Tractable Latent State Filtering for Non-Linear DSGE Models Using a Second-Order Approximation *

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1 Federal Reserve Bank of Dallas Globalizaion and Moneary Policy Insiue Working Paper No. 147 hp:// Tracable Laen Sae Filering for Non-Linear DSGE Models Using a Second-Order Approximaion * Rober Kollmann ECARES, Universié Libre de Bruxelles and CEPR June 2013 Absrac This paper develops a novel approach for esimaing laen sae variables of Dynamic Sochasic General Equilibrium (DSGE) models ha are solved using a second-order accurae approximaion. I apply he Kalman filer o a sae-space represenaion of he second-order soluion based on he pruning scheme of Kim, Kim, Schaumburg and Sims (2008). By conras o paricle filers, no sochasic simulaions are needed for he filer here--he presen mehod is hus much faser. In Mone Carlo experimens, he filer here generaes more accurae esimaes of laen sae variables han he sandard paricle filer. The presen filer is also more accurae han a convenional Kalman filer ha reas he linearized model as he rue daa generaing process. Due o is high speed, he filer presened here is suied for he esimaion of model parameers; a quasi-maximum likelihood procedure can be used for ha purpose. JEL codes: C63, C68, E37 * Rober Kollmann, European Cenre for Advanced Research in Economics and Saisics (ECARES), CP 114, Universié Libre de Bruxelles, 50 Av. Franklin Roosevel, B-1050 Brussels, Belgium rober_kollmann@yahoo.com. I hank Raf Wouers for useful discussions. Financial suppor from he Naional Bank of Belgium and from 'Acion de recherche concerée' ARC-AUWB/ /ULB-11 is graefully acknowledged. The views in his paper are hose of he auhor and do no necessarily reflec he views of he Federal Reserve Bank of Dallas or he Federal Reserve Sysem.

2 1. Inroducion Dynamic Sochasic General Equilibrium (DSGE) models ypically feaure sae variables ha canno direcly be measured empirically (such as preference shocks), or for which daa include measuremen error. A vas lieraure during he pas wo decades has aken linearized DSGE models o he daa, using likelihood-based mehods (e.g., Smes and Wouers (2007), Del Negro and Schorfheide (2011)). 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. Recen research has begun o esimae non-linear DSGE models using paricle filers; 1 hese filers infer laen saes using Mone Carlo mehods, and are hus slow compuaionally, which limis heir use o small models. This paper develops a novel deerminisic filer for esimaing laen sae variables of DSGE models ha are solved using a second-order accurae approximaion (as derived by Jin and Judd (2000), Sims (2000), Collard and Juillard (2001), Schmi-Grohé and Uribe (2004), Kollmann (2005) and Lombardo and Suherland (2007)). Tha approximaion provides he mos racable non-linear soluion echnique for medium-scale models, and has hus widely been used in macroeconomics (see Kollmann (2002) and Kollmann, Kim and Kim (2011) for deailed references). When simulaing second-order accurae model soluions, i is common o use he pruning scheme of Kim, Kim, Schaumburg and Sims (2008), under which second-order erms are replaced by producs of he linearized soluion. This paper assumes ha he pruned secondorder model is he rue daa generaing process (DGP). The mehod presened here explois he fac ha he sae equaion of he pruned sysem is linear in a sae vecor ha consiss of variables solved o second- and firs-order accuracy, and of producs of firs-order accurae variables. I apply he Kalman filer o ha sae equaion. In Mone Carlo experimens, he filer here generaes more accurae esimaes of laen sae variables han he sandard paricle filer, especially when he model has srong curvaure or when shocks are large. Imporanly, he filer here is much faser han paricle filers, as i is no based on sochasic simulaions. The presen filer is also more accurae han a convenional Kalman filer ha reas he linearized model as 1 See Fernández-Villaverde and Rubio-Ramírez (2007) and An and Schorfheide (2007) for early applicaions. 2

3 he rue DGP. 2 Due o is high speed, he filer presened here is suied for he esimaion of model parameers; a quasi-maximum likelihood procedure can be used for ha purpose. 2. Model forma and filer Model forma and second-order soluion Many widely-used DSGE models can be expressed as: EG(,, ) 0, 1 1 where E is he mahemaical expecaion condiional on dae informaion; G: R R 2n m n is a funcion, and is an nx1 vecor of endogenous and exogenous variables known a ; 1 is an mx1 vecor of innovaions o exogenous variables. In wha follows, is Gaussian: 2 N(0, ), where is a scalar ha indexes he size of shocks. The soluion of model is a "policy funcion" 1F(, 1, ), such ha EGF ( (, 1, ),, 1) 0. This paper focuses on second-order accurae model soluions, namely on second-order Taylor series expansions of he policy funcion around a deerminisic sead sae, i.e. around 0 and a poin such ha F(,0,0). Le. For a qx1 column vecor x whose i-h elemen is denoed P x vech xx x x x x x x x x x x x x x x ( ) ( ') (( ),,.., q,( ),,.., q,...,( q n ), q x,( q ) ), i x, le be a vecor consising of all squares and cross-producs of he elemens of x. 3 The second-order accurae model soluion can be wrien as F F F F P( ) F ( ) F P( ), (2) where F0, F1, F2, F11, F12, F 22 are vecors/marices ha are funcions of srucural model parameers, bu ha do no depend on (Sims (2000), Schmi-Grohé and Uribe (2004)). The firs-order accurae (linearized) model soluion is: (3) 1 F1 F2 1. The superscrip denoes a variable solved o firs-order accuracy. I is assumed ha all eigenvalues of F 1 are sricly inside he uni circle, i.e. ha he linearized model is saionary. 2 The lieraure has discussed Exended Kalman filers, i.e. Kalman filers applied o linear approximaions of nonlinear models; e.g., Harvey (1989). 3 For a square marix M, vech(m) is he column vecor obained by verically sacking he elemens of M ha are on or below he main diagonal. 3

4 Pruning As discussed above, I use he pruning scheme of Kim e al. (2008) under which second-order erms are replaced by producs of he linearized soluion--i.e. P( ) and 1 are subsiued by P and ( ) 1, respecively. Wih pruning, he soluion (2) is hus replaced by: F F F F P( ) F F P( ). (4) Noe ha P ( ) P( ) and hold, up o second-order accuracy. Thus, 1 1 (4) is a valid second-order accurae soluion. In repeaed applicaions of (2), hird and higherorder erms of sae variables appear; e.g., when 1is quadraic in, hen 2 is quaric in ; pruning removes hese higher-order erms. The moivaion for pruning is ha (2) 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 secondorder soluion (4) oo is sable. The subsequen discussion assumes ha he rue DGP is given by he pruned sysem (4). Augmened sae equaion The law of moion of P can be expressed as ( ) P( ) K P( ) K K P( ), where K11, K12, K 22 are marices ha are funcions of F 1 and F 2. Sacking his marix equaion, as well as (3) and (4) gives he following sae equaion: 2 1 F0 F1 F11 0 F2 F12 F22 P( 1) 0 0 K11 0 P( ) 0 1 K 12 ( 1) K 22 P( 1). (5) F1 F2 0 0 (5) can be wrien as: Z g G Z G G ( ) G P( ), wih Z ( ', P( )', ')', while g0, G1, G2, G 12 and G 22 are he firs o fifh coefficien vecors/marices on he righ-hand side of (5), respecively. Thus, Z G G Z u, (6) G g G E( P( e )), while where u G G ( ) G [ P( ) E( P( ))] is a serially uncorrelaed, mean zero, disurbance. Noe ha he sae equaion (6) is linear in he augmened 4

5 sae vecor Z consising of he second- and firs-order accurae variables, and of he squares and cross-producs of firs-order accurae variables. 4 Observaion equaion A =1,..,T, he analys observes n y variables ha are linear funcions of he sae vecor plus i.i.d. measuremen error ha is independen of he sae vecor, a all leads and lags: y, where is an n y xn marix and N(0, ) is an nyx1 vecor of measuremen errors; is a diagonal marix. The observaion equaion can be wrien as: The filer yz, wih (,0). (7) Le { y } 1 be he observables known a dae ; X, ( X ) and V E([ X X ][ X X ]' ) denoe he condiional mean and variance of he column vecor X,,, X given. Uncondiional means and variances are denoed by EX ( ) and VX ( )., Z Given Z,, V,, he 1 s and 2 nd condiional momens of he augmened sae vecor Z condiional on Z, we can compue momens of 1 condiional on using (6): Z 1, G0 G1 Z,, (8) V GV G ' V, wih (9) Z Z u 1, 1, 1 1, u 1, 2 2' 12 (, ) 2' 2 (, )' 12' 12 {(,,, ') } 12' 22 ( ( 1)) 22' V G V G G G G G G V G G V P G (10) (see Appendix). Z To generae Z 1, 1, V 1, 1, I apply he linear updaing equaion of he sandard Kalman filer (e.g., Hamilon (1994, ch.13)) o he sae-space represenaion (6),(7): Z Z ( y y ), wih y 1, Z 1,, (11) 1, 1 1, 1 1, 4 Aruoba, Bocola and Schorfheide (2012) esimae a pruned univariae quadraic ime series model, using paricle filer mehods. These auhors discard he erm ha is quadraic in 1 on he righ-hand side of (4). By conras, he paper here allows for non-zero coefficiens on second-order erms in, 1 and i develops a deerminisic filer ha can be applied o mulivariae models. Andreasen e al. (2013) also derive a mulivariae pruned sae equaion for second-order accurae models (I learn abou ha paper afer he presen research was compleed); hese auhors use ha equaion o derive a simulaed mehods of momens esimaor for DSGE models; by conras, he paper here focuses on filering laen sae variables. 5

6 Z Z 1 and V 1, '{ V 1, }, V V V '{ V ' } V. (12) Z Z Z Z 1 Z 1, 1 1, 1, 1, 1, The filer is sared wih he uncondiional mean and variance of 0 : 1, 1 Z Z Z0,0E( Z0), V0,0V Z0 Z Z and V 1, 1 for 0 are compued by ieraing on (8)-(12). Henceforh, I refer o his filer as he KalmanQ filer. Compuer code ha implemens KalmanQ is available on my web page. ( ); u E( Z0), V( Z 0) and V 1, can be compued exacly; see he Appendix. The linear updaing formula (11) would be an exac algorihm for compuing he condiional expecaion Z 1, 1, if Z 1 and he observables were (joinly) Gaussian (hen Z 1, 1 would be a linear funcion of he daa). This condiion is no me in he second-order model, as he disurbance u 1 of he sae equaion (6) is non-gaussian. However, as shown below, he KalmanQ filer closely racks he rue laen variables. 5 When he linearized model is he rue DGP (i.e. when F0 0, F11 0, F12 0, F22 0), hen he filer here is idenical o he convenional linear Kalman filer, and he updaing formula (11) holds exacly. In he presence of second-order model erms, KalmanQ is more accurae han a convenional Kalman filer ha assumes ha he linearized model (3) is he rue DGP; see below. Quasi-maximum likelihood esimaion of model parameers If model parameers are unknown, hen a quasi-maximum likelihood (QML) esimae of hose T y parameers can be obained by maximizing he funcion L( ) ln h( y y, 1 ( ); V, 1( )), wih respec o he vecor of unknown parameers,. Here h( y ; V) is he mulivariae normal densiy wih mean and variance V. For a given, y, 1, 1 T 1 ( ) Z ( ) is he predicion of y generaed by KalmanQ, based on dae -1 informaion, 1 y Z ;, 1, 1 V ( ) V ( ) ' is he condiional variance of y, given 1. Under condiions discussed in Hamilon (1994, ch.13), he QML QML esimaor T is asympoically normal: QML T( ) N(0,( J ( J ) J ) ), where is he rue T Wihou Gaussianiy, Z 1, 1 is a non-linear funcion of daa 1 : Z 1, 1 y 1 (, ). (11) can be viewed as a linear approximaion of his funcion: Z 1, 1 ( y 1 y), wih ( y 1, ) / y 1 y. 1 y By he Law of Ieraed Expecaions, Z 1, ( 1, 1 E Z ), and hus: Z 1, 1Z 1, ( y 1 y 1, ). Recall ha he observable y 1 is a linear funcion of Z 1 (see (7)); his may help o explain he good performance of he linear updaing rule. 6

7 parameer vecor and h ( ) h(y y ( );V ( )) z,1 0,1 0 and J plimt ( ) ( )', wih ( 0) log(h ( 0))/, 1 T T J plimt log(h ( ))/ '. 3. Mone Carlo evidence 3.1. A exbook RBC model The mehod is esed for a basic RBC model. Assume a represenaive infiniely-lived household whose dae expeced lifeime uiliy V is given by / V{ 1C 11/ N } EV 1, where C and N are consumpion and hours worked, a, respecively. 0 and 0 are he risk aversion coefficien and he (Frisch) labor supply elasiciy. is an exogenous ase (discoun facor) shock of uni uncondiional mean. 0 1 is he seady sae subjecive discoun facor. The household maximizes expeced life-ime uiliy subjec o he resource consrain C K K N K where K 1 is he end-of-period capial sock; Y K N is 1 1 (1 ), 1 oupu. 0, 1 are he capial share and he capial depreciaion rae, respecively. 0 exogenous oal facor produciviy (TFP). The household s firs-order condiions are: 1 1 E ( C 1/ C) ( 1 K 1 N 1 1 ) 1, C (1 ) K N N. 1/ The forcing variables follow independen auoregressive processes: ln( ) ln( 1 ), is ln( ) ln( 1 ),, 0, 1, where, and, are normal i.i.d. whie noises wih sandard deviaions and, respecively. The numerical simulaions discussed below assume 0.99, 4, 0.3, 0.025, 0.99; parameer values in ha range are sandard in quarerly macro models. The parameer ha scales he size of he shocks is normalized as 1. The risk aversion coefficien is se a a high value, 10, so ha he model has enough curvaure o allow for non-negligible differences beween he second-order accurae and linear model approximaions. One model varian assumes shocks ha are much larger han he shocks in sandard macro models, in order o generae big differences beween he wo approximaions: 0.20, I refer o his varian as he big shocks varian. I also consider a second small shocks varian, in which he sandard deviaions of shocks are weny ime smaller: 0.01, (convenional RBC 7

8 models assume ha he sandard deviaion of TFP innovaions is abou 1%; e.g., Kollmann (1996)). 6 The observables are assumed o be GDP, consumpion, invesmen and hours worked; measuremen error is added o he logs of hese variables. Measuremen error has a sandard deviaion of 0.04 (0.002) in he model varian wih big (small) shocks. Chris Sims MATLAB program gensys2 is used o compue firs- and second-order accurae model soluions. The model is approximaed in erms of logged variables (he sae- and observaion equaions are expressed in erms of logged variables). Prediced sandard deviaions Table 1 repors uncondiional sandard deviaions of 7 logged variables (GDP, consumpion, invesmen, capial, hours, TFP and he ase shock ) generaed by he firs- and second-order approximaions. 7 Model varians wih boh shocks, and varians wih jus one ype of shock, are considered; momens for non-hp filered variables are shown, as well as momens of HP filered variables (smoohing parameer: 1600). In he big shocks model varian, he sandard deviaions of endogenous variables are huge; e.g., wih boh shocks, he sandard deviaion of (non-hp filered) GDP is 176% (82%) under he second-order (firs-order) approximaion; GDP is hus abou wice as volaile under he second-order approximaion (han in he linearized model). 8 The capial sock, invesmen and hours worked (non-hp filered) are abou one-half more volaile under he second-order approximaion han under he linear approximaion. By conras, consumpion volailiy is similar across he wo approximaions. Consumpion is much less volaile han GDP, due o he assumed high risk aversion of he household. The preference shock () is he main source of flucuaions in he capial sock, GDP and invesmen; TFP shocks are he main drivers of consumpion. The correlaion beween he second- and firs-order approximaions of a given variable is noiceably below uniy, in he model varian wih big shocks: e.g., abou 0.7 for capial and invesmen, and 0.5 for GDP. 6 The relaive size of he TFP and ase shocks assumed here (i.e. 20-imes larger han ) ensures ha each shock accouns for a non-negligible share of he variance of he endogenous variables; see below. 7 The saisics are shown for variables wihou measuremen error. The ranking of volailiies generaed by he wo approximaions and shocks is no affeced by he presence of measuremen error. 8 HP filered variables are markedly less volaile han non-hp filered variables; however, volailiy remains much higher under he second-order approximaion han under he linear approximaion, in he big shocks varian. E.g. he sandard dev. of HP filered GDP is 47% (23%) under he second- (firs-) order approximaion. 8

9 The small shocks model varian generaes much smaller sandard deviaions of endogenous variables ha are roughly in line wih prediced momens repored in he RBC lieraure (e.g., Kollmann (1996)); e.g., he prediced sandard deviaion of HP-filered GDP and invesmen are abou 1% and 5%, respecively (wih boh shocks). Wih small shocks, i remains rue ha variables are more volaile in he second-order model han in he linearized model, however, he difference is barely noiceable. E.g., he raio of he GDP [invesmen] sandard dev. across he 2 nd /1 s order approximaions is merely [1.002]. Filer accuracy I generae 50 simulaion runs of T=500 and of T=100 periods, using he second-order (pruned) sae equaion of he RBC model. Each run is iniialized a he uncondiional mean of he sae vecor. I apply he KalmanQ filer o he simulaed series (wih measuremen error). I also use a convenional Kalman filer, referred o as KalmanL, ha reas he linearized model (3) as he rue DGP. In addiion, he sandard paricle filer--referred o a PF(p), where p is he number of paricles--as described in An and Schorfheide (2007) is applied o he pruned sae equaion (4); for he simulaion runs wih T=500 periods, 100,000 paricles are employed; for runs wih T=100 periods, versions of he PF wih 100,000 and wih 500,000 paricles are used. 9 Accuracy is evaluaed for he 7 logged variables considered in Table 1. In each simulaion run s=1,..,50, he roo mean square error (RMSE) is compued, across all (logged) 7 variables, RMSE and separaely for each individual 1 7 T i i 2 1/ 2 s, All ( 7 Ti 1 1 ( s, s,, ) ), variable i=1,..7, i RMSE where s, is he rue dae value of variable i in 1 T i i 2 1/ 2 s, i ( 1 ( s, s,, ) ), i run s, while s,, is he filered esimae (condiional expecaion) of ha variable, given he dae informaion se. Table 2 repors RMSEs ha are averaged across simulaion runs. In he Panels labeled Average RMSEs, Column shows average RMSE, across all 7 variables, s1 RMSE s, ALL, while Cols. (2)-(8) separaely show average RMSEs for each individual variable i, s1 RMSE si,. Also repored are maximum esimaion errors across all variables, periods and runs, as well as maximum esimaion errors for each variable i (across all periods and all simulaion runs); see Panels labeled Maximum Errors. These accuracy measures are 9 I apply KalmanL o de-meaned series, as he linearized model implies ha he uncondiional mean of sae variables, expressed as differences from seady sae, is zero, while variables generaed from he second-order model have a non-zero mean. The iniial paricles used for he paricle filer are drawn from a muli-variae normal disribuion whose mean and variance are se o uncondiional momens of he sae vecor. 9

10 repored for each of he filers (see rows labeled KalmanQ, PF(100,000), PF(500,000), and KalmanL ). In addiion, I repor he fracion of simulaion runs in which he KalmanQ filer generaes lower RMSEs and lower maximum esimaion error han he oher filers. Table 2 shows ha he KalmanQ filer is more accurae han he oher filers, in all (or almos all) simulaion runs his holds for boh he big shocks and small shocks model varians. The convenional KalmanL filer, is leas accurae. Average RMSEs generaed by KalmanQ are ofen orders of magniudes smaller han he RMSE s generaed by he paricle filer, and ha even when 500,000 paricles are used. E.g., for he simulaion runs of he big shocks model varian wih T=100 periods, he average RMSEs for GDP are 0.039, 0.755, and 1.488, respecively, for KalmanQ, PF(100,000), PF(500,000), and for KalmanL; he corresponding maximum errors are 0.160, , and 9.826, respecively (see Panel (a.2), Col. (2)). For he small shocks simulaion runs wih T=100 periods, he average RMSEs for GDP are , , and , for KalmanQ, PF(100,000), PF(500,000) and KalmanL respecively, while corresponding maximum errors are , , and (see Panel (b.2)). No surprisingly, all he filers are more accurae when shocks are small, and hus, he absolue accuracy differences beween he filers are smaller. However, he relaive improvemen in accuracy from using he KalmanQ filer remains sizable. Compuing ime KalmanQ, he paricle filers wih 100,000 and 500,000 paricles, and KalmanL require 0.03, 14.69, and 0.01 seconds, respecively, o filer simulaed series of T=100 periods generaed by he RBC model, on a deskop compuer wih a 64-bi operaing sysem and a 3.4 Ghz processor. For a series of T=500 periods, he corresponding compuing imes are 0.12, 73.72, and 0.04 seconds, respecively. Thus, he KalmanQ filer is abou 500 (3000) imes faser han he paricle filer wih 100,000 (500,000) paricles. For a sufficienly large number of paricles, he paricle filer is (asympoically) an exac algorihm for compuing he condiional expecaion of he sae vecor. However, he experimens in Table 2 sugges ha a very large number of paricles (above 500,000) is needed o ouperform KalmanQ; he compuaional cos of using such a large number of paricles would be subsanial. 10

11 Evaluaing he QML parameer esimaes For 20 simulaions runs of he big shocks and small shocks model varians, I compued quasimaximum likelihood (QML) esimaes of he risk aversion coefficien ( ), he labor supply elasiciy ( ), he auocorrelaions of he forcing variables (, ) and he sandard deviaions of he innovaions o he forcing variables (, ). Table 3 repors he mean and median parameer esimaes, and he sandard deviaion of he esimaes, across he sample of 20 esimaes per model varian. The parameers are ighly esimaed; mean and median parameer esimaes are close o he rue parameer values Sae equaions wih randomly drawn coefficiens Many oher Mone Carlo experimens confirmed ha he KalmanQ filer is compeiive wih he paricle filer, in erms of accuracy of he esimaed sae variables. Tables 4 and 5 consider model varians in which he coefficiens of he sae equaion (4) are drawn randomly from normal disribuions. In boh Tables, Panel (a) perains o models wih n=20 variables, while Panel (b) assumes n=7 variables; I refer o he models in Panels (a) and (b) as large models, and as small models, respecively. In boh se-ups, m 7 independen exogenous innovaions, and n 4 observables are assumed. In all varians, he sandard deviaions of he (independen) Y exogenous innovaions and of measuremen errors are se a 1%. 11 The elemens of F 0 are independen draws from N(0,1) ha are scaled by a common facor so ha he larges elemen of F 0 is 2 (0.01) in absolue value. The elemens of F 1 are independen draws from N(0,1) ha are scaled by a common facor so ha he larges eigenvalue of F 1 has an absolue value of The elemens of F 2 are independen draws from N(0,1). In one se of simulaions, referred o as srong curvaure simulaions, all elemens of F11, F12, F 22 are independen draws from N(0,1); in anoher se of simulaions wih weak curvaure, he elemens of F11, F12, F 22 are independen draws from 2 N (0,(0.01) ), so ha curvaure is much smaller, on average. For boh he large and small model varians, 50 random srong curvaure coefficien ses, and 50 random weak 10 A more deailed evaluaion of he small sample properies of he QML esimaor is lef for fuure research. (.01) I ; (.01) I. As before, he parameer ha scales he size of he shocks is normalized as m n y 11

12 curvaure coefficien ses were drawn. For each of he 200 random ses of coefficien, he model was simulaed over T=100 periods. Table 4 repors averaged sandard deviaions of model variables (elemens of ) across he draws for he large and small model varians (see Panels (a) and (b), respecively) wih srong curvaure (Col. ) and wih weak curvaure (Col. (2)). 12 The srong curvaure model varians generae much greaer average sandard deviaions (above 240%) han he weak curvaure varians (abou 10%). In srong curvaure model varians, average prediced volailiy is several ime larger han in corresponding linear srucures (in which F0, F2, F11, F12, F 22 are se o zero). By conras, in he weak curvaure varians, volailiy is only slighly higher han in corresponding linear srucures. Table 5 evaluaes he accuracy of he KalmanQ, PF(100,000) and KalmanL filers, for each of he four model classes (large/small models wih srong/weak curvaure). For each model class, he KalmanQ filer generaes he lowes average RMSEs and he lowes maximum errors. For example, for he large models, he average RMSEs of KalmanQ, PF(100,000) and KalmanL are 0.155, and 2.978, respecively, under srong curvaure, and 0.031, and 0.052, respecively under weak curvaure (see Panel (a)). 4. Conclusion This paper has developed a novel approach for he esimaion of laen sae variables in DSGE models ha are solved using a second-order accurae approximaion. By conras o paricle filers, no sochasic simulaions are needed for he filer here--he presen mehod is hus much faser han paricle filers. In Mone Carlo experimens, he filer here generaes more accurae esimaes of laen sae variables han he sandard paricle filer, especially when he model has srong curvaure or when shocks are large. The presen filer is also more accurae han a Kalman filer ha reas he linearized model as he rue DGP. Due o is high speed, he filer presened here is suied for he esimaion of model parameers; a quasi-maximum likelihood procedure can be used for ha purpose. 12 For each se of coefficiens, he sandard deviaion of each variable was compued; hen, sandard deviaions were averaged across variables and coefficien draws, for each model class (large/small models wih srong/weak curvaure). 12

13 APPENDIX: Compuing momens of he sae vecor (for filer formula) The uncondiional mean and variance of he sae vecor Z 1of he augmened sae equaion (5) are given by: 1 E( Z 1) ( I G1) G0 and V( Z 1) GV 1 ( Z 1) G1' V( u 1), respecively. Saionariy of Z 1 (which holds under he assumpion ha all eigenvalues of F 1 are sricly inside he uni circle) implies E( Z 1) E( Z), V( Z 1) V( Z). Once Vu ( 1) has been deermined, VZ ( 1) can efficienly be compued using a doubling algorihm. Noe ha ( F 0 1 )i F i 2 i and recall ha u G G G [ P( ) E( P( ))]. (A.1) E( ) 0, E( 1) 0, E(( 1) 1' ) 0, E(( 1) P( 1)') 0 hold as 1 has mean zero and is serially independen. Hence, he covariances beween he firs and second righ-hand side (rhs) erms in (A.1), and beween he second and hird rhs erms are zero. Normaliy of 1 implies ha he uncondiional mean of all hird order producs of elemens of 1 is zero (Isserlis heorem): i j k h E 11 0 for all i,j,k=1,..,m, where 1 is he h-h elemen of 1. Thus he covariance beween he firs and hird rhs erms in (A.1) oo is zero. Noe ha V( 1) V( ), wih V( P ( 1)), he variance of V( ) FV( ) F ' F F '. Thus, V( u ) G G ' G ( V( ) ) G ' G V( P( )) G ' P( ) (,,...,,,...,,...,,, ), m m m1 m1 m1 m m m can be compued as follows. For i=1,..,m, le e i be a column vecor whose i-h elemen is 1, i j i j while all oher elemens are zero. Thus 1 1' e and 1 1' e j. Hence, 1 1 1' ij1, i 1 wih ( ' '). ij 2 eie j ejei Noe ha E( 1' ij 1) race( ij ), Var( 1' ij 1) 2 race( ijij ) and Cov( 1' ij 1, 1' kl 1) 2 race( ijkl ) (normaliy of 1 is needed for he second and hird formulae); see Magnus (1978). These formulae allow o compue he variance of elemens of he vecor P( 1) and he covariance beween any wo elemens of ha vecor, which pins down all elemens of he marix V( P ( 1)). Condiional variance of sae-form disurbance To derive he formula for he condiional variance of u 1 ((10) in main ex) hese facs are used: (i) E(( 1) 1 ' ),, wih, E( ). (ii) ,,, E(( )( )' ) E(( ') ( ') ) E(( ') ) ( V '). (Noe ha V, E( ' ) E( ) E( )' E( ' ) '. ),, (iii) EP ( ( 1) 1' ) 0, EP ( ( 1)( 1)' ) 0 (due o Isserlis heorem). Thus, he condiional covariance beween he 1 s and 3 rd rhs erms in (A.1) and beween h 2 nd and 3 rd rhs erms is zero. 13

14 References An, S. and Schorfheide, Bayesian Analysis of DSGE models. Economeric Reviews 26, Andreasen, M., Fernández-Villaverde, J. and J. Rubio-Ramírez, The Pruned Sae-Space Sysem for Non-Linear DSGE Models: Theory and Empirical Applicaions. NBER WP Aruoba, S., L. Bocola and F. Schorfheide, A New Class of Nonlinear Time Series Models for he Evaluaion of DSGE Models. Working Paper, U MD and U Penn. Collard, F., Juillard, M., Perurbaion mehods for raional expecaions models. Working Paper, GREMAQ and CEPREMAP. Del Negro, M. and F. Schorfheide, Bayesian Macroeconomerics, in: The Oxford Handbook of Bayesian Economerics (J. Geweke, G. Koop and H. van Dijk, eds.), Oxford Universiy Press, Fernández-Villaverde, J. and J. Rubio-Ramírez, Esimaing Macroeconomic Models: a Likelihood Approach. Review of Economic Sudies 74, Hamilon, J., Time Series Analysis. Princeon Universiy Press. Harvey, A., Forecasing Srucural Time Series Models and he Kalman Filer. Cambridge Universiy Press. Jin, H., Judd, K., Applying PerSolv o Complee Marke RBC models. Working paper, Sanford Universiy. Kim, J., Kim, S., Schaumburg, E. and C. Sims, Calculaing and Using Second-Order Accurae Soluions of Discree-Time Dynamic Equilibrium Models, Journal of Economic Dynamics and Conrol 32, Kollmann, R., Incomplee Asse Markes and he Cross-Counry Consumpion Correlaion Puzzle, Journal of Economic Dynamics and Conrol 20, Kollmann, R., Moneary Policy Rules in he Open Economy: Effecs of Welfare and Business Cycles, Journal of Moneary Economics 49, Kollmann, R., Solving Non-Linear Raional Expecaions Models: Approximaions based on Taylor Expansions, Working Paper, Universiy of Paris XII. Kollmann, R., H.S. Kim and J. Kim, Solving he Muli-Counry Real Business Cycle Model Using a Perurbaion Mehod, Journal of Economic Dynamics and Conrol 35, Lombardo, G. and A. Suherland, Compuing Second-Order Accurae Soluions for Raional Expecaion Models Using Linear Soluion Mehods. Journal of Economic Dynamics and Conrol 31, Magnus, J., The Momens of Producs of Quadraic Forms in Normal Variables. Saisica Neerlandica 32, Schmi-Grohé, S. and M. Uribe, Solving Dynamic General Equilibrium Models Using a Second-Order Approximaion o he Policy Funcion. Journal of Economic Dynamics and Conrol 28, Sims, C., Second Order Accurae Soluion of Discree Time Dynamic Equilibrium Models. Working Paper, Economics Deparmen, Princeon Universiy Smes, F. and R. Wouers, Shocks and Fricions in US Business Cycles: A Bayesian DSGE Approach. American Economic Review 97,

15 Table 1. RBC model: prediced sandard deviaions Y C I K N (2) (3) (4) (5) (6) (7) (a) Model varian wih big shocks ( 0.20, 0.01) (a.1) Non-HP filered variables Second-order model approximaion Boh shocks Jus shock Jus shock Linearized model Boh shocks Jus shock Jus shock (a.2) HP filered variables Second-order model approximaion Boh shocks Jus shock Jus shock Linearized model Boh shocks Jus shock Jus shock (b) Model varian wih small shocks ( 0.01, ) (b.1) Non-HP filered variables Second-order model approximaion Boh shocks Jus shock Jus shock Linearized model Boh shocks Jus shock Jus shock (b.2) HP filered variables Second-order model approximaion Boh shocks Jus shock Jus shock Linearized model Boh shocks Jus shock Jus shock Noe: Sandard deviaions (sd.) of logged variables (lised above Cols. -(7)), wihou measuremen error, are shown for he RBC model in Secion 3.1. The sd. were compued using he formulae in he Appendix. Sd. are no repored in %. Panel (a) ( Big shocks ) assumes sd. of innovaions o TFP () and he ase parameer of 20% and 1%, respecively. Panel (b) ( Small shocks ) ses hese sd. a 1% and 0.05%, respecively. Rows labeled Boh shocks ; Jus shock ; and Jus shock show momens prediced wih simulaneous wo shocks; wih jus he TFP shock; and wih jus he ase shock, respecively. Panels (a.1) and (b.1) repor momens of Non-HP filered variables; Panels (a.2) and (b.2) perain o HP filered variables (smoohing parameer: 1600). Y: GDP; C: consumpion; I: gross invesmen; K: capial sock; N: hours worked. 15

16 Table 2. RBC model: accuracy of filers All variables Y C I K N (2) (3) (4) (5) (6) (7) (8) (a) Model varian wih big shocks ( 0.20, 0.01) (a.1) 50 simulaion runs wih T=500 periods Average RMSEs KalmanQ PF (100,000) KalmanL Maximum Errors KalmanQ PF (100,000) KalmanL Fracion of runs in which RMSE is lower for KalmanQ han for oher filers PF (100,000) KalmanL Fracion of runs in which maximum error is lower for KalmanQ han for oher filers PF (100,000) KalmanL (a.2) 50 simulaion runs wih T=100 periods Average RMSEs KalmanQ PF (100,000) PF (500,000) KalmanL Maximum Errors KalmanQ PF (100,000) PF (500,000) KalmanL Fracion of runs in which RMSE is lower for KalmanQ han for oher filers PF (100,000) PF (500,000) KalmanL Fracion of runs in which maximum error is lower for KalmanQ han for oher filers PF (100,000) PF (500,000) KalmanL

17 Table 2. cd. All variables Y C I K N (2) (3) (4) (5) (6) (7) (8) (b) Model varian wih small shocks ( 0.01, ) (b.1) 50 simulaion runs wih T=500 periods Average RMSEs KalmanQ PF (100,000) KalmanL Maximum Errors KalmanQ PF (100,000) KalmanL Fracion of runs in which RMSE is lower for KalmanQ han for oher filers PF (100,000) KalmanL Fracion of runs in which maximum error is lower for KalmanQ han for oher filers PF (100,000) KalmanL (b.2) 50 simulaion runs wih T=100 periods Average RMSEs KalmanQ PF (100,000) PF (500,000) KalmanL Maximum Errors KalmanQ PF (100,000) PF (500,000) KalmanL Fracion of runs in which RMSE is lower for KalmanQ han for oher filers PF (100,000) PF (500,000) KalmanL Fracion of runs in which maximum error is lower for KalmanQ han for oher filers PF (100,000) PF (500,000) KalmanL Noe: The Table repors average Roo Mean Squared Errors (RMSEs) and maximum esimaion errors of esimaed laen saes, across simulaion runs (errors are no expressed in %), for differen varians of he RBC model; accuracy is repored across all logged variables (Col. All variables ), and separaely for each logged variable (see Cols. (2)-(8) labeled Y,, ' '); see Sec for furher informaion. KalmanQ: he filer for pruned second-order models developed in his paper; PF(p): sandard paricle filer wih p paricles: KalmanL: sandard Kalman filer ha assumes ha he linearized DSGE model is he rue DGP. Panel (a) ( Big shocks ) assumes sd. of TFP and ase shock innovaions of 20% and 1%, respecively; panel (b) ( Small shocks ) ses hese sandard deviaions a 1% and 0.05%, respecively. Y: GDP; C: consumpion; I: gross invesmen; K: capial sock; N: hours worked; : TFP; : ase shock. 17

18 Table 3. RBC model: disribuion of quasi-maximum likelihood esimaes of model parameers based on he KalmanQ filer (2) (3) (4) (5) (6) (a) Model varian wih big shocks ( 0.20, 0.01) True parameer value % 1.00% Mean esimae % 0.95% Median esimae % 0.89% Sandard dev. of esimaes % 0.17% (b) Model varian wih small shocks ( 0.01, ) True parameer value % 0.05% Mean esimae % 0.18% Median esimae % 0.14% Sandard dev. of esimaes % 0.11% Noe: The Table repors rue parameer values, as well as he mean, median and sandard deviaion of quasimaximum likelihood (QML) esimaes of srucural model parameers based on he KalmanQ filer (developed in his paper), obained for 20 differen simulaion runs wih T=100 periods. Resuls are shown for he big shocks varian of he RBC model (Panel (a)), and for he small shocks varian (Panel (b)). : risk aversion coefficien (Col. ); : Frisch labor supply elasiciy (Col. (2)); [ ]: auocorrelaion of TFP [ase shock] (Cols. (3)-(4)); [ ]: sandard deviaion of TFP [ase shock] innovaion (Cols. (5)-(6)). Table 4. Models wih randomly drawn coefficiens: sandard deviaions of variables srong curvaure weak curvaure 18 (2) (a) Large models (n=20) (a.1) Non-HP filered variables Second-order model Linearized model (a.2) HP filered variables Second-order model Linearized model (b) Small models (n=7) (b.1) Non-HP filered variables Second-order model Linearized model (b.2) HP filered variables Second-order model Linearized model Noe: Panel (a) considers models wih n=20 variables ( large models ), while Panel (b) assumes n=7 variables ( small models ). The coefficiens of he sae equaion (4) are drawn randomly. In srong curvaure [ weak curvaure ] experimens see Col. [Col. (2)]--all elemens of he marices of curvaure coefficiens F11, F12, F 22 are independen draws from N(0,1) [N(0,(0.01) 2 ]. For boh he large and small model varians, 50 random srong curvaure coefficien ses, and 50 random srong curvaure coefficien ses were drawn. For each of he resuling 200 ses of coefficiens, he model was simulaed over T=100 periods. For each se of coefficiens, he sandard deviaion of each variable wihou measuremen error was compued; hen, sandard deviaions were averaged across all variables and coefficien draws, for each of he four model class (large/small models wih srong/weak curvaure). (The averaged sandard deviaions are no repored in %.)

19 Table 5. Models wih randomly drawn coefficiens: accuracy of filers srong curvaure (2) (a) Large models (n=20) Average RMSEs KalmanQ PF (100,000) KalmanL Maximum Errors KalmanQ PF (100,000) KalmanL weak curvaure Fracion of runs in which RMSE is lower for KalmanQ han for oher filers PF (100,000) KalmanL Fracion of runs in which maximum error is lower for KalmanQ han for oher filers PF (100,000) KalmanL (b) Small models (n=7) Average RMSEs KalmanQ PF (100,000) KalmanL Maximum Errors KalmanQ PF (100,000) KalmanL Fracion of runs in which RMSE is lower for KalmanQ han for oher filers PF (100,000) KalmanL Fracion of runs in which maximum error is lower for KalmanQ han for oher filers PF (100,000) KalmanL Noe: The Table repors average Roo Mean Squared Errors (RMSE) and maximum esimaion errors of esimaed laen variables produced by hree filers, across simulaion runs, for versions of sae equaion (4) wih randomly drawn coefficiens. Panel (a) ( Large models ) assumes n=20 variables; Panel (b) ( Small model ) assumes n=7 variables. In srong curvaure [ weak curvaure ] experimens (see Column 1 [2]), all elemens of he marices of curvaure coefficiens F11, F12, F 22 are independen draws from N(0,1) [N(0,(0.01) 2 ]. For boh he large and small model varians, 50 random srong curvaure coefficien ses, and 50 random weak curvaure coefficien ses were drawn. For each of he resuling 200 ses of coefficien, he model was simulaed over T=100 periods. For each se of coefficiens, he RMSE and he maximal error was compued, for each of he n esimaed laen variables; hen, RMSEs were averaged across variables and coefficien draws, for each of he four model classes (large/small models wih srong/weak curvaure); maximum errors were likewise deermined across all n variables, and across all draws, for each of he four model classes. KalmanQ: he filer for pruned second-order models developed in his paper; PF(p): sandard paricle filer wih p paricles: KalmanL: sandard Kalman filer ha assumes ha he linearized model is he rue DGP. 19