Tractable Estimation of Non-Linear DSGE Models Using Observation Equation Inversion. Robert Kollmann (*) ECARES, Université Libre de Bruxelles & CEPR

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1 Work in progress Tracable Esimaion of Non-Linear DSGE Models Using Observaion Equaion Inversion Rober Kollmann (*) ECARES, Universié Libre de Bruxelles & CEPR Firs version: February, 015 This version: July 7, 016 This paper presens a racable approach for compuing he likelihood funcion of non-linear Dynamic Sochasic General Equilibrium (DSGE) models ha are solved using second- and hird order accurae approximaions. The mehod assumes ha he number of exogenous shocks equals he number of observables. For given iniial saes, exogenous innovaions are compued recursively by invering he observaion equaion. I is hence easy o compue he sample likelihood funcion. Iniial saes and model parameers can be esimaed by maximizing he likelihood funcion. Numerical examples sugges ha he mehod provides reliable esimaes of model parameers, even for highly non-linear economies wih big shocks. By conras o paricle filers, no sochasic simulaions are needed o compue he likelihood funcion. The mehod here is, hence, much faser and i is hus suiable for he esimaion of medium-scale non-linear models. Keywords: Likelihood-based esimaion of non-linear DSGE models, higher-order approximaions, pruning, laen sae variables. JEL codes: C63, C68, E (*) I am very graeful o Tom Holden, Maeo Iacoviello, Johannes Pfeifer, Raf Wouers and Chris Sims for useful suggesions, and o conference paricipans a he Naional Bank of Belgium for helpful commens. Special hanks are also due o Johannes Pfeifer for advice abou he Dynare sofware. The research leading o hese resuls has received funding from Académie Universiaire Wallonie-Bruxelles (Acion de recherche concerée, gran ARC-AUWB/010-15/ULB-11) and from he European Communiy s Sevenh Framework Programme (FP7/ ) under gran agreemen no , Projec MACFINROBODS ( Inegraed Macro- Financial Modelling for Robus Policy Design ). Address: ECARES, CP 114, ULB; 50 Av. F. Roosevel, B-1050 Brussels, Belgium. rober_kollmann@yahoo.com 1

2 1. Inroducion During he las hree decades, Dynamic Sochasic General Equilibrium (DSGE) models have become he workhorse of macroeconomic research. These models are also invaluable ools for economic policy analysis and forecasing. Due o heir complexiy, numerical approximaions are required o solve DSGE models. The bulk of DSGE-based analysis uses linear approximaions. A fas growing recen lieraure has aken linearized DSGE models o he daa, using likelihoodbased mehods, building on conribuions by i.a. Kim (000), Schorfheide (000) and Orok (001). Lineariy (in sae variables) grealy faciliaes model esimaion, as i allows o use he sandard Kalman filer o infer laen variables and o compue sample likelihood funcions based on predicion error decomposiions. However, linear approximaions are inadequae for models wih big shocks, and hey canno capure he effec of risk on economic decisions and welfare. Non-linear approximaions are hus, for example, needed for welfare calculaions in sochasic models, or for sudying asse pricing and non-lineariies due o financial fricions and consrains. Recen research has begun o esimae non-linear DSGE models. Tha work has mainly used paricle filers, i.e. filers ha infer laen saes using Mone Carlo mehods (see Fernández- Villaverde and Rubio-Ramírez (007) and An and Schorfheide (007) for early applicaions). Paricle filers are slow compuaionally, which limis heir use o small models. This paper presens a racable esimaion mehod for non-linear DSGE models. The mehod assumes ha he number of exogenous shocks equals he number of observables (daa variables). For given iniial values of he sae variables, one hen recursively infer he exogenous innovaions by invering he observaion equaion. This mehod also produces esimaes of he rajecories of all laen sae variables. I hus allows o compue he sample likelihood funcion wihou using a simulaions-based filer. The paper uses his idea o esimae DSGE models ha are solved by second- or hird- order Taylor expansions of he decision rules around a deerminisic seady sae. 1 A challenge for observaion equaion inversion is ha approximaing decision rules include powers of innovaions o exogenous variables: given he sae variables realized a dae -1, muliple dae exogenous innovaions are consisen wih he period observables. To overcome his problem, I consider runcaed dae decision rules from which powers of he exogenous innovaions have been suppressed. The runcaed decision rules are, 1 Guerrieri and Iacoviello (014) and Deák, Holden and Mele (015) also discuss esimaion of DSGE models via observaion equaion inversion. These auhors do no consider second- or hird-order approximaed models (ha are he focus of he presen paper).

3 hence, linear in dae exogenous innovaions (he coefficiens of hose innovaions are, however, funcions of lagged sae variables), bu non-linear in lagged sae variables. Hence, i is sraighforward o inver he observaion equaion. I presen examples of DSGE models for which he runcaed decision rules are observaionally indisinguishable from decision rules ha include higher-order powers of conemporaneous exogenous innovaions. Numerical examples show ha he esimaion mehod here is fas and accurae, even for models wih srong curvaure and big shocks. The mehod requires a guess abou he iniial sae vecor. In he numerical example provided here, I use a raining sample o reduce he effec of he iniial sae vecor on he inferred exogenous innovaions. I is also possible o esimae he iniial saes, by maximizing he likelihood funcion wih respec o hose saes. The numerical DSGE-model soluion echnique (second- or hird-order approximaions) considered here is he mos racable non-linear soluion mehod for medium-scale models, and i has hus widely been used in macroeconomics (see Kollmann (00) and Kollmann e al. (011) for deailed references). When simulaing higher-order approximaed models, i is common o use he pruning scheme of Kim e al. (008), under which second-order erms are replaced by he producs of he linearized soluion (hird-order erms are replaced by producs of linearized and second-order approximaed variables). Unless he pruning algorihm is used, second-order (and hird-order) approximaed models ofen generae exploding simulaed pahs. Pruning is herefore crucial for applied work based on second-order (and hird-order) approximaed models. The paper here also uses he pruning scheme. The presen paper is complemenary o Kollmann (015a,b) who developed a racable deerminisic filer for pruned second-order and hird-order approximaed DSGE models. Tha filer explois he fac ha a pruned second-order accurae sysem is linear in an exended sae vecor ha consiss of variables solved o second- and firs-order accuracy, and of producs of firs-order accurae variables. 3 Lineariy in ha exended sae vecor allows closed-form deerminaion of he sae vecor s (condiional) mean and variance. Kollmann (015a,b) applies he linear updaing rule of he sandard Kalman filer o he pruned sae equaion. 4 The filer based on he linear updaing rule is more accurae han paricle filers (unless a very large Compuer code ha solves higher-order approximaed models is freely available; see, e.g., Sims (000), Schmi- Grohé and Uribe (004), and Adjemian e al. (014). 3 The pruned hird-order accurae sysem is linear in a sae vecor ha consiss of variables solved o hird-, secondand firs-order, of producs of firs-order accurae variables and of second-order accurae variables (see Appendix). 4 Ivashchenko (014) also develops a Kalman filer for second-order approximaed models, bu his mehod does no use he pruning scheme. 3

4 number of paricles is used), and i oo is much faser han paricle filers. A pracical issue for he Kollmann (015a,b) filer is ha he dimension of he pruned sae space increases rapidly when model size increases. As here is no need o compue he momens of he pruned sae vecor, he observaion equaion inversion mehod developed in he presen paper is faser han he Kollmann (015a,b) filer. However, he mehod here requires ha he number of exogenous shocks equals he number of observables, while he Kollmann (015a,b) mehod allows o handle siuaions in which he number of shocks exceeds he number of observables.. Model forma.1. Model and second-order accurae soluion Sandard DSGE models can be expressed as: EG(,, ) 0, (1) 1 1 where E is he mahemaical expecaion condiional on dae informaion; G: R R n m n is a funcion, and is an nx1 vecor of endogenous and exogenous variables known a ; 1 is an mx1 vecor of serially independen innovaions o exogenous variables. In wha follows, Gaussian: (0, ), where is a scalar ha indexes he size of shocks. The model soluion N is a "policy funcion" ("decision rule") 1 FX (, 1, ), where X is a vecor consising of he sae variables of he economy, i.e. of he predeermined endogenous variables (e.g., he physical capial sock) and he exogenous variables included in he vecor, i.e. X, where is a marix of zeros and ones ha selecs he sae variables among he elemens of The policy funcion has o be such ha such ha EGF ( (,, ),, 1) 0. See,. e.g., Sims (000) and Schmi-Grohé and Uribe (004). Following hese auhors, his paper focuses on Taylor series expansions of he policy funcion around a deerminisic sead sae, i.e. around 0 and a vecor such ha F(,,0). Le and xx X denoe deviaions from seady sae. The second-order accurae model soluion has he form F F x F F x x F x F, wih x. () Here denoes he Kronecker produc. F0, F1, F, F11, F1, F are marices ha are funcions of he srucural model parameers, bu do no depend on he scale of shocks (); see, Schmi-Grohé and Uribe (004). The firs-order accurae (linearized) model soluion is: 1 is 4

5 F x F. (3) (1) (1) The superscrip (1) denoes a variable solved o firs-order accuracy... The pruned second-order accurae sysem When simulaing second-order accurae models, i is common o use he pruning scheme of Kim, Kim, Schaumburg and Sims (008), under which producs of sae variables are replaced by producs of variables approximaed o lower order, i.e. x x and x 1 are replaced by x x and (1) (1) x (1) 1, respecively. Wih pruning, he soluion () is hus replaced by: (4) () () (1) (1) (1) 1 F0 F1 x F 1 F11 x x F1 x 1 F 1 1. Noe ha x x x x and (1) (1) x x holds, up o second-order accuracy. 5 Thus, (4) (1) 1 1 is a valid second-order accurae soluion. The moivaion for pruning is ha, in repeaed applicaions of (), hird and higher-order erms of sae variables appear; e.g., when 1is quadraic in, hen is quaric in ; pruning removes hese higher-order erms. The nonpruned sysem () has exraneous seady saes (no presen in he original model)--some of hese seady saes mark ransiions o unsable behavior. Large shocks can hus move he model ino an unsable region. Pruning overcomes his problem. If he firs-order soluion is sable, hen he pruned second-order soluion (4) oo is sable. The following exposiion focuses on he esimaion of he pruned second-order sysem. Exension of he mehod o hird (or higher) order approximaed sysem is sraighforward. See he Appendix..3. Inferring he exogenous innovaions from observables Assume ha, a dae +1, he economerician knows he sae vecors x, x and observes m (1) () of he elemens of he vecor vecor of observables is (or m linear combinaions of he elemens of () 1 () 1 ), i.e. he 5 (1) () ( n) For any variable a we can wrie aa R where R conains erms of order n or higher in deviaions from he (1) () (1) () (1) (1) (1) () (1) () (4) (1) (1) (3) () (1) (1) seady sae. Thus, a b ( a R )( b R ) a b a R b R R a b R and hence ( a b ) a b. This () (1) (1) () (1) logic implies ( x x ) x x and ( x 1) x 1. See, e.g., Kollmann (004; 015a,b), Lombardo and Suherland (007) and Lombardo and Uhlig (015). 5

6 z () 1 Q 1, (5) where Q is a known marix of dimension mxn. (Recall ha m and n are he number of exogenous innovaions and of endogenous variables, respecively.) (4) implies: z Q( F F x F x x ) Q( F F x F ). (6) () (1) (1) (1) As he righ-hand side of (6) includes squares of, 1 one canno uniquely solve (6) for he unknown vecor of innovaions, 1 given he daa z 1 and he saes x, x. There is no (1) () racable mehod for compuing all vecors 1 ha solve (6) when m is larger han 3 or 4. 6 In wha follows, I replace he erm F 1 1 by is expeced value F E( 1 1) in (6). 7 Thus, equaions (4),(6) are replaced by F F x F F x x F x F E( ) and (7) () () (1) (1) (1) z Q( F F x F x x F E( )) Q( F F x ). (8) () (1) (1) (1) Numerical experimens, for a range of models, show ha his modificaion only has a minor effec on he model dynamics; see below. 8 Noe ha he observaion equaion (8) is linear in : 1 z, where 1 1 () (1) (1) Q( F0 F1 x F11 x x F E( 1 1)), while is an (n x n) marix such ha (1) 1Q( F 1F1 x 1). If is non-singular, one can solve (7) for he vecor of exogenous innovaions: 1 1 z 1 ( )..3. Sample likelihood Assume ha (3),(7),(8) is he rue daa generaing process. Given he iniial sae x, x and (1) () 0 0 daa { z } T one can recursively compue he innovaions 1 { } T and he saes () 1 { i } T 1 for i=1, using (3),(6),(8). The log likelihood of he daa, condiional on x, x is: (1) () If i were possible o compue all 1 ha solve (6) hen one could pick he mos likely of hese vecors (i.e. he 1 one for which ' 1( ) 1 is smalles) as an esimae of he rue 1. 7 I hank Chris Sims for suggesing his approach. In earlier versions of he algorihm, I dropped he erm F from (6), which likewise produces an observaion equaion ha is linear in. 1 The wo approaches 1 1 produce very similar resuls. An advanage of replacing F 1 1 by is expeced value F E( 1 1) is ha he uncondiional mean of he modified process equals ha of he original process. 8 Feeding a given sequence of exogenous innovaions { } ino he sysem (3),(4) and ino he sysem (3),(7) generaes pahs for he endogenous variables ha are almos perfecly correlaed across he wo sysems. 6

7 ln L({ z } x, x ) ln( ) ln { ' ( ) ln. T (1) () T mt T One can esimae srucural model parameers (and he iniial sae), by maximizing he likelihood funcion wih respec o he parameers. 3. Applicaion: basic RBC model I illusrae he mehod for a sandard RBC model. Assume a closed economy wih a represenaive infiniely-lived household whose dae expeced lifeime uiliy V is given by V { 1 C 1 N } EV 1, where C and N are consumpion and hours worked, a, respecively. 0 and 0 are he risk aversion coefficien and he inverse of he (Frisch) labor supply elasiciy. is he subjecive discoun facor beween periods and and 0 are exogenous preference shocks: subjecive discoun facor. is a labor supply shock, while is a shock o he and equal uniy in he deerminisic seady sae. The household maximizes expeced lifeime uiliy subjec o he period resource consrain where Y and C I G Y, I are oupu, gross invesmen and exogenous governmen consumpion, respecively. The producion funcion is where Y K N 1 K is he beginning-of-period capial sock, and 0 produciviy (TFP). The law of moion of he capial sock is K (1 ). 1 K I is exogenous oal facor 0, 1 are he capial share and he capial depreciaion rae, respecively. The household s firs-order condiions are: 1 1 E ( C 1/ C) ( 1 K 1 N 1 1 ) 1, The forcing variables follow independen auoregressive processes: C (1 ) K N N. 1/ ln( / ) ln( 1 / ),, ln( G/ G) Gln( G 1 / G) G,, ln( ) ln( 1 ),, ln( ) ln( 1 ),, 7

8 wih 0,,, 1, where and G are seady sae TFP and seady sae governmen G purchases.,, G,,, and, are normal i.i.d. whie noises wih sandard deviaions, G, and. The numerical simulaions discussed below assume 0.99, 0.5, 0.3, 0.05; he seady sae raio of governmen purchases o GDP ( GY / ) is se a 0.. The auocorrelaions of all forcing variables is se a 0.98, i.e. he exogenous variables undergo G persisen flucuaions. Parameer values in ha range are sandard in (quarerly) macro models. The risk aversion coefficien is se a a high value, 10, so ha he model has enough curvaure o produce non-negligible differences beween he second- and hird-order model approximaions and he linearized model. In all model varians, I se he scalar (ha indexes he size of shocks; see Secion.1) a 1. One model varian, referred o as he small shocks varian, assumes G 1% and 0.05%. Those shock sizes (i.e. rae of ime preference shocks 40-imes smaller han he oher shocks) ensure ha each shock accouns for a nonnegligible share of he variance of he endogenous variables (see Table 1). Tha small shocks calibraion is sandard in he RBC lieraure, and i implies ha he volailiy of he endogenous variables in he model is roughly consisen wih he empirical volailiy. In he small shocks varian, he behavior of endogenous variables prediced by he second- and hird-order approximaed model is broadly similar o ha prediced by he linearized model (see he Appendix for a presenaion of he hird-order accurae model soluion). I hus also consider model varians wih much bigger shocks in hose varians, he higher-order approximaed model generaes prediced behavior ha differs noiceably from behavior in he firs-order approximaed model. In one model varian, he sandard deviaions or shocks are 5 imes greaer han in he small shocks varian ( 5%, 0.15%); I also consider a varian in which G he sandard deviaion of exogenous innovaions is 10 ime greaer ( 10%, 0.50%). G I refer o hese model varians as he big shocks varian and he very big shocks varian, respecively. I solve he model using he Dynare oolbox (Adjemian e al. (014)). The Taylor expansions of he model equaions are aken wih respec o logs of all variables. 8

9 3.1. Prediced momens Table 1 repors prediced sandard deviaions of GDP, consumpion, invesmen, hours worked and he capial sock. The prediced momens are shown for variables in logged levels, as well as for firs-differenced logged variables. In he small shocks varian, he order of approximaion does no maer much for prediced behavior. For example, he prediced sandard deviaion of GDP is 3.00% (3.09%) [3.04%] under he firs- (second-) [hird-] order accurae model approximaion. By conras, in he model varians wih big and wih very big shocks, he second- and hird-order approximaions generae markedly greaer volailiy of he endogenous variables han he linear approximaion. In he big shocks [ very big shocks ] varian he prediced volailiy of GDP rises by one quarer [doubles] when he hird-order approximaion is used, insead of he linear approximaion. Under he linear approximaion, he uncondiional means of all endogenous variables equals heir values in he deerminisic seady sae. Under he second- and hird-order approximaions, he uncondiional means can differ from he deerminisic seady sae (uncondiional means implied by he second and hird-order approximaions are idenical). In he small shocks varian, he mean of capial sock and mean GDP exceeds seady sae values by 0.81% and 0.5%, respecively. This is due o precauionary saving ha is capured by he second-order approximaion. In he big shocks [ very big shocks ] model varian, he mean capial sock and mean GDP are 0.39% and 6.6% [81.56% and 5.05%] above he deerminisic seady sae. 3.. Comparing he runcaed versions of he hird-order accurae model Table documens ha he runcaed version of he (pruned) hird-order accurae model is observaionally equivalen o he non-runcaed version (see Appendix). The correlaions of simulaed ime series across he runcaed and non-runcaed varians are very close o uniy, boh in levels and in firs differences, and ha even when shocks are very big Esimaing srucural parameers I now evaluae he abiliy of he mehod o esimae srucural model parameers, for he case of he hird-order accurae model approximaion. For each of he hree model varians, I generaed 30 simulaion runs of 5100 periods (each simulaion run was iniiaed a he uncondiional means 9

10 of he sae variables). I use he las 100 periods of each simulaion run for esimaion. As he model has four exogenous shocks, four observables are needed for esimaion. Firs differences of log GDP, consumpion, invesmen and hours worked are used as observables. I add equaions defining he firs difference of hese four variables o he model equaions. Dynare hen idenifies 9 sae variables (he capial sock, and lagged values of he four exogenous variables and of GDP, consumpion, invesmen and hours). I assume ha he iniial values of he saes x, x, x equal he uncondiional means of hese vecors. Alhough rue iniial values differ (1) () (3) from he assumed values, he inferred esimaes of he exogenous innovaions converge fas o he rue values, in mos of he simulaion runs. I hus use he firs 10 periods of each sample period (of lengh T=100) as a raining sample (i.e. he firs 10 periods are dropped in he consrucion of he likelihood funcion). I esimae he following 10 srucural parameers: he risk aversion coefficien ( ), labor supply parameer ( ), as well as he auocorrelaions and sandard deviaions of he four exogenous variables. Panel (a) of Table 3 repors he mean, median and sandard deviaion of he esimaed model parameers across he 30 simulaion runs, for he small shocks model varian (Columns (1)-(3)), he big shocks varian (Cols. (4)-(6)) and he very big shocks model varian (Cols. (7)-(9)). Table 3 shows ha he risk aversion coefficien, he auocorrelaions of he exogenous variables and he sandard deviaions of exogenous innovaions are relaively ighly esimaed: he mean and median parameer esimaes (across runs) are close o he rue parameer values, and he sandard deviaions of he parameer esimaes are mosly small. The labor supply parameer is less precisely esimaed. For he model and sample lengh considered here, one evaluaion of he likelihood funcion (for a given parameer vecor) akes 0.5 second for he hird-order accurae model (using a PC wih an Inel i7-600 processor, 3.40Ghz). By conras, one evaluaion of he likelihood akes 0.80 second when he Kollmann (015a,b) filer is used. Thus, he mehod here is markedly faser. The gain in speed is even greaer when he number of saes is bigger. For he wo-counry DSGE model wih 19 sae variables discussed in Kollmann (015b), one evaluaion of he likelihood funcion akes 1.80 second when he mehod here is used. Esimaion of a model of his size is hence feasible. The Kalman filer mehod of Kollmann (015a,b) is an order of magniude slower. 10

11 APPENDIX: Esimaion mehod for hird-order accurae approximae model soluions The hird-order accurae model soluion of he DSGE model (1) is given by: F ( F F ) x ( F F ) F x x F x F F x x x F x x F x F (A.1), where F1, F, F111, F11, F1, F are marices ha are funcions of he srucural model parameers (he remaining marices of coefficiens, F0, F1, F, F11, F1, F are idenical o he corresponding coefficiens in he second-order accurae model soluion; see ()). To apply he logic of pruning o (A.1), noe ha (3) (1) ( x) x, (3) () (1) () () (1) ( xx ) x x x ( x x ), ( x ) x, (3) () 1 1 (3) (1) (1) (1) ( xxx ) x x x, ( x x ) x x, (3) (1) (1) 1 1 ( x ) x. 9 (3) (1) The pruned hird-order accurae sysem is hus given by: F F x F x ( F F ) F { x x x ( x x )} F x F... (3) (3) (1) () (1) (1) () (1) () F x x x F x x F x F (A.) (1) (1) (1) (1) (1) (1) , where he firs- and second order accurae quaniies (1) x and () x obey (3) and (4): F x F, (3) (1) (1) (4) () () (1) (1) (1) 1 F0 F1 x F 1 F11 x x F1 x 1 F 1 1. Unless he pruning algorihm is used, hird-order approximaed models ofen generae exploding simulaed ime pahs. Pruning ensures ha (A.) is non-explosive if he firs-order sysem (3) is saionary. To permi inversion of he observaion equaion, I replace squares and cubes of 1 by heir expeced values, on he righ-hand side of (A.): F F x F x ( F F ) F { x x x ( x x )} F x F E( )... (3) (3) (1) () (1) (1) () (1) () F x x x F x x F x E( ). (A.3) (1) (1) (1) (1) (1) (1) Noe ha E( 11 1) 0, because 1 is normally disribued. Assume ha he rue daa generaing process is given by (A.3),(3),(4), and ha he economerician observes m of he 9 For variable a we can wrie a a R and a a R where (1) () () (3), ( n) R conains erms of order n or higher in deviaions from he seady sae. The produc ab can hus be expressed as a b a a a R b b b R a b a a b R hence, (1) () (1) (3) (1) () (1) (3) (1) () () (1) (1) (4) ( )( ) ( ) ; () (1) () () (1) () (1) (4), and hence ( a a )( a a ) R.) a a R (3) (1) () () (1) (1) ( ab ) a b ( a a ) b. (3) (1) (1) (1) The same logic shows ha ( a b c ) a b c. (Noe ha 11

12 hird-order accurae variables, z (3) 1 Q 1, where Q is a known marix of dimension mxn. Given iniial sae (1) () (3) x0, x0, x 0 and daa { z } T 1 one can recursively compue he innovaions { } T and he saes () 1 { i } T 1 for i=1,,3 using (3),(4) and (A.). The log likelihood of he daa, condiional on x, x, x is: (1) () (3) ln L({ z } x, x, x ) ln( ) ln { ' ( ) ln, (A.4) T (1) () (3) T mt T where is he (m x m) marix such ha ( F F ) F x F x x. () (1) (1)

13 References Adjemian, S., H. Basani, M. Juillard, F. Mihoubi, G. Perendia, J. Pfeifer, M. Rao, S. Villemo, 014. Dynare: reference manual, Version , Working Paper, CEPREMAP. An, S. and F. Schorfheide, 007. Bayesian Analysis of DSGE models. Economeric Reviews 6, Deák, S., T. Holden and A. Mele, 015. An Advanced Course On The Science and Ar of DSGE Modelling. Universiy of Surrey. Fernández-Villaverde, J. and J. Rubio-Ramírez, 007. Esimaing Macroeconomic Models: a Likelihood Approach. Review of Economic Sudies 74, Guerrieri, L. and M. Iacoviello, 014. Collaeral Consrains and Macroeconomic Asymmeries. Working Paper, Federal Reserve Board. Ivashchenko, S., 014. DSGE Model Esimaion on he Basis of Second-Order Approximaion. Compuaional Economics 43, Kim, J., 000. Consrucing and Esimaing A Realisic Opimizing Model of Moneary Policy. Journal of Moneary Economics 45, Kim, J., Kim, S., Schaumburg, E. and C. Sims, 008. Calculaing and Using Second-Order Accurae Soluions of Discree-Time Dynamic Equilibrium Models, Journal of Economic Dynamics and Conrol 3, Kollmann, R., 00. Moneary Policy Rules in he Open Economy: Effecs of Welfare and Business Cycles. Journal of Moneary Economics 49, Kollmann, R., 004. Solving Non-Linear Raional Expecaions Models: Approximaions Based on Taylor Expansions, Working Paper, Universiy of Paris XII. Kollmann, R., S. Maliar, B. Malin and P. Pichler, 011. Comparison of Numerical Soluions o a Suie of Muli-Counry Models. Journal of Economic Dynamics and Conrol 35, pp Kollmann, R., 015a. Tracable Laen Sae Filering for Non-Linear DSGE Models Using a Second-Order Approximaion and Pruning, Compuaional Economics 45, Kollmann, R., 015b. Tracable Laen Sae Filering for Non-Linear DSGE Models Using a Third- Order Approximaion, Pruning and a Kalman filer, work in progress. Lombardo, G. and A. Suherland, 007. Compuing Second-Order Accurae Soluions for Raional Expecaions Models Using Linear Soluion Mehods. Journal of Economic Dynamics and Conrol 31, Lombardo, G. and H. Uhlig, 015. A Theory of Pruning, Working Paper, BIS and Universiy of Chicago. Orok, C., 001. On Measuring he Welfare Cos of Business Cycles. Journal of Moneary Economics 47, Schmi-Grohé, S. and M. Uribe, 004. Solving Dynamic General Equilibrium Models Using a Second-Order Approximaion o he Policy Funcion. Journal of Economic Dynamics and Conrol 8, Schorfheide, F., 000. Loss Funcion-Based Evaluaion of DSGE Models. Journal of Applied Economerics 16, Sims, C., 000. Second Order Accurae Soluion of Discree Time Dynamic Equilibrium Models. Working Paper, Economics Deparmen, Princeon Universiy 13

14 Table 1. RBC model: prediced sandard deviaions (in%) Y C I N K (1) () (3) (4) (5) (a) Model varian wih small shocks ( 0.01, ) G (a.1) Variables in levels (logs) 1 s order, all shocks s order, jus shock s order, jus G shock s order, jus shock s order, jus shock nd order, all shocks rd order, all shocks (a.) Firs-differenced variables (logs) 1 s order, all shocks nd order, all shocks rd order, all shocks (b) Model varian wih big shocks ( 0.05, ) G (b.1) Variables in levels (logs) 1 s order, all shocks nd order, all shocks rd order, all shocks (b.) Firs-differenced variables (logs) 1 s order, all shocks nd order, all shocks rd order, all shocks (c) Model varian wih very big shocks ( 0.10, ) G (c.1) Variables in levels (logs) 1 s order, all shocks nd order, all shocks rd order, all shocks (c.) Firs-differenced variables (logs) 1 s order, all shocks nd order, all shocks rd order, all shocks Noe: Sandard deviaions (sd.) of logged variables (lised above Cols. (1)-(5)) are shown for he RBC model. All momens are compued based on one simulaion run of 5000 periods (he run is iniiaed a he uncondiional mean of he sae variables). Rows labeled 1 s order, nd order and 3rd order show sandard deviaions prediced by he firs-, second- and hird-order accurae model varians, respecively. Y: GDP; C: consumpion; I: gross invesmen; N: hours worked; K: capial sock. 14

15 Table. RBC model, hird-order approximaion: correlaions beween variables prediced by runcaed and non-runcaed model versions Y C I N K (1) () (3) (4) (5) (a) Model varian wih small shocks ( G 0.01, ) (a.1) Variables in levels (logs) All shocks Jus shock Jus G shock Jus shock Jus shock (a.) Firs-differenced variables (logs) All shocks Jus shock Jus G shock Jus shock Jus shock (b) Model varian wih big shocks ( G 0.05, ) (b.1) Variables in levels (logs) All shocks Jus shock Jus G shock Jus shock Jus shock (b.) Firs-differenced variables (logs) All shocks Jus shock Jus G shock Jus shock Jus shock (c) Model varian wih very big shocks ( G 0.10, ) (c.1) Variables in levels (logs) All shocks Jus shock Jus G shock Jus shock Jus shock (c.) Firs-differenced variables (logs) All shocks Jus shock Jus G shock Jus shock Jus shock Noe: Correlaions beween variables prediced by he full and runcaed hird-order models are repored. All shocks : simulaions wih all 4 shocks. Jus shocks, Jus G shocks ec. perain o simulaions in which jus one ype of shock is fed ino he model; he oher exogenous variables are se a seady sae values (model is solved assuming 4 shocks). Repored saisics are based on one simulaion run of 5000 periods (he run is iniiaed a he uncondiional mean of he sae variables). Y: GDP; C: consumpion; I: gross invesmen; N: hours worked; K: capial sock. Correlaions greaer han are repored as

16 Table 3. RBC model: esimaes of srucural parameers, 30 simulaion runs (100 periods) Model varian Model varian Model varian wih small shocks wih big shocks wih very big shocks (1) () (3) (4) (5) (6) (7) (8) (9) (a) Parameer esimaes Median Mean Sd Median Mean Sd Median Mean Sd G (%) G (%) (%) (%) Noe: The Table summarizes esimaion resuls across 30 simulaion runs of 100 periods each (he model is simulaed over 5100 periods, he las 100 periods are used for esimaion). The observables are he firs differences of logged GDP, consumpion, invesmen and hours worked. Panel (a) repors he mean, median and sandard deviaion of he esimaed model parameers across he 30 runs, for he small shocks model varian (Columns (1)-(3)), he big shocks varian (Cols. (4)-(6)) and he very big shocks varian (Cols. (7)-(9)). The rue values of he esimaed parameers are: 10, 0.5, In he G small shocks model varian, he rue sandard deviaions of exogenous innovaions are: G 1%, 0.05%. Big shocks model varian: G 5%, 0.15%. Very big shocks model varian: G10%, 0.50%. 16