Calculating and Using Second Order Accurate Solutions of Discrete Time Dynamic Equilibrium Models

CALCULATING AND USING SECOND ORDER ACCURATE SOLUTIONS OF DISCRETE TIME DYNAMIC EQUILIBRIUM MODELS JINILLKIM,SUNGHYUNKIM,ERNSTSCHAUMBURG,ANDCHRISTOPHERA.SIMS ABSTRACT. Wedescribeanalgorithmforcalculatingsecondorderapproximationstothe solutions to nonlinear stochastic rational expectations models. The paper also explains methodsforusingsuchanapproximatesolutiontogenerateforecasts,simulatedtimepaths for the model, and evaluations of expected welfare differences across different versions of a model. The paper givesconditions for local validityof the approximation that allow fordisturbancedistributionswithunboundedsupportandallowfornon-stationarityofthe solutionprocess. 1. INTRODUCTION It is now widely understood how to obtain first-order accurate approximations to the solutiontoadynamic,stochasticgeneralequilibriummodel(DSGEmodel). Suchsolutions are fairly easy to construct and useful for a wide variety of purposes. They are likely to be accurateenoughtobeabasisforfittingthemodelstodata,forexample. However,forsomepurposesfirst-orderaccuracyisnotenough. Thisistrueinparticular for comparing welfare across policies that do not have first-order effects on the model’s deterministic steady state, for example. It is also true for attempts to study asset pricing in the context of DSGE models. It is possible to assume directly that nonlinearities are Date:August3,2003. Discussions,andinsomecasesexchangeofcodetestingresults,withFabriceCollard,KennethL.Judd, Robert Kollmann, Stephanie Schmitt-Grohe´, and Martin Uribe have been useful to us. Kollmann has contributed to the Matlab code that implements the paper’s algorithm. (cid:176)c2003 by Jinill Kim, Sunghyun Kim, ErnstSchamburgandChristopherSims. Thismaterialmaybereproducedforeducationalandresearchpurposessolongasthecopiesarenotsold,eventorecovercosts,thedocumentisnotaltered,andthiscopyright noticeisincludedinthecopies. 1

SECONDORDERSOLUTION 2 themselves small in certain dimensions as a justification for use of first-order approximations in these contexts; Woodford (2002) is an example of making the necessary auxiliary assumptionsexplicit. Buttheusualrelianceonlocalapproximationbeinggenerallylocally accuratedoesnotapplytothesecontexts. It is therefore of some interest to have an algorithm available that will produce secondorder accurate approximations to the solutions to DSGE’s from a straightforward secondorder expansion of the model’s equilibrium equations, and this is an active area of recent research. KennethJuddpioneeredthisfieldbyusingperturbationmethodsinsolvingvarioustypes ofeconomicmodels1. JinandJudd(2002)describehowtocomputeapproximationsofarbitraryorderindiscrete-timerationalexpectationsmodels. Theyaimatprovidingacomplete set of regularity conditions justifying the local approximations, and they discuss methods for checking the validity of the approximations. Others also have studied perturbation methods of higher than first order including Collard and Juillard (2000), Anderson and Levin(2002),andSchmitt-Grohe´ andUribe (2002). Kim and Kim (2003a) and Sutherland (2002) have developed a bias correction method that produce the same results as the second order perturbation method for certain welfare calculations,whilerequiringlesscomputationaleffortthanafullperturbationsolution. Several papers have applied the second-order perturbation method to dynamic general equilibrium models. Kim and Kim (2003b) used the second-order solution method to analyzewelfareeffectsoftaxpoliciesinatwo-countryframework. Inparticular,theycalculate the optimal degree of response for various tax rates to TFP shocks faced by each country. Welfaregainsoftaxpoliciesaremeasuredbyconditionalwelfarechangesfromthebenchmarkcase. Kollmann(2002)hasanalyzedthewelfareeffectsofmonetarypoliciesinopen economies using the software that has been developed along with this paper, and Bergin andTchakarov(2002)haveusedittoexaminethewelfareeffectsofexchangeraterisk. 1Judd(1998). ForcontinuoustimemodelsseeGasparandJudd(1997)aswell.

SECONDORDERSOLUTION 3 This paper describes the algorithm for computing a second order approximation and shows how to apply it to calculating forecasts and impulse responses in dynamic models and to evaluating welfare in DSGE models. It points out some necessary regularity conditions for application of the method and discusses the sense in which the approximate solutionsarelocallyaccurate. While much of the paper parallels others in this rapidly growing literature, this paper makes some new contributions. The rest of the literature in most cases begins from a formulation of the problem in which a partition of variables in the model into “states” and “controls” or “co-states” is assumed known. While in smaller models such a partition is often obvious, in larger models it can be unclear how to partition the variables into states and controls. The Matlab program gensys.m, implementing the approach described in Sims (2001), accepts model specifications that do not partition the variable list into predetermined and non-predetermined variables; instead it partitions disturbances into predetermined and non-predetermined categories. This approach is more natural in systems derived from equilibrium models, in which equation disturbances often fall neatly into these categories. In such models translating the list of predetermined disturbances into a correspondinglistofpredeterminedvariables(or,wherenecessary,newpredeterminedvariables that are linear combinations of the original model’s variables) may not be easy. This paper extendsthatapproachtosecond-orderapproximations.2 The“state-free”approachofgensys.mhasthedisadvantagethatitsoutput,whilecompletelycharacterizingthedynamicsintermsoftheoriginalvariables,includesonlyitsown artificialdecompositionintostatesandco-states,whichmaybeopaque. Forsomepurposes itisimportant to haveanintuitivelyappealing decompositioninto states andco-states. We discuss how to do this, with the aid of another program, gstate.m, that uses the output of gensys.m or gensys2.m to test proposed state vectors and and to provide guidance astowhatavalidstatevectormustlooklike. 2King and Watson (1998) and Klein (2000) describe solution algorithms that handle the essentially the sameclassofmodelsasSims(2001),butpresumethatthelistofpredeterminedvariablesisgiven.

SECONDORDERSOLUTION 4 Wherethesenseinwhichaccuracyoflocalexpansionsisclaimedhasbeenmadeexplicit intheliterature,ithasforthemostpart(JinandJudd,2002,mostprominently)focusedon accuracyofthefunctionmappingstatevariablestoco-states. Ithasalsotendedtoassertas regularity conditions almost-sure boundedness of stochastic disturbances and stationarity of the dynamic model being studied. These assumptions allow strong claims to be made about approximation accuracy, but they are disquieting for most DSGE modeling applications. Modelswithunitroots,orevenmildexplosiveness,arenotuncommoninmacroeconomics, and models with near-unit roots are the rule. Often disturbance distributions with unbounded support seem more realistic than any particular truncation to bounded support. If perturbation methods break down, or are at the edge of their domain of applicability, for suchmodels,theymightseemtobeunattractiveformanyofthemodelstowhichtheyhave infactbeenapplied. In this paper we argue that boundedness of shocks and stationarity of the model are not essential to the validity of perturbation methods. For their main applications so far, perturbation methods can be shown to produce results that are in a natural sense locally accurate,withouttheinvocationofthedubiousstationarityandboundednessassumptions. Thereislittleexplicitdiscussionintheliteratureofhowtousehigherorderperturbation approximations in constructing simulations, forecasts, and welfare evaluations. We show thatsomeapparentlyobviousapproachestothesetasksinfactresultinanaccumulationof “garbage” high-order terms that can make accuracy deteriorate. We lay out an algorithm that always produces stationary second-order accurate dynamics whenever the first-order dynamicsarestable. TheMatlabcodethatwasbuiltalongwiththispaperisavailableathttp://eco-072399b. princeton.edu/yftp/gensys2/, where the current version of this paper will also befound.

SECONDORDERSOLUTION 5 2. THE GENERAL FORM OF THE MODEL Wesupposeamodelthattakestheform (1) K (w ,w ,se )+P sh =0, t t−1 t t n×1 n×1 m×1 p×1 where E h = 0 and E e = 0.3 The equations hold for t = 0,...,¥ , as does the t t+1 t t+1 E e = 0 condition. The disturbances e are exogenously given, while h is determined t t+1 t t as a function of e when the model is solved, if the solution exists and is unique. Note that because there is no assumption at all about h , it is a free vector that is likely to make 0 certainlinearcombinationsoftheequationstautologicalattheinitialdate. The scale factor s is introduced to allow us to shrink the distribution of e toward zero t as we seek a domain of validity for our local approximation. The distribution of e itself is t assumed to be constant across time t and invariant to changes in s , so that in particular it hasafixedcovariancematrix W . Theequationsystemcouldbewrittenequivalentlyas (2) Q K(w ,w ,se )=0 1 t t−1 t (3) E [Q K(w ,w ,se )]=0, t 2 t+1 t t+1 where Q is any matrix such that Q P =0 and [Q(cid:48),Q(cid:48)] is a full rank square matrix. The 1 1 1 2 “forward-shift”oftheexpectationalblockreflectstheabsenceofanyrestrictionon h . 0 We assume that the solution will imply that w remains always on a stable manifold, t definedby H(w ,s )=0andsatisfying t (4) { H (w ,s )=0,H(w ,s )=0a.s. and Q K(w ,w ,se )=0a.s.} t t+1 1 t+1 t t+1 nu ×1 n×1 ⇒ E [Q K(w ,w ,se )]=0. t 2 t+1 t t+1 3 Thisformismoregeneralthanitmightseem. SeeSims(2001)forexamplesshowinghowmodelswith explicitexpectationsoperators,includinglaggedexpectations,canbecastintothisform.

SECONDORDERSOLUTION 6 We consider expansion of the system about a deterministic steady state w¯, i.e. a point satisfying K(w¯,w¯,0) = 0. We do not need to assume the steady state is unique, so the situation arising in unit root models, where there is a continuum of steady states, is not ruledout. We also assume that the nonlinear system (1) is formulated in such a way that its firstorderexpansioncharacterizesthefirst-orderbehaviorofthedeterministicsolution. Thatis, weassumethatsolvingthefirst-orderexpansionof (1)about w¯, (5) K dw =−K dw −K se +P h , 1 t 2 t−1 3 t t as a linear system results in a unique stable saddle path in the neighborhood of the deterministic steady state. If so, this saddle path characterizes the first-order behavior of the system. WeassumefurtherthatH (w¯,0)isoffullrowrank,sothatthefirst-ordercharacter 1 ofthesaddlepathisdeterminedbythefirst-orderexpansionof H.4 Thesystem (1)hasthesecond-orderTaylorexpansionabout w¯ (6) K dw =−K dw −K se +P h 1ij jt 2ij j,t−1 3ij jt ij jt −1(K dw dw +2K dw dw +2K dw se 2 11ijk jt kt 12ijk jt k,t−1 13ijk jt kt +K dw dw +2K dw se +K s 2e e ), 22ijk j,t−1 k,t−1 23ijk j,t−1 kt 33ijk jt kt wherewehaveresortedtotensornotation. Thatis,weareusingthenotationthat (cid:229) (7) A B =C ⇔ c = a b . ijk mnjq ikmnq ikmnq ijk mnjq j where a,b,c in this expression refer to individual elements of multidimensional arrays, while A,B,C refer to the arrays themselves. As special case, for example, ordinary matrix multiplication is AB = A B and the usual matrix expression A(cid:48)BA becomes A B A . ij jk ji jk km 4 ThisassumptiononH isnotrestrictivesolongasthereisacontinuous,differentiablesaddlemanifold. Howevertherearemodels—someassetpricingmodels,forexample—inwhichthefirstorderapproximationdoesnotdeliverdeterminacy, buthigher-ordertermsdo. Thealgorithmssuggestedherecannothandle modelsofthistype.

SECONDORDERSOLUTION 7 Note that we are distinguishing the array K of first derivatives from the array K of mij mnijk secondderivativesonlybythenumberofindexingsubscriptsthetwoarrayshave. 3. REGULARITY CONDITIONS Because we are taking first and second derivatives and because we are expanding about the steady state w¯, it is clear that we require existence of first and second derivatives of K at w¯. We have also directly assumed that the first order behavior of K near w¯ determines H(·,0). In order to make our local expansion in dw, se , and e work, we will need that H(w,s )iscontinuousandtwice-differentiableinbothitsarguments. It may seem that these are all standard assumptions on the degree of differentiability of the system near w¯. Consider what emerges, though, when we split the system into expectational and non-expectational components as in (2)-(3). If we replace (3) with its second-orderexpansionandtakesomeexpectationsexplicitly,wearriveat (cid:163) (8) E Q (K dw +K dw +1(K dw dw +2K dw dw t 2 1ij j,t+1 2ij jt 2 11ijk j,t+1 k,t+1 12ijk j,t+1 k,t (cid:164) +K dw dw +K W s 2)) =0, 22ijk j,t k,t 33ijk jk and find ourselves needing to assert that e has finite second moments, which is not a local t property. That is, if e does not have second moments, shrinking s will not make se t t have finite second moments. The same point applies to (3) in its original nonlinear form. If it is to be differentiable in w and s , we will in general need to impose restrictions on t the distribution of e . Jin and Judd (2002) have an example of a model in which some t apparently natural choices of a distribution for e imply that E [Q K(w ,w ,se )] is t t 2 t+1 t t+1 discontinuousin s ats =0,eventhough K hasplentyofderivativesatthesteadystate. 4. SOLUTION METHOD Thesolutionwearelookingforcanbewrittenintheform (9) w =F∗(w ,se ,s ). t t−1 t

SECONDORDERSOLUTION 8 Because we know the saddle manifold characterized by H exists and that H (w¯,s ) has 1 full row rank n , we can use H to express n linear combinations of w’s in terms of the u u remaining n = n−n . Let the n linear combinations of w’s chosen as “explanatory” s u s variablesinthisrelationbe (10) y = F w . t t ns ×n Thenthesolution (9)canbeexpressedequivalently,inaneighborhoodof w¯,as (11) y =F F∗(w ,se ,s )=F(y ,x ,se ,s ) t t−1 t t−1 t−1 t (12) x =h(y ,s ), t t nu ×1 where (12) is just the solved version of the H = 0 equation that characterizes the stable manifold. Hereofcourse x,likey,isalinearcombinationof w’s. Theappearanceofx in(11)mayseemredundant,sincealongthesolutionpathwewill t−1 havex =h(y ,s ),butattheinitialdatethelaggedwvectormightnotsatisfythisrestriction. t t This is likely in a growth model with multiple types of capital, for example, where there may be optimal proportions of capital of different types, but no physical requirement that theinitialendowmentsareintheseproportions.5 The solution method for linear rational expectations systems described in Sims (2001) beginsbyapplyinglineartransformationstothelistofvariablesandtotheequationsystem to produce an upper triangular block recursive system. In the transformed system, the unstablerootsofthesystemareallassociatedwiththelowerrightblock,h doesnotappear t in the upper set of equations in the system,6 and the upper part of the equation system is normalizedtohavetheidentityasthecoefficientmatrixoncurrentvaluesoftheupperpart of the transformed variable matrix. In other words, by applying to the equation system the 5 Seesection5belowforfurtherdiscussionofthispoint. 6 It may not be possible in fact to eliminate h from the upper part of the system. When it is not, the t solution is not unique. The programs signal the non-uniqueness and deliver one solution, in which the h ’s aresettozerointheupperblockofthissystem.

SECONDORDERSOLUTION 9 same sequence of linear operations as applied in the earlier paper to a linear system7, we cantransform(6)to (cid:179) dy =G dx +G dv +G e +1 G dv dv +2G dv dv it 1ij jt 2ij j,t−1 3ij jt 2 11ijk jt kt 12ijk jt k,t−1 (13) (cid:180) +2G dv e +G dv dv +2G dv e +G e e 13ijk jt kt 22ijk j,t−1 k,t−1 23ijk j,t−1 kt 33ijk jt kt (cid:179) J dx =J dx +J e +P ∗h +1 J dv dv +2J dv dv 1ij jt 2ij j,t−1 3ij jt t 2 11ijk jt kt 12ijk jt k,t−1 (14) (cid:180) +2J dv e +J dv dv +2J dv e +J e e , 13ijk jt kt 22ijk j,t−1 k,t−1 23ijk j,t−1 kt 33ijk jt kt wherev =(y(cid:48) x(cid:48))(cid:48),i.e. the yandx vectorsstackedup. t t t Nowtheyandxintroducedabovemayseemtohavenoconnectiontotheyandxinterms ofwhichwewrotethesolution (11)-(12). Butthatsolutionhassecond-orderexpansion dy =F dv +F e +F s 2 it 1ij j,t−1 2ij jt 3i (15) (cid:179) (cid:180) +1 F dv dv +2F dv e +F e e 2 11ijk j,t−1 k,t−1 12ijk j,t−1 kt 22ijk jt kt (16) dx = 1M dy dy +M s 2. it 2 11ijk jt kt 2 Ofcourseifxwerechosenasanarbitrarylinearcombinationofw’s,therewouldingeneral be a first-order term in dy on the right-hand side of (16). However, we can always move t such terms to the left-hand side and then redefine x to include them. We will now proceed toshowthatthedyanddxin(15)-(16)areindeedthosein(13)-(14),andthatindeedwecan constructthecoefficientmatricesintheformerfromknowledgeofthecoefficientmatrices inthelatter. Thetermsins in(15)-(16)deservediscussion. Ascanbeseenfrom(8),theappearance of expectations operators in our system makes it depend on the distribution of e , not just on realized values of e . But there is only one term in (8) that is first-order in dw . All t+1 the other terms are second-order, or depend on dw or s 2, not s . Therefore if there were t a component of Q K dw that depended on s (rather than s 2), that term could not be 1 1 t+1 zero as the equation requires. Hence we can be sure that there is no term linear in s in 7andimplementedintheMatlabfunctiongensys.m

SECONDORDERSOLUTION 10 the second order expansion of (2)-(3), and thus none in (15)-(16). This then also rules out any term of the form s ·se also, since such a term could enter only through the cross t+1 productsindw se orthroughthedw dw terms,andwithoutafirst-ordertermin t+1 t+1 t+1 t+1 s indw ,thesecrossproductscangenerateno s ·se terms. t+1 t+1 Observe that dx in (13)-(14) must be zero to first order (except for t = −1), because t otherwise there would be an explosive component in the first order part of the solution, contradicting the stability assumption. Therefore, F is exactly G from (13). Clearly also 1 2 F =G . Thereforewehaveacompletefirst-ordersolutionfor dy anddx inhand: 2 3 . (17) dy =F dv +F e t 1 t−1 2 t . (18) dx =0. t Wefindthesecondordertermsinthefollowingsteps. Firstshift(14)forwardintimeby one (so that the left-hand side is dx ) and substitute the right-hand side of (16), shifted t+1 forward in time by 1, for the dx on the left. Then substitute the right-hand-side of (17), t+1 shifted forward by 1, for all occurrences of dy in the resulting system. Finally apply t+1 the E operator to the result. In doing this, we are dropping all the second order terms in t the solution for dy and dx when these terms themselves occur in second order terms. This makes sense because cross products involving terms higher than first order are third order or higher, and thus do not contribute to the second order expansion. Note that this means that, since dx is zero to first order, in (13)-(14) all the second-order terms in dv can be written in terms of dy alone. We will abuse notation by using the same G and J labels for thesmallersecond-ordercoefficientmatricesthatapplytodyalonethatweusein(13)-(14)

SECONDORDERSOLUTION 11 forthesecondordertermsinvolvingthefull vvector. Inthiswaywearriveat (cid:179) (cid:180) (cid:161) (cid:162) (19) J 1 M F dy F dy +M F F W s 2 +M s 2 1ij 2 11jk(cid:96) 1kr rt 1(cid:96)s st 11jk(cid:96) 2kr 2(cid:96)s rs 2j (cid:179) (cid:180) (cid:179) (cid:161) =J 1M dy dy +M s 2 +1 J F F dy dy 2ij 2 11jk(cid:96) kt (cid:96)t 2j 2 11ijk 1jr 1ks rt st (cid:162) +F F W s 2 +2J F dy dy +2J F W s 2 2jr 2ks rs 12ijk 1jr rt kt 13ijk 2jr rk (cid:180) +J dy dy +J W s 2 , 22ijk jt kt 33ijk jk Wherewehaveset Var(e )=s 2W . t Forthisequationtoholdforalldyands 2values,wemustmatchcoefficientsoncommon terms. Therefore,lookingatthe dy ·dy terms,weconcludethat t t (20) J M F F =J M +J F F +2J F +J . 1ij 11jkt 1kr 1(cid:96)s 2ij 11jrs 11ijk 1jr 1ks 12ijs 1jr 22ijk This is a linear equation, and every element of it is known except for M . The transfor- 11··· mationsthatproducedtheblock-recursivesystemwithorderedrootsguaranteethatJ ,an 2·· ordinary2×2matrix,hasallitseigenvaluesabovethecriticalstabilityvalue. Itistherefore invertible,andwecanmultiply(20)throughontheleftbyJ−1,togetasystemintheform 2 (21) AM∗F ⊗F =M∗+B. 1 1 In this equation, M∗ is the ordinary n ×n2 matrix obtained by stacking up the second and s s third dimensions of M , A=J−1J , and B is everything else in the equation that doesn’t 11··· 2 1 depend on M∗. If the dividing line we have specified between stable and unstable roots is 1+d , then our construction of the block-recursive system has guaranteed that J−1J has 2 1 all its eigenvalues ≤ 1/(1+d ), while at the same time it is a condition on the solution that all the eigenvalues of F be <1+d . To guarantee that a second-order solution exists, 1 we require that the product of the largest eigenvalue of F ⊗F , which is the square of the 1 1 largest eigenvalue of F , be less than the inverse of the largest eigenvalue of A=J−1J . If 1 2 1 d =0 this condition is automatically satisfied. Otherwise, there is an extra condition that wasnotrequiredforfindingasolutiontothelinearsystem: thesmallestunstablerootmust exceedthesquareofthelargeststableroot.

SECONDORDERSOLUTION 12 Assuming this condition holds, (21) has the form of a discrete Lyapunov or Sylvester equation that is guaranteed to have a solution. Because of the special structure of F ⊗F , 1 1 it would be very inefficient to solve this system with standard packages (like Matlab’s lyap.m),butitiseasytoexploitthespecialstructurewithadoublingalgorithmtoobtain anefficientsolutionfor M∗. With M in hand, it is easy to see from (19) that we can obtain a solution for M 11··· 2 by matching coefficients on s 2. The only slightly demanding calculation is a required inversion of J −J . But since J−1J has all its eigenvalues less than one, this J −J is 2 1 2 1 2 1 guaranteedtobenonsingular. Thenextstepistouse(16)tosubstituteforthefirst-ordertermindx ontherightof(13) t and (17)-(18) to substitute for all occurrences of dy and dx in second-order terms on the t t rightintheresultingequation. Thisproducesanequationwithdy ontheleft,andfirstand t second-ordertermsindy ande andtermsins 2 ontheright. WithM andM inhand, t−1 t 11 2 it turns out that it is only a matter of bookkeeping to read off the values of F , F , and F 12 22 3 bymatchingthemtothecollectedcoefficientsinthisequation.8 5. ANALYZING THE STATE REPRESENTATION Thegensys.mprogramproducesasoutput,amongotherthings,afirst-orderexpansion of(9),as . (22) dw =F∗dw +F∗se . t 1 t−1 2 t To find a conventional state-space representation of such a system, we can form a singular valuedecomposition    (cid:104) (cid:105) D 0 R(cid:48) R(cid:48) (23) [F∗F∗]= U V   1 2, 1 2 0 0 S(cid:48) S(cid:48) 1 2 8 This bookkeeping is not trivial to program, but it is probably best for those who need to program it to consulttheprogram,ratherthantakeupspaceherewiththebookkeeping.

SECONDORDERSOLUTION 13 where [U V] and [R S] are orthonormal matrices and D is diagonal. Any state vector z t that has the property that w is determined by z in this system will have to be of the form t t z =q U(cid:48)w . Theonlywayw affectscurrentw isviaR(cid:48)w . WhileR(cid:48) canbethesame t t t−1 t 1 t−1 1 rowrankasU,itcanalsobeless,sothatasmaller“state”vectorsummarizesthepastthan isneededtocharacterizethecurrentsituation. Also,therankofF∗ canbebelowitsnumber 1 ofnon-zerosingularvalues. Inthiscaseitmaybepossibletofindaz that,afterthesystem t has run a few periods, summarizes the past and/or characterizes the current situation yet is oflowerdimensionthantherankof D. The program gstate.m takes as input F∗ and F∗, together with an optional candidate 1 2 matrix f of coefficients that might form a state vector as z = f w . The program checks t t whetherf liesinU’sor R(cid:48)’srowspaceandreturnsU and R(cid:48) forfurtheranalysis. 1 1 Onceastate/co-staterepresentationoftheformv =Y w hasbeensettledon,whereY is t t non-singularandthev=(y(cid:48) x(cid:48))(cid:48) vectorispartitionedintostateandcostate,itisstraightforward to convert a first or second-order approximate solution from one co-ordinate system intotheanother. 6. THE LOCAL ACCURACY OF THE APPROXIMATION Oncewehaveasecond-orderaccurateapproximationtothedynamics,intheform(15)- (16),wecanmakeaclaimtolocalaccuracyofthefollowingform: (24) dv =Fˆ(dv ,se ,s )+o ((cid:107)dv ,s (cid:107)2), t+1 t t+1 p t where o means “order in probability” and Fˆ is the second-order approximation to the p dynamics. That is, the error in the approximation is claimed to converge in probability to zero, at a more rapid rate than (cid:107)dv ,s (cid:107)2 , when (cid:107)dv ,s (cid:107)2 goes to zero. This rate is the t t weakest kind of claim that can be made for a Taylor expansion. If we are willing to claim that third derivatives exist at the deterministic steady state, then we can replace the error termwithO ((cid:107)dv ,s (cid:107)3). Thisclaimdoesnotdependonstrictboundednessofthesupport p t of the distribution of e , because we are only claiming our local accuracy with a certain t (high) probability. Whatever the distribution of e , se converges in probability to zero as t t

SECONDORDERSOLUTION 14 s →0, allowing us to make this claim. Of course this is all dependent on the underlying assumption that the original nonlinear model has dynamics differentiable of sufficiently highorderins intheneighborhoodofdeterministicsteadystate,andontheexistenceand continuityoftheexpectationsthatoccurinthestatementofthemodel. This one-step-ahead “local accuracy in probability” claim obviously can be extended to a corresponding claim to accuracy n-steps-ahead for any finite n. We have made no appeal to stationarity of the system in making these claims. Of course the size of the n for which accuracy remains good at a given level of s will in general be smaller for systems that are not stationary. But the qualitative nature of the accuracy claim is no different for non-stationarysystems. Thistypeoffinite-time-span,accuracy-in-probabilityclaimisexactlywhatisappropriate for purposes of fitting a model to data — which always cover a finite time span — or for purposesofsimulatingthemodelfromgiveninitialconditionsoverafinitespanoftime. It is also exactly appropriate for the correct calculation of expected welfare, when welfare is constructed as a discounted sum of period utilities. The discounting means that accuracy oftheapproximationisunimportantaftersometimehorizoninthefuture. Finite-time-span,accuracy-in-probabilityclaimswillnotjustifyestimatingunconditional expectationsofanyfunctionsofvariablesinthemodelviasimulation. Tomaketheeffects of initial conditions die away, such simulations must cover long spans of time. If the second-order approximation is non-stationary, expectations calculated from simulations of it will of course not converge. If the true nonlinear model is non-stationary, then the true unconditionalexpectationswillingeneralnotexist,eventhoughitispossiblethatthelocal second-order approximation is stationary, so again in this case it will not be possible to estimateunconditionalexpectationsfromsimulatedpaths. Whenboththetruenonlinearmodelandthesecondorderapproximatemodelarestationaryandergodic,andthetrueunconditionalexpectationinquestionisatwice-differentiable function of s in the neighborhood of s =0, then it is possible to estimate the expectation from long simulations of the approximate model, with the estimates accurate locally in s

SECONDORDERSOLUTION 15 in the usual sense. This is true even though it may be (e.g. because of unbounded support of e ) that with probability one the path of the model repeatedly enters regions where the t local approximation is inaccurate. This is possible because as s →0 the fraction of time spentintheseregionsgoestozero,forboththetrueandtheapproximatemodel. Howeveritwilloftenbepreferabletoestimateanexpectationbyusingthesecond-order approximation analytically, expanding the function whose expectation is being taken as a Taylorseriesandapplyingthemethodsofthenextsection. It goes without saying that no theoretical result about local or asymptotic global accuracy for approximate solutions can prove that in a particular model, with particular shock variances, one method or another is more accurate than another or accurate enough for somespecificpurpose. TheemphasisbyJinandJudd(2002)oncheckingmodelvalidityis therefore appropriate. There is no uniquely best measure of solution accuracy, but by now avarietyofstringentcheckshavebeenproposed. Themethodsthathavebeenappliedmost widely(butnotwidelyenough)arebasedonevaluatingtheconditionalexpectationin(3)at a collection of values of the lagged state vector. There are important practical questions as tohowtoselectthecollectionofstatevectorvaluesatwhichoneevaluatestheexpectation andastowhatmetrictouseinmeasuringthevectorofdeviationsfromthetheoreticalzero values for these expectations. Jin and Judd suggest deterministically fixing a collection of state variable values and a set of “relative error” metrics for the expectational errors, based on economic interpretation of the model being solved. den Haan and Marcet (1994) suggest another approach, in which the state variable values are generated stochastically via simulation and the metric for evaluation of expectational errors is based on statistical detectability of the errors in a sample of relevant length. Each of these approaches has pitfalls,butisworthconsideration. Thoughthereisnowidelyunderstoodalternativetothis“Eulerequationresidual”family of accuracy checks at this point, there is probably room for further work in this area. For many purposes, the most relevant measure of accuracy is the accuracy of the solution’s approximation to the mapping from w , e , and s to w corresponding to (9). This is not t−1 t t

SECONDORDERSOLUTION 16 measured directly by the size of the Euler equation errors, but no more direct measure of theaccuracyofthismappingisatthispointavailable. 7. FORECASTING AND SIMULATION Forecasts s steps ahead, E [dw ] and Var [dw ] are the building blocks for the calcut t+s t t+s lationofimpulseresponsefunctionsaswellaswelfare. We build the forecasts from the second-order accurate dynamic model given by (15)- (16),modifiedheretoreflectourassumptionthattheinitialconditionssatisfytheequations of the model and that therefore dx = 0 to first order. We abuse notation by using the t−1 same F’s here, for the pieces of the original F matrices corresponding to dy’s, as we did fortheoriginal F matricesin(15)-(16)thatcorrespondedtothefull dv=[dy,dx]vector. . dy =F dy +F se +F s 2 t 1j j,t−1 2j j,t 3 (25) +1F dy dy +F s dy e +1F s 2e e 2 11jk j,t−1 k,t−1 12jk j,t−1 k,t 2 22jk j,t k,t . (26) dx = 1M dy dy +M s 2 t 2 11jk j,t k,t 2 Wewouldthenliketocalculate,tosecondorderaccuracy, E [dy ]andVar [y ]. t t+s t t+s Tobeginwith,notethat,sincetheconditionalmeanofdy isofsecondorder,thevarit+s ance terms Sˆ ≡Var (y ) are correct to second order accuracy when computed from the s t t+s first-ordertermsintheexpansion(25)aloneandthat,tosecond-orderaccuracy,Var (x )= t t+s 0sincedx itselfisofsecondorder. t Fors=1,itiseasytoseefrom(25)-(26)thatwehave . dyˆ =E [dy ]=F dy +F s 2 t+1 t t+1 1j j,t 3 (27) s 2 +1F dy dy + F W 2 11jk jt kt 2 22jk jk . (cid:161) (cid:162) (28) dxˆ =E [dx ]= 1M dyˆ dyˆ +Sˆ +M s 2. t+1 t t+1 2 11jk j,t+1 k,t+1 1jk 2 Theexpressionin(28)fordeterminingE [dx ]fromtheconditionalmeanandvarianceof t t+1 dy worksequallywellfordeterminingE [dx ]fromtheconditionalmeanandvariance t+1 t t+s ofdy fors>1. Thestraightforwardapproachtodeterminingdyˆ anddxˆ istoapply t+s t+s t+s

SECONDORDERSOLUTION 17 (27) recursively, computing dyˆ from dyˆ and Sˆ , etc. This procedure is in fact t+s t+s−1 k−1 second-orderaccurate,butitintroduceshigherordertermsintotheexpansion. Forexample, since dˆy contains quadratic terms in dy , and (27) makes dyˆ quadratic in dyˆ , in a t+1 t t+2 t+1 simple recursive computation dyˆ becomes quartic in dy . These extra high-order terms t+2 t do not in general increase accuracy of the approximation, as they do not correspond to higher order coefficients in a Taylor series expansion of the true dynamic system, and in practiceoftenleadtoexplosivetimepathsfor dyˆ . t+s Toseewhatgoeswrong,considerthesimpleunivariatemodel y =r y +a y2 +e , t t−1 t−1 t where|r |<1anda >0. Thoughthismodelislocallystableaboutitsuniquedeterministic steady state of y¯ = 0, it has a second steady-state, at (1−r )/a . If x exceeds the other steady state, it will tend to diverge. This is likely to be a generic problem with quadratic expansions—theywillhaveextrasteadystatesnotpresentintheoriginalmodel,andsome ofthesesteadystatesarelikelytomarktransitionstounstablebehavior. Sincetheuniquelocaldynamicsarestableinaneighborhoodofthesteadystate,itwillbe desirabletochooseamongstthesecondorderaccurateexpansionsonethatimpliesstability. Derivingsufficientconditionsonthesupportofe toguaranteenonexplosivenessunderthe t iterative scheme (27)-(28) is in general a non-trivial task and therefore it is useful to have available an algorithm which generates non-explosive forecasts and simulations without imposingexplicitconditionsonthesupportofe . Themerefactthatthegeneratedforecasts t are stable of course does not imply superior accuracy in general, especially when shocks arenotbounded. However,stationaritywillingeneralimplythat,foragivenneighborhood U of the steady state and a given time horizon T, we can restrict s in such a way as to maketheprobabilityofleaving U intime T arbitrarilysmall. Obtainingastablesolutionbasedon(27)canbeachievedbypruningouttheextraneous high-order terms in each iteration by computing the projections of the second order terms basedona first-order expansion,dy¯ ofE [dy ],asfollows: t+1 t t+1

SECONDORDERSOLUTION 18 . dyˆ =F dyˆ +F s 2 t+s 1j j,t+s−1 3 (29) (cid:161) (cid:162) +1F dy¯ dy¯ +Sˆ + s 2 F W 2 11jk j,t+s−1 k,t+s−1 k−1,jk 2 22jk jk . (cid:161) (cid:162) (30) dxˆ = 1M dy¯ dy¯ +Sˆ +M s 2 t+s 2 11jk j,t+s k,t+s s,jk 2 . (31) dy¯ =F dy¯ t+s 1j j,t+s−1 (32) Sˆ =s 2F W F +F Sˆ F . ij,s 2ik k(cid:96) 2j(cid:96) 1ik k(cid:96),s−1 1j(cid:96) Using these equations recursively results in a dyˆ series which, by construction, is t+s quadratic in dy for all s. Furthermore, when the eigenvalues of F are less than one in t 1 absolute value, the first order accurate solution dy¯ is stable and hence so is the squared t+s (cid:161) (cid:162) process dy¯ dy¯ . It follows that dyˆ must be stable as well.9 Note that the F j,t+s k,t+s t 12 component of the second order expansion — the coefficients of the interactions between dy ande —donotenterthisrecursionatall. t−1 t The same issues arise if the aim is to generate simulated time paths, rather than simply conditional expectations and variances of future variables. For this purpose, we can intro- (1) (2) duce the notation dy and dy for first and second order accurate simulated time paths, t+s t+s respectively. Arecursive,non-explosive,“pruned”simulationschemeisthengivenby dy (2) = . F dy (2) +F se +F s 2 t+s 1·j j,t+s−1 2·j j,t+s 3 (33) +1F dy (1) dy (1) +s F y (1) e + s 2 F e e 2 11·jk j,t+s−1 k,t+s−1 12·jk j,t+s−1 k,t+s 2 22·jk j,t+s k,t+s (34) dx (2) = . 1M dy (1) dy (1) +M s 2 t+s 2 11·jk j,t+s k,t+s 2 (35) dy (1) = . F dy (1) +F se , t+s 1·j j,t+s−1 2·j j,t+s where the F terms that could be ignored in forming conditional expectations have neces- 12 sarily returned for generation of accurate simulations. By preventing buildup of spurious higher-order terms, we make stability of the simulation over a long time path more likely, 9 Thesamematrixeigenvalueconditionsareatissuehereasinsection4’sdiscussionofexistenceofthe solutionto(21)

SECONDORDERSOLUTION 19 while at the same time preserving second-order accuracy of the mapping from initial variablevaluesy ,x ,shockse ,...,e ,and s tothesimulatedvaluesy (2) ,...,y (2) . t t t+1 t+s t+1 t+s It can help in understanding these recursions to append the vector dy(1)⊗dy(1) to dy(2) andusematrixnotation:     (2) (2) dy dy (36)  t+1 =Q  t +Q s 2+x 1 2 t+1 (1) (1) (1) (1) (dy ⊗dy ) (dy ⊗dy ) t+1 t+1 t t with   F 1(F∗) (37) Q = 1 2 11  1 0 (F ⊗F ) 1 1   F +1F W (38) Q = 3 2 22·jk jk  2 (F ⊗F )vec(W ) 2 2   se +s F e dy (1) + s 2 F (e e −W ) (39) x t = t 23· (cid:161) jk k,t j,t−1 2 33·jk jt (cid:162) kt jk . s 2 e ⊗e −(F ⊗F )vec(W ) t t 2 2 The F∗ in the definition of Q (37) is a matrix with number of rows equal to the length of 11 1 y and with the second and third dimensions of the array vectorized into a row vector — so it is an n ×n2 matrix. Note that Q is upper block triangular and is stable exactly when s s 1 the eigenvalues of F are less than one in absolute value. Note also that, to second order 1 accuracy,   s 2W 0 Var(x )= . t 0 0 Calculations of conditional and unconditional first and second moments can therefore be carried out using (36) as if it were an ordinary first order VAR. This can be an aid to understanding, or to computation in small models, though for larger systems it is likely to beimportantforcomputationalefficiencytotakeaccountofthespecialstructureofthe Q j matricesin(36).

SECONDORDERSOLUTION 20 8. WELFARE One can easily produce cases where the second-order approximation is necessary to get an accurate evaluation of certain aspects of the model. Utility-based welfare calculation is one case. For example, calculating welfare effects of various monetary and fiscal policies or welfare effects of changes in economic environment such as financial market structure should include second-order or even higher-order terms in order to get an accurate measure. Kim and Kim (2003a) present an example of how inaccurate the linearized solution can be in calculating welfare using a two-country model. Using the linearized solution, welfareofautarkycanappeartobehigherthanthatofthecompletemarkets,solelybecause of the inaccuracy of the linearization method. Another application in which second-order approximation is important is examination of asset price behavior in DSGE’s. Linearized solutions will imply equal expected returns on all assets. Second order solutions will generate correct risk premia, though generally to analyze time variation in risk premia will requirehigherthansecond-orderaccuracy. Equation (36) makes it relatively straightforward to see how to carry out a second-order accuratewelfarecalculation.Welfareisdefinedasadiscountedsumofexpectedutility. Let the period utility function be given by u : Rns → R.10 Then the utility conditional on an 10 Ofcourseofteningrowthmodelsutilityisafunctionofconsumption,whichisnotaconventionalstate variable. Tousetheformulationwedevelophere,then,consumption(anxvariable)hastobereplacedbythe correspondingcomponentofh(y,s ). Also,becauseweworkentirelyintermsofy,wearenotcoveringthe casewheretheinitialdistributionofwdoesnotlieonthesaddlepath. Themethodswedescribeherecanbe expandedtocoverthiscaseandtoallowxtoenteru,atthecostofsomeincreaseintheburdenofnotation.

SECONDORDERSOLUTION 21 initialdistributionof y withmeanandvariance (m ,S )is 0 (cid:34) (cid:35) ¥ U(m ,S )=E (cid:229) b tu(y ) ≈ 0 t t=0 (40) (cid:34) (cid:35) u(y¯) +E (cid:229) ¥ b t (cid:161) (cid:209) u(y¯)dy (2) +1vec((cid:209) 2u(y¯))(cid:48)(dy (1) ⊗dy (1) ) (cid:162) ⇒ 1−b 0 t 2 t t t=0 (cid:104) (cid:105) u(y¯) U(m ,S )= + (cid:209) u(y¯) 1vec((cid:209) 2u(y¯))(cid:48) 1−b 2    (41) m ·[I−b Q ]−1 +b (1−b )−1Q s 2 1 2 vec(S +mm (cid:48)) If we are interested only in unconditional expected u, we can arrive at the correct formula bymultiplying (41)throughby 1−b andtakingthelimitas b →1,givingus (cid:104) (cid:105) (42) E[u(y t )]=u(y¯)+ (cid:209) u(y¯) 1vec((cid:209) 2u(y¯))(cid:48) (I−Q 1 )−1Q 2 s 2. 2 Note that in (41) we make no use, explicitly or implicitly, of F . Also note that though 12 thematrix I−b Q appearsintheformulainverted,theutilitycalculationonlyrequires 1 (cid:104) (cid:105) (cid:209) u(y¯) 1vec((cid:209) 2u(y¯))(cid:48) ·(I−b Q )−1, 1 2 whose computation is only an equation-solving problem, not a full inversion;11 furthermore, this part of the computation does not need to be repeated as m and S are varied. Finally, note that (42) uses only (I−Q )−1Q , regardless of the form of u. This is again 1 2 an equation-solving problem. So if we are interested only in unconditional expectations, even in unconditional expectations of many different functions u, the computation of a fullsecond-ordercorrectionmaybemuchsimplerthancalculationofthefullsecond-order expansionofthedynamics. 11 Thoughforann×nmatrixAbothsolvingAx=bforxandcomputingA−1 areO(n3)operations, the latterissubstantiallymoretimeconsuming. InMatlabinversiontakesroughlytwicethetime.

SECONDORDERSOLUTION 22 It is these simplifications, applied to particular models, that are the insights provided by thepapersthathaveputforward“bias-correction”methodsformakingsecond-orderaccurate expected welfare computations in DSGE models (Kim and Kim, 2003a; Sutherland, 2002). We should note that there is a situation in which second-order accurate evaluations of welfare can avoid entirely the need for a second-order expansion of the model solution. If (cid:209) u(y¯) = 0, as would be true if the deterministic steady-state sets y to the value that maximizes u(y), then only the lower blocks of Q and Q enter the solution, as can be 1 2 seen from (41) or (42). As can be seen from (37) and (39), these blocks contain F and 1 F only, not any terms from the second-order solution. Of course in most problems with 2 discounting, even an optimal solution will not maximize static welfare u(y) in the steady state, so this result will not apply. Also, even where the solution has been computed to maximizestaticperiodwelfareu,theresultdependsonhavingasecondorderexpansionof uintermsofthestatevectory. Whentheproblemhasbeenformulated(asinusualgrowth models) with a non-state variable (e.g. consumption) appearing in the utility function, the second-orderexpansion of the utility function in terms of y may require use of the secondordersolutionfor x asafunctionof y.12 8.1. Conditional vs. Unconditional welfare. From the discussion in the preceding section it is apparent that evaluating expected welfare based on unconditional E[u(y)] is a more straightforward task than evaluating the conditional expectation of discounted expectedutilityatagivendate.13 Itisthereforenotsurprisingthatmanyexistingpapershave 12 Rotemberg and Woodford (1997) is an example of a context where use of the first-order solution for welfareanalysisisjustifiedbyspecialregularityconditions.Thepaperevaluatedwelfareusingunconditional expectation of period utility. Regularity conditions required to justify use of the first-order solution in the paper’smodelincludeanassumptionthatsomeotherpolicychangeperfectlyoffsetssecond-ordereffectsof monetary policy on the mean level of output and an assumption that monetary policy is the only source of inefficientfluctuationsinprices. 13 Woodford(2002)discussesthedifferencesbetweenunconditionalandconditionalwelfareincalculating welfareeffectsofmonetarypolicies.

SECONDORDERSOLUTION 23 used unconditional welfare for evaluating policies. Examples include Clarida, Gal´ı, and Gertler (1999), Rotemberg and Woodford (1997, 1999), Sutherland (2002) and Kollmann (2002). Therearestrongobjectionsinprincipletouseoftheunconditionalwelfarecriterion. We know that it takes time for one steady state to reach another steady state and unconditional welfare neglects the welfare effects during the transitional period. It is therefore generally notinfactoptimal,inproblemswithdiscounting,tousepoliciesthatmaximizetheunconditionalexpectationofone-periodwelfare. Thisisnotanewpoint—itisthesamepointas thenon-optimalityofdrivingtherateofreturntozeroinagrowthmodel—andithasbeen recognizedintheDSGEliteraturein,e.g, KimandKim (2003b),and Woodford(2002). Because unconditional welfare can often be computed easily, using the “bias correction” shortcut, it is important to note that using unconditional welfare can give nonsensical results. Kim and Kim (2003a) construct a two-country DSGE model and compute risk-sharing gains from autarky to the complete-markets economy using a second-order approximation method. Welfare is defined as conditional welfare and the results show that there are positive welfare gains from autarky to the complete-markets economy. But the unconditional welfare measure can for certain parameter values produce the paradoxical resultthatautarkygeneratesahigherlevelofwelfarethanthecompletemarkets. The use of conditional welfare does not imply that results necessarily are tied to some particular initial state. One can condition on a distribution of values for the initial state. Thecriticalpointisthatwhencomparingtwopoliciesorequilibriaoneshouldusethesame distribution for the initial state for each. When there is no time-inconsistency problem the optimalpolicywillhavethepropertythatnomatterwhatinitialdistributionisspecifiedfor the state, it will produce a higher conditional expectation of welfare than any other policy. However, when comparing a collection of policies that are not optimal, one may find that rankingsofpoliciesvarywiththeassumeddistributionoftheinitialstate. Whenthereisatime-inconsistencyproblem,theoptimalpolicygenerallydependsonthe initialconditions,evenifwerestrictattentiontopolicyrulesthatareafixedmappingfrom

SECONDORDERSOLUTION 24 state to actions. Using a conditional expectation as the welfare measure does not avoid this problem. One attempt to get around this issue is the suggestion in, e.g., Giannoni and Woodford (2002) that policy should follow the rule that would prevail under commitment inthelimitastheinitialconditionsrecedeintothepast. This“timelessperspective”policy can be implemented by treating the Lagrange multipliers on private sector Euler equations as“states”,andthenmaximizingconditionalexpecteddiscountedutility. Thetimelessperspective policy is a useful benchmark, but it cannot resolve the fundamental problem of time inconsistency. A policy-maker who can make believable commitments will not want to choose the timeless-perspective solution, while one that cannot make believable commitments cannot implement the timeless-perspective solution. As a normative suggestion, the timeless perspective depends on the idea that it will be easier to convince the public of a commitment to the timeless-perspective policy than of commitments to other types of policies, but this is likely to depend on the nature of the policy optimization problem and ontheparticularinitialconditionsfacedatthetimethepolicyisimplemented. 9. CONCLUSION Use of perturbation methods to improve analysis of DSGE models is still in its early stages. Programs that automate computations for models higher than second order are just beginning to emerge. Methods of dealing with the kinds of singularities that show up in economic models — for example the indeterminacy of asset allocations in standard portfolioproblemswhenvariancesarezero—arestillnotwidelyunderstood. Andwehave only begun to get a feel for where these methods are useful and what their limitations are. Real progress is being made, however, in an atmosphere that is both competitive enough to be stimulating and cooperative enough that researchers located around the world are benefitingfromeachothers’insights.

SECONDORDERSOLUTION 25 REFERENCES ANDERSON, G., AND A. LEVIN (2002): “A user-friendly, computationally-efficient algorithm for obtaining higher-order approximations of non-linear rational expectations models,”Discussionpaper,BoardofGovernorsoftheFederalReseerve. BERGIN, P. R., AND I. TCHAKAROV (2002): “Does Exchange Rate Risk Matter for Welfare? AQuantitativeInvestigation,”Discussionpaper,UniversityofCaliforniaatDavis, http://www.econ.ucdavis.edu/faculty/bergin/. CLARIDA, R., J. GAL´I, AND M. GERTLER (1999): “The science of monetary policy: A newKeynesianperspective,” JournalofEconomicLiterature,37,1661–1707. COLLARD, F., AND M. JUILLARD (2000): “Perturbation Methods for Rational Expectations Models,” Discussion paper, CEPREMAP, Paris, fabrice.collard@ cepremap.cnrs.fr. DEN HAAN, W. J., AND A. MARCET (1994): “Accuracy in Simulations,” The Review of EconomicStudies,61(1),3–17. GASPAR, J., AND K. L. JUDD (1997): “Solving large-scale rational expectations models,” MacroeconomicDynamics,1,44–75. GIANNONI, M. P., AND M. WOODFORD (2002): “OptimalInterest-RateRules: I.General Theory,” Discussion paper, Columbia University and Princeton University, http:// www.princeton.edu/˜woodford. JIN, H., AND K. L. JUDD (2002): “Perturbation Methods for General Dynamic Stochastic Models,”Discussionpaper,StanfordUniversity, judd@hoover.stanford.edu. JUDD, K. L. (1998): NumericalMethodsinEconomics.MITPress,Cambridge,Mass. KIM, J., AND S. KIM (2003a): “Spurious Welfare Reversals in International Business CycleModels,”JournalofInternationalEconomics,60,471–500. (2003b): “Welfare Effects of Tax Policy in Open Economies: Stabilization and Cooperation,” Discussion paper, University of Virginia and Tufts University, http: //www.tufts.edu/˜skim20.

SECONDORDERSOLUTION 26 KING, R. G., AND M. WATSON (1998): “The Solution of Singular Linear Difference Systems Under Rational Expectations,” International Economic Review, 39(4), 1015– 1026. KLEIN, P.(2000): “UsingthegeneralizedSchurformtosolveamultivariatelinearrational expectationsmodel,” JournalofEconomicDynamicsandControl,24(10),1405–1423. KOLLMANN, R. (2002): “Monetary Policy Rules in the Open Economy:Effects on Welfare and Business Cycles,” Journal of Monetary Economics, 49, 989–1015, http: //www1.wiwi.uni-bonn.de/users/rkollmann/www/. ROTEMBERG, J. J., AND M. WOODFORD (1997): “An Optimization-Based Econometric FrameworkfortheEvaluationofMonetaryPolicy,” NBERMacroAnnual,12,297–345. (1999): “Interestraterulesinanestimatedsticky-pricemodel,”inMonetaryPolicy Rules,ed.byJ.B.Taylor.UniversityofChicagoPress,Chicago. SCHMITT-GROHE´, S., AND M. URIBE (2002): “Solving dynamic general equilibrium models using a second-order approximation to the policy function,” Discussion paper, RutgersUniversityandUniversityofPennsylvania. SIMS, C. A. (2001): “SolvingLinearRationalExpectationsModels,”ComputationalEconomics,20(1-2),1–20, http://www.princeton.edu/˜sims/. SUTHERLAND, A. (2002): “A simple second-order solution method for dynamic generalequilibriummodels,”Discussionpaper,UniversityofSt.Andrews,http://www. st-andrews.ac.uk/˜ajs10/home.html. WOODFORD, M. (2002): “Inflation Stabilization and Welfare,” Contributions to Macroeconomics,2(1),Article1. FEDERAL RESERVE BOARD, TUFTS UNIVERSITY, NORTHWESTERN UNIVERSITY, PRINCETON UNI- VERSITY E-mailaddress: sims@princeton.edu