DSGE Model Nonlinearities Frank Schorfheide University of - PowerPoint PPT Presentation

DSGE Model Nonlinearities Frank Schorfheide University of Pennsylvania, CEPR, NBER, PIER June 2017 JEDC Plenary Lecture 2017 SCE/CEF Conference Papers and software available at https://web.sas.upenn.edu/schorf/

DSGE Model Nonlinearities Large body of recent work on DSGE model nonlinearities: stochastic volatility; effective lower bound on nominal interest rates; occasionally-binding financial constraints; general nonlinear dynamics in macro-financial models; (...) F. Schorfheide DSGE Model Nonlinearities

Three Tasks Model Solution 1 Model Estimation 2 Model Assessment 3 I will provide an overview of some of my recent collaborative research in these areas. F. Schorfheide DSGE Model Nonlinearities

Task 1 – Model Solution F. Schorfheide DSGE Model Nonlinearities

Nonlinear Model Solution Reference: B. Aruoba, P. Cuba-Borda, and F. Schorfheide (2017): “Macroeconomic Dynamics Near the ZLB: A Tale of Two Countries,” Review of Economic Studies , forthcoming. Perturbation solutions capture some nonlinearities but not all − → not well suited for occasionally-binding constraints. Example: ZLB/ELB for nominal interest rates � ψ 1 � Y t � � π t � ψ 2 � 1 − ρ R t e ǫ R , t } , R ρ R R t = max { 1 , R ∗ R ∗ t = r π ∗ t − 1 . π ∗ Y ∗ t Two Challenges: capture “kinks” in decision rules; 1 solution needs to be accurate in region of state-space that is relevant according to model 2 AND according to data. Other issue in paper: multiplicity of equilibria, sunspots ... F. Schorfheide DSGE Model Nonlinearities

Challenge 1 – Kinks Consider decision rule π ( S t ), states S t = ( R t − 1 , y ∗ t − 1 , d t , g t , z t , ǫ R , t ) “Stitch” two functions for each decision rule (endogenous “seam”):  f 1 π ( S t ; Θ) if R ( S t ) > 1  π ( S t ; Θ) = f 2 π ( S t ; Θ) if R ( S t ) = 1  j are linear combinations of a complete set Chebyshev polynomials up to 4 th order, with f i weights Θ. F. Schorfheide DSGE Model Nonlinearities

Sample Decision Rules - Small-Scale NK Model for U.S. Interest Rate In . ation 3 4 2 2 1 0 0 -6 -4 -2 0 -6 -4 -2 0 ^ g ^ g Output Consumption 1 2.5 0 2 -1 1.5 -2 1 -3 -4 0.5 -6 -4 -2 0 -6 -4 -2 0 g ^ g ^ F. Schorfheide DSGE Model Nonlinearities

Challenge 2 – Accuracy Where it Matters Choose Θ to minimize sum squared residuals from the (intertemporal) equilibrium conditions over particular grid of points in state space 1.31 0.51 0.02 1.29 0.48 0.01 y − 1 1.27 0.45 0 g z 1.25 0.42 −0.01 1.23 0.39 −0.02 1 1.01 1.02 1.03 1 1.01 1.02 1.03 1 1.01 1.02 1.03 R − 1 R − 1 R − 1 −3 8 x 10 0.2 4 0.1 ǫ R 0 d 0 −4 −0.1 −8 −0.2 1 1.01 1.02 1.03 1 1.01 1.02 1.03 R − 1 R − 1 Ergodic Distribution(s=1) Ergodic Distribution(s=1) Ergodic Distribution(s=0) Ergodic Distribution(s=0) 2000−2008 2000−2008 2009(s=0) 2009(s=0) 2009−2015(s=1) 2009−2015(s=1) F. Schorfheide DSGE Model Nonlinearities

Task 2 – Model Estimation F. Schorfheide DSGE Model Nonlinearities

Model Estimation – A Plug for Bayesian Inference... p ( Y | θ ) p ( θ ) p ( θ | Y ) = � p ( Y | θ ) p ( θ ) d θ Treat uncertainty with respect to shocks, latent states, parameters, and model specifications uncertainty symmetrically. Condition inference on what you know (the data Y ) instead of what you don’t know (the parameter θ ). Make optimal decision conditional on observed data. Large set of computational tools available. F. Schorfheide DSGE Model Nonlinearities

Model Estimation Bayesian inference is implemented by sampling draws θ i from the posterior p ( θ | Y ). Posterior samplers require evaluation of likelihood function: θ − → model solution − → state-space representation − → p ( Y | θ ). State-space representation − → p ( Y , S | θ ): u t ∼ F u ( · ; θ ) y t = Ψ( s t , t ; θ ) + u t , ǫ t ∼ F ǫ ( · ; θ ) . s t = Φ( s t − 1 , ǫ t ; θ ) , In order to obtain p ( Y | θ ) = � T t =1 p ( y t | Y 1: t − 1 , θ ) we need to integrate out latent states S from p ( Y , S | θ ) − → use filter: Initialization: p ( s t − 1 | Y 1: t − 1 , θ ) Forecasting: p ( s t | Y 1: t − 1 , θ ), p ( y t | Y t − 1 ) Updating: p ( s t | y t , Y 1: t − 1 ) = p ( s t | Y 1: t ). F. Schorfheide DSGE Model Nonlinearities

Particle Filtering Particle Filtering: represent p ( s t − 1 | Y 1: t − 1 ) by { s j t − 1 , W j t − 1 } M j =1 such that M � 1 � h ( s j t − 1 ) W j t − 1 ≈ h ( s t − 1 ) p ( s t − 1 | Y 1: t − 1 ) ds t − 1 . M j =1 Example: Bootstrap particle filter Mutation/Forecasting: turn s j s j s j t ∼ p ( s t | s j t − 1 into ˜ t : sample ˜ t − 1 ). W j ˜ s j t ) W j Correction/Updating: change particle weights to: t ∝ p ( y t | ˜ t − 1 . s j t , ˜ W j t } M j =1 into { s j t , W j t = 1 } M Selection (Optional): Resample to turn { ˜ j =1 . Problem: naive forward simulation of Bootstrap PF leads to uneven particle weights − → inaccurate likelihood approximation! F. Schorfheide DSGE Model Nonlinearities

Tempered Particle Filter Reference: E. Herbst and F. Schorfheide (2017): “Tempered Particle Filtering,” NBER Working Paper , 23448. Construct a sequence “bridge distributions” with inflated measurement errors. Define � − 1 | Σ u ( θ ) | − 1 / 2 exp φ d / 2 p n ( y t | s t , θ ) ∝ 2( y t − Ψ( s t , t ; θ )) ′ n � × φ n Σ − 1 u ( θ )( y t − Ψ( s t , t ; θ )) , φ 1 < φ 2 < . . . < φ N φ = 1 . Bridge posteriors given s t − 1 : p n ( s t | y t , s t − 1 , θ ) ∝ p n ( y t | s t , θ ) p ( s t | s t − 1 , θ ) . Bridge posteriors given Y 1: t − 1 : � p n ( s t | Y 1: t ) = p n ( s t | y t , s t − 1 , θ ) p ( s t − 1 | Y 1: t − 1 ) ds t − 1 . Traverse these bridge distributions with “static” Sequential Monte Carlo method (Chopin, 2002). References in stats lit: Godsill and Clapp (2001), Johansen (2016) F. Schorfheide DSGE Model Nonlinearities

Bridge Posteriors: p n ( s t | Y 1: t ), n = 1 , . . . , N φ 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1.0 0.8 1 0.6 0 φ n − 1 0.4 − 2 0.2 − 3 0.0 − 4 F. Schorfheide DSGE Model Nonlinearities

Distribution of Log-lh Approx Error – Great Recession Sample 0.30 TPF ( r ∗ = 2), M = 40000 0.25 0.20 TPF ( r ∗ = 3), M = 40000 Density 0.15 TPF ( r ∗ = 2), M = 4000 0.10 TPF ( r ∗ = 3), M = 4000 0.05 BSPF , M = 40000 0.00 − 350 − 300 − 250 − 200 − 150 − 100 − 50 0 F. Schorfheide DSGE Model Nonlinearities

Putting it All Together Once a reasonably accurate likelihood approximation has been obtained, it can be embedded in a posterior sampler. The Full Monty is a real pain: see Gust, C., E. Herbst, D. Lopez-Salido, and M. E. Smith (2017): “The Empirical Implications of the Interest-Rate Lower Bound,” American Economic Review , forthcoming. Potential shortcuts: less accurate model solution; cruder state extraction / likelihood approximation; non-likelihood-based parameterization of model. Schorfheide, Song, Yaron (2017): slight short-cut in model solution − → conditionally-linear state-space representation − → efficient particle filter approximation of likelihood − → full Bayesian estimation. F. Schorfheide DSGE Model Nonlinearities

Part III – Model Assessment F. Schorfheide DSGE Model Nonlinearities

Can the nonlinearities in DSGE models correctly reproduce the nonlinearities in the data? Model Assessment (see Fernandez-Villaverde, Rubio-Ramirez, and Schorfheide (2016): “Solution and Estimation of DSGE Models,” Handbook of Macroeconomics , Vol 2., Elsevier) : Relative fit: comparison with other models. Absolute fit: can the DSGE reproduce salient features of the data? Violation of over-identifying restrictions? Linear VARs have been useful benchmark / reference model for the evaluation of linearized DSGE models: testing over-identifying restrictions; model odds comparison of model-implied autocovariances, spectra, impulse responses (...) No obvious benchmark for evaluation of nonlinear models: generalized autoregressive models? bilinear models? ARCH-M? LARCH? regime-switching models? time-varying coefficient models? threshold autoregressions? smooth transition autoregressions? F. Schorfheide DSGE Model Nonlinearities

Assessing Nonlinearities Can Be Delicate... An Example Many papers argue that it is important to incorporate stochastic volatility in DSGE models. Important nonlinearity or device to capture rare events like Great Moderation, Great Recession? Diebold, Schorfheide, Shin (2016): Evaluate forecast performance of DSGE model with stochastic volatility versus structural break in volatility Posterior Mean Structural Shock Volatilities / Final Data Vintage F. Schorfheide DSGE Model Nonlinearities

Assessing Nonlinearities Can Be Delicate... An Example: DSS (2016) Coverage Rates of 70% Interval Forecasts, h = 1 , ..., 8 Log Predictive Scores, h = 4 F. Schorfheide DSGE Model Nonlinearities