DSGE methods. Introduction to Dynare via Clarida, Gali, and Gertler (1999) Willi Mutschler, M.Sc.

Transcription

1 DSGE mehods Inroducion o Dynare via Clarida, Gali, and Gerler (1999) Willi Muschler, M.Sc. Insiue of Economerics and Economic Saisics Universiy of Münser willi.muschler@uni-muenser.de Summer 2014 Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

2 Previously... Insigh Theory and inuiion behind he Smes/Wouers model as he workhorse DSGE model. Derivaion of he srucural form and log-linearizaion. DSGE model consiss of se of Euler equaions, i.e. firs-order opimaliy condiions, ransiion equaions for srucural shocks and innovaions, observable variables and measuremen errors which can be cas ino a nonlinear sysem of expecaional difference equaions. Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

3 Inroducion o Dynare Dynare compues he soluion of deerminisic models (arbirary accuracy), compues firs, second and hird order approximaion o soluion of sochasic models, esimaes (maximum likelihood or Bayesian approach) parameers of DSGE models, compues opimal policy (Ramsey-opimal), performs global sensiiviy analysis and local idenificaion of a model, solves problems under parial informaion, esimaes BVAR and Markov-Swiching Bayesian VAR models esimaes DSGE-VAR models esimaes Markov-Swiching DSGE models... Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

4 Simple model: Clarida, Gali, and Gerler (1999) Household Household preferences are given by ( ) max E 0 β log C exp(τ ) N1+φ C,N,B φ =0 C denoes consumpion, N denoes employmen or marke work, φ > 0 denoes a labor supply parameer, 0 < β < 1 is discoun parameer τ is a preference shock The budge consrain of he household is τ = ρ τ τ 1 + ε τ, ε τ iidn(0, σ τ 2 ) (1) P C + B +1 W N + R 1B + T, T denoes (lump-sum) axes and profis, P denoes price level, W denoes nominal wage rae B +1 denoes bonds purchased a ime, which deliver a non-sae-coningen rae of reurn, R, in period + 1. Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

5 Simple model: Clarida, Gali, and Gerler (1999) Household Opimaliy yields: Inraemporal opimaliy: Marginal cos of working (in consumpion unis) equals marginal benefi (he real wage) exp(τ )C N φ = W P Ineremporal Euler equaion: Uiliy coss of consumpion foregone wih corresponding benefi Π +1 gross inflaion from o R = βe (2) C C +1 Π +1 Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

6 Simple model: Clarida, Gali, and Gerler (1999) Compeiive firms Represanaive, compeiive firm produces a homogeneous oupu good, Y, using he following Dixi-Sigliz echnology: [ 1 ] ε ε 1 Y = di, ε > 1, 0 Y ε 1 ε i, Y i, denoes he i h inermediae good, i (0, 1). ε is degree of subsiuion beween inpus (love of variey) Compeiive firm akes he price of final oupu good, P, and he prices of inermediae goods, P i,, as given and chooses Y and Y i, o maximize profis. Firs-order condiion: Y i, = Y ( Pi, P ) ε wih P = ( 1 Y i, is he demand curve for he producer of Y i,. 0 ) 1 P 1 ε 1 ε i, di Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

7 Simple model: Clarida, Gali, and Gerler (1999) Inermediae firms The i h inermediae good firm is a monopolis for Y i, and uses labor, N i,, o produce oupu using he following producion funcion: Y i, = A N i,, a = log(a ), A 0 = Ā = 1 a = ρ a a 1 + ε a, ε a iidn(0, σ a 2 ) (3) is he firs difference operaor a has a uni roo represenaion, ε a is hus a shock on he growh rae of echnology A The i h firm ses prices subjec o Calvo fricions. In paricular, { P wih probabiliy 1 θ P i, =, wih probabiliy θ P i, 1 P denoes he price chosen by he 1 θ firms ha can reopimize heir price a ime. Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

8 Simple model: Clarida, Gali, and Gerler (1999) Inermediae firms i h producer is compeiive in labor markes, pays W (1 ν) for one uni of labor. ν represens a subsidy which eliminaes he monopoly disorion on labor in he seady sae: 1 ν = (ε 1) /ε. Define real marginal coss s = dcos dworker doupu dworker = (1 ν) W P exp(a ) = (1 ν)cexp(τ)nφ exp(a ) Each of he 1 θ firms ha can opimize price, choose P o maximize profis: wih: E j=0 period +j profis sen o household {}}{ β j θ j µ +j [ P Y i,+j P +j s +j Y i,+j ] }{{}}{{} revenues cos µ +j : marginal value of dividends o household= u c,+j /P +j θ j firm cares only abou saes in which i can reopimize Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

9 Simple model: Clarida, Gali, and Gerler (1999) Inermediae firms and aggregae oupu Opimaliy (Calvo-Yun algebra) yields: p = P ( ) E P+j ε j=0 (βθ)j ε s P ε 1 +j K = P ( ) E P+j ε 1 = F j=0 (βθ)j P ( 1 θπ ε 1 ) 1 1 ε 1 θ (4) where ε K = (1 ν) ε 1 exp(τ )N φ C exp(a ) + βθe Π ε +1K +1, (5) F = 1 + βθπ ε 1 +1 F +1 (6) Define P = Aggregae oupu: [ ] P ε i, di ε, law of moion of price disorion: ( ) [ p P ε ( := 1 θπ ε 1 ) ε ] 1 ε 1 θπ ε = (1 θ) + (7) P 1 θ p 1 Y = p exp(a )N = C (8) Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

10 Simple model: Clarida, Gali, and Gerler (1999) Seady sae Seady-Sae Π ss = 1, R ss = 1 β, p ss = 1, 1 F ss = K ss = 1 βθ, Y ss = C ss = N ss = 1, τ ss = a ss = a ss = 0 Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

11 Simple model: Clarida, Gali, and Gerler (1999) Summary We have 8 equaions for 10 unknowns (C, p, N, Π, K, F, R, ν) Need o pin down policy variables ν and R! Two (in principle equivalen) ways: Ramsey opimal policy (Solve Lagrangian w.r. policy variables) Taylor rule (no monopoly power, no inflaion) Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

12 Simple model: Clarida, Gali, and Gerler (1999) Ramsey equilibrium Ramsey equilibrium (Efficien Allocaion) The efficien equilibrium associaed wih he opimal moneary policy is characerized by zero inflaion, Π = 1, a each dae and for each realizaion of a & τ consumpion and employmen correspond o heir firs bes levels C and N saisfy he resource consrain: C = exp (a ) N marginal rae of subsiuion beween consumpion and labor equals he marginal produc of labor: C exp (τ ) N ϕ = exp (a ) Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

13 Simple model: Clarida, Gali, and Gerler (1999) Ramsey equilibrium No monopoly power: No price disorion ( : p 1 ν = ε 1 ε = 1) yields no inflaion (Eff1) Π = p = 1 (Eff2) p 1 Efficien (firs-bes) allocaions: Y N = C = exp ( ) τ 1 + φ = exp(a )N ( = exp a ) τ 1 + φ (Eff3) (Eff4) Naural ineres rae back ou of Euler equaion: R = 1 ( ) β Eexp τ+1 τ a φ (Eff5) Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

14 Simple model: Clarida, Gali, and Gerler (1999) Ramsey equilibrium Log-linearizaion around seady-sae yields for he efficien allocaion: π n c r := log(π ) log(π ss) = 0, := log(n ) log(n ss) = τ 1 + φ, := log(c ) log(c ss) = a τ 1 + ϕ := log(r τ +1 τ ) log(r ss) = E a +1 E 1 + ϕ, = log(y ) log(y ss) =: y, indicaes a variable corresponding o he Ramsey equilibrium, i.e. naural raes, lower case leers are log of he corresponding variable minus log of corresponding seady-sae, e.g. x = log(x ) log(x ss ) Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

15 Simple model: Clarida, Gali, and Gerler (1999) Closing he model: Taylor rule Mos models are exended by a Taylor rule ha is designed such ha in seady-sae inflaion is zero, e.g. r = αr 1 + (1 α) [φ ππ + φ xx ] If he Taylor rule includes he Ramsey equilibrium, we call i opimal moneary policy: Taylor-principle r = r + α(r 1 r 1) + (1 α) [φ ππ + φ xx ] The moneary auhoriy reacs o a change in inflaion by implemening a bigger change in ineres raes ((1 α)φ π > 1). Technical requiremen for sable and convergen soluion (Blanchard-Khan) If (1 α)φ π 1: shocks ha raise inflaion resul in lower real ineres raes and higher oupu, which furher fuels he iniial increase in inflaion unsable explosive spiral indeerminacy Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

16 Simple model: Clarida, Gali, and Gerler (1999) Log-linearized model and Taylor-rule Linearizing abou he seady-sae he model equaions are given by π = βe π +1 + κx (Phillips curve) x = E x +1 (r E π +1 r ) (IS equaion) r = αr 1 + (1 α) [φ ππ + φ xx ] (baseline Taylor-rule) a = ρ a a 1 + ε a τ = ρ τ τ 1 + ε τ r (echnological shock) (preference shock) = ρ a a + 1 ρτ τ (naural ineres rae) 1 + ϕ y = x x 1 + a τ τ φ (oupu growh) wih x = y y (oupu gap) y = a 1 τ 1 + φ (naural oupu) (1 θ)(1 βθ)(1 + ϕ) κ = θ (auxiliary parameer) Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

17 Simple model: Clarida, Gali, and Gerler (1999) Pracicing Dynare Exercise: CGG-Model wih Dynare 1 Insall Malab and Dynare, open cgg.mod, ry o undersand he code, run i. Inerpre he Dynare oupu. 2 Compue he impulse response funcion of he model o a echnology shock and a preference shock for he nex 7 periods. 3 Given he IRF, indicae wheher he economy over- or underresponds due o he shocks, relaive o heir naural response. Wha is he economic inuiion in each case? (Hin: Use plos.m) 4 Do he calculaions wih φ π = Explain he error message and give economic inuiion behind his. Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

18 Simple model: Clarida, Gali, and Gerler (1999) Pracicing Dynare Reurn o φ π = Explain why i is ha when he moneary policy rule is replaced by he naural equilibrium, i.e. r = r, he soluion is indeerminae. 6 Now replace he moneary policy rule by r = r + α(r 1 r 1) + (1 α) [φ π π + φ x x ] Explain why his Taylor rule uniquely suppors he naural equilibrium. Calibrae he model o a more empirically relevan paramerizaion: φ x = 0.15, α = 0.8, ρ = Simulae he model for 1000 periods. Save he middle 100 observaions of y and π ino an Excel-file as well as ino a ma-file. Plo he pah of oupu growh. Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19

19 Simple model: Clarida, Gali, and Gerler (1999) Pracicing Dynare Now le s esimae he coefficiens as well as he sandard errors of he sochasic process (ρ a, ρ τ, σ a, σ τ ) 8 via maximum likelihood: Use 1000 observaions o verify ha everyhing works fine and sar far from he rue values. 9 via maximum likelihood: Use only 50 observaions and sar a he rue values. Do he resuls change? 10 via Bayesian mehods: Use he invered gamma disribuion (mean o rue values, sandard deviaion 10) as he prior on he wo sandard errors and he bea disribuion (mean o rue values, sandard deviaion o 0.4) as he prior on he wo auocorrelaions. Use 50 observaions for he esimaion, 1000 MCMC replicaions, one MCMC chain, and 1.5 for he scale parameer. Willi Muschler (Insiue of Economerics) DSGE mehods Summer / 19