How Well Does a Moneary Dynamic Equilibrium Model Accoun for Chilean Daa? Robero Duncan Cenral Bank of Chile March 2002 Absrac The aim of his paper is o know how well a money-in-he-uiliy funcion model wih a Taylor rule is able o mach Chilean daa, specially some moneary sylized facs. A dynamic sochasic general equilibrium model is formulaed, solved and calibraed o evaluae is abiliies o replicae he main feaures of he Chilean economy in he 986-2000 period. In paricular, I focus he aenion on a possible explanaion o wha is called in he lieraure he price puzzle, he co-movemen beween ineres rae and inflaion. The soluion of he model is adequaely achieved hrough a perurbaion mehod (second-order approximaion). A posiive ransiory policy ineres rae shock causes: () a emporary (insignifican) diminishmen in oupu, (2) a decrease in real money balances, and (3) a emporary increase in inflaion rae. These findings are relaively consisen wih hose obained from impulse-response funcions esimaed for Chile. Therefore, he heoreical model proposed is able o explain and reproduce he co-movemen beween ineres rae and inflaion. This is caused by a Fisher effec and srenghened by he presence of a Taylor rule ha depends posiively on inflaion deviaion. Keywords: Dynamic equilibrium models, real business cycles, calibraion. JEL classificaion: E32, E40, E50. This paper is he firs draf of my M.A. disseraion. I am very graeful o my advisor, Rómulo Chumacero. I also hank Paricia Toledo for helpful suggesions. Of course, remaining errors are enirely of my own. The views expressed herein are hose of he auhor and do no necessarily represen hose of he Cenral Bank of Chile. Economis of he Research Uni of he Cenral Bank of Chile. E-mail address: rduncan@bcenral.cl.
. Inroducion Almos wo decades afer he influenial work by Kydland and Presco (982), he lieraure on moneary real-business-cycle models has had he chance o prove is abiliy o accoun for regulariies on developed counries daa. Neverheless, here are virually no sudies ha have aemped o do so for emerging Lain American economies 2. The aim of his paper is o know how well a money-in-uiliy funcion model wih a Taylor rule is able o mach some paricular moneary sylized facs from Chilean daa beween 986 and 2000. Specially, I focus he aenion on a heoreical explanaion o wha is called in he lieraure he price puzzle (he co-movemen beween ineres rae and inflaion rae), fac ha is found in many VAR-ype esimaions for Chile and oher economies. I find ha a posiive ransiory policy ineres rae shock causes a emporary (no significan) diminishmen in oupu, a ransiory decrease in real money balances, and a emporary increase in inflaion rae. These findings are relaively consisen wih impulseresponse funcions obained from a 5-variable VAR esimaed for Chile. Therefore, he heoreical model proposed is able o explain and reproduce he co-movemen beween ineres rae and inflaion (he so-called price puzzle ). This co-movemen is caused by a Fisher effec, ha is, an increase in he nominal ineres rae generaes an increase in inflaion leaving he real ineres rae virually invarian. Besides, his effec is srenghened by a moneary policy expressed by a Taylor rule ha depends posiively on inflaion deviaion. An analogous explanaion is also found in some recen sudies in he economic lieraure (see Monne and Weber, 200; and Alvarez, Lucas, and Weber, 200). The srucure of his sudy is organized as follows. Secion 2 provides an overview of he sudies relaed o real business cycle (RBC) models calibraed for he Chilean economy, highlighing he main characerisics of he models, mehods of soluion, objecives, and See, for insance, Chrisiano and Eichenbaum (992), Cooley and Hansen (995), Chrisiano e al (997), McCallum and Nelson (997), for U.S.; Dhar and Millard (2000), for U.K.; Folkersma (998), for Neherlands. 2 Perhaps Chile is he excepion, see Acuña and Oyarzún (200) and Bergoeing and Soo (2002). RBC models for he Chilean economy wihou moneary variables were formulaed and calibraed by Quiroz
resuls. In secion 3, I presen a brief descripion of he Chilean regulariies during he 986-200 period on he basis of he mos imporan resuls of esimaing a Vecor Auoregression model. This ask is accomplished o provide a saisical characerizaion of he Chilean economy and obain impulse-response funcions ha will be used as a meric o compare hem wih hose simulaed by he heoreical model. Thus, in secion 4, a dynamic sochasic general equilibrium model is formulaed, solved, and calibraed considering he presence of disorionary axes in an open-economy. The soluion of he model is adequaely achieved using a perurbaion mehod (second-order-approximaion) proposed by Schmi-Grohé and Uribe (200). Secion 5 describes he resuls of calibraing he model and examines wheher i is capable of replicaing he VAR impulse-response funcions. Concluding remarks and nex seps for fuure research are provided in he las secion. 2. Previous Sudies abou RBC Models for he Chilean Economy The calibraion of RBC models in Chile sared a he beginning of he nineies (see able ) wih he work by Quiroz e al (99) who used he Kydland and Presco s (982) framework o replicae several second momens of Chilean oupu and invesmen rae 3 (series from 977 o 990). They found ha his model could replicae sample volailiies bu had problems wih auocorrelaions. Simulaneously, Quiroz (99) formulaed a wo-good small open economy model wih coss of adjusmen of labor o replicae some regulariies exhibied by he real exchange rae during 977-90 period. He concluded ha he model was able o mach he real exchange rae volailiy and is negaive correlaion wih wages, he price of copper, and capial inflows, bu is high auocorrelaion remained unexplained. Acuña and Oyarzún (200) is one of he firs papers ha included moneary variables o an RBC framework and analyzed he role of moneary shocks on Chilean daa. Using Cooley and Hansen s (989) cash-in-advance model, he resuls of heir calibraion showed similariies wih acual daa in he co-movemen of he simulaed variables (excep capial and money socks), bu showed difficulies o replicae several volailiies (GDP, employmen, (99), Quiroz e al (99), Bergoeing and ohers (200), and Chumacero and Fuenes (2002). 3 Defined as invesmen as a percenage of oupu. 2
Table. Characerisics of RBC Models Calibraed for he Chilean Economy Auhors (year) Quiroz e al (99) Quiroz (99) Acuña and Oyarzún (200) Bergoeing e al (200) Chumacero and Fuenes (2002) Bergoeing and Soo (2002) Theoreical Framework DSGE model. Infiniely-lived agen, -good, closed economy. Based on Kydland and Presco (982). Time-o-build resricions were incorporaed. DSGE facor model. Infinielylived agen, 2-good, small open economy. Four secors and cos of adjusmen of labor. Based on Corbo (985). DSGE model. Infiniely-lived agen, -good, closed economy. Based on Cooley and Hansen (989) cash-in-advance model. DGE model. Infiniely-lived agen, -good, closed economy. Basic growh model wih income ax. DSGE model. Infiniely-lived agen, 2-good, small open economy. Includes relaive price of invesmen and income axes. Based on Greenwood, Hercowiz, and Krusell (2000). DSGE models (5 specificaions). Infiniely-lived agen, 2-good, closed, cash-inadvance economy, wih labor and wage rigidiies. Based on Cooley and Hansen (989) and Feaures of he Calibraion Filer: No repored Mehod of soluion: Linear quadraic Meric: Sandard deviaions, conemporaneous cross-correlaions, and (firs-hird) auo-correlaions. Filer: No repored Mehod of soluion: Linear quadraic Meric: Sandard deviaions, conemporaneous cross-correlaions, and (firs-fourh) auo-correlaions. Filer: Hodrick and Presco (997) Mehod of soluion: Linear quadraic wih disorions Meric: Sandard deviaions, (conemporaneous-fifh) crosscorrelaions. Filer: No used Mehod of soluion: Linear quadraic Meric: Decomposiion of average annual changes in real oupu per worker. Filer: No used Mehod of soluion: Perurbaion mehod (second-order approximaion) Meric: Se of VAR coefficiens and impulse-response funcions of real GDP. Filer: Hodrick and Presco (997) Mehod of soluion: Linear quadraic Meric: Sandard deviaions and cross/auocorrelaions. McGraan (994). Noes: DSGE denoes dynamic sochasic general equilibrium, DGE denoes dynamic general equilibrium. Source: Auhor s elaboraion. Daa Frequency and Span Quarerly daa, 977.-990.4 Quarerly daa, 977.-990.4 Quarerly daa, 986.-2000. Annual daa, 98-2000 Annual daa, 960-2000 Quarerly daa, 986.-2000. Objecive /Resuls Objecive: To replicae regulariies (several sample momens) Good fi: volailiy of oupu and invesmen, and firs auocorrelaion of invesmen. Difficulies o replicae: cross correlaions and auocorrelaions of oupu Conrary signs: hird auocorrelaions of oupu and invesmen. Objecive: To replicae empirical regulariies of he real exchange rae (volailiy, conemporaneous cross-correlaions, and auo-correlaions). Good fi: volailiy of oupu and invesmen, and conemporaneous cross-correlaion wih wages, price of copper and foreign capial inflows. Difficulies o replicae: RER auocorrelaions. Objecive: To replicae regulariies (several sample momens) and assess he role of money in Chilean business cycles. Good fi: direcion of variables excep capial sock and money. Difficulies o replicae: GDP, employmen, prices, inflaion, and produciviy volailiy; phase shif of all variables excep consumpion, prices, and produciviy; oupu-money correlaion. Objecive: To explain he Chilean and Mexican recoveries (985-2000). Tax policy was imporan bu i canno explain more han a small fracion of he differences in boh counries recoveries. Good fi: average annual changes in real oupu per working-age person. Difficulies o replicae: work and capial effor in he early eighies. Objecive: To replicae regulariies (several impulse-response funcions) and assess he deerminans of growh in Chilean economy. Good fi: response of oupu o a shock on erms of rade, fiscal expendiures as a percenage of GDP (fiscal disorions) and relaive price of equipmen wih respec o consumpion. Objecive: To replicae regulariies (several sample momens) and assess he role of moneary and fiscal variables in Chilean business cycles. Good fi: prices and oupu volailiy, consumpion volailiy and is correlaion wih oupu. Difficulies o replicae: correlaions: oupuprice level (and inflaion), hours worked-average produciviy. Correlaions of money and oher variables no repored.
prices, and produciviy) and phase shifs (see able ). They concluded ha inroducing an erraic moneary rule improves he abiliy of he model o reproduce consumpion behavior. Bergoeing e al (200) wondered wheher ax policy changes could explain he differen recoveries in Chile and Mexico during he pas wo decades. They used a basic RBC model wih income axes o conclude ha even hough ax policy was imporan, i could only explain more han a small fracion of he differences in boh counries recoveries. Recenly, Chumacero and Fuenes (2002) formulaed a small-open economy model ha included he relaive price of invesmen and income axes. Their objecive was o assess he deerminans of growh of he Chilean economy beween 960 and 2000. They found a close fi among VAR impulse-response funcions of GDP and hose of simulaed oupu when here is a shock on erms of rade, fiscal disorions (fiscal expendiures as a percenage of GDP) and he relaive price of equipmen. Finally, based on Cooley and Hansen (989) and McGraan (994), he work by Bergoeing and Soo (2002) used five specificaions of RBC models (wih cash-in-advance, labor, and wage rigidiies) o replicae several empirical regulariies and assess he role of moneary and fiscal variables in Chilean business cycles. One of heir specificaions achieved close fi in prices and oupu volailiy, consumpion volailiy and is correlaion wih oupu, bu has some difficulies o replicae he correlaions of oupu-price level, oupu-inflaion, and hours worked-average produciviy. Also, hey found ha he inclusion of wage rigidiies does no conribue significanly o mach he daa. 3. Sylized Facs: VAR Impulse-Response Funcions In his secion I esimae a Vecor Auorregresion (VAR) model o characerize he Chilean economy during he period of sudy and obain impulse-response funcions and confidence inervals ha will be used as a meric of comparison wih hose from he model o be presened in he nex secion. 3
As menioned before, he daa consiss of monhly series from 986.0 o 2000.2, so he recen period of nominalizaion of he moneary policy by he Cenral Bank of Chile (CBC) is no aken ino consideraion. The purpose is o compare he daa and he simulaed series from he heoreical model during he period when he CBC had a unique moneary policy (in his case, an UF-indexed moneary policy). The VAR esimaed herein is a five-variable model ha also considers a rend and seasonal dummy variables. The variables used are: he log of he erms of rade (TOT), he log of (gross) UF 4 -indexed policy ineres rae 5 (R), he log of (gross) inflaion rae (INF), he log of M in real erms (M), and he log of he Monhly Aciviy Index of Chile or IMACEC (Y). Appendix A shows he sources of he series. I is imporan o say ha hese variables were chosen as he empirical counerpar of he variables explained by he heoreical model o be presened in he nex secion. The seps ha will be aken o esimae he VAR are he following. Firs, I will compue informaion crieria, such as he Schwarz or Hannan-Quinn crieria, o deermine he opimal number of lags of he VAR. Second, I will es he saionariy of he represenaion, checking if he eigenvalues are inside he uni circle or no. Third, I will verify if residuals presen a Normal disribuion or no. Deparures from Normaliy imply ha he confidence inervals should be consruced hrough a boosraping echnique as long as he residuals are whie-noise process. Finally, I will compue he VAR impulse-response funcions and, accordingly, heir confidence inervals. From he calculaion of he ess menioned above I conclude ha he opimal lag lengh should be wo following Hannan-Quinn informaion crierion. Despie he Schwarz crieria preferred a lag lengh of one, he confidence inervals and he impulse-response funcions in his case do no differ significanly from hose when he Hannan-Quinn crierion is used. Since all he eigenvalues are inside he uni circle, he chosen sysem 4 UF denoes Unidad de Fomeno. I is a uni of accoun used for commercial and financial ransacions in Chile. 5 The ineres rae paid on 90-day bonds issued by he CBC (or Tasa Pagarés Reajusables del Banco Cenral) from 986 o 995, and he Policy Ineres Rae (Tasa de Políica Monearia) from 995 o 2000. Boh raes are UF-indexed. 4
presens covariance saionariy. Even hough he residuals are whie-noise process, hey show imporan deparures from Normaliy. Hence, he confidence inervals of impulseresponse funcions are calculaed using boosraping. The ordering followed in he esimaion of he VAR model is he one presened above. Also, i mus be said ha he confidence inervals of impulse-response funcions are almos invarian o alernaive orderings. Besides, following Pesaran and Shin (998), a generalized decomposiion of he variance-covariance marix -where impulse-response analysis is invarian o he ordering of he variables- was performed and he resuls were very similar. Figure shows he main impulse-response funcion derived from he VAR model using a Cholesky decomposiion. On he oucomes obained from he esimaion I can conclude ha: Firs, he oupu level ends o decline when i faces a emporary shock on he (UFindexed) policy ineres rae. This effec is saisically significan beween he fifh and he eigheenh monh. However, i is imporan o say ha a one-lagged VAR, following he Schwarz crierion, is able o show a non-significan decrease in oupu when here is a shock on he ineres rae. So his resul can be summarized as a diminishmen (slighly significan or non-significan) in oupu level. Second, in response o a ransiory ineres-rae shock, he inflaion rae rises. This incremen is saisically significan beween he hird and sixh monh. This is he socalled price puzzle in he economic lieraure. Tha is, if a posiive innovaion in he ineres rae is seen as a resricive moneary policy, hen inflaion rae should increase (insead of decreasing) according o he predicions of a sandard Mundell-Fleming model. This sylized fac is also found by oher sudies for Chile. 6 This resul is robus o he use of any ordering or number of opimal lags. 6 See Parrado (200) for a brief discussion abou he price puzzle in he Chilean economy. 5
Figure. Impulse-Response Funcions from he VAR Model 0.2% 0.0% -0.2% Impulse-Response Funcion of Oupu o a Shock on Ineres Rae 0 9 28 37 46-0.4% -0.6% -0.8% -.0% Lower Band Response of Oupu Upper Band 0.3% Impulse-Response Funcion of Inflaion Rae o a Shock on Ineres Rae 0.0% 0.08% 0.05% 0.03% 0.00% -0.03% 0 9 28 37 46-0.05% Lower Band Response of Inflaion Upper Band 0.5% Impulse-Response Funcion of Money o a Shock on Ineres Rae 0.0% -0.5% 0 9 28 37 46 -.0% -.5% -2.0% Lower Band Response of Money Upper Band
Third, a policy rae shock implies a negaive effec on real money balances, hus here is no evidence of liquidiy puzzle. This effec is saisically significan for more han a year and is also invarian o he use of any ordering or number of opimal lags. The confidence inervals of he impulse-response funcions will be used as a meric of comparison o es he capabiliy of he heoreical model presened in he nex secion o mach Chilean daa. Basically, I am ineresed in finding a heoreical framework capable of explaining and replicaing he facs shown before: he co-movemen beween inflaion and ineres raes and he effecs of an ineres rae shock on oupu and money. 4. The Model In his secion I describe he main characerisics of he proposed model. The general feaures o be considered are: household s uiliy funcion depends on consumpion, real money holdings, and leisure; a Taylor rule followed by he moneary auhoriy; demand for nominal and UF-indexed bonds; he presence of echnological and fiscal expendiure shocks; consan disorionary axes; and open economy. Households Consider an economy of an infiniely-lived agen ha opimizes an uiliy funcion which depends on real privae consumpion c, real money balances m, and leisure l : E β u 0 = ( c, m, l ), () where 0<β< is he subjecive discoun facor and E{.} he expecaion operaor. The represenaive household s consrain is: 6
c i b b U m ( τ L ) w L ( τ U U U b ( R ) D, U K π ) r K q y 2 m T π ( R) b π (2) where i denoes real invesmen in period, b is he real sock of bonds, b U is he UFindexed real sock of bonds, τ L and τ K are (consan) axes on labor and capial income, w denoes real wage, L represens he level of employmen 7, r represens real cos of capial, K is sock of physical capial, q is he relaive price of exporable goods o imporable goods or erms of rade, T denoes real lump-sum ransfers, π is he inflaion rae, R represens nominal (ne) ineres rae, R is UF-indexed (ne) ineres rae, D are firm profis, and U denoes he value of one Unidad de Fomeno (UF) in period, ha evolves according o: U υ ( π ) ( π ), υ = U (3) where υ is equal o 9/30 and represens he number of days of he monh ha he UF growh in period depends on inflaion in period - since he res of days (2/30) i depends on inflaion in period. Tha is, UF growh in is a weighed geomeric mean of he inflaion rae in and he inflaion rae in -. I assume ha here are wo goods produced in his economy; he firs good (y, or imporable good) is produced domesically and can be impored, bu he second one (y 2, or exporable good) is no consumed domesically and i is supposed o be consan. Nex, assume he following uiliy funcion ha depends on he logs of consumpion, money, and employmen: u ( c m, l ) = logc log m ηlog ( L )., φ (4) 7 The household is endowed each period wih one uni of ime, which i divides beween leisure -L and work L. 7
Capial accumulaion has he following law of moion: ( ) K i, K = δ (5) where δ is he rae of capial depreciaion. The law of moion of he exogenous erms of rade is: q 2 ( ) q ρ q ε ; ε iid ( 0, σ ) ; q > 0 ; 0< ρ. = q 0 q q q q 0 q < ρ (6) To finalize he descripion of he economy, I suppose a (lagged) Taylor rule ha depends posiively on he oupu and inflaion deviaions from seady-sae values plus an auorregresive erm: y y π π ( θ ) R θ log θ log θ R ε ; R, θ, θ > 0, 0<. = 3 0 * 2 * 3 R 0 2 θ3 < R (7) where y denoes real oupu level, y * is he seady-sae oupu level, π * represens he seadysae inflaion rae, and ε R is a zero-mean shock wih variance σ 2 R. Noice ha R 0 corresponds o he long-run (or seady-sae) ineres rae. In a decenralized equilibrium, he agen maximizes () subjec o (2)-(7). Accordingly, he firs-order condiions are: c λ = 0 (8) φ m λ β E λ π = 0 (9) 8
η ( L ) λ ( τ L ) w = 0 (0) λ λ ( ) = 0 β R E π () U U λ λ ( ) = 0 β R E U π (2) [ τ ) r ( )] 0. λ β λ ( K δ = E (3) Noice ha since he nominal and UF-indexed bonds are risk-free asses, R and R U are known in period, hus hey are placed ou of he expecaion operaor. Besides, observe ha equaion (0) and () imply he arbirage condiion beween he asses: U R (4) U ( ) = ( R ). U equaion (4). Given he laws of moion (3) and (7), we can obain he law of moion for R U using Firms The represenaive firm maximizes is profi given by equaion (5), D = y w L r K (5) subjec o a reurns-o-scale echnology: α α z ( L, z ) = A K L e ; A > 0, 0< α, y = F K, 0 0 < (6) where z is a echnological shock ha follows an auorregresive process: 9
0 ( ), 0 ; 0,, 2 < < = z z z z z iid z z ρ σ ε ε ρ (7) and ε z is a zero-mean shock wih variance σ 2 z. Thus, he firm maximizes (5) subjec o (6)-(7), obaining he following firs-order condiions: 0 0 = z r e K L A α α (8) 0 ) ( 0 = z w e L K A α α (9) Public Secor The governmen budge consrain is: ( ) ( ), U U U K L b U U R b b R b m m K r L w T g π π π τ τ = (20) where g is he exogenous governmen expendiure. The model also considers a saionary law of moion for he fiscal policy: ( ) ( ), 0 ; 0 ; 0, ; 0 2 0 < < > = g g g g g g g iid g g g ρ σ ε ε ρ ρ (2) and ε g is a zero-mean shock wih variance σ 2 g. The Economy Finally, equaion (2) and (20) imply ha aggregae demand equals producion in boh secors:
c i g = y q y 2. (22) Summing up, he parameers of he model are β, φ, η, δ, α, τ L, τ K, hose relaed o he exogenous auorregresive processes (ρ z, ρ g, ρ q, A 0, g 0, q 0, y 2, σ 2 z, σ 2 g, σ 2 q), and hose relaed o he Taylor Rule (R 0, θ, θ 2, θ 3 y σ 2 R). The sae variables are K, b, b U, g, z, q. The conrollable sae variables are R, R U, π. And he conrol variables are c, L, and m. The soluion of he model in seady sae is presened in Appendix B. 5. Calibraion and Resuls 5.. Parameerizaion This secion describes he parameerizaion of he model. As menioned above, here are 22 parameers ha appear in he equaions ha characerize behavior around seady sae. Previous works for Chilean daa have assigned values o some of hese parameers. Table 2 summarizes some of he mos frequen values used for common parameers for Chilean economy and, as a reference, some values for US economy. For example, i is known ha he capial-share parameer values used before for Chilean daa were beween 0.33 and 0.6, even hough mos of hem were in he 0.33-0.40 range, as able 2 shows. Also i can be said ha parameers such as he subjecive discoun facor, he auorregresive coefficien of he echnological shock, he capial depreciaion rae have similar values -once hey are compared in he same frequency 8 - along he lieraure. 8 Remember ha he calibraion mus be done in erms of monhly daa, hus, for insance, a monhly subjecive discoun facor of 0.996 corresponds o a annual value of 0.953.
Table 2. Parameers Used in Previous Sudies Auhors Counry and Preferences and Technology Period of Sudy b f a r z s z d McGraan (994) US, 947-987 0.985... 0.397 NC 0.098 0.0226 Cooley and Hansen (995) Quiroz and ohers (99) Acuña and Oyarzún (200) Bergoeing and ohers (200) Chumacero and Fuenes (2002) Bergoeing and Soo (2002) US, 954-99 0.989... 0.4 0.95 0.007 0.09 Chile, 977-990 NR... NR 0.999 0.02 0 Chile, 986-2000 0.9859... 0.4 0.99 0.078 0.025 Chile, 98-2000 0.98... 0.6 NR NR 0.08 Chile, 960-2000 0.98... /3 0.73 0.04 0.06 Chile, 986-2000 0.9787... 0.37-0.4 0.98 0.099 0.02 Walsh (998) US, (NR) 0.989 0.005 0.40 0.95 0.0089 0.09 Exporable Secor and Fiscal Policy r q s q l k g 0 r g s g McGraan (994) US, 947-987...... NP NP 70.99 NC 0.078 Bergoeing and ohers (200) Chile, 98-2000......... 0.5-0.2......... Chumacero and Fuenes (2002) Chile, 960-2000 0.892 0.4 0.25 0.25 NR 0.895 0.024 Bergoeing and Soo (2002) Chile, 986-2000...... NP NP 0.089 0.76 0.097 Noes: NR denoes no repored. NP denoes ha i is no considered as parameer in he sudy. NC means no comparable, since he auhor(s) did no used an AR() process. Therefore, I assume hree crieria o assign values o each parameer of he model: The firs crierion is o use some of he sandard parameer values given in previous lieraure for Chile (according o able 2). 2
Table 3. Parameerizaion of he Model Parameer Symbol Value Crieria of Choice Subjecive discoun facor β 0.996 Previous lieraure. The value implies a seady-sae real ineres rae of 5% Uiliy sensiiviy o money φ 0.005 Source: Walsh (998) Uiliy sensiiviy o leisure η.2 Calibraion of seady-sae labor beween 0.3 and 0.4 Capial share α 0.35 Previous lieraure (beween 0.3-0.4) and calibraion of invesmen (as a percenage of GDP) Technological consan A 0 0.9 Calibraion of share of consumpion on GDP Technological-AR coefficien ρ z 0.9 Calibraion of impulse-response funcions and previous lieraure Technological volailiy σ z 0.000 Calibraion of impulse-response funcions and GDP volailiy Depreciaion rae δ 0.09/2 Previous lieraure and calibraion of invesmen (as a percenage of GDP) Seady-sae erms of rade q 0.072 AR() esimaes (daa: 986.0-2000.2) Terms-of-rade-AR coefficien ρ q 0.962 AR() esimaes (daa: 986.0-2000.2) Volailiy σ q 0.027 AR() esimaes (daa: 986.0-2000.2) Capial axes τ l 0.25 Source: Chumacero and Fuenes (2002) Labor axes τ k 0.25 Source: Chumacero and Fuenes (2002) Calibraion of share of governmen Seady-sae governmen g expendiure 0.22 expendiures and invesmen (as a percenage of GDP) Governmen-AR coefficien ρ g 0.76 Source: Bergoeing and Soo (2002) Governmen Expend. volailiy σ g 0.008 Source: Bergoeing and Soo (2002) Taylor Rule consan R 0 0.006 Calibraion of seady-sae inflaion around 2.64% Taylor Rule AR coefficien θ 0.67 Calibraion of impulse-response funcions Oupu deviaion coefficien θ 2 0. Calibraion of impulse-response funcions Inflaion deviaion coefficien θ 3 0.2 Calibraion of impulse-response funcions Taylor Rule volailiy σ r 0.68e-4 Calibraion of impulse-response funcions Noes: AR() denoes firs-order auoregression process. The second crierion is o find he parameer value necessary o mach some seady-sae values for Chilean economy (such as he seady-sae consumpion as a percenage of GDP, he seady-sae inflaion rae, and so on). 3
The las crierion is o adjus he parameer values o allow he model mach he meric of comparison (he confidence inervals of he impulse-response funcions). Thus, able 3 repors he parameer values assumed for he calibraion and he respecive crierion used in each case. 5.2. Main Resuls The soluion of he model is achieved using a perurbaion mehod (second-orderapproximaion) developed by Schmi-Grohé and Uribe (200). This mehod consiss of a second order approximaion o he policy funcions of he dynamic equilibrium model. As he auhors sae, he inconvenience wih he use of firs order approximaion mehods is ha hey are no well suied o handle quesions such as welfare comparisons across alernaive sochasic or policy environmens. Aside from ha, he problem of linearized decision rules o evaluae second-order approximaions o he objecive funcion is ha some second-order erms of such objecive funcion are ignored. Therefore, such problems do no arise wih he use of second or higher order approximaions. Figure 2 presens he responses of a ransiory shock on he log of (gross) ineres rae. The shock given is posiive so i represens a resricive moneary policy. The incremen is 0.5% (50 basis poins) of he annual policy rae (or 0.04% in monhly erms). The following resuls are found: The posiive shock on he ineres rae has a ransiory negaive bu no significan effec on oupu level. The increase in ineres rae implies -by he arbirage condiion beween he physical capial and he financial capial markes- an incremen in he cos of capial. This generaes a reducion in he demand for capial (ha is only parially overweighed by he increase in labor due o a subsiuion effec) and, consequenly, he fall in oupu. The insignifican fall on oupu is probably because of he absence of rigidiies (such as non-flexibiliies in he good marke or he labor marke). 4
Figure 2. Impulse-Response Funcions from he Theoreical Model Response of Oupu o a Posiive Shock on Ineres Rae (0.5% annual) 0.E00 0 9 28 37 46-2.E-3-4.E-3 Response of Inflaion Rae o a Posiive Shock on Ineres Rae (0.5% annual) 0.05% 0.03% 0.00% 0 9 28 37 46 Response of Money o a Posiive Shock on Ineres Rae (0.5% annual) 0% 0 9 28 37 46-2% -4% -6% -8%
Second, he policy shock causes a ransiory increase in he inflaion rae. As was menioned in secion 3, his is called in he empirical lieraure he price puzzle. Nowihsanding, his effec has a sraighforward explanaion according o he heoreical model proposed here: an increase in he ineres rae produces a similar effec on inflaion leaving real ineres virually invarian due o a Fisher effec. Tha is, he Fisher equaion implies ha higher ineres raes are associaes wih higher inflaion raes, exacly he relaionship shown in figure 2 and, also, in figure. The explanaion o he co-movemen of ineres rae and inflaion has been given in previous sudies bu wihou a specific applicaion for a paricular economy. Monne and Weber (200) presens a discussion ha reconciles he posiive relaionship (Fisher view) beween ineres rae and inflaion and he negaive relaionship beween hose variables (liquidiy view). Also, Alvarez, Lucas, and Weber (200) raised an analogous explanaion wih he inclusion of segmened markes. In he model presened above, he reason is ha he nominal ineres rae has effec only on inflaion bu no on he real ineres rae. Besides, his effec is srenghened in he model because of he assumpion of a Taylor rule ha depends posiively on inflaion deviaion from seady sae. Third, he posiive policy shock produces a emporary decrease in real money holdings hrough he demand for money funcion. Tha is, he agens respond wih higher demand for nominal bonds (and, consequenly, hey lower heir money holdings) when hey face a posiive ineres shock. On his poin, i should be underlined ha values for he inflaion and oupu deviaion coefficiens higher han hose assumed in able 2 (θ 2 =0. and θ 3 =0.2), generaes explosive equilibrium or indeerminacy, resuls ha are consisen wih hose found by Chrisiano and Gus (999). This opic could be a maer of ineres and fuure research for Chilean economy. 5
5.3. How Well is he Maching? In his secion I compare he responses of he variables from he empirical model (he VAR) and hose from he heoreical model when hey face an equivalen posiive emporary policy rae shock. As before, he ineres of he analysis is on he responses of oupu, inflaion and money. Figure 3 repors he confidence inervals of he impulse-response funcions (he upper and lower bands) and he response of each variable from he heoreical model. The shock consiss of an increase of 0.03% in he ineres rae (0.36% in annual erms). As i can be seen, he maching is relaively well, bu wih some observaions. The response of oupu given by he heoreical model is no as significan as he real response showed by he daa. The response of inflaion rae is inside he confidence inervals from he second period. Finally, he response of real money balances is negaive as in he VAR, bu is rajecory falls ou of he bands from he firs o he sevenh period. 6. Conclusions The goal of his paper was o find ou how well a money-in-uiliy funcion model wih Taylor rule could accoun for some moneary sylized facs from Chilean daa during he 986-2000 period. Basically, I focused he aenion on a heoreical explanaion o wha is called in he lieraure he price puzzle (he co-movemen beween ineres rae and inflaion rae), fac ha is found in many VAR-ype esimaions for Chile and oher economies. As seen in secion 2, he previous works ha consider real business cycle models wih moneary variables, pracically ignore or are only capable of explaining a few feaures of moneary relaionships, generally some second momens. This paper formulaes, solves and calibraes a dynamic sochasic general equilibrium model for he Chilean economy beween 986 and 2000. The soluion of he model is adequaely achieved using a perurbaion mehod (second-order-approximaion) proposed by 6
Figure 3. VAR Confidence Inervals and Impulse-Response Funcions from he Theoreical Model Response of Oupu o a Shock on Ineres Rae 0.0% 0 9 28 37 46-0.5% -.0% Lower Band Response of Oupu Upper Band Response of Inflaion Rae o a Shock on Ineres Rae 0.0% 0.05% 0.00% 0 9 28 37 46-0.05% Lower Band Response of Inflaion Upper Band Response of Money o a Shock on Ineres Rae 2.0% 0.0% -2.0% 0 9 28 37 46-4.0% -6.0% Lower Band Response of Money Upper Band
Schmi-Grohé and Uribe (200). The meric consiss of confidence inervals of impulseresponse funcions from a 5-variable VAR. These variables were chosen as he empirical counerpar of he variables explained by he heoreical model presened in secion 4. I find ha a posiive ransiory policy ineres rae shock causes a emporary (no significan) fall of oupu. From a heoreical viewpoin, he increase in ineres rae implies an incremen in he cos of capial, and, consequenly, a reducion in he demand for capial and oupu. The diminishmen is consisen wih he sign of he impulse-response funcion from de VAR esimaed in secion 3. Even hough his effec is saisically significan in ha case, i is imporan o say ha one-lagged VAR, following he Schwarz crierion, is able o show a non-significan decrease in oupu when here is a shock on he ineres rae. The policy shock causes a ransiory increase in he inflaion rae. As was menioned in secion 3, his is called in he empirical lieraure he price puzzle. According o he heoreical model proposed here, his effec has a sraighforward explanaion: an increase in he ineres rae produces a similar effec on inflaion leaving real ineres virually invarian due o a Fisher effec. Therefore, he heoreical model proposed is able o explain and reproduce he co-movemen beween ineres rae and inflaion. Besides, his effec is srenghened by a moneary policy expressed by a Taylor rule ha depends posiively on inflaion deviaion. This explanaion is also found in some recen sudies in he economic lieraure (Monne and Weber, 200; and Alvarez, Lucas, and Weber, 200). Finally, a ransiory incremen in ineres raes decreases real money balances. This effec is relaively consisen wih he impulse-response funcions obained from he VAR from he eighh period of analysis. Nex seps for fuure research consider he es of alernaive Taylor rules, he inclusion of a more exacing meric, and he addiion of price rigidiies ha could be able o allow real variables respond in significan magniude. 7
References Alvarez, F., R. Lucas, and W. Weber. 200. Ineres Raes and Inflaion. Unpublished paper. Universiy of Chicago. Acuña, A., and C. Oyarzún. 200. Money and Real Flucuaions: Calibraing a Cash-in- Advance Model for he Chilean Economy. Unpublished paper. Universidad de Concepción. Benne, H., and R. Valdés. 200. Terms of Trade in Chile. Working Paper 98. Cenral Bank of Chile. Bergoeing, R., P. Kehoe, T. Kehoe, and R. Soo. 200. A Decade Los and Found: Mexico and Chile in he 980s. Documeno de Trabajo 07. Saniago: Cenral Bank of Chile. Bergoeing, R., and R. Soo. 2002. Tesing Real Business Cycle Models in an Emerging Economy. Unpublished paper. Chrisiano, L., and M. Eichenbaum. 992. Curren Real Business Cycle Theories and Aggregae Labor Marke Flucuaions. American Economy Review 82(3): 430-450. Chrisiano, L., M. Eichenbaum, and C. Evans. 997. Sicky Price and Limied Paricipaion Models of Money: A Comparison. European Economic Review 4(6): 20-49. Chrisiano, L., and C. Gus. 999. Taylor Rules in a Limied Paricipaion Model. Naional Bureau of Economic Research Working Paper 707 (March). Chumacero, R., and R. Fuenes. 2002. On he Deerminans of he Chilean Economic Growh. Working Paper 34. Cenral Bank of Chile. 8
Cooley, T., and G. Hansen.989. The Inflaion Tax in a Real Business Cycle Model. American Economic Review :733-48. Cooley, T., and G. Hansen.995. Money and he Business Cycle. In: Froniers of Business Cycle Research, edied by T. Cooley. Princeon, New Jersey: Princeon Universiy Press. Corbo, V. 985. Inernaional Prices, Wages, and Inflaion in an Open Economy: A Chilean Model. Review of Economics and Saisics, 57: 564-73. Dhar, S., and S. Millard. 2000. A Limied Paricipaion Model of he Moneary Transmission Mechanism in he Unied Kingdom. London: Bank of England. Folkersma, C.K. 998. Nominal wage conracs, adjusmen coss and real persisence of moneary shocks. Nederlandsche Bank Research Memorandum WO&E 566. Kydland, F.E., and E. Presco. 982. Time o Build and Aggregae Flucuaions. Economerica 50(6): 345-70. Greenwood, J., Z. Hercowiz, and P Krusell. 2000. The Role of Invesmen-Specific Technological Change in he Business Cycle. European Economic Review 44: 9-5. Hodrick, R., and E. Presco.997. Poswar U.S. Business Cycles: An Empirical Invesigaion. Journal of Money, Credi and Banking 29. McCallum, B., and E. Nelson. 997. An Opimizing IS-LM Specificaion for Moneary Policy and Business Cycle Analysis. Naional Bureau of Economic Research Working Paper 5875. McGraan, E. 994. The Macroeconomic Effec of Disorionary Taxaion. Journal of Moneary Economics 33(3): 573-60. 9
Monne, C., and W. Weber. 200. Money and Ineres Raes. Federal Reserve Bank of Minneapolis Quarerly Review 25(4): 2-3. Parrado, E. 200. Effec of Foreign and Domesic Moneary Policy in a Small Open Economy: The Case of Chile. Working Paper 08. Cenral Bank of Chile. Pesaran, H., and Y. Shin. 998. Generalized Impulse Response Analysis in Linear Mulivariae Models. Economic Leers, 58: 7-29. Quiroz,. J. 99. Las regularidades empíricas del ipo de cambio real en Chile: un enfoque de ciclos de negocios reales. Revisa de Análisis Económico 6 (2): 47-78. Quiroz,. J., F. Bernasconi, R. Chumacero, and C. Revoredo. 99. Modelos y Realidad: enseñando economía en los novena. Revisa de Análisis Económico 6 (2): 79-03. Schmi-Grohé, S.,and M. Uribe. 200. Solving Dynamic General Equilibrium Models Using a Second Order Approximaion o he Policy Funcion. Discussion Paper 2963. Cenre for Economic Policy Research, London. Walsh, C. 998. Moneary Theory and Policy. Cambridge: Massachuses Insiue of Technology. 20
Appendix A. Daa and Sources Variable Definiion Source Y INF Log of Monhly Economic Aciviy Index of Chile or IMACEC Log of (gross) inflaion rae (or growh of Consumer Price Index) Cenral Bank of Chile Cenral Bank of Chile M Log of M deflaed by he Consumer Price Index Cenral Bank of Chile R Log of (gross) UF-indexed ineres rae paid on 90- day bonds issued by he Cenral Bank of Chile or PRBC Cenral Bank of Chile TOT Log of erms of rade Benne and Valdés (200) Appendix B. Seady-Sae Equilibrium of he Model In seady sae, equaion (6) implies he seady-sae ne ineres rae: * R = R0. (23) Subsiuion of equaion (23) in () generaes he seady-sae inflaion rae: * ( ) * = β R π, (24) where he variables wih aserisk denoe seady-sae values. Rearranging equaion (3) and using equaion (8) and (8) yields: K * ( ) * αa0 β τ L α * = L ω α = 0 L, β( δ ) (25) ω αa0 β( τ = where: β( δ ) ) L 0 > 0. 2
Using (8), (0) and (25), and rearranging one obains an expression for he seadysae consumpion ha depends on he seady-sae employmen: c * ( α) A ( ) 0 A0 L α α β τ α * α * ( L ) = 0 ( L ), ( ) ω ω η β δ = α (26) where: ω ( α) A0 = > η 0. Equaions (22), (25), (26), he seady-sae level of invesmen from equaion (5), and he seady-sae governmen expendiure from equaion (2), allow one find he seadysae level of employmen: α * ω0 ω g0 q0 y2 L =. α (27) ε ω 0 α ( A ω δω ) 0 0 Wih equaion (27) one can ge he seady-sae capial sock, invesmen, consumpion, and producion. Finally, equaion (8), (9), (26) and (28) generae he seadysae money balances: * * * R m = φ c. * (28) R Noice ha he demand for money depends posiively on consumpion and negaively on he nominal ineres rae. 22