Estimating a Nonlinear New Keynesian Model with the Zero Lower Bound for Japan

Transcription

1 Estimating a Nonlinear New Keynesian Model with the Zero Lower Bound for Japan Hirokuni Iiboshi 1, Mototsugu Shintani 2, Kozo Ueda 3 August EEA-ESEM 1 Tokyo Metropolitan University 2 University of Tokyo 3 Waseda University IIboshi et al. DSGE w/ ZLB August EEA-ESEM 1 / 33

2 Primary Motivation Quantitative macroeconomic (DSGE) model Challenge: effectively zero lower bound <ZLB, hereafter>) on the nominal interest rate Solving a REE (for a given parameter set) is hard. Estimation is even harder. Japan s lost decades We estimate a nonlinear DSGE (New Keynesian) model with the ZLB for Japan. Nonlinear (not log-linearized) Stochastic (not perfect foresight, no absorbing state) ZLB for more than two decades Albeit small-scale We investigate monetary policy natural rate of interest. IIboshi et al. DSGE w/ ZLB August EEA-ESEM 2 / 33

3 Monetary policy Under the ZLB, what kind of policy has the BOJ been conducting? Commitment? How different is the IRF owing to the ZLB? Natural rate of interest Did the natural rate of interest decline during the lost decades? Wicksell (1936), Krugman (1998), Woodford (23) Now negative? And why? Demand or supply shock? IIboshi et al. DSGE w/ ZLB August EEA-ESEM 3 / 33

4 Our Method Time iteration with linear interpolation (TL) Recommended by Richter, Throckmorton, and Walker (214) To solve REE Within the class of policy function iterations Sequential Monte Carlo squared (SMC 2 ) Developed by Chopin, Jacob, and Papaspiliopoulos (213) and applied to DSGE models by Herbst and Schorfheide (215) To estimate Evaluate L given a certain parameter set by generating particles of endogenous variables (particle filter). Draw the posterior distribution of parameters by sampling the particles of parameter sets (Sequential Monte Carlo or SMC). IIboshi et al. DSGE w/ ZLB August EEA-ESEM 4 / 33

5 Literature Review Empirical methods Richter, Throckmorton, and Walker (214): TL Chopin, Jacob, and Papaspiliopoulos (213), Herbst and Schorfheide (215) and Fernández-Villaverde, Rubio-Ramírez, and Schorfheide (216): SMC 2 Model with the ZLB Hirose and Sunakawa (215) and Hirose and Inoue (216): data generation w/ ZLB and estimation w/o ZLB Boneva, Braun and Waki (216) and Nakata (217): simulation in a calibrated model Kulish, Morley, and Robinson (217): assuming that the duration of ZLB τ t is perfectly foresighted in each period t, they estimate τ t and time-varying policy functions given estimated τ t. Aoki and Ueno (212) Kim and Pruitt (217): make use of forward rate curves Natural rate of interest Concept: Wicksell (1936), Woodford (23) Recent decline: Krugman (1998), Laubach and Williams (23, 216), Andrés, López-Salido, and Nelson (29), Hall (211), Barsky, Justiniano, and Melosi (214), Ikeda and Saito (214), Cúrdia (215), Cúrdia et al. (215), Bank of Japan (216), Hirose and Sunakawa (217), Holston, Laubach, and Williams (217) IIboshi et al. DSGE w/ ZLB August EEA-ESEM 5 / 33

6 Two closest papers. Gust et al. (216) and Richter and Throckmorton (216) estimate DSGE models with ZLB Differences Japan, where the ZLB matters for a long period, not the US SMC 2, not MCMC (PFMH) Gust et al. (216) and Richter and Throckmorton (216) need to introduce very high measurement error for feasibility, but we do not. Not a medium-sized DSGE model like Gust et al. (216) IIboshi et al. DSGE w/ ZLB August EEA-ESEM 6 / 33

7 Key Findings Monetary policy Nonlinear estimation is crucial to draw implications for monetary policy. The past experience of recessions to bring the nominal interest rate down to zero is carried over to today s monetary policy. forward guidance (commitment) policy. IRFs to a monetary policy shock are very different depending on the model for a monetary policy rule as well as a sign of the shock. The natural rate of interest often negative since the mid-199s caused mainly by weak demand shocks. The ZLB does not produce a bias for the estimated natural rate of interest. The effect of biased parameters on the natural rate of interest is canceled out by the effect of biased shocks on it. IIboshi et al. DSGE w/ ZLB August EEA-ESEM 7 / 33

8 Outline of Today s Talk Introduction (done) Model (quickly) Estimation method Estimation results Monetary policy Natural rate of interest Concluding remarks IIboshi et al. DSGE w/ ZLB August EEA-ESEM 8 / 33

9 Canonical Simple Model Simple 3-equation NK model with the ZLB The representative household, monopolistic firms, and the central bank 3 kinds of models for monetary policy 3 exogenous shocks: preference, technology, and monetary policy IIboshi et al. DSGE w/ ZLB August EEA-ESEM 9 / 33

10 The representative household The representative household maximizes { C 1 σ β j Zt+j b t+j j= 1 σ (A t+j) 1 σ χlt+j 1+ω 1 + ω subject to the budget constraint: C t + B t P t W t l t + R t 1B t 1 P t + T t, }, where the preferance shock Z b t is I() and obeys log(z b t ) = ρ b log(z b t 1) + ɛ b t. IIboshi et al. DSGE w/ ZLB August EEA-ESEM 1 / 33

11 Final-good Firms { 1 Y t = } ε Y ε 1 ε f,t df ε 1 Y f,t = ( Pf,t P t ) ε Y t, Intermediate-Good Firms (monopolisitc firms) The production function is Y f,t = A t l f,t, where the technology shock A t is I(1) and µ a t log(a t /A t 1 ) γ a obeys µ a t = ρ a µ a t 1 + ɛ a t. With Rotemberg-type price stickiness, firm f sets P f,t+j so as to maximize E t β j Λ ( t+jzt+j b P f,t+j j= Λ t Zt b W t+j φ ( ) ) 2 Pf,t+j π Y f,t+j. P t+j A t+j 2 P f,t+j 1 IIboshi et al. DSGE w/ ZLB August EEA-ESEM 11 / 33

12 Monetary Policy with Model 1: or Model 2: R t = (R t 1) ρr ( R t = (R t 1 ) ρr ( R t = max (1, R t ), ( r π π ) ( t ψπ Yt /A t π Y t /A t ( r π π ) ( t ψπ Yt /A t π Y t /A t ) ψy ) 1 ρr e ɛr t, ) ψy ) 1 ρr e ɛr t, where the monetary policy shock ɛt r is i.i.d with zero mean. * Compared with Model 2, Model 1 involves strong commitment for future policy. Because Rt can be below zero and depends on Rt 1, the experience of adverse shocks tie the hands of the central bank for long periods. IIboshi et al. DSGE w/ ZLB August EEA-ESEM 12 / 33

13 Solution, Estimation, and the Advantage of the SMC 2 Solution Solve the rational expectation equilibrium of our model. Time iteration with linear interpolation (TL) Richter, Throckmorton, and Walker (214) Within the class of policy function iteration methods TL provides the best balance between speed and accuracy. In our model, given parameters θ, the policy function is in the form of X t = X (µ a t, Zt b, ɛt r, Rt 1 ) Time iteration until equations between X t and E t (X t+1 ) are satisfied at every node. Then, locally approximate the policy functions with linear interpolation. Compared to global approximation methods such as the projection method using the Chebyshev polynomial basis, linear interpolation is considered to perform better in an environment where the ZLB produces kinks in the policy functions. IIboshi et al. DSGE w/ ZLB August EEA-ESEM 13 / 33

14 Repeat 2 to 4 for N φ = 1 stages. It takes about a week using a 32-core (Intel Xeon E5-2698v3) IIboshi et al. DSGE w/ ZLB August EEA-ESEM 14 / 33 Estimation SMC 2 developed by Chopin, Jacob, and Papaspiliopoulos (213) and Herbst and Schorfheide (215) 4 steps 1 Initialization: draw N θ = 1, 2 particles for parameters θ 2 Correction: given θ, compute the likelihood ˆp(Y t θ) and incremental weight w Policy function iteration using the TL: X t = X (µ a t, Zt b, ɛt r, Rt 1 ) Draw N S = 4, particles for shocks (µ a t, Zt b, ɛt r ) Generate the path of Ŷ t Compare with actual Y t and compute ˆp(Y t θ) using the measurement error Kalman filter cannot be used p(y t θ) replaced by ˆp(Y t θ) with a sufficiently large number of particles wrt shocks 3 Selection: resample θ and w based on w 4 Mutation: propagate θ and w using an MH algorithm

15 Advantage of the SMC 2 over the PFMH Gust et al. (217) and Richter and Throckmorton (216) use the Particle Filters combined with Metropolis Hastings (PFMH) SMC 2 more reliable posterior inference no need for big measurement errors easily parallelized Graphic illlustration in our estimation Estimating σ IIboshi et al. DSGE w/ ZLB August EEA-ESEM 15 / 33

16 A scatter plot where each dot represents a particle for the value of parameter σ (horizontal axis) and its posterior likelihood (vertical axis). -25 Log Likelihood Particles Particle with maximum log lik Particle for median What will happen if we use the MCMC in this example? Trapped at the outlier. The SMC 2 can resolve this problem. Because the SMC 2 uses more than two particles for parameter candidates and allocates weight w on each particle corresponding to its likelihood, particles are much less likely to be stuck at the outlier. IIboshi et al. DSGE w/ ZLB August EEA-ESEM 16 / 33

17 Data Japan 1983:2Q to 216:2Q The beginning period coincides with that for the output gap data. 3 variables (benchmark) y t : The real per-capita GDP growth rate alternative: the output gap constructed by the Bank of Japan π t : The CPI inflation rate (consumption tax adjusted) R t : The overnight call rate (divided by 4) IIboshi et al. DSGE w/ ZLB August EEA-ESEM 17 / 33

18 Our Data (q/q,%) GDP growth 5 (q/q,%) 2 Inflation (q,%) Nominal interest 3-1 (%) Output gap IIboshi et al. DSGE w/ ZLB August EEA-ESEM 18 / 33

19 Specifications (prior, etc) Measurement errors of y t, π t, and R t : the size of.5%,.25%, and.5% of their standard deviations, respectively. In Gust et al. (216) and Richter and Throckmorton (216), it is.25 5% and.1 3%, respectively. Calibrated parameters: β =.995, χ = 1, ε = 6. Prior distribution (κ (ε 1) (ω + σ) /(φπ )) Parameter Mean S.D. Shape σ Normal γ a.5 Normal ω 3.5 Normal κ.5.6 Normal π.5 Normal Parameter Mean S.D. Shape ρ r.5.2 Beta ψ π Normal ψ y Normal ω 3.5 Normal ρ a.5.2 Beta σ a, σ b, σ r.2 5 (d.f.) Inv Gamma IIboshi et al. DSGE w/ ZLB August EEA-ESEM 19 / 33

20 Estimation Results Model 1 Model 2 Model w/o ZLB Parameter Mean (95% low, high) Mean (95% low, high) Mean (95% low, high) σ 1.4 (1.289, 1.55) (1.472, 1.584) 1.37 (1.4, 1.83) γ a -.28 (-.119,.5).123 (.68,.172) (-.539, -.31) ω (2.298, 2.674) (2.98, 3.413) (3.71, 3.287) κ.55 (.5,.62).53 (.5,.55).47 (.45,.49) π.36 (-.117,.619).5 (-.148,.274) (-.728, -.246) r.464 (.343,.574).691 (.67,.764).67 (-.55,.181) ρ r.521 (.475,.611).685 (.644,.721).214 (.182,.24) ψ π (1.627, 1.745) (1.739, 1.811) 1.59 (1.453, 1.553) ψ y.15 (.91,.123).113 (.98,.125).133 (.127,.141) ρ a.254 (.129,.446).21 (.96,.292).122 (.93,.147) ρ b.75 (.693,.82).754 (.728,.776).74 (.689,.788) σ a (.96, 1.367) 1.32 (1.146, 1.524) (1.675, 1.859) σ b (1.435, 2.318) (2.18, 2.442) (1.191, 1.558) σ r (1.14, 1.673) (1.22, 1.349).921 (.89, 1.12) Likelihood IIboshi et al. DSGE w/ ZLB August EEA-ESEM 2 / 33

21 Summary of the Estimation Results Model 1 is selected. Tie the hand of the BOJ Support for the commitment effect (power of forward guidance) Estimation biases: neglecting the ZLB leads to a smaller γ a, π, and ρ r. a lower steady-state level for the natural rate of interest and the inflation target. IIboshi et al. DSGE w/ ZLB August EEA-ESEM 21 / 33

22 Notional Nominal Interest Rate 2.5 quarterly, % R t R * (Model 1) t R * (Model 2) t IIboshi et al. DSGE w/ ZLB August EEA-ESEM 22 / 33

23 IRFs to a Monetary Policy Shock (ɛ r t = ±.1) Model 1: monetary easing is effective even at the ZLB, and monetary tightening yields a smaller response..2 t 1985: Q1 Pos: Model 1 Pos: Model 2 Pos: Model w/o ZLB.2 t 21: Q1 Neg: Model 1 Neg: Model 2 Neg: Model w/o ZLB [ % ] R t 1985: Q1 1 R t 21: Q1 [ % ] R * t 1985: Q1 1 R * t 21: Q1 [ % ] horizon horizon IIboshi et al. DSGE w/ ZLB August EEA-ESEM 23 / 33

24 Probability of the ZLB, 2% Inflation, and Deflation The probability of the ZLB is considerably high, that is, around 6% and 4% for h = 1 and 8, respectively, when r t is around %. Significantly higher comparing with 7.1% reported by Gust et al. (217) for US Natural rate (rt ) Prob ( R t+h = % rt ) Prob ( π t+h > 2% rt ) Prob ( π t+h < % rt ) for h = 1 for h = 2 for h = 4 for h = 8 h for 1 h 8 rt 3% 7.% 14.35% 23.35% 31.65% 6.2% 76.6% 2% rt < 3% 12.64% 19.59% 26.37% 33.8% 47.16% 81.28% 1% rt < 2% 25.97% 28.62% 32.6% 35.61% 28.44% 88.1% % rt < 1% 52.4% 44.56% 39.94% 38.17% 15.36% 94.11% 1% rt < % 61.86% 51.74% 43.58% 39.43% 1.4% 97.48% rt < 1% 69.6% 58.83% 51.6% 41.63% 7.3% 99.7% rt = r 35.12% 36.79% 37.52% 37.29% 15.16% 94.26% IIboshi et al. DSGE w/ ZLB August EEA-ESEM 24 / 33

25 Natural Rate of Interest Log-Linearized Model (for an illustrative purpose) π t π = βe (1 σ)γa E t [π t+1 π ] + (ε 1) (ω + σ) φπ (y t y t ), [ y t yt = E t y t+1 yt+1 1 ( Rt r π σ r π π t+1 π π r t r )] r, and the monetary policy rule, where the natural rate of interest rt by [ ] rt = r 1 + σρ a µ a t + (1 ρ b )logzt b ). is given The inflation rate and output gap are influenced by the natural rate of interest, which, in turn, is influenced by preference and technology shocks. Steady state r = e σγa /β. IIboshi et al. DSGE w/ ZLB August EEA-ESEM 25 / 33

26 Natural Rate of Interest Often negative during the lost decades. Steady state r = e σγa /β is.46% (quarterly). Preference shock zt b is most important. 4 Natural Rate of Interest.1 Shocks [ quarterly % ] : baseline : only technology shock : only discount factor shock : only monetary policy shock : technology shock : discount factor shock : monetary policy shock IIboshi et al. DSGE w/ ZLB August EEA-ESEM 26 / 33

27 Natural Rate of Interest: Model Comparison (1) Not much different. Contrast with Hirose and Sunakawa (217) quarterly, % Model 1 Model 2 Model w/o ZLB IIboshi et al. DSGE w/ ZLB August EEA-ESEM 27 / 33

28 Why Hardly Differs between Model 1 and Other Models? Differences come from the presence of the ZLB, estimated parameters, and estimated shocks. Thus, we change one of the three as Model w/o ZLB (parameters and shocks) but now with the ZLB Model w/o ZLB but with estimated parameters in Model 1, analogous to Hirose and Sunakawa (217) Model w/o ZLB but with estimated shocks in Model 1 Then, we simulate the path of the natural rate of interest. IIboshi et al. DSGE w/ ZLB August EEA-ESEM 28 / 33

29 The effect of different estimated parameters offset that of different estimated shocks. The ZLB itself does not affect the natural rate. This suggests the importance of estimating parameters and shocks simultaneously. quarterly, % Model Model w/ ZLB Model w/ Model 1 Parameters Model w/ Model 1 Shocks IIboshi et al. DSGE w/ ZLB August EEA-ESEM 29 / 33

30 Model Comparison 2 Laubach-Williams (23) Estimate the backward-looking IS and Phillips curves jointly using the Kalman filter Calculate the ex ante real interest rate by estimating the one-year ahead inflation expectation from a univariate AR(3) model The natural rate of interest given by r t = g t + z t, (1) They assume that both g t and z t obey an I(1) process (i.e., the natural output is I(2)), while both log(a t /A t 1 ) and logzt b in our model obey to an I() process. Hodrick Prescott Filter λ = 1, 6 IIboshi et al. DSGE w/ ZLB August EEA-ESEM 3 / 33

31 Natural Rate of Interest: Model Comparison (2) Not much different from that by Laubach and Williams (23) 4 3 Benchmark (Model 1) Laubach-Williams (one-sided) HP Filter quarterly, % IIboshi et al. DSGE w/ ZLB August EEA-ESEM 31 / 33

32 Robustness Constrained linear model Linerized model but with the ZLB Richter and Throckmorton (216) argue that this is a good approximation. However, for Japan, it is not. Maybe because of long ZLB periods Use of the output gap data The preference and technology shocks are hard to be separated. IIboshi et al. DSGE w/ ZLB August EEA-ESEM 32 / 33

33 Concluding Remarks Future work Richer model (intrinsic persistence) US data Regime-switching model IIboshi et al. DSGE w/ ZLB August EEA-ESEM 33 / 33