Piecewise Linear Continuous Approximations and Filtering for DSGE Models with Occasionally-Binding Constraints

Transcription

1 The views expressed in this presentation are those of the authors and do not necessarily reflect those of the Board of Governors or the Federal Reserve System. Piecewise Linear Continuous Approximations and Filtering for DSGE Models with Occasionally-Binding Constraints S. Borağan Aruoba 1 Pablo Cuba-Borda 2 Kenji Higa-Flores 1 Frank Schorfheide 3 Sergio Villalvazo 3 1 University of Maryland 2 Federal Reserve Board 3 University of Pennsylvania February 26, MFM Winter Meeting

2 Overview Goals of Research Project: 1 Develop algorithm to construct piecewise linear and continuous solutions to models with occasionally-binding constraints. 2 Develop an efficient particle filter tailored to the structure of the solution. Why: Many recent macro-financial models feature occ.-binding constraints: financial, ZLB,... It is computationally very costly to solve and estimate such models (e.g., Bocola (2016, JPE); Gust, Herbst, Lopez-Salido, and Smith (2017, AER)) Solution & estimation of models with multiple occ.-binding constraints has been elusive. More Broadly: Wide adoption of such models require shortcuts to keep computations fast and feasible. Extreme: assume constraint binds always/never/for fixed period; don t fit model to data. Our approach: slightly less accurate solution to enable efficient likelihood evaluation.

3 Background

4 Background (Model Solution) Aruoba, Cuba-Borda, and Schorfheide (2018, REStud) Macroeconomic Dynamics Near the ZLB: A Tale of Two Countries Interest Rate In.ation ^g ^g 1 Output 2.5 Consumption ^g ^g

5 Background (Filtering) Nonlinear State-Space Model Measurement Eq. : y t = Ψ(s t ; θ) + u t, u t F u ( ; θ) State Transition : s t = Φ(s t 1, ɛ t ; θ), ɛ t F ɛ ( ; θ). Objects of interest: Estimates of states: p(s t Y 1:t, θ) Likelihood function: p(y 1:T θ) = T t=1 p(yt Y1:t 1, θ). Particle filtering (sequential Monte Carlo): represent p(s t Y 1:t, θ) by particle swarm {st, j Wt j } M j=1 such that 1 M M j=1 h(sj t)wt j SLLN,CLT E[h(s t ) Y 1:t, θ].

6 Background (Particle Filtering) Iteration t: starting from {s j t 1, W j t 1 }M j=1 1 Mutation: Draw s j t g t ( s t s j t 1 ). 2 Correction: Compute incremental weights and update/normalize weights w j t = p( s t s j t 1 ) g t ( s t s j t 1 )p(y t s j t, θ), W j t w j t W j t 1. 3 Selection: Resample particles to equalize weights. Potential Implementations: Bootstrap particle filter (BSPF): g t ( s t s j t 1 ) = p( s t s j t 1 ). Conditionally-optimal particle filter (COPF): g t ( s t s j t 1 ) p(y t s t )p( s t s j t 1 ). Key Issue: BSPF (solid) can be VERY inaccurate; COPF (dashed) is infeasible in many nonlinear models

7 Current Project

8 Solution Method (Partial Model) Example of occasionally binding constraint. State X = (ɛ R, g, R 1 ), log-linearized ZLB: { } R(X) = max (1 ρ)ψπ(x) + ρr 1 + ɛ R, log(rπ ) }{{} h(x) Postulate decision rule of the form: { a0 + a π(x) = 1 ɛ R + a 2 g + a 3 R 1 if h(x) > log(rπ ) a 4 + a 5 ɛ R + a 6 g + a 7 R 1 otherwise Define threshold function: R 1 (ɛ R, g) = δ 0 + δ 1 ɛ R + δ 2 g, such that h(x) log(rπ ) if R 1 R 1(ɛ R, g) h(x) = log(rπ ) if R 1 < R 1(ɛ R, g). NOTE: for now we abstract from (a) other decision rules and (b) other equilibrium conditions!

9 Solution Method (Partial Model) Impose continuity of π(x) for R 1 = R 1 (ɛ R, g): yields δ 0, δ 1, δ 2. Impose continuity of R(X) at R 1 = R 1 (ɛ R, g): yields a 4, a 5, a 6. Recall the model: { (1 ρ)ψπ(x) + ρr 1 + ɛ R(X) = R if R 1 > R 1 (ɛ R, g) log(rπ ) otherwise { a0 + a π(x) = 1 ɛ R + a 2 g + a 3 R 1 if R 1 > R 1 (ɛ R, g) a 4 + a 5 ɛ R + a 6 g + a 7 R 1 otherwise R 1(ɛ R, g) = δ 0 + δ 1 ɛ R + δ 2 g. In sum, imposing piecewise linearity and continuity we have: parameters: r, π, ψ free decision rule coefficients: a 0, a 1, a 2, a 3, a 7. Note: in a full model we have (a) more decision rules and (b) other equilibrium conditions to pin down free coefficients.

10 Solution Method Interpretation True decision rules f 0 (s) satisfy a functional equation: H [f 0 (s), s; θ] = 0, s S. We consider PLC decision rules with finite-dim parameter ζ: g(s; ζ) G. No convergence of g(s; ζ) to f 0 (s). Therefore, define pseudo-optimal coefficient vector ζ. Let π(s; ζ ) be the ergodic distribution of s if agents use decision rules g(s; ζ ). Define ζ as the solution to ζ = argmin ζ H ( g(s; ζ), s; θ) 2 W (ζ ) π(s; ζ )ds. Two interpretations: 1 Approximation to nonlinear decision rule. 2 Boundedly rational behavior of the agents.

11 Solution Stylized Example Equilibrium condition: H[f ( ), s] = f ( ) s 2 = 0 for all s. Optimal decision: f 0 (s) = s 2. Approximate decision rule: g(s; ζ) = I{s 0}ζ u s I{s < 0}ζ l s Ergodic distribution: Solve for π(s; ζ) = ζ l + ζ u I{ ζ l 1 s ζ u + 1} ζ = argmin ζ H ( g(s; ζ), s; θ) 2 W π(s; ζ ).

12 Algorithm S(0) Compute decision rules d θ (s) for log-linear approximation and use to simulate model using set of innovations ɛ. = Preliminary grid S (0) θ (d θ, ɛ, θ). S(1) Let ˆζ (0) = argmin ζ H ( g(s; ζ), s; θ) 2 W s S (0) θ (d θ,ɛ,θ) Simulate model based on decisional rules g (0) θ ( ) = g(s; ˆζ (0) (θ, ɛ)) using innovations ɛ. = New grid S (1) (0) θ (g θ, ɛ, θ). S(k) Define ˆζ (k) = arg min ζ s S (k) (k 1) θ (g θ,ɛ,θ) H ( g(s; ζ), s; θ) 2 W Simulate model based on decisional rules g (k) θ ( ) = g(s; ˆζ (k) (θ, ɛ)) using innovations ɛ. = New grid S (k+1) θ (g (k) θ, ɛ, θ).

13 Filtering Approach (Partial Model) Measurement Equation: Let y t = [π o t, R o t ] and s t = [π t, R t, g t ], then y t = Ψ(s t ; θ) + u t πo t = π t + u π,t R o t = R t + u R,t State-transition Equation: s t = Φ(s t 1, ɛ t ; θ) R t = R(X t ) = R(R t 1, g t 1, ɛ g,t, ɛ R,t ) π t = π(x t ) = π(r t 1, g t 1, ɛ g,t, ɛ R,t ) g t = ρ g g t + ɛ g,t Recall, conditionally-optimal proposal: g t ( s t s j t 1 ) p(y t s t )p( s t s j t 1 ).

14 Filtering Approach (Partial Model) Density p( s t s j t 1 ) is truncated normal. Rewrite the kink condition R t 1 R t 1 as η 1,t = δ 1 ɛ R,t + δ 2 ɛ g,t R t 1 δ 0 δ 2 ρ g g t 1 = ζ t 1 With a bit of algebra, we obtain canonical form: { Φ 1 s t = 0 + Φ 1 1 s t 1 + Φ 1 ηη t if η 1,t ζ(s t 1 ) Φ Φ2 1 s t 1 + Φ 2 ηη t otherwise After more tedious algebra, we find g t( s t s j t 1 ) is a mixture of truncated normals direct sampling; a formula for importance weight p( s t s j t 1 )/ g t( s t s j t 1 ).

15 Application: Small-Scale New Keynesian Model

16 Application to Small-Scale NK Model More Equilibrium Conditions: 1 interest rate rule, 2 consumption Euler equation, 3 price-setting equation. More Decision Rules: 1 inflation π(x) parameterized piecewise linear function, 2 output y(x) parameterized piecewise linear function, 3 consumption c(x) determined by agg resource constraint, π(x), y(x).

17 Equilibrium Conditions E t 1 1 ν (1 χ hc τ t y 1/η t ( ) ( 1 β e ˆd t+1 ˆdt ct+1 ) τ Rt ct γπ t+1 eẑt+1 ( ) ( ) φ(π t π ) [(1 b)π t + bπ ] + φβ e ˆd t+1 ˆdt ct+1 [ ct 1 c t g φ eĝt 2 (πt π ) 2] y t R t max { 0, R ρ R t 1 (πt/π )(1 ρ R )ψ 1 (y t/γy t 1) (1 ρ R )ψ 2 exp(ɛ R,t ) } ẑ t ρ z ẑ t 1 σ z ɛ z,t ˆd t ρ d ˆdt 1 σ d ɛ d,t ĝ t ρ g ĝ t 1 σ g ɛ g,t ) τ y t+1 (π t+1 π )π t+1 yt = 0

18 Performance of Model Solution Algorithm Compute the pseudo-optimal ζ using simulated grids with M G = 10, 000 grid points, iterating the algorithm k = 10 steps. Compute 100 ˆζ (k) ζ 2 / ζ 2 for k = 5. (Run times in seconds). Cluster Grid Size M G Time-Sep. Grid Size M G T G , ,000 1, (322) (257) (242) (10) (36) (68) 5, (233) (123) (115) (12) (38) (73) 10, (412) (123) (137) (11) (38) (73)

19 Welfare Loss from Bounded Rationality We simulate a long sequence of observations from the NK model using: nonlinear decision rules; piecewise linear and continuous decision rules; linear decision rules. Welfare differentials: to make households as well off as under the fully nonlinear decision rules Piecewise linear: 0.013% Linear: 0.061%

20 Performance of the Filter Fix θ 0 and generate a single sample of observations. Generate draws θ i from the prior. For each θ i run the particle filters N sim = 100 times. We compute the likelihood function for T = 100 observations. We are using M = 1, 000 particles for the BSPF and M = 220 particles for the COPF, which approximately equalizes the run times of the filters.

21 Distribution of Run Times BSPF runtime (seconds) COPF runtime (seconds)

22 Distribution of Log Likelihood Approximation for θ i Small Model-Data Distance (20% ME) Large Model-Data Distance (2% ME) 0.25 Bootstrap PF Cond.Opt. PF 0.25 Bootstrap PF Cond.Opt. PF

23 Accuracy Std Dev (COPF) Small Model-Data Distance (20% ME) Large Model-Data Distance (2% ME) (l( i )) COPF L( 0 ) COPF - L( i ) COPF (l( i )) COPF L( 0 ) COPF - L( i ) COPF

24 Accuracy Comparison Std Dev Differential (BSPF - COPF) Small Model-Data Distance (20% ME) Large Model-Data Distance (2% ME) (l( i )) BSPF - (l( i )) COPF 10 5 (l( i )) BSPF - (l( i )) COPF L( 0 ) COPF - L( i ) COPF L( 0 ) COPF - L( i ) COPF

25 Testing the MCMC Algorithm Fix θ 0 and generate a single sample of observations. Sample size T = 200 Number of particles for COPF is M = 5, 000 and for BSPF is M = 1, 000, 20% ME. Number of MCMC draws N 20, 000. Computational time approximately 13 sec per draw (8 cores w/ parallelized solution and filter). Configuration of solution: T G = 10, 000, M G = 100, clustered-grid algorithm.

26 Trace Plots of κ i COPF BSPF 0.25 kappa 0.25 kappa

27 Autocorrelations of Draws COPF ACF ACF of κ i s All Parameters 0.98 Autocorrelation function of kappa COPF BSPF Autocorrelations Lag 20 Lag BSPF ACF

28 Extensions

29 Extensions Gertler-Karadi Model: we can solve a baseline model that combines a New Keynesian structure with endogenous balance sheet constraints. We are currently fine-tuning the solution to perform quantitative excercises. Two occasionally-binding constraints: we are currently exploring the parameterization of decision rules with two types of kinks.