A Specification Test for Linear Dynamic Stochastic General Equilibrium Models

Transcription

1 Journal of Saisical and Economeric Mehods, vol.1, no.2, 2012, ISSN: (prin), (online) Scienpress Ld, 2012 A Specificaion Tes for Linear Dynamic Sochasic General Equilibrium Models Kenichi Tamegawa 1 Absrac In his paper, we inroduce he procedure of a specificaion es for linear dynamic sochasic general equilibrium (DSGE) models. Given a parameerized DSGE model, we can empirically find omied variables and check wheher he model s srucure is correc. Mahemaics Subjec Classificaion: 62J07 Keywords: DSGE model, Specificaion es 1 Inroducion In his paper, we inroduce a simple model-specificaion es for linear dynamic sochasic general equilibrium (DSGE) models. Given a parameerized DSGE model, one can empirically find omied variables ha should have been included in he model and check wheher he model s srucure is correc. We add anoher approach o he series of mehods ha evaluae DSGE models, for example, DeJong e al. [1], Schorfheide [3], Smes and Wouers [4], and Fernández-Villaverde and Rubio-Ramirez [2]. The firs sep of our es is o obain reduced shocks from a given parameerized DSGE model. Theoreically, hese shocks are spanned by srucural shocks. The curren srucural shocks should no include any informaion ha is available up o 1 School of Commerce, Meiji Universiy, amegawa@kisc.meiji.ac.jp Aricle Info: Received : April 4, Revised : May 26, 2012 Published online : July 30, 2012

2 66 A Tes for Linear Dynamic Sochasic General Equilibrium Models he previous period. If he curren reduced shocks are correlaed wih ha informaion, hey are spanned no only by he curren srucural shocks bu also by he previous informaion, as explained in Secion 3. In his case, we can conclude ha a given DSGE model may be miss-specified. Concreely, he null hypohesis is ha a given DSGE model is correcly specified. This hypohesis can be esed by regressing reduced shocks on some lagged variables ha are excluded from he model and lagged endogenous variables. This paper is organized as follows. Secion 2 shows how reduced shocks are obained from a given DSGE model. Secion 3 inroduces our es procedure. Secion 4 presens an example of our es. Finally, we conclude our paper in Secion 5. 2 Obaining Reduced Shocks In his secion, we seup a linear DSGE model and ge reduced shocks from he model and acual daa. Consider he following model: AE y ] By Cy D 0, (1) [ 1 1 where y denoes a vecor of endogenous variables and denoes a vecor of srucural shocks. Coefficien marices A, B, C, and D are conformable and are parameerized by a user in advance of our es. If model (1) has a unique soluion, is reduced form is wrien as y Py 1 S. (2) The reduced shocks denoed by e can be recovered wih P and he acual daa as follows: e y Py 1. (3) The srucural shocks migh be serially correlaed and hen he model s srucure migh explicily include E ], bu even if his is he case, obaining P ( and herefore [ 1 e ) is independen of his serial correlaion. 3 Tes Procedure In he previous secion, we could recover he reduced shocks e by (3). Under he null hypohesis ha a given DSGE model is correcly specified, e should no correlae wih any economic variables available up o he previous period, which is denoed by 1. If he model is incorrecly specified, e correlae wih 1 as follows. Firs, assume ha a given DSGE model is

3 Kenichi Tamegawa 67 misspecified and he rue model is described by y plus he omied variables x as follows: y ~ y 1 P S ~ x x, 1 where P ~ and S ~ are he rue coefficien marices. In his case, ~ ~ ~ y P y 1 P 12 x 1 S 1 ~ ~ ~ Py 1 P Py 1 P 12 x 1 S 1 Therefore, he reduced shocks ( e ) in he false model can be expressed as follows: ~ ~ ~ e P Py 1 P 12 x 1 S 1. (4) Thus, if he underlying model is misspecified, is reduced shocks correlae wih he omied variables. Furhermore, if he model canno correcly capure he ~ parial effecs of y 1 on y, ha is, P P 0, he reduced shocks would correlae wih y 1 (This means ha he model should explicily include lagged variables). Therefore, in our es, we can find wo ypes of misspecificaion: he ~ ~ omied variable if P 12 0 and he modeling error if P P 0. In our es, we simply regress e on y 1 and x 1. This can be done using ordinary leas squares (OLS) esimaion under he regulariy condiions such as ha e, y 1, x 1 is joinly saionary and ergodic. In his siuaion, sandard es saisics such as he - es can be applied. If some variables of he OLS esimaor are saisically significan, he null hypohesis is rejeced, and we can conclude ha a given DSGE model is misspecified. Alhough of course in pracice, he rue variables are no known, researchers have a hypohesis ha cerain variable is imporan. Then, using our procedure, hey can deec wheher he variable is really imporan. 4 Example Consider he following real business cycle model wih he producion funcion 1 Y e K 1L, where Y, K, L and are oupu, capial sock, labor, and an i.i.d. random variable wih a mean of 0, respecively. A emporal-uiliy funcion is log( C ) where C denoes consumpion, and capial accumulaion funcion K ( 1 ) K 1 Y C. The problem o be solved by a social planner is

4 68 A Tes for Linear Dynamic Sochasic General Equilibrium Models Max i E0 log( Ci ), i0 s.. K ( 1 ) K 1 Y C, 1 Y e K 1L. Seing L 1 for simpliciy and linearizing he model around he non-sochasic seady saes, we have Kˆ (1 ) Kˆ ( Kˆ ) Y / K Cˆ C K, 1 1 / E [ Cˆ ] ˆ ( 1) ˆ 1 C K (1 (1 )), where ^ denoes he deviaion from he seady saes and Y, K, and C denoe he seady-sae values. We calibrae he parameer as follows: Y / K C / K Under his parameerizaion, he linearized model above has he following reduced form: Kˆ Kˆ C C. ˆ ˆ Table 1: Resul of model specificaion es Parameer esimaes Sanndard error Reduced shock for capial Own lag K *** C G ** Residual for consumpion Own lag *** K *** C G "***" and "**" indicae significance a he 1% and 5% levels, respecively.

5 Kenichi Tamegawa 69 Using he procedure shown in he previous secion, we can es wheher he above model is correcly specified. In paricular, we consider governmen invesmen denoed by G as an omied variable. Our es is done by regressing he residual shock e from (3) on K ˆ 1, C ˆ 1, and G ˆ 1, following he previous secion. 2 Furhermore, o capure he serial correlaion of srucural shocks, we add he own lag of reduced shocks e 1 o he regressors. The resul is shown in Table 1. In he regression for residuals of capial, he coefficiens of capial lag and governmen invesmen lag are significan a he 5% and 1% criical levels, respecively. In he regression for residuals of consumpion, he coefficiens of boh he own lag and capial lag are significan a he 1% level. Therefore, our es indicaes ha he above model s srucure should be correced and ha i incorporaes governmen invesmen. 5 Concluding Remarks In his paper, we inroduce he model specificaion es for a parameerized DSGE model and apply i o a simple real business cycle model as an example. Our es procedure is consruced on he basis of a simple idea: if reduced shocks are spanned by srucural shocks, as he heory requires, he curren reduced shocks do no correlae wih any informaion up o he previous period. Therefore, he correlaion beween he curren reduced shocks and he previous economic informaion is a sign of misspecificaion. Using a simple OLS esimaion, we can easily check his correlaion. ACKNOWLEDGEMENTS. I am graeful o an anonymous referee and Shin Fukuda for heir helpful commens. 2 The daa for { K ˆ },{ C ˆ }, and { G ˆ } are consruced from he Japanese Sysem of Naional Accouns quarerly daa (from 1980:Q1 o 2010:Q4) and is HP-filered daa.

6 70 A Tes for Linear Dynamic Sochasic General Equilibrium Models References [1] D. DeJong, B. Ingram and C. Whieman, A Bayesian approach o dynamic macroeconomics, Journal of Economerics, 98, (2000), [2] J. Fernández-Villaverde and J. Rubio-Ramírez, Comparing dynamic equilibrium models o daa: A Bayesian approach, Journal of Economerics, 123, (2004), [3] F. Schorfheide, Loss funcion-based evaluaion of DSGE models, Journal of Applied Economerics, 15, (2000), [4] F. Smes and R. Wouers, An esimaed dynamic sochasic general equilibrium model of he Euro area, Journal of European Economic Associaion, 20, (2003),