Dynamics of Open Economy Business Cycle Models: Understanding the Role of the Discount Factor *

Transcription

1 Dynamics of Open Economy Business Cycle Models: Undersanding he Role of he Discoun Facor * Sunghyun Henry Kim ** and M. Ayhan Kose ** Absrac: This paper examines he dynamic implicaions of differen preference formulaions in open economy business cycle models ih incomplee asse markes. In paricular, e sudy o preference formulaions: a ime separable preference formulaion ih a fixed discoun facor, and a ime non-separable preference srucure ih an endogenous discoun facor. We analyze he momen implicaions of o versions of an oherise idenical open economy model one ih a fixed discoun facor and he oher ih an endogenous discoun facor and sudy impulse responses o produciviy and orld real ineres rae shocks. Our resuls sugges ha business cycle implicaions of he o models are quie similar under convenional parameer values. We also find he approximaion errors associaed ih he soluions of hese o models are of he same magniude. Keyords: business cycle dynamics, fixed discoun facor, endogenous discoun facor, nonsaionariy. JEL classificaion: F4, E3 * We ould like o hank William A. Barne, an anonymous Associae Edior, and o anonymous referees for providing very deailed and helpful commens hich significanly improved he paper. We are graeful o Marianne Baxer, William Blankenau, Mario Crucini, Narayana Kocherlakoa, Rober Kollmann, Jim Nason, Fabrizio Perri, and Kei-Mu Yi for heir helpful suggesions. An earlier version of his paper as presened a Tufs Universiy, 999 Compuing in Economics and Finance Conference a Boson College, 2 Mides Macroeconomics Meeings a Universiy of Ioa. We ould like o hank seminar and conference paricipans for heir feedback. The usual disclaimer applies. ** Graduae School of Inernaional Economics and Finance, Brandeis Universiy, Walham MA, (Kim): hkim@brandeis.edu, (Kose): akose@brandeis.edu, phone (Kim): (78) , phone (Kose): (78)

2 . Inroducion There have been o popular preference formulaions employed in infinie-horizon open economy Real Business Cycle (RBC) models under incomplee asse markes: ime separable preferences ih a fixed discoun facor, and ime non-separable preferences ih an endogenous discoun facor. The former formulaion is he sandard one ha is idely used in closed and open economy RBC models, under complee and incomplee asse markes. Hoever, i is by no ell knon ha ih his formulaion, hen he models are solved using he usual linear approximaion mehods, i is no possible o generae saionary sae variables and a ell-defined ealh disribuion in an open economy seing under incomplee markes. 2 This problem has led several researchers o employ he laer formulaion, hich generaes ell-defined seady sae dynamics in hese models. Ho can e jusify he fac ha a number of researchers have employed he fixed discoun facor formulaion despie he ell-knon problems associaed ih seady sae dynamics in open economy RBC models under incomplee asse markes? Three poenial reasons emerge for he idespread use of his formulaion. Firs, hile he issue of he exisence of a ell-defined long-run ealh disribuion migh be imporan for some economic experimens, here migh be no reason o believe ha hese o preference formulaions generae significanly differen model dynamics along he dimensions examined by ypical RBC sudies. To be more specific, i migh be he case ha he models ih hese o preference formulaions produce very similar momens and impulse responses, hich are he main ineress of hese sudies. Second, considering ha mos models in his lieraure are solved using linear approximaion mehods, poenial addiional accuracy gains from employing an endogenous discoun facor migh be insignifican. In oher ords, hile he rue soluion of he fixed discoun facor model, hich e call FDM (exac), and he rue soluion of he endogenous discoun facor model, hich e call EDM (exac), migh exhibi some differences, i is no clear heher he approximae soluions of hese o models, hich e call FDM (approximae) and EDM (approximae), are differen under incomplee markes. Third, some feaures of he endogenous discoun facor formulaion migh no be desirable in a represenaive agen business cycle model. For example, he endogenous discoun facor formulaion implies ha agens become more impaien as hey become ealhier, and here is a seady sae uiliy level ha relies on a predeermined saving arge (see Senhadji (995) and Daniel (997)). Anoher undesirable Baxer and Crucini (995), Baxer (995), Rebelo and Vegh (995), Correia, Neves and Rebelo (992, 995), Kollman (996, 998, 2), Crucini (999), Arvaniis and Mikkola (996), van Wincoop (996), van Wincoop and Marrinan (999), Kouparisas (997), Sadka and Yi (996), Blankenau, Kose, and Yi (2), and Kose and Riezman (2) employ he sandard ime separable preferences ih a fixed discoun facor in heir models. Mendoza (99, 995), Karayalcin (995), Uribe (997), Schmi-Grohe (998), and Cook and Devereux (2) use he Uzaa-Epsein ype ime nonseparable preferences ih an endogenous discoun facor. 2 As e discuss in deail in secion 2, here are some mehods o deal ih hese problems in open economy models under incomplee markes.

3 propery of he EDM is ha i is no possible o examine he sae dependen seady sae level of ne foreign asses because he EDM forces consumpion o behave in such a ay ha he ne foreign asse posiion goes back o is iniial seady sae. Moreover, since a number of macroeconomic ime series are nonsaionary, i migh no be advanageous o have a model generaing saionary variables (see Correia, e al. 995). 3 Despie he ide use of hese o preference formulaions, here has ye been no rigorous examinaion of heir impac on he cyclical dynamics of open economy RBC models. The objecive of his paper is o provide a comprehensive comparison of he behavior of an open economy RBC model ih incomplee asse markes under he fixed discoun facor ih ha under he endogenous discoun facor. In paricular, e consider o versions of an oherise idenical small open economy RBC model, one ih a fixed discoun facor (FDM), and he oher ih an endogenous discoun facor (EDM), and sudy he momen implicaions and impulse responses of hese models. We provide empirical suppor for he hree reasons above and argue for he similariy beeen FDM (exac) and EDM (exac) based on he similariy beeen FDM (approximae) and EDM (approximae), hich e sysemaically sudy in his paper. 4 Secion 2 sars ih a brief discussion of he problem associaed ih he seady sae dynamics in open economy RBC models ih incomplee asse markes. Then, e presen he o models and heir calibraion and parameerizaion. In secion 3, e provide empirical suppor for he hree jusificaions described above. We analyze he policy funcions generaed by he o models in secion 3.. Our discussion in secions 2 and 3. makes ransparen he argumens regarding he deerminisic and sochasic seady sae dynamics of he o models. We find ha hile he ne foreign asse series in he FDM follos a uni roo process, his series exhibis a near uni roo behavior in he EDM. More imporanly, he coefficiens in he policy funcions generaed by he o models display minimal differences under convenional parameer values. In secion 3.2, e examine he second momen implicaions peraining o business cycles and find ha he o models generae almos idenical business cycle momens. Secion 3.3 presens he impulse responses o produciviy and he orld ineres rae shocks. While some small quaniaive differences exis beeen he impulse responses produced by he o models, qualiaive implicaions are idenical. In secion 3.4, e compare he disribuions of variables produced by he o models using he Kolmogorov-Smirnov es saisic. The es resuls sugges ha hese empirical disribuions are no saisically differen. In secion 3.5, e evaluae he approximaion errors associaed 3 Daniel (997) noes ha i is inuiively unappealing o have ealhier agens ho are more impaien han less ealhy agens in his uiliy formulaion. Senhadji (995) criicizes he use of endogenous discoun facor since he represenaive agen mus save enough o reach a saving arge in order o aain a fixed uiliy level deermined by he seady sae. Correia, e al. (995), afer noing ha consumpion, he ne expors, and he ne foreign asses are nonsaionary in he daa, claim ha he saionariy of hese variables induced by he non-separable preference formulaion is no necessarily a desirable propery. (p. ) 4 To be more specific, our ulimae objecive is o argue for he similariy beeen he exac soluions of hese o models based on he similariy beeen heir approximae soluions. Our findings sugges ha he approximaion errors are quie small implying ha he resuls from hese approximae soluions can poenially apply o he exac soluions of hese models. 2

4 ih he soluions of hese models. We find ha hile he soluion of he FDM resuls in slighly larger approximaion errors han ha of he EDM, he errors are quie small. We provide a brief conclusion in secion The Model Before geing ino he deails of he model economies, e provide a brief discussion of he problem associaed ih he seady sae. Consider he sandard ime separable preference formulaion in an infiniehorizon open economy model ih incomplee asse markes. In a deerminisic seing, he seady sae or he long-run ealh depends on he iniial condiions of he economy and he seady sae is compaible ih any level of ne foreign asses. 5 In a sochasic environmen, since he ne foreign asse series follos a uni roo process, he model generaes nonsaionary variables implying he absence of a ell-defined sochasic seady sae. 6 In oher ords, cerain model variables do no reurn o heir iniial seady sae values hen he model is subjeced o a emporary shock. Hence, he long-run ealh of he economy changes ih he sae of naure, i.e. he long-run ealh disribuion is no ell defined. Researchers have developed several mehods ha can resolve he seady sae issues associaed ih he fixed discoun facor formulaion. 7 As menioned above, one popular alernaive is o employ he Uzaa- Epsein ype ime non-separable preferences ih an endogenous discoun facor. Unlike he fixed discoun facor model, he endogenous discoun facor model generaes saionary ne foreign asse series and, in urn, oher sae variables become saionary. Moreover, he endogenous discoun facor model generaes a elldefined sochasic seady sae a hich, under cerain condiions, a unique level of ne foreign asses is aained implying ha he long-run ealh disribuion of he economy is also ell defined. 5 We consider he case hen he ime preference is equal o he orld ineres rae. Oherise, no deerminisic seady sae exiss. If he orld ineres rae is greaer (smaller) han he ime preference rae, agens permanenly accumulae (deplee) foreign asses. These issues are firs discussed in Helpman and Razin (982). Mendoza and Tesar (998) provide a deailed discussion of he seady sae issues in a deerminisic model ih incomplee asse markes. 6 The sochasic seady sae of a variable is defined as is expeced mean. The fac ha our approximae soluion feaures a uni roo in asse holdings does no necessarily imply ha he exac soluion of he model also has a uni roo. The sochasic seady sae of he model migh be ell defined, bu i is no possible o capure his seady sae by he approximae soluion mehod. I is no possible o provide an exac soluion of he incomplee marke models ih he fixed discoun facor formulaion unless he sae space of asse holdings is bounded. 7 Heahcoe and Perri (2) impose a quadraic adjusmen cos on bond holdings o produce saionary asse series. Cardia (99) uses he uncerain lifeime approach advanced by Blanchard (985) in her small open economy model. Harjes (997) assumes ha he orld real ineres rae depends on he ne foreign asses, and Bruno and Porier (995) assume ha ne foreign asses negaively affec he households uiliy in heir small open economy models. Senhadji (998) considers a se-up ih a donard sloping expor demand funcion. I is also possible o produce saionary equilibria in models ih incomplee asse markes by inroducing limis on he level of asse holdings (explici bounds as in Hugge (993) or implici bounds as in Levine and Zame (996, 999) and Kubler and Schmedders (2)), or by inroducing endogenous solvency consrains (see Alvarez and Jermann (999)), or enforcemen consrains (see Kehoe and Perri (2)). 3

5 To evaluae he impac of he discoun facor on he dynamics of business cycles, e sudy a small open economy RBC model in hich agens produce an inernaionally radable good using labor and capial. 8 There are o shocks in he model: a produciviy shock and a orld real ineres rae shock. Agens have access o orld financial markes in hich hey can buy and sell one-period risk free bonds a a sochasic orld real ineres rae. We solve he model using a linear approximaion mehod and choose he parameer values ha provide boh versions of he model have he same seady sae. We calibrae he model o Canada, an economy ha has been exensively sudied in he small open economy RBC lieraure. In secion 2., e presen he model ih a fixed discoun facor. In secion 2.2, e analyze he same model ih an endogenous discoun facor. Secion 2.3 explains he derivaion of he seady sae and calibraion of he model. 2.. The Fixed Discoun Facor Model The opimizaion problem of he represenaive agen is he folloing: max c, i, n, b U ( c, n ) = E n c β σ = σ, σ >, >, () s o c i + b = y + b ( r ),. + + α (2) y = z k α n, i (3) k+ = ( δ ) k + φ( ) k, k in he momenary uiliy funcion c is consumpion, n is labor hours, is he ineremporal elasiciy of subsiuion in labor supply, β is he fixed discoun facor, and σ is he household s coefficien of relaive risk aversion. Our momenary uiliy formulaion implies ha he elasiciy of subsiuion associaed ih leisure is zero. 9 In resource consrain (), i is invesmen, b is he ne foreign asses a he end of he period, r - is he sochasic orld real ineres rae from period - o period, and y is oupu. k is he domesic capial sock a he beginning of he period, z is produciviy shock, and α governs he share of income accruing o 8 See Baxer (995) and Backus, Kehoe, and Kydland (995) for a survey of open economy RBC models and heir use in sudying he sources and ransmission of inernaional business cycles. 9 This uiliy funcion is inroduced by Greenood, Hercoiz, and Huffman (988) and is idely used in he open economy RBC models. Correia, e al. (995) and Crucini (999) compare he dynamic implicaions of his uiliy funcion ih hose of he Cobb-Douglas uiliy funcion, boh ih a fixed discoun facor. We do no adop he Cobb- Douglas preference srucure because as shon by Correia, e al. (995) a small open economy model ih he Cobb- Douglas uiliy funcion is unable o mach he volailiy of consumpion and he counercyclical behavior of ne expors in real daa. 4

6 labor in consrain (2) ha describes he producion funcion. In consrain (3), δ denoes depreciaion rae and φ (.) represens he sandard adjusmen cos funcion ih φ (.) >, φ (.) >, and φ (.) < (see Baxer and Crucini (993)). Subsiuing (2) ino (), he firs order condiions of his opimizaion problem are given as n (4) c : c = λ, σ (5) n : n = αy, i (6) i : λ = µ φ ( ), k y+ i+ (7) k + : µ = βe { λ+ ( α) + µ + g( )}, k k + + i+ i+ i+ i+ here g( ) = ( δ ) + φ( ) φ ( ), k k k k (8) b : λ = βe λ+ ( + r ), here λ and µ are he Lagrange mulipliers associaed ih he consrains () and (3). Since his problem canno be solved analyically, e find an approximae soluion using he approximaion mehod of King, Plosser and Rebelo (988) The Endogenous Discoun Facor Model This par describes he same model ih a ime non-separable preference formulaion in hich he discoun facor is endogenous. 2 The opimizaion problem of he represenaive agen is max c, n, i, b U ( c, n ) = E n c γ σ = σ, σ >, >, We do no consider oher shocks such as governmen spending shocks, or ne foreign ransfers shocks since Correia, e al. (995) convincingly argue ha hese shocks are no able o generae significan business cycle dynamics in a small open economy model. Deailed derivaion of he seady sae and he linearized firs order condiions of he model are available upon reques. 2 This preference formulaion as firs inroduced by Uzaa (968), and furher developed by Epsein (983, 987). See Obsfeld (99) for a heoreical analysis of dynamics of his ype of small open economy model. 5

7 6 subjec o he same consrains () - (3) above. The endogenous discoun facor, γ, is defined as (9), ) ln( exp + = = n c τ τ τ θ γ θ >. The discoun facor depends on he level of consumpion and labor inpu in he previous periods. θ denoes he elasiciy of he discoun facor ih respec o uiliy. The funcional form of he endogenous discoun facor implies ha an increase (decrease) in curren consumpion (labor inpu) decreases he eighs assigned o all fuure uiliy, and, in urn, he agen becomes more impaien. 3 Using (9), e define an auxiliary variable ψ, he ime value of discouned fuure uiliy from dae + onards: (). ) ( k k k k n c E γ γ σ ψ σ + = = The firs order condiions of he opimizaion problem above are: () : c, + + = n c n c θ ψ λ σ (2) : n, y n α = (3) : i ), ( k i µ φ λ = (4) : + k )}, ( ) ( { = k i g k y E n c µ α λ µ θ (5) : b ). ( ) ( r E n c + + = + λ λ θ Using equaion (), e rie he folloing la of moion for he auxiliary variable ψ : 3 Epsein (983) shos ha under cerain condiions his preference formulaion generaes a unique invarian limiing disribuion of he sae variables. Our parameerizaion also mees hose condiions.

8 (6) E n ) ( c ψ E n+ ) σ σ + θ ( ψ + ) = ( + c. We solve he model using he same approximaion mehod Calibraion We choose he parameer values o ensure ha e have he same seady sae for boh models. The variables ihou ime subscrips refer o he seady sae values of he corresponding variables. The hree imporan seady sae parameers o be discussed here are r, nx/y (ne expor/oupu), and θ, since e aain he same seady sae by using he same values for r and nx/y and by endogenously deermining he value of θ in he EDM. In he FDM, given r, he seady sae version of equaion (8) deermines only he discoun facor β. Therefore, he number of endogenous variables is less han he number of seady sae equaions by one. This implies ha any level of foreign asses is compaible ih he iniial seady sae of he model. Folloing he sandard approach in he lieraure (see for example, Baxer and Crucini (993), and Correia, e al. (995)), e dra he value of nx/y (ne expors/oupu) from daa o solve his problem because nfa/y (ne foreign asses/oupu) is uniquely deermined by nx/y and r. Unlike he FDM, he EDM does no suffer from he indeerminacy problem because he number of endogenous variables is equal o he number of seady sae equaions: i.e. he model generaes a unique seady sae level of he ne foreign asses. Hoever, his hinges on he assumpion ha he value of θ is knon in advance. Using he value of nx/y, hich is dran from he daa, e pin don he value of θ endogenously using he seady sae version of equaion (5). This guaranees ha all he variables including he discoun raes in he o models have he same seady sae values. This is also he sandard calibraion mehod used in he sudies employing he endogenous discouning facor. We calibrae he srucural parameers o correspond o he exising RBC lieraure and o be consisen ih he long-run feaures of Canadian economy. Table presens he calibraed values of parameers. We se he quarerly seady sae orld real ineres rae a.2 % hich is he average rae calculaed using he U.S. 3-monh T-Bill rae deflaed ih he CPI inflaion. Folloing Mendoza (99), he elasiciy of subsiuion,, is se o.455. The risk aversion parameer, σ, is se o.5, hich is an inermediae case beeen he commonly used values of 2 and (see Schmi-Grohe (998)). Folloing Mendoza (99) and Schmi-Grohe (998), he share of labor income in he producion, α, is se o.68. The quarerly depreciaion rae, δ, is se o.25, a idely used value in he RBC lieraure. 7

9 Our benchmark value of nx/y is equal o. 4 To examine he sensiiviy of our resuls, e experimen ih differen values of nx/y, hich is equal o he ineres paymens-oupu raio (-rb/y), ranging from -.5 o.5 per quarer. 5 Since θ is an increasing funcion of nx/y a an exponenial rae, θ ranges from.74 o The corresponding value of θ is.8 hen nx/y is equal o. The adjusmen cos parameers are chosen so ha he seady sae of he model is same as he one ihou adjusmen coss. This implies ha φ ( i / k) = i / k and φ '( i / k) =. The seady sae value of i / k is equal o he depreciaion rae, δ. The elasiciy of he marginal adjusmen cos funcion, η = ( φ / φ )( i / k), is se o, o mach he volailiy of invesmen in he daa (see Baxer and Crucini (993)). The exogenous shocks, z and r, follo AR() processes ih (7) zˆ ρ ˆ + ε = zz z (8) rˆ ρ ˆ + ε = rr r here ε z and ε r 2 are assumed o follo normal disribuions ih mean and variance σ z and σ 2 r. We se he sandard deviaions of he produciviy and ineres rae shocks a.65 % and. % o mach he volailiy of oupu in he daa. The persisence parameers of shocks, ρ y and ρ r, are esimaed and hey are equal o.95 and Resuls We sudy he dynamic implicaions of he FDM and he EDM on five dimensions: firs, e compare he policy funcions generaed by he o models. Second, e sudy heir second momen implicaions. Third, e analyze he impulse responses o produciviy and ineres rae shocks. Fourh, e compare he disribuions of he hree variables, hich are nonsaionary in he FDM, in he o models. We also formally es heher he disribuions produced by hese o models are saisically differen. Finally, e esimae he approximaion errors generaed by he o models considering ha he FDM generaes nonsaionary variables ha migh induce larger approximaion errors han hose in he EDM. 3.. Policy funcions Table 2 presens he coefficiens of policy funcions for consumpion, asse holdings and he ne expors of he o models. We concenrae on he coefficiens of he endogenous sae variables considering ha impulse responses in he nex secion illusrae he differences in he coefficiens associaed ih shocks. 4 The average value of nx/y is near zero (less han.3% per quarer) for Canada. 5 This range is ide enough o cover mos realisic cases. The observed rade balance rarely exceeds ±5% of oupu per quarer for he OECD counries. 8

10 Asse holdings follo a uni roo process in he FDM, i.e. he coefficien of asse holdings, b 2, is equal o. The uni roo propery implies ha a shock in he curren period has a permanen impac and he long-run ealh of he economy depends on he shock realizaions. This also induces consumpion and he ne expors o be nonsaionary. While he coefficiens of he policy funcion for consumpion (a and b ) are alays larger in he EDM han hose in he FDM, differences beeen he coefficiens are very small. In oher ords, consumpion is more responsive o he sae variables in he EDM compared o he FDM. As he elasiciy of he endogenous discoun facor θ decreases, differences in he coefficiens of he policy funcions disappear. As θ decreases, he discoun facor in he EDM responds less o he changes in uiliy, herefore he variables in he EDM behave as if he discoun facor is almos fixed. For example, hen θ is equal o.74 (nx/y = 5%), he policy funcions generaed by he o models become almos idenical and he ne foreign asse series in he EDM follos a near uni roo process (b 2 =.9999 in he EDM). In oher ords, our findings sugges ha he momens produced by he EDM converge o hose produced by he FDM hen he elasiciy of he endogenous discoun facor, θ, ges arbirarily small. The policy funcions of oupu, labor hours, invesmen, and capial sock generaed by he o models are idenical and invarian o he value of θ. These variables do no depend on he asse holdings, i.e. he coefficiens associaed ih asse holdings are zero in boh models. Therefore, even in he FDM, hese four variables, unlike consumpion and he ne expors ha depend on he previous period's ne foreign asse holdings, become saionary. This resuls from he momenary preference srucure here labor hours depend only on he curren oupu and here is no ineremporal subsiuion involving labor. This implies ha labor hours become saionary, hich induces oupu, invesmen, and capial sock o be saionary as ell. We also examine he sensiiviy of policy funcions coefficiens o changes in oher parameers of he model such as σ and. We find ha our resuls are robus o hose changes Second Momen Implicaions One of he imporan objecives of he RBC research program is o consruc models ha are able o replicae cerain momens of he daa. In his secion, e compare he business cycle momens generaed by each model. If he discrepancies beeen he momens generaed by he o models are small, hen his suggess ha he FDM consiues a reasonable alernaive o he EDM for business cycle analysis. We simulae he model for periods ih our benchmark parameerizaion and repor he average momens over 3 simulaions. All resuls refer o he momens of Hodrick-Presco (HP(6)) filered variables (see Hodrick and Presco (997)). Panel A in able 3 repors he second momens generaed by produciviy shocks. The resuls sugges ha here is no saisically significan difference beeen he momens produced by he o models. As expeced, invesmen is he mos volaile variable, and oupu is more volaile han consumpion. All model 6 The value of θ used in he lieraure ih he EDM ranges from. o. depending on he model specificaion. 9

11 variables are procyclical excep he ne expors and he ne foreign asses. Boh models predic ha correlaions of labor hours and consumpion ih oupu are equal o as i is implied by he preference srucure. In panel B, e presen he simulaion resuls ih boh produciviy and he orld real ineres rae shocks. The momens produced by he o models are again quie similar. Adding ineres rae shocks does no change he momens of any of he variables excep invesmen, he ne expors, and he ne foreign asses. While increasing he volailiies of hese hree variables, ineres rae shocks dampen he correlaions of hese variables ih oupu. We also check he sensiiviy of our resuls o he changes in θ and find ha hese resuls are quie robus. We also examine he abiliy of he models in maching he main characerisics of Canadian business cycles. Our findings sugges ha boh FDM and EDM are successful in replicaing he main feaures of Canadian business cycles. 7 All he resuls in his secion sugges ha i is almos impossible o differeniae business cycle saisics generaed by he FDM and he EDM Impulse Responses Consrucing models ha can generae dynamic responses ha are compaible ih hose in he daa is anoher imporan objecive of he RBC research program. This secion compares he impulse responses o produciviy and he orld real ineres rae shocks. We consider he benchmark case ih nx/y = and examine he sensiiviy of impulse responses o changes in he persisence of shocks. 8 Figure a presens he impulse responses o a emporary produciviy shock, a % increase in produciviy a he iniial period ih ρ z =. Oupu, consumpion and labor inpu iniially increase, hile invesmen and, in urn, capial remain unchanged. The ne expors iniially rise because consumpion increases less han oupu. The agen is able o mainain a higher consumpion level since she receives ineres income from is posiive foreign asse holdings afer he second period. The impulse responses produced by he o models are quie similar: impulse responses of oupu, labor hours, capial, and invesmen are idenical, hile hose of consumpion, he ne expors-oupu raio, and he ne foreign asses-oupu raio exhibi lile difference. In he FDM, as prediced by policy funcions, consumpion, he ne foreign asses and he ne expors exhibi nonsaionary behavior and do no reurn o he iniial seady sae, hile he oher four variables converge o he iniial seady sae. In he EDM, all variables sloly converge o he iniial seady sae. 7 Since e se he sandard deviaions of shocks o mach he volailiy of oupu, he models are able o replicae he oupu volailiy. The models slighly undersae he volailiies of consumpion, labor supply, and he ne exporsoupu raio. The models exaggerae he correlaions of consumpion, labor supply and he ne expors ih oupu. While he persisence of he ne expors-oupu raio is loer in he models han ha in he daa, he models slighly underpredic he persisence of oher variables. Hoever, all hese differences are marginal. 8 We experimen ih oher values of nx/y and find ha impulse responses exhibi similar dynamics.

12 Figure b presens he case ih a persisen produciviy shock (ρ z =.95). We focus only on he hree variables ha are nonsaionary in he FDM since impulse responses of oher saionary variables are same as hose in he EDM. To sudy he sensiiviy of our resuls o he duraion of simulaions, e examine he impulse responses for periods (25 years) and 4 periods ( years). 9 The model economy responds o a posiive produciviy shock by increasing invesmen, consumpion, labor hours, and oupu. The ne expors decrease a impac because he agen borros from he res of he orld o increase is capial sock and, in urn, uilizes he increase in produciviy. In oher ords, he pro-borroing effec iniially dominaes he pro-saving effec inducing a fall in he ne expors. As he agen sars accumulaing foreign asses, he ne expors increase, bu hen decrease in he long run. The iniial increase in consumpion is slighly larger in he EDM han in he FDM. This can be explained by he impaience effec in he EDM: as he curren consumpion rises, he agen discouns fuure uiliy more, inducing a furher increase in he curren consumpion. The speed of convergence in he EDM is exremely slo because he ne foreign asses follo a near uni roo process. In fac, ih hese parameer values, i akes more han 4 periods ( years) for he variables in he EDM o reurn o heir iniial seady sae. Figures ih periods sugges ha he dynamic responses of he model variables are almos idenical. Impulse responses ih 4 periods indicae ha here are some minor quaniaive differences beeen he dynamic responses of he o models. Our sensiiviy analysis suggess ha he impulse responses produced by he o models become similar as θ decreases, as prediced by he policy funcions. None of he variables reurns o he iniial seady sae hen he produciviy shocks are permanen (ρ z = ) in figure c. The message of figure c is same as ha of he earlier figures: he qualiaive responses of he o models are idenical and here are only sligh differences in he quaniaive resuls. Unlike in he cases ih emporary and persisen shocks, in he case of permanen shocks, he agen, insead of accumulaing foreign asses, borros from abroad by issuing bonds and enjoys a permanen increase in consumpion. 2 One ineresing observaion is ha he ne foreign asses in he EDM do no reach he ne seady sae even afer 4 periods and gro over 6% of oupu in his period. This unrealisic predicion can be inerpreed as anoher unappealing propery of he EDM. We nex examine he impulse responses of he model variables o a.25 % increase in he orld real ineres rae in figures 2a - 2c. Figure 2a repors our resuls ih a emporary ineres rae shock (ρ r = ). Since capial sock is predeermined in he period of impac, labor supply does no respond immediaely, and oupu remains consan. Changes in invesmen and consumpion in he firs period rigger changes in 9 We limi our analysis o 4 periods ( years) since he ime span used in mos RBC papers is shorer han years. 2 We also sudy he endomen economy version of he FDM ha shos slighly differen impulse response dynamics. Wih a permanen shock, he FDM behaves like an auarky economy: he ne foreign asses do no change and consumpion is alays equal o oupu. The ne foreign asses in he EDM exhibi he same behavior as in he producion economy case. Deailed resuls of he endomen economy model are available upon reques.

13 oupu, labor inpu and capial sock in he folloing period. As in he case of produciviy shocks, responses of oupu, labor hours, capial, and invesmen are idenical in he o models, hile he behavior of he ne foreign asses is slighly differen. In figures 2b and 2c, e analyze he impulse responses of he variables ha are nonsaionary in he FDM. When he ineres rae shock is persisen (ρ r =.7) in figure 2b, none of he previous resuls changes significanly. Figure 2c depics he impulse responses o a permanen (ρ r = ) orld ineres rae shock. As hese graphs sho, all variables become nonsaionary ih permanen ineres rae shocks. Consumpion and he ne foreign asses in he FDM increase indefiniely because β is alays larger han /(+r), hile hese variables approach o he ne seady sae in he EDM. In conclusion, hile here are some quaniaive differences, he o models produce similar qualiaive responses excep hen shocks are permanen Disribuion of model variables To furher examine he dynamic implicaions of hese models, e analyze he disribuions of he model variables. We simulae he o models 5 imes ih boh shocks, and record he values of consumpion, he ne expors-oupu raio, and he ne foreign asses-oupu raio in he h and 4 h periods in each simulaion. 2 Figure 3 presens he hisograms of hese variables. While here are some differences in he hisograms across he o models, hese differences are quie small, especially a he h period. Table 4 repors he sample saisics of he hree variables in each model: mean, sandard deviaion, skeness and kurosis. These saisics sugges ha he o disribuions share almos idenical saisical properies. To formally es heher he o disribuions are saisically differen, e compue he Kolmogorov-Smirnov es saisic ha measures he maximum disance beeen he cumulaive densiy funcions of each series generaed by he o models. 22 Table 4 repors he es saisics. The saisics sho ha excep he disribuion of he ne foreign asse series in he 4h period, he disribuions of variables in he o models are no saisically differen a he % level Approximaion errors Since boh models are solved using he same linear approximaion mehod, he resuls in he previous secions may be valid only for he linearized versions of he FDM and he EDM,FDM (approximae) and EDM (approximae). Tha is, he exac soluions of hese o models, FDM (exac) and EDM (exac), may no produce similar dynamics. This is an imporan issue considering ha e linearize he model around an iniial seady sae hich he economy eiher converges o very sloly (in he EDM) or does no converge a 2 These simulaions are performed ih he benchmark parameerizaion. Since e are ineresed in levels, e do no filer he series. The disribuions of oupu, labor hours, invesmen, and capial sock produced by he o models, hich are no repored in he figure, are idenical as prediced by he policy funcions. 22 See Spanos (986) and Serfling (98) for heoreical background on his es saisic. 2

14 all (in he FDM) in response o a shock. 23 This secion focuses on he accuracy of he linearized soluions and examines heher he associaed approximaion errors are large enough o affec our previous resuls. We also compare he size of approximaion errors from he o models. If he approximaion errors are small, hen e can claim ha he resuls repored in he previous secions are valid for he FDM (exac) and EDM (exac) as ell. Approximaion errors are calculaed as follos: firs, using he benchmark parameerizaion, e simulae he linearized model and compue he value of each variable a a cerain period (every h period up o he 4 h period). Then, e plug hese values ino he original budge consrain and calculae he residual difference beeen he RHS and LHS of he budge consrain, (9) c i + b = y + b r ). + ( + Approximaion error is defined as he raio of he absolue value of his residual o oupu. 24 Approximaion errors arise from neglecing higher order erms in he linearizaion of he firs order condiions and he budge consrain. There is an addiional source of approximaion error in he FDM since he ne foreign asse series follo a uni roo process. Table 5 repors he mean of approximaion errors calculaed a every h period up o he 4 h period over 5 simulaions. 25 The firs panel repors he resuls ih produciviy shocks only, hile he second panel presens he resuls ih boh shocks. The able shos ha he FDM produces larger approximaion errors han he EDM in all cases, alhough differences are quie small. 26 This resul is compaible ih he observaion ha, in he FDM, he nonsaionariy drives he variables aay from he iniial seady sae. This also explains hy he approximaion errors increase as he lengh of simulaion increases. Anoher finding is ha he absolue magniude of approximaion errors is negligible, less han a -3 % of oupu, hen he models are simulaed ih produciviy shocks. When boh produciviy and ineres rae shocks presen, he size of approximaion errors increases up o.7 % of oupu. These findings sugges ha, even hough he approximaion errors in he FDM are larger han hose in he EDM, he size of approximaion errors is quie small if he models are subjeced o produciviy shocks only. 23 Correia, e al. (995) also observe his problem and noe ha "he accuracy of linear approximaions in models ih inegraed variables, such as ours, is sill an open quesion" (p. 95). The resuls e repor here also shed ligh on his issue. 24 Our mehod is similar o he one in Baxer (99) ho uses he Euler equaions o calculae he approximaion errors. In paricular, she defines he approximaion error as he difference beeen he lef- and righ-hand sides of he Euler equaion here she uses he approximae soluion o calculae he values of he variables. 25 We are unable o use he Den Haan-Marce (994) saisic o evaluae he accuracy of approximaion since his mehod only orks ih saionary variables. 26 Using an open economy RBC model, Kim and Kim (999) sho ha hile he linear approximaion is quie accurae in measuring he second momens, i produces inaccurae resuls in he firs momens such as elfare. 3

15 We also examine he business cycle dynamics of fixed and endogenous discoun facor formulaions in models ih complee markes. In paricular, e solve a closed economy RBC model and a o-counry RBC model ih complee markes. The resuls sugges ha he models ih fixed and endogenous discoun facors produce similar business cycle saisics and impulse responses Conclusion We examine he dynamic implicaions of differen preference formulaions in an open economy RBC model ih incomplee asse markes. In paricular, e consider o versions of an oherise idenical small open economy RBC model one ih a fixed discoun facor and he oher ih an endogenous discoun facor. Our empirical examinaion reveals five imporan resuls. Firs, hile he ne foreign asse series in he FDM follos a uni roo process, his series exhibis a near uni roo behavior in he EDM. Hence, he coefficiens in he policy funcions generaed by he o models exhibi minor differences. Second, he EDM and he FDM generae almos idenical business cycle momens. Third, hile here are small quaniaive differences beeen he impulse responses produced by he o models, heir qualiaive implicaions are same. Fourh, he disribuions of variables generaed by simulaions of he o models are no saisically differen. Finally, he approximaion errors from he soluions of he o models are of he same magniude and he errors are minimal, especially ih produciviy shocks. Undersanding he saisical properies of models ih nonsaionary variables, such as he model ih he fixed discoun facor presened here, is an imporan research opic considering ha hese ypes of models have been idely used in he dynamic macroeconomics lieraure. This paper emphasizes he quaniaive implicaions of he fixed discoun facor formulaion by comparing hese implicaions ih hose of he endogenous discoun facor model. The resuls sugges ha he FDM does no generae any resuls ha are significanly differen from hose produced by he EDM. More imporanly, business cycle dynamics produced by he o models are almos idenical. The endogenous discoun facor model exhibis some significanly imporan undesirable properies, such as he impaience effec and implausible dynamics generaed by permanen produciviy shocks. In sum, our resuls raionalize he use of he fixed discoun facor in he sudies aiming o undersand business cycle dynamics in open economies. I migh be ineresing o examine he effecs of preference srucure on some oher aspecs such as elfare issues and policy implicaions. I is possible ha he elfare implicaions or he long-erm impac of paricular policies may depend on he preference formulaion in open economy models ih incomplee asse markes. 27 We ould like o hank he Associae Edior for suggesing his exercise. Deailed examinaion of he o models and he resuls of he exercise are available upon reques.. 4

16 References Alvarez, F., and U. Jermann, 999, Quaniaive asse pricing implicaions of endogenous solvency consrains, NBER Working Paper No Arvaniis, A., and A. Mikkola, 996, Asse-marke srucure and inernaional rade dynamics, American Economic Revie Papers and Proceedings, Backus, D., Kehoe, P. and F. Kydland, 995, Inernaional business cycles: Theory and evidence, in: Thomas Cooley, ed., Froniers of Business Cycle Research, Princeon Universiy Press. Baxer, M., 99, Approximaing subopimal dynamic equilibria, Journal of Moneary Economics 27, Baxer, M., 995, Inernaional rade and business cycles, in: G. Grossman and K. Rogoff, eds., Handbook of inernaional economics, Amserdam: Norh Holland. Baxer, M., and M. Crucini, 993, Explaining saving-invesmen correlaions, American Economic Revie 83, Baxer, M., and M. Crucini, 995, Business cycles and he asse srucure of foreign rade, Inernaional Economic Revie 36, Blanchard, O., 985, Deb, deficis, and finie horizons, Journal of Poliical Economy 93, Blankenau, W., M. A. Kose, and K. Yi, 2, Can orld real ineres raes explain business cycles in a small open economy?, Journal of Economic Dynamics and Conrol 25, Bruno, C., and F. Porier, 995, A small open economy real business cycle model: he French economy case, in Pierre-Yves Henin, ed., Advances in Business Cycle Research, Springer-Verlag, Cardia, E., 99, The dynamics of small open economy in response o moneary, fiscal, and produciviy shocks, Journal of Moneary Economics 28, Cook, D. and M. Devereux, 2, The macroeconomic effecs on inernaional financial panics, mimeo, Universiy of Briish Columbia. Correia, I., Neves, J. and S. Rebelo, 992, Business cycles in Porugal: Theory and evidence, in J. Amaral, D. Lucena, and A. Mello, eds., The Poruguese Economy Toards 992. Kluer. Correia, I., Neves, J. and S. Rebelo, 995, Business cycles in a small open economy, European Economic Revie 39, Crucini, M., 999, Inernaional co-movemen: Is heory ahead of inernaional business cycle measuremen?, Working Paper, Ohio Sae Universiy. Daniel, B., 997, Precauionary saving and persisen curren accoun imbalance, Journal of Inernaional Economics 42, Den Haan, W., and A. Marce, 994, Accuracy in simulaions, The Revie of Economic Sudies 6,

17 Epsein, L., 983, Saionary cardinal uiliy and opimal groh under uncerainy, Journal of Economic Theory 3, Epsein, L., 987, A simple dynamic general equilibrium model, Journal of Economic Theory 4, Greenood, J., Hercoiz, Z. and G. Huffman, 988, Invesmen, capaciy uilizaion and he real business cycle, American Economic Revie 78, Harjes, T., 997, Real business cycles in an open economy: an applicaion o Germany, Welirschafliches Archiv 33, Heahcoe, J. and F. Perri, 2, Financial auarky and inernaional business cycles, Journal of Moneary Economics, forhcoming. Helpman, E., and A. Razin, 982, Dynamics of a floaing exchange rae regime, Journal of Poliical Economy 9, Hodrick, R., and E. Presco, 997, Posar U. S. business cycles: an empirical invesigaion, Journal of Money, Credi, and Banking 29, -6. Hugge, M., 993, The risk free rae in heerogeneous-agens, incomplee insurance economies, Journal of Economic Dynamics and Conrol 7, Karayalcin, C., 995, Capial income axaion and elfare in a small open economy, Journal of Inernaional Money and Finance 4, Kehoe, P. and Perri, F., 2, Inernaional business cycles ih endogenous incomplee markes, FRB Minneapolis Saff Repor No Kim, J., and S.H. Kim, 999, Spurious elfare reversals in inernaional business cycle models, mimeo, Brandeis Universiy. King, R., C. Plosser, and S. Rebelo, 988, Producion, groh, and business cycles I: The basic neoclassical model, Journal of Moneary Economics 2, Kollmann, R., 995, Incomplee asse markes and he cross-counry correlaion puzzle, Journal of Economic Dynamics and Conrol 2, Kollmann, R., 998, U.S. rade balance dynamics: he role of fiscal policy and produciviy shocks, Journal of Inernaional Money and Finance 7, Kollmann, R., 2, Explaining inernaional comovemens of oupu and asse reurns: he role of money and nominal rigidiies, Journal of Economic Dynamics and Conrol, forhcoming. Kose, M. A., and R. Riezman, 2, Trade shocks and macroeconomic flucuaions in Africa, Journal of Developmen Economics, forhcoming. Kouparisas, M., 997, Norh-Souh financial inegraion and business cycles, orking paper, Federal Reserve Bank of Chicago. Kubler, F. and K. Schmedders, 2, Incomplee markes, ransiory shocks, and elfare, mimeo, Sanford Universiy. 6

18 Levine, D., and W. Zame, 996, Deb consrains and equilibrium in infinie horizon economies ih incomplee markes, Journal of Mahemaical Economics 26, 3-3. Levine, D. and W. Zame, 999, Does marke incompleeness maer?, mimeo, UCLA. Mendoza, E., 99, Real business cycles in a small open economy, American Economic Revie 8, Mendoza, E., 995, The erms of rade, he real exchange rae, and economic flucuaions, Inernaional Economic Revie 36, -37. Mendoza, E., and L. Tesar, 998, The inernaional ramificaions of ax reforms: supply-side economics in a global economy, American Economic Revie 88, Obsfeld, M., 99, Ineremporal dependence, impaience, and dynamics, Journal of Moneary Economics 26, Rebelo, S. and C. A. Vegh, 995, Real effecs of exchange-rae-based sabilizaion: an analysis of compeing heories, in: B. Bernanke and J. Roemberg, eds., NBER macroeconomics annual, Cambridge and London: MIT Press, Rebelo, S., 997, Wha happens hen counries peg heir exchange raes? The real side of moneary reforms, CEPR Discussion Paper No Sadka, J. C. and K. Yi, 996, Consumer durables, permanen erms of rade shocks, and he recen U.S. rade deficis, Journal of Inernaional Money and Finance 5, Schmi-Grohé, S., 998, The inernaional ransmission of economic flucuaions, Journal Inernaional Economics 44, Serfling, R., 98, Approximaion Theorems of Mahemaical Saisics, John Wiley and Sons. Senhadji, A., 995, Sources of deb accumulaion in a small open economy, orking paper, Washingon Universiy. Senhadji, A., 998, Dynamics of he rade balance and he erms-of-rade in LDCs: The S-curve, Journal of Inernaional Economics 46, 5-3. Spanos, A., 986, Saisical Foundaions of Economeric Modelling, Cambridge Universiy Press. Uribe, M., 997, Exchange-rae-based inflaion sabilizaion: The iniial real effecs of credible plans, Journal of Moneary Economics 39, Uzaa, H., 968, Time preference, he consumpion funcion and opimum asse holdings, in: J. Wolfe, ed., Value, Capial, and Groh: Papers in Honor of Sir John Hicks, Edinburgh Universiy Press, van Wincoop, E., 996, A muli-counry real business cycles model ih heerogeneous agens, Scandinavian Journal of Economics 98, van Wincoop, E., and J. Marrinan, 999, Saving invesmen relaionships: consisen ih perfec capial mobiliy?, orking paper, Federal Reserve Bank of Ne York. 7

19 Table Parameers of he model Parameer Descripion Value Preferences r Seady sae real ineres rae.2% Ineremporal elasiciy of subsiuion in labor supply.455 σ Coefficien of relaive risk aversion.5 Technology α Share of labor income.68 δ Depreciaion rae 2.5% η Elasiciy of marginal adjusmen cos funcion η = ( φ / φ ) /( i / k) nx/y Seady sae ne expors o oupu raio Shocks ρ z Persisence of echnology shock.95 σ z Sandard deviaion of echnology shock.625% ρ r Persisence of ineres rae shock.7 σ r Sandard deviaion of ineres rae shock.% Corr(ε z,ε r ) Correlaion beeen echnology and ineres rae shocks See secion 2.3 for deails. 8

20 Table 2 Policy funcions cˆ a bˆ a nx ˆ a yˆ = a nˆ a iˆ a kˆ + a b b 2 b3 kˆ b4 b bˆ 5 b6 b 7 + shocks Parameer Coefficien FDM EDM Coefficien FDM EDM nx/y = -.5 a b θ =.74 a b a b Nx/y =. θ =.8 a b a b a b Nx/y =.5 θ =.88 a b a b a b All cases a b 4 a b 5 a b 6 a b 7 FDM: Fixed Discoun Facor Model, EDM: Endogenous Discoun Facor Model. NX/Y is he iniial seady sae value of he ne expors-oupu raio. θ denoes he elasiciy of he discoun facor ih respec o uiliy. See secion 3. for deails. 9

21 Oupu.5 (.5) Consumpion.9 (.) Hours.3 (.) Invesmen 4.37 (.42) NFA/Y.3 (.22) NX/Y.35 (.4) Table 3 Business cycle momens Panel A: Produciviy shocks Volailiy (%) (σ) Relaive Volailiy Correlaion ih oupu Auocorrelaion FDM EDM FDM EDM FDM EDM FDM EDM.5 (.5). (.).3 (.) 4.37 (.42).32 (.22).36 (.4)... (.) (.) (.) (.) (.4) (.4). (.). (.). (.).97 (.) -.65 (.4) -.78 (.4) Oupu.5 (.6) Consumpion.9 (.) Hours.3 (.) Invesmen 5.3 (.42) NFA/Y 2.2 (.32) NX/Y.8 (.6) Panel B: Produciviy and ineres rae shocks Volailiy (%) (σ) Relaive Volailiy Correlaion ih oupu Auocorrelaion FDM EDM FDM EDM FDM EDM FDM EDM.5 (.6). (.).3 (.) 5.3 (.42) 2. (.3).76 (.5)... (.) (.) (.) (.4) (.3) (.9). (.). (.). (.).78 (.4) -.44 (.3) -.34 (.9) FDM: Fixed Discoun Facor Model, EDM: Endogenous Discoun Facor Model. All momens are averages over 3 simulaions here each simulaion consiss of periods. All variables, excep NFA/Y and NX/Y, are logged and HP (6) filered. NX (ne expors) and NFA (ne foreign asses) are normalized by oupu, and HP (6) filered. Numbers in he parenhesis are sandard errors. See secion 3.2 for deails. 2

22 Table 4 Sample saisics of empirical disribuions h Period Variable Saisic FDM EDM Consumpion (K-S saisic =.62) NFA/Y (K-S saisic =.24) NX/Y (K-S saisic =.42) Mean Sandard Dev. Skeness Kurosis Mean Sandard Dev. Skeness Kurosis Mean Sandard Dev. Skeness Kurosis 4 h Period Variable Saisic FDM EDM Consumpion (K-S saisic =.6) NFA/Y (K-S saisic =.454)* NX/Y (K-S saisic =.36) Mean Sandard Dev. Skeness Kurosis Mean Sandard Dev. Skeness Kurosis Mean Sandard Dev. Skeness Kurosis The disribuions are dran from 5 simulaions ih boh shocks, here each simulaion consiss of 4 periods. K-S saisic denoes he Kolmogorov-Smirnov saisic for esing he difference beeen o cumulaive densiy funcions. If he es saisic is greaer han he criical value, hen he o disribuions are significanly differen (noed ih *). % criical value is.326 for all cases. See secion 3.4 for deails. 2