Review of Economic Dynamics

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1 Review of Economic Dynamics 15 (2012) Conens liss available a SciVerse ScienceDirec Review of Economic Dynamics Compuing DSGE models wih recursive preferences and sochasic volailiy Dario Caldara a, Jesús Fernández-Villaverde b,c,d,e,, Juan F. Rubio-Ramírez f,g,e,wenyao b a Federal Reserve Board, Washingon, DC 20551, Unied Saes b Universiy of Pennsylvania, 160 McNeil, 3718 Locus Walk, Philadelphia, PA 19104, Unied Saes c NBER, Unied Saes d CEPR, Unied Kingdom e FEDEA, Spain f Duke Universiy, 213 Social Sciences Building, Box 90097, Durham, NC , Unied Saes g Federal Reserve Bank of Alana, Unied Saes aricle info absrac Aricle hisory: Received 20 Sepember 2011 Available online 17 Ocober 2011 JEL classificaion: C63 C68 E32 Keywords: DSGE models Recursive preferences Perurbaion This paper compares differen soluion mehods for compuing he equilibrium of dynamic sochasic general equilibrium (DSGE) models wih recursive preferences such as hose in Epsein and Zin (1989, 1991) and sochasic volailiy. Models wih hese wo feaures have recenly become popular, bu we know lile abou he bes ways o implemen hem numerically. To fill his gap, we solve he sochasic neoclassical growh model wih recursive preferences and sochasic volailiy using four differen approaches: secondand hird-order perurbaion, Chebyshev polynomials, and value funcion ieraion. We documen he performance of he mehods in erms of compuing ime, implemenaion complexiy, and accuracy. Our main finding is ha perurbaions are compeiive in erms of accuracy wih Chebyshev polynomials and value funcion ieraion while being several orders of magniude faser o run. Therefore, we conclude ha perurbaion mehods are an aracive approach for compuing his class of problems Elsevier Inc. All righs reserved. 1. Inroducion This paper compares differen soluion mehods for compuing he equilibrium of dynamic sochasic general equilibrium (DSGE) models wih recursive preferences and sochasic volailiy (SV). Boh feaures have become very popular in finance and in macroeconomics as modeling devices o accoun for business cycle flucuaions and asse pricing. Recursive preferences, as hose firs proposed by Kreps and Poreus (1978) and laer generalized by Epsein and Zin (1989, 1991) and Weil (1990), are aracive for wo reasons. Firs, hey allow us o separae risk aversion and ineremporal elasiciy of subsiuion (EIS). Second, hey offer he inuiive appeal of having preferences for early or laer resoluion of uncerainy (see he reviews by Backus e al., 2004, 2007, and Hansen e al., 2007, for furher deails and references). SV generaes heeroskedasic aggregae flucuaions, a basic propery of many ime series such as oupu (see he review by Fernández-Villaverde and Rubio-Ramírez, 2010), and adds exra flexibiliy in accouning for asse pricing paerns. In fac, in an influenial paper, We hank Michel Juillard for his help wih compuaional issues and Larry Chrisiano, Dirk Krueger, Pawel Zabczyk, and paricipans a several seminars for commens. Beyond he usual disclaimer, we mus noe ha any views expressed herein are hose of he auhors and no necessarily hose of he Board of Governors of he Federal Reserve Sysem or he Federal Reserve Bank of Alana. Finally, we also hank he NSF for financial suppor. * Corresponding auhor a: Universiy of Pennsylvania, 160 McNeil, 3718 Locus Walk, Philadelphia, PA 19104, Unied Saes. addresses: dario.caldara@frb.gov (D. Caldara), jesusfv@econ.upenn.edu (J. Fernández-Villaverde), juan.rubio-ramirez@duke.edu (J.F. Rubio-Ramírez), wenyao@econ.upenn.edu (W. Yao) /$ see fron maer 2011 Elsevier Inc. All righs reserved. doi: /j.red

2 D. Caldara e al. / Review of Economic Dynamics 15 (2012) Bansal and Yaron (2004) have argued ha he combinaion of recursive preferences and SV is he key for heir proposed mechanism, long-run risk, o be successful a explaining asse pricing. Bu despie he populariy and imporance of hese issues, nearly nohing is known abou he numerical properies of he differen soluion mehods ha solve equilibrium models wih recursive preferences and SV. For example, we do no know how well value funcion ieraion (VFI) performs or how good local approximaions are compared wih global ones. Similarly, if we wan o esimae he model, we need o assess which soluion mehod is sufficienly reliable ye quick enough o make he exercise feasible. More imporan, he mos common soluion algorihm in he DSGE lieraure, (log-) linearizaion, canno be applied, since i makes us miss he whole poin of recursive preferences or SV: he resuling (log-) linear decision rules are cerainy equivalen and do no depend on risk aversion or volailiy. This paper aemps o fill his gap in he lieraure, and herefore, i complemens previous work by Aruoba e al. (2006), in which a similar exercise is performed wih he neoclassical growh model wih CRRA uiliy funcion and consan volailiy. We solve and simulae he model using four main approaches: perurbaion (of second and hird order), Chebyshev polynomials, and VFI. By doing so, we span mos of he relevan mehods in he lieraure. Our resuls provide a srong guess of how some oher mehods no covered here, such as finie elemens, would work (raher similar o Chebyshev polynomials bu more compuaionally inensive). We repor resuls for a benchmark calibraion of he model and for alernaive calibraions ha change he variance of he produciviy shock, he risk aversion, and he ineremporal elasiciy of subsiuion. In ha way, we sudy he performance of he mehods boh for cases close o he CRRA uiliy funcion wih consan volailiy and for highly non-linear cases far away from he CRRA benchmark. For each mehod, we compue decision rules, he value funcion, he ergodic disribuion of he economy, business cycle saisics, he welfare coss of aggregae flucuaions, and asse prices. Also, we evaluae he accuracy of he soluion by reporing Euler equaion errors. We highligh four main resuls from our exercise. Firs, all mehods provide a high degree of accuracy. Thus, researchers who say wihin our se of soluion algorihms can be confiden ha heir quaniaive answers are sound. Second, perurbaions deliver a surprisingly high level of accuracy wih considerable speed. Boh second- and hird-order perurbaions perform remarkably well in erms of accuracy for he benchmark calibraion, being compeiive wih VFI or Chebyshev polynomials. For his calibraion, a second-order perurbaion ha runs in a fracion of a second does nearly as well in erms of he average Euler equaion error as a VFI ha akes en hours o run. Even in he exreme calibraion wih high risk aversion and high volailiy of produciviy shocks, perurbaion works a a more han accepable level. Since, in pracice, perurbaion mehods are he only compuaionally feasible mehod o solve he medium-scale DSGE models used for policy analysis ha have dozens of sae variables (as in Smes and Wouers, 2007), his finding has an oumos applicabiliy. Moreover, since implemening second- and hird-order perurbaions is feasible wih off-he-shelf sofware like Dynare, which requires minimum programming knowledge by he user, our findings may induce many researchers o explore recursive preferences and/or SV in furher deail. Two final advanages of perurbaion are ha, ofen, he perurbed soluion provides insighs abou he economics of he problem and ha i migh be an excellen iniial guess for VFI or for Chebyshev polynomials. Third, Chebyshev polynomials provide a errific level of accuracy wih reasonable compuaional burden. When accuracy is mos required and he dimensionaliy of he sae space is no oo high, as in our model, hey are he obvious choice. Fourh, we were disappoined by he poor performance of VFI, which, compared wih Chebyshev, could no achieve a high accuracy even wih a large grid. This suggess ha we should relegae VFI o solving hose problems where nondiffereniabiliies complicae he applicaion of he previous mehods. The res of he paper is organized as follows. In Secion 2, we presen our es model. Secion 3 describes he differen soluion mehods used o approximae he decision rules of he model. Secion 4 discusses he calibraion of he model. Secion 5 repors numerical resuls and Secion 6 concludes. Appendix A provides some addiional deails. 2. The sochasic neoclassical growh model wih recursive preferences and SV We use he sochasic neoclassical growh model wih recursive preferences and SV in he process for echnology as our es case. We selec his model for hree reasons. Firs, i is he workhorse of modern macroeconomics. Even more complicaed New Keynesian models wih real and nominal rigidiies, such as hose in Woodford (2003) or Chrisiano e al. (2005), are buil around he core of he neoclassical growh model. Thus, any lesson learned wih i is likely o have a wide applicabiliy. Second, he model is, excep for he form of he uiliy funcion and he process for SV, he same es case as in Aruoba e al. (2006). This provides us wih a se of resuls o compare o our findings. Three, he inroducion of recursive preferences and SV make he model boh more non-linear (and hence, a challenge for differen soluion algorihms) and poenially more relevan for pracical use. For example, and as menioned in he Inroducion, Bansal and Yaron (2004) have emphasized he imporance of he combinaion of recursive preferences and ime-varying volailiy o accoun for asse prices. The descripion of he model is sraighforward, and we jus go hrough he deails required o fix noaion. There is a represenaive household ha has preferences over sreams of consumpion, c, and leisure, 1 l, represened by a recursive funcion of he form: [ U = max (1 β) ( c υ (1 l 1 υ) ) c,l + β ( E U ) 1 ]. (1)

3 190 D. Caldara e al. / Review of Economic Dynamics 15 (2012) The parameers in hese preferences include β, he discoun facor, υ, which conrols labor supply, γ, which conrols risk aversion, and: = 1 γ 1 ψ 1 where ψ is he EIS. The parameer is an index of he deviaion wih respec o he benchmark CRRA uiliy funcion (when = 1, we are back in ha CRRA case where he inverse of he EIS and risk aversion coincide). The household s budge consrain is given by: c + i + b R f = w l + r k + b where i is invesmen, R f is he risk-free gross ineres rae, b is he holding of an unconingen bond ha pays 1 uni of consumpion good a ime + 1, w is he wage, l is labor, r is he renal rae of capial, and k is capial. Asse markes are complee and we could have also included in he budge consrain he whole se of Arrow securiies. Since we have a represenaive household, his is no necessary because he ne supply of any securiy is zero. Households accumulae capial according o he law of moion k = (1 δ)k + i where δ is he depreciaion rae. The final good in he economy is produced by a compeiive firm wih a Cobb Douglas echnology y = e z k ζ l1 ζ where z is he produciviy level ha follows: z = λz 1 + e σ ε, ε N (0, 1). Saionariy is he naural choice for our exercise. If we had a deerminisic rend, we would only need o adjus β in our calibraion below (and he resuls would be nearly idenical). If we had a sochasic rend, we would need o rescale he variables by he produciviy level and solve he ransformed problem. However, in his case, i is well known ha he economy flucuaes less han when λ<1, and herefore, all soluion mehods would be closer, limiing our abiliy o appreciae differences in heir performance. The innovaion ε is scaled by an SV level σ, which evolves as: σ = (1 ρ)σ + ρσ 1 + ηω, ω N (0, 1) where σ is he uncondiional mean level of σ, ρ is he persisence of he processes, and η is he sandard deviaion of he innovaions o σ. Our specificaion is parsimonious and i inroduces only wo new parameers, ρ and η. A he same ime, i capures some imporan feaures of he daa (see a deailed discussion in Fernández-Villaverde and Rubio-Ramírez, 2010). The combinaion of an exponen in he produciviy process (e σ ) and a level in he evoluion of σ generaes ineresing non-linear dynamics. Anoher imporan poin is ha, wih SV, we have wo innovaions, an innovaion o echnology, ε, and an innovaion o he sandard deviaion of echnology, ω. Finally, he economy mus saisfy he aggregae resource consrain y = c + i. The definiion of equilibrium is sandard and we skip i in he ineres of space. Also, boh welfare heorems hold, a fac ha we will exploi by jumping back and forh beween he soluion of he social planner s problem and he compeiive equilibrium. However, his is only o simplify our derivaions. I is sraighforward o adap he soluion mehods described below o solve problems ha are no Pareo opimal. Thus, an alernaive way o wrie his economy is o look a he value funcion represenaion of he social planner s problem in erms of is hree sae variables, capial k,produciviyz, and volailiy, σ : [ V (k, z,σ ) = max (1 β) ( c υ (1 l 1 υ) ) c,l s.. c + k = e z k ζ l1 ζ + (1 δ)k z = λz 1 + e σ ε, ε N (0, 1) σ = (1 ρ)σ + ρσ 1 + ηω, ω N (0, 1). Then, we can find he pricing kernel of he economy m = V / c V / c. Now, noe ha: + β ( E V (k, z,σ ) ) ] 1 V c = (1 β)v 1 υ (cυ (1 l ) 1 υ ) c

4 D. Caldara e al. / Review of Economic Dynamics 15 (2012) and: V = β V 1 c ( E V ) 1 1 E (V γ 1 (1 β)v υ (1 β)(cυ (1 l ) 1 υ ) ) where in he las sep we use he resul regarding V / c forwarded by one period. Canceling redundan erms, we ge: m = V / c V / c = β ( c c ) υ( ) 1( ) 1 l 1 l c ( )(1 υ) ( ) V 1 1. (2) E V This equaion shows how he pricing kernel is affeced by he presence of recursive preferences. If = 1, he las erm, ( ) V 1 1 E V is equal o 1 and we ge back he pricing kernel of he sandard CRRA case. If 1, he pricing kernel is wised by (3). We idenify he ne reurn on equiy wih he marginal ne reurn on invesmen: R k = ζ ez k ζ 1 l1 ζ δ wih expeced reurn E [R k ]. 3. Soluion mehods We are ineresed in comparing differen soluion mehods o approximae he dynamics of he previous model. Since he lieraure on compuaional mehods is large, i would be cumbersome o review every proposed mehod. Insead, we selec hose mehods ha we find mos promising. Our firs mehod is perurbaion (inroduced by Judd and Guu, 1992, 1997, and nicely explained in Schmi-Grohé and Uribe, 2004). Perurbaion algorihms build a Taylor series expansion of he agens decision rules. Ofen, perurbaion mehods are very fas and, despie heir local naure, highly accurae in a large range of values of he sae variables (Aruoba e al., 2006). This means ha, in pracice, perurbaions are he only mehod ha can handle models wih dozens of sae variables wihin any reasonable amoun of ime. Moreover, perurbaion ofen provides insighs ino he srucure of he soluion and on he economics of he model. Finally, linearizaion and log-linearizaion, he mos common soluion mehods for DSGE models, are paricular cases of a perurbaion of firs order. We implemen a second- and a hird-order perurbaion of our model. A firs-order perurbaion is useless for our invesigaion: he resuling decision rules are cerainy equivalen and, herefore, hey depend on ψ bu no on γ or σ.in oher words, he firs-order decision rules of he model wih recursive preferences coincide wih he decision rules of he model wih CRRA preferences wih he same ψ and σ for any value of γ or σ. We need o go, a leas, o second-order decision rules o have erms ha depend on γ or σ and, hence, allow recursive preferences or SV o play a role. Since he accuracy of second-order decision rules may no be high enough and, in addiion, we wan o explore ime-varying risk premia, we also compue a hird-order perurbaion. As we will documen below, a hird-order perurbaion provides enough accuracy wihou unnecessary complicaions. Thus, we do no need o go o higher orders. The second mehod is a projecion algorihm wih Chebyshev polynomials (Judd, 1992). Projecion algorihms build approximaed decision rules ha minimize a residual funcion ha measures he disance beween he lef- and righ-hand side of he equilibrium condiions of he model. Projecion mehods are aracive because hey offer a global soluion over he whole range of he sae space. Their main drawback is ha hey suffer from an acue curse of dimensionaliy ha makes i challenging o exend i o models wih many sae variables. Among he many differen ypes of projecion mehods, Aruoba e al. (2006) show ha Chebyshev polynomials are paricularly efficien. Oher projecion mehods, such as finie elemens or parameerized expecaions, end o perform somewha worse han Chebyshev polynomials, and herefore, in he ineres of space, we do no consider hem. Finally, we compue he model using VFI (Epsein and Zin, 1989, show ha a version of he conracion mapping heorem holds in he value funcion of he problem wih recursive preferences). VFI is slow and i suffers as well from he curse of dimensionaliy, bu i is safe, reliable, and well undersood. Thus, i is a naural defaul algorihm for he soluion of DSGE models Perurbaion We describe now each of he differen mehods in more deail. We sar by explaining how o use a perurbaion approach o solve DSGE models using he value funcion of he household. We are no he firs o explore he perurbaion of value funcion problems. Judd (1998) already presens he idea of perurbing he value funcion insead of he equilibrium condiions of a model. Unforunaely, he does no elaborae much on he opic. Schmi-Grohé and Uribe (2005) employ a (3)

5 192 D. Caldara e al. / Review of Economic Dynamics 15 (2012) perurbaion approach o find a second-order approximaion o he value funcion ha allows hem o rank differen fiscal and moneary policies in erms of welfare. However, we follow van Binsbergen e al. (2009) in heir emphasis on he generaliy of he approach. 1 To illusrae he procedure, we limi our exposiion o deriving he second-order approximaion o he value funcion and he decision rules of he agens. Higher-order erms are derived analogously, bu he algebra becomes oo cumbersome o be developed explicily (in our applicaion, he symbolic algebra is underaken by Mahemaica, which auomaically generaes Forran 95 code ha we can evaluae numerically). Hopefully, our seps will be enough o allow he reader o undersand he main hrus of he procedure and obain higher-order approximaions by herself. Firs, we rewrie he exogenous processes in erms of a perurbaion parameer χ, z = λz 1 + χe σ ε σ = (1 ρ)σ + ρσ 1 + χηω. When χ = 1, which is jus a normalizaion, we are dealing wih he sochasic version of he model. When χ = 0, we are dealing wih he deerminisic case wih seady sae k ss, z ss = 0, and σ ss = σ. Also, i is convenien for he algebra below o define a vecor of saes in differences wih respec o he seady sae, s = (k k ss, z 1, ε, σ 1 σ ss, ω, χ), where s i is he i-h componen of his vecor a ime for i {1,...,6}. Then, we can wrie he social planner s value funcion, V (s ), and he decision rules for consumpion, c(s ), invesmen, i(s ), capial, k(s ), and labor, l(s ), as a funcion of s. Second, we noe ha, under differeniabiliy assumpions, he second-order Taylor approximaion of he value funcion around s = 0 (he vecorial zero) is: where: V (s ) V ss + V i,ss s i V ij,sss i s j 1. Each erm V...,ss is a scalar equal o a derivaive of he value funcion evaluaed a 0: V ss V (0), V i,ss V i (0) for i {1,...,6}, and V ij,ss V ij (0) for i, j {1,...,6}. 2. We use he ensors V i,ss s i = 6 i=1 V i,sss i, and V ij,ss s i s j = 6 6 i=1 i=1 V ij,sss i, s j,, which eliminae he symbol 6 i=1 when no confusion arises. We can exend his noaion o higher-order derivaives of he value funcion. This expansion could also be performed around a differen poin of he sae space, such as he mode of he ergodic disribuion of he sae variables. In Secion 5, we discuss his poin furher. Fernández-Villaverde e al. (2010) show ha many of hese erms V...,ss are zero (for insance, hose implied by cerainy equivalence in he firs-order componen). More direcly relaed o his paper, van Binsbergen e al. (2009) demonsrae ha γ does no affec he values of any of he coefficiens excep V 66,ss and also ha V 66,ss 0. This resul is inuiive, since he value funcion of a risk-averse agen is in general affeced by uncerainy and we wan o have an approximaion wih erms ha capure his effec and allow for he appropriae welfare ranking of decision rules. Indeed, V 66,ss has a sraighforward inerpreaion. A he deerminisic seady sae wih χ = 1 (ha is, even if we are in he sochasic economy, we jus happen o be exacly a he seady-sae values of all he oher saes), we have: V (0, 0, 0, 0, 0, 1) V ss V 66,ss. Hence 1 2 V 66,ss is a measure of he welfare cos of he business cycle, ha is, of how much uiliy changes when he variance of he produciviy shocks is a seady-sae value σ ss insead of zero (noe ha his quaniy is no necessarily negaive). This erm is an accurae evaluaion of he hird order of he welfare cos of business cycle flucuaions because all of he hird-order erms in he approximaion of he value funcion eiher have coefficien values of zero or drop when evaluaed a he deerminisic seady sae. This cos of he business cycle can easily be ransformed ino consumpion equivalen unis. We can compue he percenage decrease in consumpion τ ha will make he household indifferen beween consuming (1 τ )c ss unis per period wih cerainy or c unis wih uncerainy. To do so, noe ha he seady-sae value funcion is jus V ss = c υ ss (1 l ss) 1 υ, which implies ha: 1 The perurbaion mehod is relaed o Benigno and Woodford (2006) and Hansen and Sargen (1995). Benigno and Woodford presen a linear-quadraic (LQ) approximaion o solve opimal policy problems when he consrains of he problem are non-linear (see also Levine e al., 2007). This allows hem o find he correc local welfare ranking of differen policies. Our perurbaion can also deal wih non-linear consrains and obains he correc local approximaion o welfare and policies, bu wih he advanage ha i is easily generalizable o higher-order approximaions. Hansen and Sargen (1995) modify he LQ problem o adjus for risk. In ha way, hey can handle some versions of recursive uiliies. Hansen and Sargen s mehod, however, requires imposing a igh funcional form for fuure uiliy and o surrender he assumpion ha risk-adjused uiliy is separable across saes of he world. Perurbaion does no suffer from hose limiaions.

6 D. Caldara e al. / Review of Economic Dynamics 15 (2012) c υ ss (1 l ss) 1 υ V 66,ss = ( (1 τ )c ss ) υ(1 lss ) 1 υ or: Then: V ss V 66,ss = (1 τ ) υ V ss. [ τ = V 66,ss V ss ] 1 υ. We are perurbing he value funcion in levels of he variables. However, here is nohing special abou levels and we could have done he same in logs (a common pracice when linearizing DSGE models) or in any oher funcion of he saes. These changes of variables may improve he performance of perurbaion (Fernández-Villaverde and Rubio-Ramírez, 2006). By doing he perurbaion in levels, we are picking he mos conservaive case for perurbaion. Since one of he conclusions ha we will reach from our numerical resuls is ha perurbaion works surprisingly well in erms of accuracy, ha conclusion will only be reinforced by an appropriae change of variables. 2 The decision rules can be expanded in he same way. For example, he second-order approximaion of he decision rule for consumpion is, under differeniabiliy assumpions: c(s ) c ss + c i,ss s i c ij,sss i s j where we have followed he same derivaive and ensor noaion as before. As wih he approximaion of he value funcion, van Binsbergen e al. (2009) show ha γ does no affec he values of any of he coefficiens excep c 66,ss. This erm is a consan ha capures precauionary behavior caused by risk. This observaion ells us wo facs. Firs, a linear approximaion o he decision rule does no depend on γ (i is cerainy equivalen), and herefore, if we are ineresed in recursive preferences, we need o go a leas o a second-order approximaion. Second, given some fixed parameer values, he difference beween he second-order approximaion o he decision rules of a model wih CRRA preferences and a model wih recursive preferences is a consan. This consan generaes a second, indirec effec, because i changes he ergodic disribuion of he sae variables and, hence, he poins where we evaluae he decision rules along he equilibrium pah. These argumens demonsrae how perurbaion mehods can provide analyic insighs beyond compuaional advanages and help in undersanding he numerical resuls in Tallarini (2000). In he hird-order approximaion, all of he erms on funcions of χ 2 depend on γ. Following he same seps, we can derive he decision rules for labor, invesmen, and capial. In addiion we have funcions ha give us he evoluion of oher variables of ineres, such as he pricing kernel or he risk-free gross ineres rae R f. All of hese funcions have he same srucure and properies regarding γ as he decision rule for consumpion. In he case of funcions pricing asses, he second-order approximaion generaes a consan risk premium, while he hirdorder approximaion creaes a ime-varying risk premium. Once we have reached his poin, here are wo pahs we can follow o solve for he coefficiens of he perurbaion. The firs procedure is o wrie down he equilibrium condiions of he model plus he definiion of he value funcion. Then, we ake successive derivaives in his augmened se of equilibrium condiions and solve for he unknown coefficiens. This approach, which we call equilibrium condiions perurbaion (ECP), ges us, afer n ieraions, he n-h-order approximaion o he value funcion and o he decision rules. A second procedure is o ake derivaives of he value funcion wih respec o saes and conrols and use hose derivaives o find he unknown coefficien. This approach, which we call value funcion perurbaion (VFP), delivers afer (n + 1) seps, he (n + 1)-h-order approximaion o he value funcion and he n-h-order approximaion o he decision rules. 3 Loosely speaking, ECP underakes he firs sep of VFP by hand by forcing he researcher o derive he equilibrium condiions. The ECP approach is simpler bu i relies on our abiliy o find equilibrium condiions ha do no depend on derivaives of he value funcion. Oherwise, we need o modify he equilibrium condiions o include he definiions of he derivaives of he value funcion. Even if his is possible o do (and no paricularly difficul), i amouns o solving a problem ha is equivalen o VFP. This observaion is imporan because i is easy o posulae models ha have equilibrium condiions where we canno ge rid of all he derivaives of he value funcion (for example, in problems of opimal policy design). ECP is also faser from a compuaional perspecive. However, VFP is only rivially more involved because finding he (n + 1)-horder approximaion o he value funcion on op of he n-h-order approximaion requires nearly no addiional effor. 2 This commen beges he quesion, neverheless, of why we did no perform a perurbaion in logs, since many economiss will find i more naural han using levels. Our experience wih he CRRA uiliy case (Aruoba e al., 2006) and some compuaions wih recursive preferences no included in he paper sugges ha a perurbaion in logs does slighly worse han a perurbaion in levels. 3 The classical sraegy of finding a quadraic approximaion of he uiliy funcion o derive a linear decision rule is a second-order example of VFP (Anderson e al., 1996). A sandard linearizaion of he equilibrium condiions of a DSGE model when we add he value funcion o hose equilibrium condiions is a simple case of ECP. This is done, for insance, in Schmi-Grohé and Uribe (2005).

7 194 D. Caldara e al. / Review of Economic Dynamics 15 (2012) The algorihm presened here is based on he sysem of equilibrium equaions derived using he ECP. In Appendix A, we derive a sysem of equaions using he VFP. We ake he firs-order condiions of he social planner. Firs, wih respec o consumpion oday: V c μ = 0 where μ is he Lagrangian muliplier associaed wih he resource consrain. Second, wih respec o capial: μ + E μ ( ζ e z k ζ 1 l1 ζ + 1 δ) = 0. Third, wih respec o labor: 1 υ υ c (1 l ) = (1 ζ)ez k ζ l ζ. Then, we have E m (ζ e z k ζ 1 l1 ζ + 1 δ) = 1 where m was derived in Eq. (2). Noe ha, as explained above, he derivaives of he value funcion in (2) are eliminaed. Once we subsiue for he pricing kernel, he augmened equilibrium condiions are: V [(1 β) ( c υ (1 l 1 υ) ) E [β ( c c c + β ( E V (k, z ) ) ] 1 = 0 ) 1( ) V 1 1 (ζ e z E V k ζ 1 l1 ζ + 1 δ)] 1 = 0 1 υ υ (1 l ) = (1 ζ)ez k ζ l ζ = 0 ( ) c 1( V E β c E V c + i e z k ζ l1 ζ = 0 k i (1 δ)k = 0 ) 1 1 R f 1 = 0 plus he law of moion for produciviy and volailiy. Noe ha all he endogenous variables are funcions of he saes and ha we drop he max operaor in fron of he value funcion because we are already evaluaing i a he opimum. Thus, a more compac noaion for he previous equilibrium condiions as a funcion of he saes is: F (0) = 0 where F : R 6 R 8. In seady sae, m ss = β and he se of equilibrium condiions simplifies o: V ss = c υ ss (1 l ss) 1 υ ( ζ 1 ζk ss l 1 ζ ss + 1 δ ) = 1/β 1 υ υ c ss (1 l ss ) = (1 ζ)kζ ssl ζ ss R f ss = 1/β c ss + i ss = k ζ ssl 1 ζ ss i ss = δk ss a sysem of 6 equaions on 6 unknowns, V ss, c ss, k ss, i ss, l ss, and R f ss ha can be easily solved (see Appendix A for he derivaions). This seady sae is idenical o he seady sae of he real business cycle model wih a sandard CRRA uiliy funcion and no SV. To find he firs-order approximaion o he value funcion and he decision rules, we ake firs derivaives of he funcion F wih respec o he saes s and evaluae hem a 0: F i (0) = 0 for i {1,...,6}. This sep gives us 48 differen firs derivaives (8 equilibrium condiions imes he 6 variables of F ). Since each dimension of F is equal o zero for all possible values of s, heir derivaives mus also be equal o zero. Therefore, once we subsiue

8 D. Caldara e al. / Review of Economic Dynamics 15 (2012) he seady-sae values and forge abou he exogenous processes (which we do no need o solve for), we have a quadraic sysem of 36 equaions on 36 unknowns: V i,ss, c i,ss, i i,ss, k i,ss, l i,ss, and R f for i {1,...,6}. One of he soluions is an i,ss unsable roo of he sysem ha violaes he ransversaliy condiion of he problem and we eliminae i. Thus, we keep he soluion ha implies sabiliy. To find he second-order approximaion, we ake derivaives on he firs derivaives of he funcion F, againwihrespec o he saes and he perurbaion parameer: F ij (0) = 0 for i, j {1,...,6}. This sep gives us a new sysem of equaions. Then, we plug in he erms ha we already know from he seady sae and from he firs-order approximaion and we ge ha he only unknowns lef are he second-order erms of he value funcion and oher funcions of ineres. Quie convenienly, his sysem of equaions is linear and i can be solved quickly. Repeaing hese seps (aking higher-order derivaives, plugging in he erms already known, and solving for he remaining unknowns), we can ge any arbirary order approximaion. For simpliciy, and since we checked ha we were already obaining a high accuracy, we decided o sop a a hird-order approximaion (we are paricularly ineresed in applying he perurbaion for esimaion purposes and we wan o documen how a hird-order approximaion is accurae enough for many problems wihou spending oo much ime deriving higher-order erms) Projecion Projecion mehods ake basis funcions o build an approximaed value funcion and decision rules ha minimize a residual funcion defined by he augmened equilibrium condiions of he model. There are wo popular mehods for choosing basis funcions: finie elemens and he specral mehod. We will presen only he specral mehod for several reasons: firs, in he neoclassical growh model he decision rules and value funcion are smooh and specral mehods provide an excellen approximaion. Second, specral mehods allow us o use a large number of basis funcions, wih he consequen high accuracy. Third, specral mehods are easier o implemen. Their main drawback is ha since hey approximae he soluion wih a specral basis, if he decision rules display a rapidly changing local behavior or kinks, i may be difficul for his scheme o capure hose local properies. Our arge is o solve he decision rule for labor and he value funcion {l, V } from: [ u c, β ( E V ) 1 1 [ ( )( 1) ( ] H(l, V ) = E V u c, ζ e z k ζ 1 l1 ζ + 1 δ)] V [ (1 β) ( c υ ( )) 1 l υ ( ) 1 + βe V ] = 0 where, o save on noaion, we define V = V (k, z, σ ) and: u c, = 1 γ Then, from he saic condiion c = υ (cυ (1 l ) 1 υ ). c υ 1 υ (1 ζ)ez k ζ l ζ (1 l ) and he resource consrain, we can find c and k. Specral mehods solve his problem by specifying he decision rule for labor and he value funcion {l, V } as linear combinaions of weighed basis funcions: l(k, z j,σ m ; ρ) = i V (k, z j,σ m ; ρ) = i ρ l ijm ψ i(k ) ρ V ijm ψ i(k ) where {ψ i (k)} i=1,...,nk are he n k basis funcions ha we will use for our approximaion along he capial dimension and ρ = {ρ l ijm, ρ V ijm } i=1,...,n k ; j=1,..., J; m=1,...,m are unknown coefficiens o be deermined. In his expression, we have discreized he sochasic processes σ for volailiy and z for produciviy using Tauchen s (1986) mehod as follows. Firs, we have a grid of M poins G σ ={e σ 1, e σ 2,...,e σ M } for σ and a ransiion marix Π M wih generic elemen πi, M j = Prob(eσ = e σ j e σ = e σ i ). The grid covers 3 sandard deviaions of he process in each direcion. Then, for each M poin, we find a grid wih J poins G m z ={zm 1, zm 2,...,zm J } for z and ransiion marices Π J,m wih generic elemen π J,m i, j = Prob(z m = zm j zm = z m i ). Again, and condiional on e σ m, he grid covers 3 sandard deviaions in each direcion. Values for he decision rule ouside he grids G σ and G m z can be approximaed by inerpolaion. We make he grids for z depend on he level of volailiy m o adap he accuracy of Tauchen s procedure o each condiional variance (alhough his forces us o inerpolae when we swich variances). Since we se J = 25 and M = 5, he approximaion is quie accurae along he produciviy axis (we explored oher choices of J and M o be sure ha our choice was sensible).

9 196 D. Caldara e al. / Review of Economic Dynamics 15 (2012) A common choice for he basis funcions are Chebyshev polynomials because of heir flexibiliy and convenience. Since heir domain is [ 1, 1], we need o bound capial o he se [k, k], where k (k) is chosen sufficienly low (high) o bind only wih exremely low probabiliy, and define a linear map from hose bounds ino [ 1, 1]. Then, we se ψ i (k ) = ψ i (φ k (k )) where ψ i ( ) are Chebyshev polynomials and φ k (k ) is he linear mapping from [k, k] o [ 1, 1]. By plugging l(k, z j, σ m ;ρ) and V (k, z j, σ m ;ρ) ino H(l, V ), we find he residual funcion: R(k, z j,σ m ; ρ) = H ( l(k, z j,σ m ; ρ), V (k, z j,σ m ; ρ) ). We deermine he coefficiens ρ o ge he residual funcion as close o 0 as possible. However, o do so, we need o choose a weigh of he residual funcion over he space (k, z j, σ m ). A collocaion poin crierion delivers he bes rade-off beween speed and accuracy (Fornberg, 1998) by making he residual funcion exacly equal o zero in {k i } n k i=1 roos of he n k-horder Chebyshev polynomial and in he Tauchen poins (also, by he Chebyshev inerpolaion heorem, if an approximaing funcion is exac a he roos of he n k -h-order Chebyshev polynomial, hen, as n k, he approximaion error becomes arbirarily small). Therefore, we jus need o solve he following sysem of n k J M 2equaions: R(k i, z j,σ m ; ρ) = 0 for any i, j,m collocaion poins on n k J M 2 unknowns ρ. We solve his sysem wih a Newon mehod and an ieraion based on he incremen of he number of basis funcions. Firs, we solve a sysem wih only hree collocaion poins for capial and 125 (125 = 25 5) poins for echnology. Then, we use ha soluion as a guess for a sysem wih more collocaion poins for capial (wih he new coefficiens being guessed o be equal o zero) and ierae on he procedure. We sop he ieraion when we have 11 polynomials in he capial dimension (herefore, in he las sep we solve for 2750 = coefficiens). The ieraion is needed because oherwise he residual funcion is oo cumbersome o allow for direc soluion of he 2750 final coefficiens Value funcion ieraion Our final soluion mehod is VFI. Since he dynamic algorihm is well known, our presenaion is mos brief. Consider he following Bellman operaor: [ TV(k, z,σ ) = max (1 β) ( c υ (1 l 1 υ) ) c,l s.. c + k = e z k ζ l1 ζ + (1 δ)k z = λz 1 + e σ ε, ε N (0, 1) σ = (1 ρ)σ + ρσ 1 + ηω, ω N (0, 1). + β ( E V (k, z,σ ) ) ] 1 To solve for his Bellman operaor, we define a grid on capial, G k ={k 1, k 2,...,k M }, a grid on volailiy and on he produciviy level. The grid on capial is jus a uniform disribuion of poins over he capial dimension. As we did for projecion, we se a grid G σ ={e σ 1, e σ 2,...,e σ M } for σ and a ransiion marix Π M for volailiy and M grids G m z ={zm 1, zm 2,...,zm J } for z and ransiion marices Π J,m using Tauchen s (1986) procedure. The algorihm o ierae on he value funcion for his grid is: 1. Se n = 0 and V 0 (k, z, σ ) = c υ ss (1 l ss) 1 υ for all k G k and all z G z. 2. For every {k, z, σ }, use he Newon mehod o find c, l, k ha solve: c = υ 1 υ (1 ζ)ez k ζ l ζ (1 l ) (1 β)υ (cυ (1 l ) 1 υ ) = β ( ( ) E V n ) 1 1 [( ) E V n γ ] V n 1, c c + k = e z k ζ l1 ζ + (1 δ)k. 3. Consruc V n+1 from he Bellman equaion: ( V n+1 = (1 β) ( c υ ( 1 l ) 1 υ ) + β ( E ( V ( k, z,σ ) )) 1 ). 4. If V n+1 V n V n 1.0e 7,henn = n + 1 and go o 2. Oherwise, sop. To accelerae convergence and give VFI a fair chance, we implemen a muligrid scheme as described by Chow and Tsisiklis (1991). We sar by ieraing on a small grid. Then, afer convergence, we add more poins o he grid and recompue

10 D. Caldara e al. / Review of Economic Dynamics 15 (2012) Table 1 Calibraed parameers. Parameer β υ ζ δ λ logσ ρ η Value he Bellman operaor using he previously found value funcion as an iniial guess (wih linear inerpolaion o fill he unknown values in he new grid poins). Since he previous value funcion is an excellen grid, we quickly converge in he new grid. Repeaing hese seps several imes, we move from an iniial 23,000-poin grid ino a final one wih 375,000 poins (3000 poins for capial, 25 for produciviy, and 5 for volailiy). 4. Calibraion We now selec a benchmark calibraion for our numerical compuaions. We follow he lieraure as closely as possible and selec parameer values o mach, in he seady sae, some basic observaions of he U.S. economy (as we will see below, for he benchmark calibraion, he means of he ergodic disribuion and he seady-sae values are nearly idenical). We se he discoun facor β = o generae an annual ineres rae of around 3.6 percen. We se he parameer ha governs labor supply, = 0.357, o ge he represenaive household o work one-hird of is ime. The elasiciy of oupu o capial, ζ = 0.3, maches he labor share of naional income. A value of he depreciaion rae δ = maches he raio of invesmen-oupu. Finally, λ = 0.95 and log σ = are sandard values for he sochasic properies of he Solow residual. For he SV process, we pick ρ = 0.9 and η = 0.06, o mach he persisence and sandard deviaion of he heeroskedasic componen of he Solow residual during he las 5 decades. (See Table 1.) Since we wan o explore he dynamics of he model for a range of values ha encompasses all he esimaes from he lieraure, we selec four values for he parameer ha conrols risk aversion, γ, 2, 5, 10, and 40, and wo values for EIS ψ, 0.5, and 1.5, which bracke mos of he values used in he lieraure (alhough many auhors prefer smaller values for ψ, we found ha he simulaion resuls for smaller ψ s do no change much from he case when ψ= 0.5). We hen compue he model for all eigh combinaions of values of γ and ψ, hais{2, 0.5}, {5, 0.5}, {10, 0.5}, and so on. When ψ = 0.5 and γ = 2, we are back in he sandard CRRA case. However, in he ineres of space, we will repor only a limied subse of resuls ha we find are he mos ineresing ones. We pick as he benchmark case he calibraion {γ,ψ,log σ, η}={5, 0.5, 0.007, 0.06}. These values reflec an EIS cenered around he median of he esimaes in he lieraure, a reasonably high level of risk aversion, and he observed volailiy of produciviy shocks. To check robusness, we increase, in he exreme case, he risk aversion, he average sandard deviaion of he produciviy shock, and he sandard deviaion of he innovaions o volailiy o {γ,ψ,log σ, η}={40, 0.5, 0.021, 0.1}. This case combines levels of risk aversion ha are in he upper bound of all esimaes in he lieraure wih huge produciviy shocks. Therefore, i pushes all soluion mehods o heir limis, in paricular, making life hard for perurbaion since he ineracion of he large precauionary behavior induced by γ and large shocks will move he economy far away from he deerminisic seady sae. We leave he discussion of he effecs of ψ = 1.5 for he robusness analysis a he end of he nex secion. 5. Numerical resuls In his secion we repor our numerical findings. Firs, we presen and discuss he compued decision rules. Second, we show he resuls of simulaing he model. Third, we repor he Euler equaion errors. Fourh, we discuss he effecs of changing he EIS and he perurbaion poin. Finally, we discuss implemenaion and compuing ime Decision rules Fig. 1 plos he decision rules of he household for consumpion and labor in he benchmark case. In he op wo panels, we plo he decision rules along he capial axis when z = 0 and σ = σ. The capial inerval is cenered on he seady-sae level of capial of 9.54 wih a widh of ±40%, [5.72,13.36]. This inerval is big enough o encompass all he simulaions. We also plo, wih wo verical lines, he 10 and 90 percenile of he ergodic disribuion of capial. 4 Since all mehods provide nearly indisinguishable answers, we observe only one line in boh panels. I is possible o appreciae miniscule differences in labor supply beween second-order perurbaion and he oher mehods only when capial is far from is seady-sae level. Monooniciy of he decision rules is preserved by all mehods. 5 We mus be cauious, however, mapping differences in choices ino differences in uiliy. The Euler error funcion below provides a beer view of he welfare consequences of differen approximaions. 4 There is he echnical consideraion of which ergodic disribuion o use for his ask, since his is an objec ha can only be found by simulaion. All over he paper, we use he ergodic simulaion generaed by VFI. We checked ha he resuls are robus o using he ergodic disribuions coming from he oher mehods. 5 Similar figures could be ploed for oher values of z and σ. We omi hem because of space consideraions.

11 198 D. Caldara e al. / Review of Economic Dynamics 15 (2012) Fig. 1. Decision rules, benchmark calibraion. In he boom wo panels, we plo he decision rules as a funcion of σ when k = k ss and z = 0. Boh VFI and Chebyshev polynomials capure well precauionary behavior: he household consumes less and works more when σ is higher. In comparison, i would seem ha perurbaions do a much poorer job a capuring his behavior. However, his is no he righ inerpreaion. In a second-order perurbaion, he effec of σ only appears in a erm where σ ineracs wih z. Since here we are ploing a cu of he decision rule while keeping z = 0, ha erm drops ou and he decision rule ha comes from a second-order approximaion is a fla line. In he hird-order perurbaion, σ appears by iself, bu raised o is cube. Hence, he slope of he decision rule is negligible. As we will see below, he ineracion effecs of σ wih oher variables, which are hard o visualize in wo-dimensional graph, are all ha we need o deliver a saisfacory overall performance. We see bigger differences in he decision rules as we increase he risk aversion and he variance of innovaions. Fig. 2 plos he decision rules for he exreme calibraion following he same convenion han before. In his figure, we change he capial inerval where we compue he op decision rules o [3, 32] (roughly 1/3 and 3 imes he seady-sae capial) because, owing o he high variance of he calibraion, he equilibrium pahs flucuae hrough much wider ranges of capial. We highligh several resuls. Firs, all he mehods deliver similar resuls in our original capial inerval for he benchmark calibraion. Second, as we go far away from he seady sae, VFI and he Chebyshev polynomial sill overlap wih each oher (and, as shown by our Euler error compuaions below, we can roughly ake hem as he exac soluion), bu second- and hird-order approximaions sar o deviae. Third, he decision rule for consumpion approximaed by he hird-order perurbaion changes from concaviy ino convexiy for values of capial bigger han 15. This phenomenon (also documened in Aruoba e al., 2006) is due o he poor performance of local approximaion when we move oo far away from he expansion poin and he polynomials begin o behave wildly. Numerically, his issue is irrelevan because he problemaic region is visied wih nearly zero probabiliy Simulaions Applied economiss ofen characerize he behavior of he model hrough saisics from simulaed pahs of he economy. We simulae he model, saring from he deerminisic seady sae, for 10,000 periods, using he decision rules for each of he eigh combinaions of risk aversion and EIS discussed above. To make he comparison meaningful, he shocks are common across all pahs. We discard he firs 1000 periods as a burn-in o eliminae he ransiion from he deerminisic seady sae of he model o he middle regions of he ergodic disribuion of capial. This is usually achieved in many fewer

12 D. Caldara e al. / Review of Economic Dynamics 15 (2012) Fig. 2. Decision rules, exreme calibraion. periods han he ones in our burn-in, bu we wan o be conservaive in our resuls. The remaining observaions consiue a sample from he ergodic disribuion of he economy. For he benchmark calibraion, he simulaions from all of he soluion mehods generae almos idenical equilibrium pahs (and herefore we do no repor hem). We focus insead on he densiies of he endogenous variables as shown in Fig. 3 (remember ha volailiy and produciviy are idenical across he differen soluion mehods). Given he low risk aversion and SV of he produciviy shocks, all densiies are roughly cenered around he deerminisic seady-sae value of he variable. For example, he mean of he disribuion of capial is only 0.2 percen higher han he deerminisic value. Also, capial is nearly always beween 8.5 and This range will be imporan below o judge he accuracy of our approximaions. Table 2 repors business cycle saisics and, because DSGE models wih recursive preferences and SV are ofen used for asse pricing, he average and variance of he (quarerly) risk-free rae and reurn on capial. Again, we see ha nearly all values are he same, a simple consequence of he similariy of he decision rules. The welfare cos of he business cycle is repored in Table 3 in consumpion equivalen erms. The compued coss are acually negaive. Besides he Jensen s effec on average produciviy, his is also due o he fac ha when we have leisure in he uiliy funcion, he indirec uiliy funcion may be convex in inpu prices (agens change heir behavior over ime by a large amoun o ake advanage of changing produciviy). Cho and Cooley (2000) presen a similar example. Welfare coss are comparable across mehods. Remember ha he welfare cos of he business cycle for he second- and hird-order perurbaions is he same because he hird-order erms all drop or are zero when evaluaed a he seady sae. When we move o he exreme calibraion, we see more differences. Fig. 4 plos he hisograms of he simulaed series for each soluion mehod. Looking a quaniies, he hisograms of consumpion, oupu, and labor are he same across all of he mehods. The ergodic disribuion of capial pus nearly all he mass beween values of 6 and 15. This considerable move o he righ in comparison wih Fig. 3 is due o he effec of precauionary behavior in he presence of high risk aversion, large produciviy shocks, and high SV. Capial also visis low values of capial more han in he benchmark calibraion because of large, persisen produciviy shocks. In any case, he ranslaion is more pronounced o he righ han o he lef. Table 4 repors business cycle saisics. Differences across mehods are minor in erms of means (noe ha he mean of he risk-free rae is lower han in he benchmark calibraion because of he exra accumulaion of capial induced by

13 200 D. Caldara e al. / Review of Economic Dynamics 15 (2012) Fig. 3. Densiies, benchmark calibraion. Table 2 Business cycle saisics benchmark calibraion. c y i R f (%) R k (%) Mean Second-order perurbaion Third-order perurbaion Chebyshev polynomial Value funcion ieraion Variance (%) Second-order perurbaion Third-order perurbaion Chebyshev polynomial Value funcion ieraion Table 3 Welfare coss of business cycle benchmark calibraion. 2nd-order per. 3rd-order per. Chebyshev Value funcion e( 5) e( 5) e( 5) e( 5) precauionary behavior). In erms of variances, he second-order perurbaion produces less volailiy han all oher mehods. This suggess ha a second-order perurbaion may no be good enough if we face high variance of he shocks and/or high risk aversion. A hird-order perurbaion, in comparison, eliminaes mos of he differences and delivers nearly he same implicaions as Chebyshev polynomials or VFI. Finally, Table 5 presens he welfare cos of he business cycle. Now, in comparison wih he benchmark calibraion, he welfare cos of he business cycle is posiive and significan, slighly above 1.1 percen. This is no a surprise, since we have boh a large risk aversion and produciviy shocks wih an average sandard deviaion hree imes as big as he observed one. All mehods deliver numbers ha are close.

14 D. Caldara e al. / Review of Economic Dynamics 15 (2012) Fig. 4. Densiies, exreme calibraion. Table 4 Business cycle saisics benchmark calibraion. c y i R f (%) R k (%) Mean Second-order perurbaion Third-order perurbaion Chebyshev polynomial Value funcion ieraion Variance (%) Second-order perurbaion Third-order perurbaion Chebyshev polynomial Value funcion ieraion Table 5 Welfare coss of business cycle exreme calibraion. 2nd-order per. 3rd-order per. Chebyshev Value funcion e e e e Euler equaion errors While he plos of he decision rules and he compuaion of densiies and business cycle saisics ha we presened in he previous subsecion are highly informaive, i is also imporan o evaluae he accuracy of each of he procedures. Euler equaion errors, inroduced by Judd (1992), have become a common ool for deermining he qualiy of he soluion mehod. The idea is o observe ha, in our model, he ineremporal condiion: u c, = β ( E V ) 1 ( (γ 1)(1 ) ) 1 E V u c, R(k, z,σ ; z,σ ) (4)

15 202 D. Caldara e al. / Review of Economic Dynamics 15 (2012) Fig. 5. Euler equaion error, benchmark calibraion. where R(k, z, σ ; z, σ ) = 1 + ζ e z k ζ 1 l1 ζ δ is he gross reurn of capial given saes k, z, σ, and realizaions z and σ should hold exacly for any given k and z. However, since he soluion mehods we use are only approximaions, here will be an error in (4) when we plug in he compued decision rules. This Euler equaion error funcion EE i (k, z, σ ) is defined, in consumpion erms: EE i (k, z,σ ) = 1 [ ( ( ) β E V i ) 1 1 (( ) (γ 1)(1 ) E V i υ ( 1 l i u i c, R(k,z,σ ;z,σ ) ) ) (1 υ) c i ] 1 υ 1 This funcion deermines he (uni free) error in he Euler equaion as a fracion of he consumpion given he curren saes and soluion mehod i. Following Judd and Guu (1997), we can inerpre his error as he opimizaion error incurred by he use of he approximaed decision rule and we repor he absolue errors in base 10 logarihms o ease inerpreaion. Thus, avalueof 3 means a $1 misake for each $1000 spen, a value of 4 a $1 misake for each $10,000 spen, and so on. Fig. 5 displays a ransversal cu of he errors for he benchmark calibraion when z = 0 and σ = σ. Oher ransversal cus a differen echnology and volailiy levels reveal similar paerns. The firs lesson from Fig. 5 is ha all mehods deliver high accuracy. We know from Fig. 3 ha capial is nearly always beween 8.5 and In ha range, he (log 10) Euler equaion errors are a mos 5, and mos of he ime hey are even smaller. For insance, he second- and hirdorder perurbaions have an Euler equaion error of around 7 in he neighborhood of he deerminisic seady sae, VFI of around 6.5, and Chebyshev an impressive 11/ 13. The second lesson from Fig. 5 is ha, as expeced, global mehods (Chebyshev and VFI) perform very well in he whole range of capial values, while perurbaions deeriorae as we move away from he seady sae. For second-order perurbaion, he Euler error in he seady sae is almos four orders of magniude smaller han on he boundaries. Third-order perurbaion is around half an order of magniude more accurae han second-order perurbaion over he whole range of values (excep in a small region close o he deerminisic seady sae). There are wo complemenary ways o summarize he informaion from Euler equaion error funcions. Firs, we repor he maximum error in our inerval (capial beween 60 percen and 140 percen of he seady sae and he grids for produciviy and volailiy). The maximum Euler error is useful because i bounds he misake owing o he approximaion. The second procedure for summarizing Euler equaion errors is o inegrae he funcion wih respec o he ergodic disribuion of capial and produciviy o find he average error. We can hink of his exercise as a generalizaion of he Den Haan Marce es (Den Haan and Marce, 1994). The op-lef panel in Fig. 6 repors he maximum Euler error (darker bars) and he inegral of he Euler error for he benchmark calibraion. Boh perurbaions have a maximum Euler error of around 2.7, VFI of 3.1, and Chebyshev, an impressive 9.8. We read his resul as indicaing ha all mehods perform adequaely. Boh perurbaions have roughly he same inegral of he Euler error (around 5.3), VFI a slighly beer 6.4, while Chebyshev polynomials do fanasically well a 10.4 (he average loss of welfare is $1 for each $500 billion). Bu even an approximaion wih an average error of $1 for each $200,000, such as he one implied by hird-order perurbaion, mus suffice for mos relevan applicaions. We repea our exercise for he exreme calibraion. As we did when we compued he decision rules of he agens, we have changed he capial inerval o [3, 32]. The op-righ panel in Fig. 6 repors maximum Euler equaion errors and heir inegrals. The maximum Euler equaion error is large for perurbaion mehods while i is raher small using Chebyshev polynomials. However, given he very large range of capial used in he compuaion, his maximum Euler error provides a.