Bayesian Estimation of DSGE Models: Lessons from Second-order Approximations Sungbae An Singapore Management University Bank Indonesia/BIS Workshop: STRUCTURAL DYNAMIC MACROECONOMIC MODELS IN ASIA-PACIFIC ECONOMIES 3 June 28
Motivation Dynamic Stochastic General Equilibrium (DSGE) model Likelihood-based estimation with log-linearization: DYNARE!! For welfare analysis, first-order accurate approximation is not sufficient: (Kim and Kim, 23)
Example: Welfare Reversal Two countries, i =, 2 No dynamics Preference: CRRA u(c i ) = C τ i τ Technology: stochastic endowment Z i = exp(u i ) where u i is iid with { with prob. /2 u i = with prob. /2
Autarchy: C A i Example: Welfare Reversal = Z i Complete market, risk-sharing: C M i = Z + Z 2 2 Deterministic steady-state: Z = C = Log-linearization: ĉ M i = 2 (ẑ + ẑ 2 ) C M i = Z Z 2 Second-order approximation: ĉ M2 i = 2 (ẑ + ẑ 2 ) + 8 (ẑ ẑ 2 ) 2 ( ) i = C M i exp 8 (log Z log Z 2 ) 2 C M2
Example: Welfare Reversal Welfare: Expected utility k A : M : M : M2 : E [ u ( )] C k i τ =.5 τ = 2 2 u(e) + 2 u(e ).2553.543 4 u(e) + 4 u(e ) + ( ) e + e 2 u.3698.956 2 4 u(e) + 4 u(e ) + u ().276.275 2 4 u(e) + 4 u(e ) + ( ) 2 u e /2.47.748
Motivation Second-order approximation: DYNARE!! Again!! Likelihood-based estimation of second-order approximated models is not trivial. Short cut: () Estimation based on linearized model (2) Plugging the linear estimates into the second-order approximated model to do policy analysis Can this practice be misleading?
New Keynesian DSGE Model Representative household Monopolistically competitive firms ( Y t = Y t (j) ν dj ) ν Nominal rigidities: quadratic adjustment cost AC t (j) = ϕ 2 ( ) Pt (j) 2 P t (j) π Y t (j) No capital accumulation Linear production with perfectly competitive factor market
New Keynesian DSGE Model Monetary policy: Interest feedback rule with nominal interest rate target R t = R ρ R t R ρ R R t = rπ ( π t π ) ψ t exp(ɛ R,t) ( ) ψ2 Yt Fiscal policy: Passive with exogenously driven government spending Three shocks: Technology, government spending, and monetary policy Y t
Symmetric Equilibrium After detrending, in log-deviation (from steady-state) form: [ ] = E t e τ(ĉ t+ ĉ t )+ ˆR t ˆπ t+ ẑ t+ = ν ( ) νϕπ 2 e τĉ t + ( e ˆπ t ) [( ) e ˆπ t + ] 2ν 2ν [ (e ˆπ β E t+ t ) ] e τ(ĉ t+ ĉ t )+(ŷ t+ ŷ t )+ ˆπ t+ eĉt ŷ t = e ĝ t ϕπ2 g ( e ˆπ t ) 2 2 ˆR t = ρ R ˆR t + ( ρ R )ψ ˆπ t + ( ρ R )ψ 2 (ŷ t ĝ t ) + ɛ R,t ĝ t = ρ g ĝ t + ɛ g,t ẑ t = ρ z ẑ t + ɛ z,t
Representation The solution of the rational expectations system takes the form s t = Φ(s t, ɛ t ; θ) where s t = [ŷ t, ĉ t, ˆπ t, ˆR t, ɛ R,t, ĝ t, ẑ t ] ɛ t = [ɛ R,t, ɛ g,t, ɛ z,t ]
Log-linearization and Rational Expectation Log-linearized economy ĉ t = E t [ĉ t+ ] τ ˆπ t = βe t [ ˆπ t+ ] + κĉ t ŷ t = ĉ t + ĝ t where κ = τ ν νπ 2 ϕ. ( ˆR t E t [ ˆπ t+ ] E t [ẑ t+ ] ) Without rigidities where ν = and ϕ =, κ tends to infinity. The solution of the log-linearized rational expectations system s t = Φ (s) s t + Φ (ɛ) ɛ t
Second-Order Approximation Judd (998), Collard and Juillard (2), Jin and Judd (22), Schmitt-Grohe and Uribe (24), Kim, Kim, Schaumburg, and Sims (25), Swanson, Anderson, and Levin (25), Klein (25) Partition s t into endogenous and exogenous variables, [x t, z t ] The solution of the second-order approximated rational expectations system z t = Ψ (z) z t + ɛ t x j,t = Ψ () j w t + where w t = [x t, z t ]. ( Ψ () j + w tψ (2) j w t )
State-Space Representation: Transition Linear transition from log-linearization: where s t = [x t, z t ] s t = Φ (s) s t + Φ (ɛ) ɛ t Quadratic transition from second-order approximation: z t = Ψ (z) z t + ɛ t x j,t = Ψ () j w t + where w t = [x t, z t ] ( Ψ () j + w tψ (2) j w t )
State-Space Representation: Measurement Output growth rate, inflation, and nominal interest rate YGR t = γ (Q) + (ŷ t ŷ t + ẑ t ) + u y,t INFL t = π (A) + 4 ˆπ t + u π,t INT t = π (A) + r (A) + 4γ (Q) + 4 ˆR t + u r,t Linear measurement: y t = Θ () + Θ (s) s t + u t
Filter and Likelihood Initialization at the beginning of t: p ( s t Y t ) Prediction/Forecasting: ( p s t Y t ) = Filtering/Updating by Bayes Theorem: Likelihood evaluation: ( p (s t s t ) p s t Y t ) ds t p ( s t Y t) = p (y t s t ) p ( s t Y t ) p (y t Y t ) ln L(θ Y) = ln p (y θ) + T t=2 ( ) ln p y t Y t, θ
Particle Filter Use Monte Carlo method to integrate out the latent variables Pioneering works: Gordon, Salmon, and Smith (993) Kitagawa (996) Economics literature: Stochastic volatility: Pitt and Shephard (999), Kim, Shephard, and Chib (998), Duan and Fulop (27) Neoclassical growth model: Fernandez-Villaverde and Rubio-Ramirez (24,25,26)
Prior Specification Parameter Density Mean Std. Description τ Gamma 2.5 inverse EIS ν Beta..5 degree of imperfect competition κ Gamma.3. Phillips curve, κ = τ ( ) ν π 2 ϕ ν c/y Beta.85.5 consumption-output ratio ψ Gamma.5.25 responsiveness to inflation ψ 2 Gamma.5.25 responsiveness to output gap ρ R Beta.5.2 policy inertia r (A) Gamma.55.25 steady state real interest rate π (A) Gamma 3 2 target inflation γ (Q) Normal.8.25 steady state output growth rate ρ g, ρ z, σ R, σ g, and σ z
Posterior Distribution Log marginal data densities: Modified harmonic mean estimator by Geweke (999) Linear/Kalman Quadratic/Particle 323.984 322.2 Information on ν and c/y Steady state parameters, r (A), π (A), and γ (Q) Policy parameters, ψ and ψ 2
Prior Linear/Kalman Posterior Quadratic/Particle Posterior.9.9.9 c/y.8 c/y.8 c/y.8.7.7.7.6 2 3 τ.6 2 3 τ.6 2 3 τ κ.5 κ.5 κ.5..2 ν..2 ν..2 ν ψ 2.5 ψ 2.5 ψ 2.5.5 2 2.5 ψ.5 2 2.5 ψ.5 2 2.5 ψ
Prior Linear/Kalman Posterior Quadratic/Particle Posterior.8.8.8 γ (Q).6 γ (Q).6 γ (Q).6.4.4.4.2.2.2 2 4 6 π (A) 2 4 6 π (A) 2 4 6 π (A) 6 6 6 5 5 5 π (A) 4 3 π (A) 4 3 π (A) 4 3 2 2 2.2.4.6.8 r (A).2.4.6.8 r (A).2.4.6.8 r (A)
Posterior Distribution Prior Posterior Linear/Kalman Quadratic/Particle Mean 9% Interval Mean 9% Interval Mean 9% Interval τ 2.4 [.75, 2.793] 2.352 [.486, 3.28] 2.565 [.757, 3.269] ν. [.22,.75].3 [.25,.79]. [.52,.7] κ.3 [.38,.452].582 [.389,.763].52 [.36,.678] c/y.85 [.77,.93].85 [.773,.93].847 [.776,.994] ψ.58 [.9,.97].779 [.47, 2.47].749 [.456, 2.27] ψ 2.5 [.,.866].565 [.23,.978].48 [.4,.86] ρ R.499 [.48,.823].88 [.76,.863].86 [.755,.853] r (A).548 [.6,.98].56 [.86,.797].4 [.59,.632] π (A) 2.992 [.225, 5.765] 3.47 [2.772, 4.25] 4.282 [3.43, 5.22] γ (Q).8 [.38,.97].653 [.45,.93].84 [.63,.8] ρ g.8 [.65,.959].956 [.927,.987].959 [.932,.985] ρ z.66 [.429,.99].937 [.95,.969].943 [.97,.976] σ R.5 [.2,.79].95 [.65,.229].25 [.69,.242] σ g.26 [.537,.996].682 [.589,.784].658 [.596,.79] σ z.5 [.28,.796].8 [.46,.23].76 [.44,.26]
Identification Likelihood contour from simulated data κ = τ ( ) ν π 2 ϕ ν.4 Linear/Kalman Likelihood 375.4 Quadratic/Particle Likelihood.35 375.3 4.25 4.35.3.25 4 375.2.2 344 ν.5. 35 35 ν.5. 35.5 375 4 2 4 6 8 φ.5 344 35 375 4 2 4 6 8 φ
Expected Utility as Welfare Measures Infinite sum of the discount utilities [ ( )] W = E β t c τ t χ H y t τ t= [ ( )] [ = E β t c τ ] t χ H E β t y t τ t= = U χ H V Approximations in recursive form t= U t = c τ t + βe t U t+, V t = y t + βe t V t+ τ
Welfare Cost Two sets of parameters, reference and alternative, implies different levels of welfare. The fraction of consumption in the alternative to be increased (or to be given up) to achieve the welfare level of the reference. ( Wt R = E t ) ] β A s ( + λ)c A [ t+s /A A τ A t+s τ A χ A HHt+s A s= Welfare differentials Fix nonpolicy parameters Differentials relative to baseline policy parameter
Welfare Cost Linear/Kalman Posterior Quadratic/Particle Posterior.5.5.5.2.4.6.8 2.5.2.4.6.8 2 ψ ψ...5.5.5.5..5.5 ψ 2..5.5 ψ 2
Generalized Impulse Responses Impulse responses in a linear model Symmetry Scalability Path-independency: past and future Koop, Pesaran, and Potter (996): Generalized Impulse Response GI(n) = E [x t+n ɛ t, Ω t ] E [x t+n Ω t ]
Generalized Impulse Responses Consider a nonlinear process X t = f (X t ) + ɛ t and two alternative shock processes which differ only in time t, that is, ɛ k = ɛ k for all k t. Then conventional impulse response at time t + 2 can be written as X t+2 X t+2 = f [f {f (X t ) + ɛ t } + ɛ t+ ] which implies the path-dependence. f [f {f (X t ) + ɛ t} + ɛ t+ ]
Impulse Responses Monetary Policy.. Output.2 2 3.5 Inflation 2 3 Interest Rate.4.2 2 3 Government Spending.5 2 3 x 4 2 3 x 4 2 3 5.5.5 Technology 4 3 2.5.5 2 3 2 3 2 3
Impulse Responses For Nonlinear Processes.5 Output 2 Inflation 2 Interest Rate Monetary Policy.5 5 2 5 2 5 Government Spending 2 2 5..5.5. 5..5.5. 5 5 5 5 Technology 5 5 5 5 5 5
Conclusion Particle filter for second-order approximated DSGE model Stylized monetary model in line with Woodford (23) U.S. quarterly data More identification on structural parameters Better fit as measured by the marginal data density Bias corrections on parameter estimates Aggressive estimates on welfare cost Unveils neglected impulse response dynamics