0.019(0.037) 0.020(0.039) Note: GLB,20AandRSSrefertotheglobal,second-orderandriskysteadystatesolutions,respectively. σ(·)denotes thecoefficientofvariationforvariablesinlevelsandthestandarddeviationforvariablesinratios(nx/y,b/yandthe leverageratiolev/rat.). TheresultswereobtainedusingMatlab2016ainaLinuxclusterwith128gbofRAM,two10coreIntel(R)Xeon(R)CPUE5-2690v2@3.00GHzprocessors,andaSamsungSSD840512GBharddrive.Thenumber ofCPUscalledbytheparallelcomputingtoolboxwassettominimizeexecutiontime. Executiontimesincludeelapsed timeuptothesolutionofdecisionrules.Weusedthesamesoftwareandhardwareforallthecomputationsreportedin thepaper.ExecutiontimesarenotreportedfortheRSSsolutionwithβR<1becauseweusedanalyticsolutions.Euler equationerrorsanddecisionruledifferencesarecomputedforall(b,z)pairsinthestatespaceoftheGLBsolution. Decision rule differences in the last two rows are differences between the local and GLB solutions in percent of the latter.Wereportmeanandmaximum(maximuminbrackets)differencesconditionalonbondvaluesthathavepositive probabilityintheergodicdistributionoftheGLBsolution. 57

Table6: EndowmentEconomyModelwithIncomeandInterest-RateShocks StdDevofIntRate(percent) 0.0 0.5 1.0 1.5 2.0 2.5 Globalcalibrated E(b/y) -0.413 -0.411 -0.408 -0.403 -0.396 -0.384 σ(c)/σ(y) 0.992 0.977 1.009 1.058 1.126 1.214 σ(b)/σ(y) 7.461 7.169 7.465 8.009 8.874 10.311 ρ(y,nx) 0.729 0.681 0.617 0.527 0.415 0.298 ρ(nx) 0.543 0.540 0.542 0.546 0.551 0.559 ρ(b) 0.977 0.973 0.975 0.976 0.978 0.981 GlobalwithNDL E(b/y) -10.778 -9.249 -7.445 -5.991 -4.875 -3.956 σ(c)/σ(y) 9.747 7.375 6.962 6.189 5.563 4.906 σ(b)/σ(y) 1.682 2.418 4.232 5.771 7.194 8.374 ρ(y,nx) 0.684 0.457 0.343 0.308 0.297 0.301 ρ(nx) 0.858 0.880 0.924 0.931 0.927 0.914 ρ(b) 0.999 0.998 0.998 0.998 0.998 0.997 FullRSSw. βR¯ <1 E(b/y) -11.21 -9.098 -7.182 -5.577 -4.226 -3.075 σ(c)/σ(y) 12.484 11.171 9.672 8.209 6.745 5.322 σ(b)/σ(y) 19.067 38.394 49.967 53.952 52.038 45.600 ρ(y,nx) 0.315 0.077 0.011 -0.021 -0.044 -0.066 ρ(nx) 0.933 0.987 0.993 0.994 0.992 0.986 ρ(b) 0.999 0.999 0.999 0.999 0.999 0.999 PartialRSSw. baselineψ E(b/y) -0.451 -0.426 -0.279 -0.018 0.381 0.942 σ(c)/σ(y) 1.617 1.645 1.773 2.085 2.894 4.969 σ(b)/σ(y) 40.078 43.072 71.486 1327.807 94.562 71.228 ρ(y,nx) 0.567 0.560 0.531 0.469 0.357 0.217 ρ(nx) 0.823 0.823 0.830 0.856 0.910 0.965 ρ(b) 0.999 0.999 0.999 0.999 0.999 0.999 2OAw. baselineψ E(b/y) -0.285 -0.319 -0.179 0.056 0.384 0.806 σ(c)/σ(y) 1.594 1.612 1.664 1.747 1.857 1.990 σ(b)/σ(y) 62.327 55.583 100.480 313.282 47.421 23.101 ρ(y,nx) 0.572 0.568 0.555 0.536 0.512 0.485 ρ(nx) 0.819 0.816 0.809 0.798 0.785 0.771 ρ(b) 0.999 0.999 0.999 0.999 0.999 0.999 PartialRSSw. targetedψ E(b/y) -0.506 -0.507 -0.505 -0.501 -0.495 -0.487 σ(c)/σ(y) 0.997 1.016 1.068 1.150 1.254 1.375 σ(b)/σ(y) 6.576 6.571 6.657 6.805 7.022 7.315 ρ(y,nx) 0.708 0.695 0.663 0.619 0.571 0.523 ρ(nx) 0.582 0.580 0.576 0.570 0.564 0.559 ρ(b) 0.977 0.977 0.977 0.977 0.977 0.977 2OAw. targetedψ E(b/y) -0.502 -0.505 -0.502 -0.498 -0.492 -0.484 σ(c)/σ(y) 1.001 1.020 1.073 1.157 1.264 1.391 σ(b)/σ(y) 6.647 6.612 6.694 6.833 7.030 7.287 ρ(y,nx) 0.705 0.693 0.660 0.615 0.564 0.514 ρ(nx) 0.583 0.581 0.577 0.572 0.566 0.561 ρ(b) 0.977 0.977 0.977 0.977 0.977 0.977 Note:ThevariabilityandpersistenceofendowmentshocksarekeptasinTable3.Thecorrelationbetweenendowment andinterest-rateshocksissetto−0.669,forallcolumnswiththeexceptionofthefirstcolumnforwhichthecorrelation issetto0.GLB,20AandRSSrefertotheglobal,second-orderandrisky-steadystatesolutions,respectively. 58

Table7: CalibrationoftheRBC&SuddenStopsModels Notation Parameter/Variable Value 1. Commonparameters σ Coefficientofrelativeriskaversion 2.0 R Grossworldinterestrate 1.0857 α Laborshareingrossoutput 0.592 γ Capitalshareingrossoutput 0.306 η Importedinputsshareingrossoutput 0.102 δ Depreciationrateofcapital 0.088 ω Laborexponentintheutilityfunction 1.846 φ Workingcapitalconstraintcoefficient 0.2579 a Investmentadjustmentcostparameter 2.75 τ Consumptiontax 0.168 κ Collateralconstraintcoefficient 0.20 ydss GDPatthedeterministicsteadystate 396 2. RBCglobalsolutionparameters β Discountfactor 0.920 ϕ Ad-hocdebtlimitasashareofydss −0.758 3. RBClocalsolutionparameters CommonParameters β Discountfactor 0.9211 bdss/ydss NFA/GDPatthedeterministicsteadystate −0.758 BaselineCalibration ψ InessentialDEIRcoefficient 0.001 TargetedCalibration ψ DEIRcoefficientfor2OA 0.0109 ψ DEIRcoefficientforRSS 0.008 4. SuddenStopsglobalsolutionparameters β Discountfactor 0.920 ϕ Ad-hocdebtlimitasashareofydss −0.505 bdss/ydss NFA/GDPatthedeterministicsteadystate −0.192 5. SuddenStopslocalsolutionparameters DynareOBCwithβR<1 β Discountfactor 0.920 bdss/ydss NFA/GDPatthedeterministicsteadystate −0.192 DynareOBCwithDEIR β Discountfactor 0.9211 ψ InessentialDEIRcoefficient 0.001 bdss/ydss NFA/GDPatthedeterministicsteadystate 0.015 Note:2OAandRSSdenotethesecond-orderandrisky-steadystatesolutions,respectively.FortheSuddenStopsmodel, theGLBsolutionhastwocreditconstraints,namelyϕandthecollateralconstraint. Creditisconstrainedatthedeterministicsteadystate,sinceβR<1,butϕissetlowenoughsothatthecollateralconstraintbindsfirst. 59

Table8: Long-runMoments: RBCmodel BaselineCalibration TargetedCalibration GLB 2OA RSS 2OA RSS ψ = na 0.001 0.001 0.0109 0.008 Averages E(y) 393.847 396.990 396.190 397.050 397.210 E(c) 264.021 294.900 342.850 259.180 265.420 E(I) 67.53 68.008 67.747 68.035 68.063 E(nx/y) 0.045 -0.042 -0.185 0.065 0.046 E(b/y) -0.372 0.732 2.559 -0.620 -0.397 E(lev.rat.) -0.286 -0.237 -1.100 0.400 0.295 E(υ) 42.649 42.938 42.852 42.946 42.975 E(L) 18.433 18.519 18.499 18.521 18.528 VariabilityrelativetovariabilityofGDP σ(y) 0.040 0.039 0.039 0.041 0.040 σ(c)/σ(y) 1.291 1.910 1.412 1.268 1.212 σ(I)/σ(y) 3.386 3.467 3.493 3.320 3.388 σ(nx/y)/σ(y) 0.885 1.293 1.212 0.712 0.731 σ(b/y)/σ(y) 7.589 13.824 12.909 3.758 4.269 σ(lev.rat.)/σ(y) 3.614 6.549 6.084 1.849 2.053 σ(υ)/σ(y) 1.481 1.496 1.504 1.463 1.482 σ(L)/σ(y) 0.596 0.600 0.600 0.596 0.598 CorrelationswithGDP ρ(y,c) 0.773 0.631 0.509 0.929 0.904 ρ(y,I) 0.640 0.632 0.628 0.661 0.648 ρ(y,nx/y) -0.227 -0.278 0.026 -0.476 -0.381 ρ(y,b/y) 0.090 0.200 -0.160 0.511 0.343 ρ(y,lev.rat.) 0.112 -0.206 0.150 -0.532 -0.366 ρ(y,υ) 0.834 0.831 0.830 0.839 0.835 ρ(y,L) 0.995 0.995 0.995 0.995 0.995 First-orderautocorrelations ρ(y) 0.830 0.824 0.820 0.841 0.853 ρ(b) 0.996 0.999 0.998 0.996 0.996 ρ(c) 0.885 0.940 0.918 0.873 0.862 ρ(I) 0.516 0.511 0.509 0.519 0.513 ρ(nx/y) 0.711 0.850 0.843 0.555 0.563 ρ(lev.rat.) 0.997 0.999 0.998 0.991 0.995 ρ(υ) 0.780 0.777 0.774 0.788 0.782 ρ(L) 0.808 0.808 0.799 0.819 0.810 Performancemetrics Timeinsec. 61.0 37.8 40.6 37.9 39.6 ratiorel. toGLB 1.0 0.620 0.666 0.621 0.649 Max. Abs. bEulereq. error 1.17E-07 1.33E-07 6.21E-04 1.43E-07 1.13E-03 Max. Abs. kEulereq. error 3.84E-16 4.52E-07 8.92E-05 3.95E-07 6.59E-05 Decisionrulediffb 0.089(0.546) 0.089(0.546) 0.081(0.505) 0.081(0.505) Decisionrulediffk 0.017(0.051) 0.017(0.051) 0.017(0.049) 0.017(0.049) Decisionrulediffc 0.019(0.066) 0.019(0.066) 0.019(0.077) 0.019(0.077) SeeNotetoTable5. 60

Table9: Long-runMoments: SuddenStopsmodel GLB DynareOBC-βR<1 DynareOBC-DEIR Averages E(y) 393.619 391.390 395.230 E(c) 274.123 269.610 279.970 E(I) 67.481 66.714 67.897 E(nx/y) 0.015 0.025 0.000 E(b/y) 0.015 -0.100 0.206 E(lev.rat.) -0.102 -0.157 -0.011 E(υ) 42.617 42.263 42.712 E(L) 18.426 18.364 18.469 VariabilityrelativetovariabilityofGDP σ(y) 0.039 0.032 0.032 σ(c)/σ(y) 1.023 0.937 1.207 σ(I)/σ(y) 3.383 3.492 3.777 σ(nx/y)/σ(y) 0.746 0.927 1.262 σ(b/y)/σ(y) 4.980 3.703 9.595 σ(lev.rat.)/σ(y) 2.340 1.705 4.498 σ(υ)/σ(y) 1.495 1.632 1.612 σ(L)/σ(y) 0.599 0.571 0.569 CorrelationswithGDP ρ(y,c) 0.842 0.823 0.557 ρ(y,I) 0.641 0.309 0.224 ρ(y,nx/y) -0.117 0.176 0.223 ρ(y,b/y) -0.120 0.027 -0.054 ρ(y,lev.rat.) -0.111 0.008 -0.056 ρ(y,υ) 0.832 0.777 0.775 ρ(y,L) 0.994 0.987 0.986 First-orderautocorrelations ρ(y) 0.825 0.752 0.754 ρ(b) 0.990 0.980 0.995 ρ(c) 0.829 0.826 0.910 ρ(I) 0.500 0.470 0.502 ρ(nx/y) 0.601 0.456 0.651 ρ(lev.rat.) 0.992 0.988 0.996 ρ(υ) 0.777 0.753 0.756 ρ(L) 0.801 0.761 0.774 Prob.(µ>0) 2.58 51.80 71.06 Performancemetrics Timeinsec. 268.0 243.5 187.4 Max. Abs. bEulereq. error 2.62E-04 na na Max. Abs. kEulereq. error 4.25E-16 na na Note:SeeNotetoTable5. 61

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(cid:75)(cid:90)(cid:69)(cid:11)(cid:21)(cid:17)(cid:27)(cid:15)(cid:3)(cid:85)(cid:15)(cid:3)(cid:69)(cid:12) (cid:32) (cid:20)(cid:17)(cid:21)(cid:27)(cid:21) Figure1: Thefirst-ordercoefficientofNFAdecisionrulesastheelasticityoftheDEIRfunctionvaries (cid:29)(cid:32) (cid:19)(cid:17)(cid:19)(cid:19)(cid:19)(cid:15)(cid:3)(cid:19)(cid:17)(cid:19)(cid:19)(cid:20)(cid:17)(cid:17)(cid:19)(cid:17)(cid:28) (cid:20) (cid:19)(cid:17)(cid:28)(cid:23) (cid:53)(cid:72)(cid:71)(cid:3)(cid:76)(cid:86)(cid:3)(cid:73)(cid:82)(cid:85)(cid:3)(cid:71)(cid:72)(cid:87)(cid:3)(cid:54)(cid:54)(cid:3)(cid:11)(cid:21)(cid:50)(cid:36)(cid:3)(cid:70)(cid:68)(cid:86)(cid:72)(cid:12) (cid:19)(cid:17)(cid:27)(cid:27) (cid:37)(cid:79)(cid:88)(cid:72)(cid:3)(cid:76)(cid:86)(cid:3)(cid:73)(cid:82)(cid:85)(cid:3)(cid:53)(cid:54)(cid:54) (cid:42)(cid:85)(cid:72)(cid:81)(cid:81)(cid:3)(cid:76)(cid:86)(cid:3)(cid:73)(cid:82)(cid:85)(cid:3)(cid:69)(cid:86)(cid:86)(cid:32)(cid:19) (cid:19)(cid:17)(cid:27)(cid:21) (cid:75)(cid:90)(cid:69)(cid:11)ψ(cid:15)(cid:3)(cid:85)(cid:15)(cid:3)(cid:16)(cid:19)(cid:17)(cid:24)(cid:20)(cid:12) (cid:19)(cid:17)(cid:26)(cid:25) (cid:41)(cid:82)(cid:85)(cid:3)(cid:83)(cid:86)(cid:76)(cid:31)(cid:32)(cid:19)(cid:17)(cid:20)(cid:24)(cid:15)(cid:3)(cid:75)(cid:66)(cid:90)(cid:3)(cid:76)(cid:86)(cid:3) (cid:75)(cid:90)(cid:69)(cid:11)ψ(cid:15)(cid:3)(cid:85)(cid:15)(cid:3)(cid:16)(cid:19)(cid:17)(cid:23)(cid:20)(cid:12) (cid:68)(cid:69)(cid:82)(cid:88)(cid:87)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:86)(cid:68)(cid:80)(cid:72)(cid:3)(cid:73)(cid:82)(cid:85)(cid:3)(cid:21)(cid:50)(cid:36) (cid:19)(cid:17)(cid:26) (cid:82)(cid:85)(cid:3)(cid:53)(cid:54)(cid:54)(cid:4) (cid:75)(cid:90)(cid:11)ψ(cid:15)(cid:3)(cid:85)(cid:12) (cid:19)(cid:17)(cid:25)(cid:23) 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(cid:12)(cid:152)(cid:11)(cid:16)(cid:69)(cid:152) (cid:14) (cid:14) (cid:20)(cid:12)(cid:64) (cid:75)(cid:90)(cid:69)(cid:68)(cid:83)(cid:11) (cid:15)(cid:3)(cid:85)(cid:15)(cid:3)(cid:69)(cid:12)(cid:29)(cid:32) Note:Thisfigureshowshowthefirst-orderc (cid:21) o (cid:152)(cid:72) e (cid:91) ffi (cid:83)(cid:11) c (cid:69) ie (cid:152) nt (cid:12) oftheNFAdecisionru (cid:75)(cid:90) le (cid:69) s (cid:68) , (cid:83)ρ(cid:11) b (cid:19)((cid:17)ψ(cid:19)(cid:19),(cid:20)b∗(cid:15)(cid:3)(cid:85)) (cid:15) , (cid:3)(cid:19) v (cid:12) a (cid:32) rie (cid:20) s (cid:17)(cid:19) w (cid:22) ithψ(cid:75) f (cid:90) or (cid:69)(cid:11) t (cid:19) h (cid:17) r (cid:19) e (cid:19) e (cid:20)(cid:15)(cid:3)(cid:85)(cid:15)(cid:3)(cid:19)(cid:12) (cid:32) (cid:19)(cid:17)(cid:28)(cid:27)(cid:25) valuesofb∗:-0.51(deterministicsteadystate),-0.41(riskysteadystate)andzero. 63

Figure2: AverageNFAintheendowmenteconomyasthevariabilityandpersistenceofoutputrise BaselineCalibration(ψ = 0.001) 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 1 2 3 4 5 6 7 8 Standard Deviation of Output (%) noitisoP AFN naeM a. Effect of Variability 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Autocorrelation of Output GLB 2OA RSS noitisoP AFN naeM b. Effect of Persistence GLB 2OA RSS TargetedCalibration(ψ = 0.0469) 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 1 2 3 4 5 6 7 8 Standard Deviation of Output (%) noitisoP AFN naeM c. Effect of Variability -0.3 -0.35 -0.4 -0.45 -0.5 -0.55 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Autocorrelation of Output GLB 2OA RSS noitisoP AFN naeM d. Effect of Persistence GLB 2OA RSS Note:GLBreferstoglobalsolution,2OAreferstosecond-ordersolution,RSSreferstorisky-steadystatesolution. 64

Figure3: EndowmentModelImpulseResponseFunctionstoaNegativeIncomeShock a. NFA/Output Baseline b. NFA/Output Targeted 0 0 s tn -2 s tn -2 io GLB io GLB p p e g a -4 2 R O S A S e g a -4 2 R O S A S tn FOA tn FOA e e c re -6 c re -6 P P -8 -8 0 10 20 30 40 0 10 20 30 40 Time Time c. Consumption Baseline d. Consumption Targeted 0 0 -0.5 GLB -0.5 GLB tn e 2OA tn e 2OA c re RSS c re RSS P -1 FOA P -1 FOA -1.5 -1.5 0 10 20 30 40 0 10 20 30 40 Time Time e. NX/Output Baseline f. NX/Output Targeted 1 1 s 0 s 0 tn tn io GLB io GLB p p e-1 2OA e-1 2OA g a RSS g a RSS tn e-2 FOA tn e-2 FOA c c re re P P -3 -3 0 10 20 30 40 0 10 20 30 40 Time Time g. Output 0 -1 GLB tn e c re -2 2 R O S A S P FOA -3 -4 0 10 20 30 40 Time Note:GLB,1OA,2OAandRSSdenoteglobal,first-order,second-orderandrisky-steadystatesolutions,respectively. GLBimpulseresponsesareforecastfunctionsoftheequilibriumMarkovprocessesoftheendogenousvariableswith initialconditionssettoE[b]andavalueofzequaltoaone-standard-deviationshock. 65

Figure4: SpectralDensityFunctionsintheEndowmentEconomyModel BaselineCalibration(ψ = 0.001) TargetedCalibration(ψ = 0.0469) a. Bonds b. Bonds 100 10-2 10-1 m10-2 m10-3 urtcepS10-3 urtcepS 10-4 10-4 10-5 10-5 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB 2OA RSS GLB 2OA RSS c. NX d. NX 10-3 m m10-4 urtcepS urtcepS 10-4 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB 2OA RSS GLB 2OA RSS e. Consumption f. Consumption 10-2 10-3 10-3 m urtcepS 10-4 m urtcepS 10-4 10-5 10-5 10-6 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB 2OA RSS GLB 2OA RSS Note: These graphs show parametric estimates of spectral density functions. GLB, 2OA and RSS denote the global, second-orderandrisky-steadystatesolution,respectively. 66

Figure5: RBCImpulseResponseFunctionstoaNegativeTFPshock: BaselineCalibration 2 0 -2 -4 -6 0 20 40 60 80 100 .P.P a. NFA/GDP 0 -1 -2 -3 0 20 40 60 80 100 tnecreP b .Consumption 2 1 0 0 20 40 60 80 100 .P.P c. NX/GDP 0 -1 -2 0 20 40 60 80 100 tnecreP d. Capital 0 -5 -10 0 20 40 60 80 100 tnecreP e. Investment 0 -1 -2 0 20 40 60 80 100 tnecreP f. Labor 0 -1 -2 0 20 40 60 80 100 tnecreP g. Intermediate Input 0 -1 -2 0 20 40 60 80 100 tnecreP h. GDP GLB 1OA RSS 2OA Note: GLB,1OA,2OA,RSSrefertotheglobal, first-order, second-orderandrisky-steadystatesolution, respectively. GLBimpulseresponsesareforecastfunctionsoftheequilibriumMarkovprocessesoftheendogenousvariableswith initialconditionssettoE[b],E[k]andavalueofTFPequaltoaone-standard-deviationshock. Variablesareplottedas percentdeviationsfromlong-runmeans,withtheexceptionofNFAandnetexports,whichareplottedasdifferences relativetotheirlong-runmeans(sincethesevariablesaremeasuredasGDPratios,andhencearealreadyinpercent). 67

Figure6: RBCImpulseResponseFunctionstoaNegativeTFPShock: TargetedCalibration 0 -2 -4 -6 0 20 40 60 80 100 .P.P a. NFA/GDP 0 -1 -2 -3 0 20 40 60 80 100 tnecreP b .Consumption 1 0.5 0 0 20 40 60 80 100 .P.P c. NX/GDP 0 -1 -2 0 20 40 60 80 100 tnecreP d. Capital 0 -5 -10 0 20 40 60 80 100 tnecreP e. Investment 0 -1 -2 0 20 40 60 80 100 tnecreP f. Labor 0 -1 -2 0 20 40 60 80 100 tnecreP g. Intermediate Input 0 -1 -2 0 20 40 60 80 100 tnecreP h. GDP GLB 2OA RSS 1OA Note:GLB,1OA,2OA,RSSrefertotheglobal,first-order,second-orderandrisky-steadystatesolution,respectively. 68

Figure7: SpectralDensityFunctionsfortheRBCmodel: BaselineCalibration a. NFA b. Consumption 105 103 104 m urtcepS 103 m urtcepS 102 102 101 101 100 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB 2OA RSS GLB 2OA RSS c. NX d. Capital 103 103 m urtcepS 102 m urtcepS 102 101 101 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB 2OA RSS GLB 2OA RSS e. Investment f. Labor 10-1 m m urtcepS urtcepS 101 10-2 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB 2OA RSS GLB 2OA RSS g. Intermediate Input h. GDP 101 102 m m urtcepS 100 urtcepS 101 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB 2OA RSS GLB 2OA RSS Note: Thesegraphsshowparametricestimatesofspectraldensityfunctions. GLB,2OA,andRSSrefertotheglobal, second-orderandrisky-steadystatesolution,respectively. 69

Figure8: SpectralDensityFunctionsfortheRBCmodel: TargetedCalibrations a. NFA b. Consumption 104 103 102 m m urtcepS 102 urtcepS 101 101 100 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB 2OA RSS GLB 2OA RSS c. NX d. Capital 103 102 m m urtcepS urtcepS 102 101 101 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB 2OA RSS GLB 2OA RSS e. Investment f. Labor 10-1 m m urtcepS urtcepS 101 10-2 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB 2OA RSS GLB 2OA RSS g. Intermediate Input h. GDP 101 102 m m urtcepS 100 urtcepS 101 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB 2OA RSS GLB 2OA RSS Note: Thesegraphsshowparametricestimatesofspectraldensityfunctions. GLB,2OA,andRSSrefertotheglobal, second-orderandrisky-steadystatesolution,respectively. 70

Figure9: PerfectforesightpathsforDynareOBC 0.8 1.05 0.6 0.4 1 0.2 0.95 0 100 150 200 250 100 150 200 250 1.02 0.1 1 0 0.98 -0.1 0.96 -0.2 0.94 -0.3 0.92 140 150 160 170 180 140 150 160 170 180 71

Figure10: SuddenStopsModel: ImpulseResponseFunctionstoaNegativeTFPShock 0 -2 -4 0 10 20 30 40 50 .P.P a. NFA/GDP 0 -1 -2 0 10 20 30 40 50 tnecreP b .Consumption 1 0.5 0 -0.5 0 10 20 30 40 50 .P.P c. NX/GDP 0 -0.5 -1 0 10 20 30 40 50 tnecreP d. Capital 0 -5 -10 0 10 20 30 40 50 tnecreP e. Investment 0 -0.5 -1 -1.5 0 10 20 30 40 50 tnecreP f. Labor 0 -1 -2 0 10 20 30 40 50 tnecreP g. Intermediate Input 0 -1 -2 0 10 20 30 40 50 tnecreP h. GDP GLB DynareOBC DynareOBC DEIR 72

Figure11: SpectralDensityFunctionsfortheSuddenStopsModel a. NFA b. Consumption 105 103 104 102 m urtcepS 103 m urtcepS 102 101 101 100 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB DynareOBC DynareOBC DEIR GLB DynareOBC DynareOBC DEIR c. NX d. Capital 103 103 m urtcepS 102 m urtcepS 102 101 101 100 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB DynareOBC DynareOBC DEIR GLB DynareOBC DynareOBC DEIR e. Investment f. Labor 10-1 m m urtcepS 101 urtcepS 10-2 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB DynareOBC DynareOBC DEIR GLB DynareOBC DynareOBC DEIR g. Intermediate Input h. GDP 101 102 m m urtcepS 100 urtcepS 101 10 7 5 3 10 7 5 3 Period of cycle (years) Period of cycle (years) GLB DynareOBC DynareOBC DEIR GLB DynareOBC DynareOBC DEIR 73