"Assessing DSGE Model Nonlinearities " Andrea Prestipino - PowerPoint PPT Presentation

"Assessing DSGE Model Nonlinearities " Andrea Prestipino NYU April 2014

Motivation � Identify nonlinearities and evaluate nonlinear DSGE – State-space model S t = � ( S t � 1 ; w t � 1 ; � ) Y e t = M ( S t ; v t ; � ) – Statistical model � � Y s Y s t = f t � 1 ; u t � First order approximation – State-space model t = � 1 ( � ) S t � 1 + H ( � ) w t S 1 Y e t = A ( � ) + B ( � ) S t + v t – Statistical model Y s t = CY s t � 1 + u t

Motivation � What reference model for second order approxia- tion? – QAR � How to use this model to evaluate DSGE? – Posterior predictive checks

Quadratic Autoregressive Model (QAR) � Let � � y � y � t = f t � 1 ; !u t where u t � N (0 ; 1) � Second order approximation t + !y (1) + ! 2 y (2) y t = y 0 t t � So that � � y � y � t � � y = f y t � 1 � � y + f u !u t � � 2 + f y;u � � +1 y � y � 2 f y;y t � 1 � � y t � 1 � � y !u t +1 2 f u;u ( !u t ) 2 + higher order terms � Substitute y t and match coe¢cients

QAR � The resulting approximation is t � 1 + (1 + �s t � 1 ) �u t + 1 y ) + � 2 s 2 2 � 3 ! 2 u 2 y t = � 0 + � 1 ( y t � 1 � � t s t = � 1 s t � 1 + �u t � Unique steady state and non-explosive if j � 1 j < 1 � Not true for "standard" approximation y ) 2 y t � � ^ y = � 1 (^ y t � � y ) + � 2 (^ y t � �

Why QAR? � State dependent IRFs � y t + h j u t = 1 � � E t � y t + h � IRF t ( h ) = E t IRF t (0) = � (1 + �s t � 1 ) � � q 1 � � 2 IRF t (1) = � � 1 (1 + �s t � 1 ) + 2 � 1 � 2 1 s t � 1 � Conditional Heteroskedasticity V t � 1 [ y t ] = (1 + �s t � 1 ) 2 � 2

How to use QAR? � Estimate QAR � Estimate 2nd order approximation to DSGE � Use posterior on DSGE parameters to get a posterior predictive distribution on QAR estimates � Check how far the actual QAR estimate lies in the tail of this distribution

QAR: Estimation � Computing � � p ( Y 0: T ; �; s 0 ) = p Y 1: T j y 0 ; s 0 ; � p ( y 0 ; s 0 j � ) p ( � ) � Factorize likelihood T � � � � Y p Y 1: T j y 0 ; s 0 ; � = p y t j y 0: t � 1 ; s 0 ; � t =1 � Computed recursively using p ( y t j y t � 1 ; s t � 1 ) � N s t = g ( y t ; y t � 1 ; s t � 1 )

QAR Estimation � Initialization " # " #! � y � yy � ys p ( y 0 ; s 0 j � ) = N ; � sy � ss � d � Substitute in steady state at t = � T � � Find � � � � � � � � � � � � ; cov ( s 2 s 2 E s j ; E y j ; V s j ; V y j ; cov s j ; y j j ; y j ) ; V j as a function of their lagged values using the QAR law of motions

QAR Estimation � Priors: GDP Growht Wage Growth In‡ation Fed Funds Rate � 0 N ( : 48 ; 2) N (1 : 18 ; 2) N (2 : 38 ; 2) N (2 : 50 ; 2) N T ( : 36 ; : 5) N T ( � : 02 ; : 5) N T (0 : 00 ; : 5) N T (0 : 66 ; : 5) � 1 IG (1 : 42 ; 4) IG ( : 82 ; 4) IG (1 : 87 ; 4) IG ( : 58 ; 4) � � 2 N (0 ; 0 : 1) N (0 ; 0 : 1) N (0 ; 0 : 1) N (0 ; 0 : 1) N (0 ; 0 : 1) N (0 ; 0 : 1) N (0 ; 0 : 1) N (0 ; 0 : 1) � � Pre-sample information to parametrize priors

QAR Estimation � RWM Algorithm: � Use prior to get a Cov matrix for parameters � � Produce 100k draws using proposal density ^ � = � t + U t U t � N (0 ; �) � Use last 50k to compute � 0 � Produce 60k draws using new proposal density � 0 ; � 0 � � = � t + U 0 U 0 ^ t � N t

DSGE � New Keynesian DSGE with asymmetric price and wage adjust- ment costs � 4 exogenous shocks: tfp; markup; government; monetary pol- icy. � Approximate solution using "standard" method � Bayesian estimation using RWM and particle …lter

Particle Filter � The goal is to approximate Z � � � � � � � � � Y t � 1 ; � � Y t � 1 ; � p y t = p ( y t j s t ; � ) p s t ds t n o N n o N s i s i – Start from p ( s 0 j � ) to draw i =1 ; Assume we have 0 t � 1 i =1 which approximate p ( s t � 1 j Y t � 1 ; � ) – p ( s t j Y t � 1 ; � ) is approximated by Z � � � � � Y t � 1 ; � p ( s t j Y t � 1 ; � ) = p ( s t j s t � 1 ; � ) p s t � 1 ds t � � � X 1 � � s i � p s t t � 1 ; � N

Particle Filter � n o N � � � s i � s i – Drawing i =1 from p approxiamates p ( s t j Y t � 1 ; � ) ~ � s t t � 1 ; � t hence Z � � � � � � � � � X ds t � 1 � � � � Y t � 1 ; � � Y t � 1 ; � s i = p ( y t j s t ; � ) p � ~ p y t s t p y t t ; � N n o N s i – Finally get an approximation i =1 of p ( s t j Y t ; � ) by draw- t n o N s i ing with replacement from i =1 with pmf given by ~ t � � � � s i p y t � ~ t ; � � i � � � t = P p � s i y t � ~ t ; �

Posterior predictive checks � Draw � i from posterior of the DSGE parameters n o Y i � Simulate Data from the DSGE and obtain median � T � : T estimate of QAR parameters S i Examine how far the median estimate from actual US data � lie in the tail of the empirical distribution of S i

Estimation of QAR(1,1) Model on U.S. Data – Φ 2 Wage Growth GDP Growth 0.2 0.2 0.1 0.1 φ 2 φ 2 0 0 −0.1 −0.1 −0.2 −0.2 60−83 60−07 60−12 84−07 84−12 60−83 60−07 60−12 84−07 84−12 In fl ation Federal Funds Rate 0.2 0.2 0.1 0 φ 2 φ 2 0 −0.2 −0.1 −0.2 −0.4 60−83 60−07 60−12 84−07 84−12 60−83 60−07 60−12 84−07 84−12 φ 0 + φ 1 ( y t − 1 − φ 0 ) + φ 2 s 2 = t − 1 + (1 + γ s t − 1 ) σ u t y t i.i.d. = φ 1 s t − 1 + σ u t ∼ N (0 , 1) s t u t

Estimation of QAR(1,1) Model on U.S. Data – γ Wage Growth GDP Growth 0.3 0.3 0.2 0.2 0.1 γ γ 0 0.1 −0.1 0 −0.2 −0.1 60−83 60−07 60−12 84−07 84−12 60−83 60−07 60−12 84−07 84−12 In fl ation Federal Funds Rate 0.4 0.3 0.4 0.2 γ γ 0.2 0.1 0 0 −0.1 60−83 60−07 60−12 84−07 84−12 60−83 60−07 60−12 84−07 84−12 φ 0 + φ 1 ( y t − 1 − φ 0 ) + φ 2 s 2 = t − 1 + (1 + γ s t − 1 ) σ u t y t i.i.d. = φ 1 s t − 1 + σ u t ∼ N (0 , 1) s t u t

Log Marginal Data Density Differentials: QAR(1,1) versus AR(1) Wage Growth GDP Growth 20 20 15 15 10 10 5 5 0 0 −5 −5 60−83 60−07 60−12 84−07 84−12 60−83 60−07 60−12 84−07 84−12 In fl ation Federal Funds Rate 20 80 15 60 10 40 5 20 0 0 −5 −20 60−83 60−07 60−12 84−07 84−12 60−83 60−07 60−12 84−07 84−12

Posterior Predictive Checks: 1960-2007 Sample φ 0 φ 1 10 1 0.5 5 0 0 GDP Wage Infl FFR GDP Wage Infl FFR φ 2 γ 0.2 0.2 0 0 −0.2 −0.2 GDP Wage Infl FFR GDP Wage Infl FFR σ 3 2 1 0 GDP Wage Infl FFR ◮ QAR estimates from actual and model-generated data are similar. ◮ Only interest rates exhibit noticeable differences. ◮ Except for wage and inflation ˆ γ , nonlinearities are generally weak.

Posterior Predictive Checks: 1984-2007 Sample φ 0 φ 1 15 1 10 0.5 5 0 0 GDP Wage Infl FFR GDP Wage Infl FFR φ 2 γ 0.2 0.2 0 0 −0.2 −0.2 GDP Wage Infl FFR GDP Wage Infl FFR σ 2 1 0 GDP Wage Infl FFR ◮ Model does not generate nonlinearity (ˆ φ 2 ) in GDP dynamics.

Effect of Adjustment Costs on Nonlinearities: 1960-2007 Sample φ 2 γ 0.15 0.3 0.25 0.1 0.2 0.05 0.15 0 0.1 −0.05 0.05 −0.1 0 −0.05 Wage Infl Wage Infl No asymmetric costs is ψ p = ψ w = 0 (light blue); high asymmetric costs is ψ p = ψ w = 300 (dark blue). Large dots correspond to posterior median estimates based on U.S. data.