"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 St = (St1; wt1; ) Y e

t = M (St; vt; )

– Statistical model Y s

t = f

• Y s

t1; ut

• First order approximation

– State-space model S1

t = 1 () St1 + H()wt

Y e

t = A () + B () St + vt

– Statistical model Y s

t = CY s t1 + ut

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

t = f

• y

t1; !ut

• where ut N (0; 1)

Second order approximation yt = y0

t + !y(1) t

+ !2y(2)

t

So that y

t

y = fy

• y

t1

y

• + fu!ut

+1 2fy;y

• y

t1

y

2 + fy;u

• y

t1

y

• !ut

+1 2fu;u (!ut)2 + higher order terms Substitute yt and match coe¢cients

QAR

The resulting approximation is yt = 0 + 1 (yt1 y) + 2s2

t1 + (1 + st1) ut + 1

23!2u2

t

st = 1st1 + ut Unique steady state and non-explosive if j1j < 1 Not true for "standard" approximation ^ yt y = 1 (^ yt y) + 2 (^ yt y)2

Why QAR?

State dependent IRFs IRFt (h) = Et

yt+hjut = 1 Et yt+h

• IRFt (0) = (1 + st1)

IRFt (1) =

• 1 (1 + st1) + 212

q

1 2

1st1

• Conditional Heteroskedasticity

Vt1 [yt] = (1 + st1)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 (Y0:T; ; s0) = p

• Y1:Tjy0;s0;
• p (y0; s0j) p ()

Factorize likelihood p

• Y1:Tjy0;s0;
• =

T

Y

t=1

p

• ytjy0:t1;s0;
• Computed recursively using

p (ytjyt1; st1) N st = g (yt; yt1; st1)

QAR Estimation

Initialization p (y0; s0j) = N

"

y d

#

;

"

yy ys sy ss

#!

Substitute in steady state at t = T Find E

• sj
• ; E
• yj
• ; V
• sj
• ; V
• yj
• ; cov
• sj; yj
• ; cov(s2

j; yj); V

• s2

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 N (:48; 2) N (1:18; 2) N (2:38; 2) N (2:50; 2) 1 NT (:36; :5) NT (:02; :5) NT (0:00; :5) NT (0:66; :5)

• 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 + Ut Ut N (0; ) Use last 50k to compute 0 Produce 60k draws using new proposal density ^ = t + U0

t

U0

t N

• 0; 0

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 p

• yt
• Y t1;
• =

Z

p (yt jst; ) p

• st
• Y t1;
• dst

– Start from p (s0 j) to draw

n

si

• N

i=1 ; Assume we have

n

si

t1

• N

i=1

which approximate p (st1 jYt1; )

– p (st jYt1; ) is approximated by p (st jYt1; ) =

Z

p (st jst1; ) p

• st1
• Y t1;
• dst
• 1

N

X

p

• st
• si

t1;

Particle Filter

• – Drawing

n

~ si

t

• N

i=1 from p

• st
• si

t1;

• approxiamates p (st jYt1; )

hence

p

• yt
• Y t1;
• =

Z

p (yt jst; ) p

• st
• Y t1;
• dst 1

N

X

p

• yt
• ~

si

t;

• – Finally get an approximation

n

si

t

• N

i=1 of p (st jYt; ) by draw-

ing with replacement from

n

~ si

t

• N

i=1 with pmf given by

i

t =

p

• yt
• ~

si

t;

• P p
• yt
• ~

si

t;

Posterior predictive checks

Draw i from posterior of the DSGE parameters Simulate Data from the DSGE

n

Y i

T :T

• and obtain median

estimate of QAR parameters Si

• Examine how far the median estimate from actual US data

lie in the tail of the empirical distribution of Si

Estimation of QAR(1,1) Model on U.S. Data – Φ2

60−83 60−07 60−12 84−07 84−12 −0.2 −0.1 0.1 0.2

GDP Growth

φ2

60−83 60−07 60−12 84−07 84−12 −0.2 −0.1 0.1 0.2

Wage Growth

φ2

60−83 60−07 60−12 84−07 84−12 −0.2 −0.1 0.1 0.2

Inflation

φ2

60−83 60−07 60−12 84−07 84−12 −0.4 −0.2 0.2

Federal Funds Rate

φ2

yt = φ0 + φ1(yt−1 − φ0) + φ2s2

t−1 + (1 + γst−1)σut

st = φ1st−1 + σut ut

i.i.d.

∼ N(0, 1)

Estimation of QAR(1,1) Model on U.S. Data – γ

60−83 60−07 60−12 84−07 84−12 −0.2 −0.1 0.1 0.2 0.3

GDP Growth

γ

60−83 60−07 60−12 84−07 84−12 −0.1 0.1 0.2 0.3

Wage Growth

γ

60−83 60−07 60−12 84−07 84−12 −0.1 0.1 0.2 0.3 0.4

Inflation

γ

60−83 60−07 60−12 84−07 84−12 0.2 0.4

Federal Funds Rate

γ

yt = φ0 + φ1(yt−1 − φ0) + φ2s2

t−1 + (1 + γst−1)σut

st = φ1st−1 + σut ut

i.i.d.

∼ N(0, 1)

Log Marginal Data Density Differentials: QAR(1,1) versus AR(1)

60−83 60−07 60−12 84−07 84−12 −5 5 10 15 20

GDP Growth

60−83 60−07 60−12 84−07 84−12 −5 5 10 15 20

Wage Growth

60−83 60−07 60−12 84−07 84−12 −5 5 10 15 20

Inflation

60−83 60−07 60−12 84−07 84−12 −20 20 40 60 80

Federal Funds Rate

Posterior Predictive Checks: 1960-2007 Sample

GDP Wage Infl FFR 5 10

φ0

GDP Wage Infl FFR 0.5 1

φ1

GDP Wage Infl FFR −0.2 0.2

φ2

GDP Wage Infl FFR −0.2 0.2

γ

GDP Wage Infl FFR 1 2 3

σ

◮ 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

GDP Wage Infl FFR 5 10 15

φ0

GDP Wage Infl FFR 0.5 1

φ1

GDP Wage Infl FFR −0.2 0.2

φ2

GDP Wage Infl FFR −0.2 0.2

γ

GDP Wage Infl FFR 1 2

σ

◮ Model does not generate nonlinearity (ˆ

φ2) in GDP dynamics.

Effect of Adjustment Costs on Nonlinearities: 1960-2007 Sample

Wage Infl −0.1 −0.05 0.05 0.1 0.15

φ2

Wage Infl −0.05 0.05 0.1 0.15 0.2 0.25 0.3

γ

No asymmetric costs is ψp = ψw = 0 (light blue); high asymmetric costs is ψp = ψw = 300 (dark blue). Large dots correspond to posterior median estimates based on U.S. data.