DSGE Models in a Liquidity Trap and Japan s Lost Decade

DSGE Models in a Liquidity Trap and Japan s Lost Decade Koiti Yano Economic and Social Research Institute ESRI International Conference 2009 June 29, 2009 1 / 27

Definition of a Liquidity Trap Terminology The definition of a Liquidity Trap : A liquidity trap is defined as a situation in which the short-term nominal interest rate is zero. (Gauti B. Eggertsson, (2008), Liquidity Trap, New Palgrave Dictionary of Economics.) 2 / 27

Terminology How stable structural parameters are? The motivation of this paper: How stable structural parameters are? Fernandez-Villaverde and Rubio-Ramirez (2007b) showed the serious doubts on new Keynesian models and Calvo pricing because they found the strong correlation between inflation and the Calvo parameter and the instability of the coefficients of the Taylor rule. They adopted time-varying-parameter approach on Calvo parameters and Taylor parameters. In this paper, we estimate most parameters as time-varying-parameters. 3 / 27

Terminology New Keynesian DSGE models and the liquidity trap Estimating parameters of DSGE models under the non-negativity constraint of a short-term nominal interest rate. Ŷ t = ρ y E t Ŷ t+1 + (1 ρ y )Ŷt 1 σ(î t E t ˆπ t+1 ) + ɛ y t ˆπ t = ρ π E t ˆπ t+1 + (1 ρ π )ˆπ t 1 + κŷt + ɛ π t (1) î t = max[ (r s + π s ), ρ i î t 1 + (1 ρ i )(φ Y Ŷ t + φ πˆπ t ) + ɛ i t] ˆX denotes the deviation from the steady state of a variable X. 4 / 27

Nonlinear, non-gaussian, and non-stationary state space model Nonlinear, non-gaussian, and non-stationary state space model A nonlinear, non-gaussian, and non-stationary state space model is given by x t = f t (x t 1, v t, ξ s ), Y t = h t (x t, ɛ t, ξ o ), (2) where Y t is an observation vector, x t is a state vector, ɛ t and v t are noise vectors, and ξ s and ξ o are unknown parameter vectors. To estimate x t, given Y t (t = {1 T }). 5 / 27

Names of the filter Self-organizing state space model Very similar filters are proposed simultaneously and independently. Kitagawa (1996): Gordon et al. (1993): Bootstrap filter Doucet et al. (2001): Particle filter Others: Sequential Monte Carlo method Today, I use the name the. 6 / 27

(1) Self-organizing state space model The is base on the Bayes theorem. The theorem is given by It is rewritten by P(x t Y 1:t ) P(Y t x t )P(x t Y 1:(t 1) ) (3) P(x t Y 1:t ) P(Y t x t )P(x t x t 1 )P(x t 1 Y 1:(t 1) ) (4) Using it, one estimates unknown state vector x t. 7 / 27

(2) Self-organizing state space model Kitagawa (1996) and Gordon et al. (1993) propose the Monte Carlo filter which estimates x t. x m t p(x t x m t 1, ξ s ), wt m = r(ψ(y t, x m t )) ψ, y t E[p(x t y t )] M wt m δ(x t x m t ). m=1 (5) The particles xt m are resampled with sampling probabilities proportional to wt 1,, wt M. 8 / 27

Augmented state vector Self-organizing state space model Kitagawa (1998) proposes an augmented state vector (ASV) z t and a self-organizing state space model. ASV is defined as follows. z t = [ xt θ ], (6) In the method, states and parameters in ASV are estimated simultaneously by MCPF. 9 / 27

Self-organizing state space model (1) Self-organizing state space model A self-organizing state space model is given by z t = F (z t 1, v t, ξ s ), Y t = H(z t, ɛ t, ξ o ), (7) where and [ ] f (xt 1, v F (z t 1, v t, ξ s ) = t, ξ s ), θ H(z t, ɛ t, ξ o ) = h(x t, ɛ t, ξ o ), θ = [ξ s, ξ o ]. 10 / 27

Self-organizing state space model (2) Self-organizing state space model In practice, invariant parameters are estimated using time-varying-parameter approach. In the TVP approach, the random walk prior is assumed. θ t = θ t 1 + v 2,t, (8) where v 2,t q(v 2,t Σ ξs ). Note that Σ ξs is a diagonal matrix. 11 / 27

Self-organizing state space model -varying parameters and deep parameters 1. The time-varying-parameter approach is practically often used in state space modeling to estimate parameters, for example, Kitagawa (1998) and Liu and West (2001). 2. Even if we assume the random walk priors, it does not indicate that the deep parameters of DSGE models are time-varying. 3. Adopting it makes a great advantage that the structural changes of parameters are detected naturally. 4. Thus, it is suitable to analyze how stable structural parameters are. 12 / 27

(1) The linearized model: Ĉ t = h 1 + h Ĉt 1 + 1 1 + h E tĉt+1 1 1 + h E t ˆît ˆπ t+1 + ɛc,t. (9) ˆπ t = 1 1 + β ˆπ t 1+ β 1 + β E t ˆπ t+1 + (1 ξ p)(1 βξ p ) ˆ(1 α)ŵt +αˆr K Ẑ t +ɛπ,t. ξ p (1 + β) (10) Ŵ t = σ C (Ŷ t hŷ t 1 ) + σ LˆL t, (11) ˆL t = Ŵ t + ˆr K + ˆK t, (12) ˆQ t = E t [î t π t+1 ] + Î t = 1 1 + β Ît 1 + β 1 + β Ît+1 + ν 1 + β ˆQ t + ɛ I,t, (13) 1 δ 1 δ + r E ˆQ r K K t t+1 + 1 δ + r E K t + ɛ Q,t, (14) 13 / 27

(2) ˆK t = (1 δ) ˆK t 1 + δî t, (15) and where Ŷ t = Ψ C Ĉ t + Ψ I Î t + Ψ G Ĝ t, (16) Ŷ t = Ẑ t + α ˆK t + (1 α)ˆl t, (17) Ĝ t = ρ G Ĝ t 1 + ɛ G,t, (18) Ẑ t = ξ Z Ẑ t 1 + ɛ Z,t, (19) î t = max[ (r s + π s ), ρ i î t 1 + (1 ρ i )(φ Y Ŷ t + φ π ˆπ t ) + ɛ i t] (20) x t = [E t Ĉ t+1, E t ˆπ t+1, E tˆr K t+1, E t Î t+1, E t ˆQ t+1, Ŷ t, Ĉ t, î t, ˆπ t, Ŵ t, ˆr K t.ẑt, ˆL t, ˆK t, Ît, ˆQ t, Ĝt] t 14 / 27

(3) Braun and Waki (2006) develop an algorithm for computing perfect foresight equilibria for DSGE models in a liquidity trap. In this paper, we adopt their algorithm in our estimation method. Structural linear rational expectations models with regime switching are given by { Γ0 x t = Γ 1 x t 1 + Ψz t + Πη t + C, if 0 t S or t > T Γ 0 x t = Γ 1 x t 1 + Ψ z t + Πη t + C, if S < t T, (21) 15 / 27

(4) The measurement equation of the model is Y t = Y s + Hx, (22) where Y t = [YGR t, CGR t, IGR t, WGR t, INFL t, LGR t, INT t ] t, Y s = [Y s, Y s, Y s, Y s, π s, L s, r s + π s ] t, and v t = ( ɛ v Y,t, ɛv C,t, ɛv π,t, ɛ v I,t ɛv W,t, ɛv L,t, ɛv i,t) T N(0, Σv,t ) with Σ v,t = diag((σ v Y,t )2 (σ v C,t )2, (σ v I,t )2, (σ v π,t) 2, (σ v W,t )2, (σ v L,t )2, (σ v i,t )2 ) 16 / 27

-varying parameters The list of parameters which are estimated as time-varying parameters: θ t = [h t, ξ p,t, σ L,t, νξ Z,t, Ψ C,t, Ψ I,t, Ψ G,t, ρ i,t, φ Y,t, φ π,t, ρ G,t, σ C,t, σ π,t, σ I,t, σ Q,t, σ i,t, σ G,t, σ Z,t, Y s t, π s t, L s t, r s t, σ v Y,t, σv C,t, σv π,t, σ v I,t, σv W,t, σv L,t, σv i,t]. (23) Following Sugo and Ueda (2008), we calibrate four parameters: β = 0.99, α = 0.3, δ = 0.06, and r K = 1/β 1 + δ. 17 / 27

Y s π s varying parameter 0.0 1.0 2.0 3.0 varying parameter 1 0 1 2 r s L s varying parameter 0 1 2 3 4 5 6 varying parameter 1.0 0.0 0.5 1.0 1.5 Figure: -varying Steady State

Y^ t π^t ^ i t Endogenous variable 1.0 0.5 0.0 0.5 1.0 1.5 Endogenous variable 0.3 0.2 0.1 0.0 0.1 0.2 0.3 Endogenous variable 0.3 0.2 0.1 0.0 0.1 0.2 W^ t L^t Z^t Endogenous variable 1.5 0.5 0.0 0.5 1.0 1.5 2.0 Endogenous variable 1.0 0.5 0.0 0.5 1.0 Endogenous variable 1.0 0.5 0.0 0.5 1.0 Figure: Endogenous variable

h ξ p σ L varying parameter 0.0 0.2 0.4 0.6 0.8 1.0 varying parameter 0.0 0.2 0.4 0.6 0.8 1.0 varying parameter 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 ν ξ z ρ G varying parameter 0.8 1.0 1.2 1.4 varying parameter 0.0 0.2 0.4 0.6 0.8 1.0 varying parameter 0.0 0.2 0.4 0.6 0.8 1.0 Figure: -varying parameter

ρ i φ Y φ π varying paramter 0.1 0.2 0.3 0.4 0.5 varying paramter 0.1 0.2 0.3 0.4 varying paramter 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 Figure: -varying parameter of the Taylor rule

Y^ t Output gap 4 0 2 4 6 Y^ t Output gap 4 0 2 4 6 Figure: Output Gap (Hodrick-Prescott filter/christiano-fitzgerald filter

Log-likelihoods of models Using the log-likelihood of a model, we compare DSGE models. Table: Log-likelihood of model Model Log-likelihood ξ s,1 ξ s,2 Standard Model -2488.100 0.318 0.068 Model without inflation indexation -2567.952 0.288 0.089 Model without habit formation -2518.378 0.345 0.098 23 / 27

Findings of this paper 1. The target rate of inflation is too low in the 1990s and the 2000s, and it causes the deflation in the Japanese economy. 2. The analysis shows that the growth rate of natural output declines in the late 1990s, however, it becomes as high as about 0.5% in the mid-2000s. 3. There exist several structural changes of structural parameters. Our results, however, are different from Fernandez-Villaverde and Rubio-Ramirez (2007b) because the structural changes are not correlated with inflation and business cycles. 4. We conclude that our estimate of output gap is relatively coincide with the estimates which are calculated by the HP/CF filters 24 / 27

How stable are structural parameters? Our conclusion is a little bit different from the serious doubts of Fernandez-Villaverde and Rubio-Ramirez (2007b). We agree with them that there exist some structural changes of structural parameters from 1981:Q1 to 2008:Q4. Our estimates, however, indicate that the structural changes are not strongly correlated with inflation and business cycles. In particular, severe structural changes of σ L and ν might point out that the models of the perfect competitive labor markets and Tobin s q are imperfect. Our results only suggest the necessity of more investigations on new Keynesian DSGE models. 25 / 27

Conclusion 1. Our method is proposed to analyze how stable structural parameters are. Adopting it makes a great advantage that the structural changes of parameters are detected naturally. 2. The second advantage of our method is that we are able to estimate new Keynesian DSGE models in a liquidity trap [Krugman (1998)] because nonlinear, non-gaussian, and non-stationary state space models allow model switching. 3. Moreover, we estimate time-varying trends of macroeconomic data: real output, an inflation rate, and an real interest rate. 26 / 27

Future study 1. Estimate second-order approximated DSGE models 2. A Unified Approach : Unifying TV-SVAR and time-varying analysis of DSGE models. 3. Now estimating a new Keynesian, small open economy model, a new Keynesian model with liquidity constraint households. 4. In -varying analysis of DSGE models based on Sequential Monte Carlo Methods, (manuscript), we estimate the US economy (from 1980 to 2008). 27 / 27

Model (1) Households The utility function of household j is given by E j 0 t=0 Subject to β t[ log(c j,t hc t 1 ) + Υ ( M j,t ) Ψ ] L L 1+σ L j,t, (24) P t 1 + σ L C j,t +I j,t + M j,t + B j,t P t P t W t L j,t + M j,t 1 +rt K K j,t +(1+i t 1 ) B j,t 1 +Π j,t, P t P t (25) 28 / 27

Model (2) Final goods sector Final goods are constructed from the intermediate goods. Y t = [ 1 The demand of Y j,t is given The aggregate price is 0 Y j,t = P t = [ 1 0 (Y j,t ) 1 1+λp dj ]1+λ p, (26) 1+λp ( ) Pj,t λp Yt, (27) P t (P j,t ) 1 λp dj ] λ p. (28) 29 / 27

Model (3) Intermediate goods sector In the sticky prices model, proposed by Calvo (1983), a fraction 1 ξ p of all firms re-optimize their nominal prices while the remaining ξ p fraction of all firms do no re-optimize their nominal prices. Following Christiano, Eichenbaum, and Evans (2005), firms that cannot re-optimize their price index to lagged inflation. E t l=0 P j,t = π t 1 P j,t 1, (29) (βξ p ) l[ P ] j,t X tl MC t+l Y j,t+l, P t+l subject to Y j,t = 1+λp ( ) Pj,t λp Yt, P t (30) P t = [(1 ξ p (P t ) 1 p. 1 λp + ξ p (π t 1 P t 1 ) 1 1 λp ]1 λ (31) 30 / 27

Model (4) Production function, technology, and marginal costs Production function is Technology progress is Marginal costs are Y j,t = Z t K α j,tl 1 α j,t, (32) log Z t = (1 ξ Z ) log Z + ξ Z log Z t 1 + ɛ Z,t, (33) MC j,t = 1 Z t ( α α (1 α) (1 α) W 1 α t (r K t ) α) (34) 31 / 27

Model (5) Market Clearing The nonlinear Taylor rule is given by i t = max [ 0, i ρ i t 1 (Y φ Y t The market Clearing Condition is π φ π t ) (1 ρ i ) e ɛ i t ], (35) Y t = C t + I t + G t, (36) where Y t = [ 1 0 (Y j,t) 1 1+λp dj ] 1+λ p, Ct = [ 1 0 (C j,t) 1 1+λp dj ] 1+λ p, and I t = [ 1 0 (I j,t) 1 1+λp dj ] 1+λ p 32 / 27

Our Algorithm 1. In time t, generate z t based on the results at time t 1. 2. Using particles, the linear rational expectations system is solved to obtain the state transition equation. 3. If a particle implies indeterminacy (or non-existence of a stable rational expectations solution), then the weight of the particle, w m t, is set to zero. 4. If Θ 1 or Θ 1 is not invertible, the particle is discarded (See Braun and Waki (2006) Algorithm in appendix). 5. If a unique stable solution exists, then the weight of the particle is calculated. 6. Resampling particles with sampling probabilities proportional to wt 1,, wt M. 7. Replace t with t + 1. 8. Go to 1. 33 / 27

-varying Calvo parameter ξ p varying parameter 0.0 0.2 0.4 0.6 0.8 1.0 1986 1988 1990 1992 1994 1996 1998 Figure: -varying Calvo Parameter 34 / 27