Advanced dynamic models

Martin Ellison

University of Warwick and CEPR

Bank of England, December 2005

More complex models

Impulses

Propagation

Fluctuations

Frisch-Slutsky paradigm

Shocks may be correlated

Impulses

Can add extra shocks to the model

ttt

ttttt

ttttttt

vi

uxE

gEixEx

ˆˆ

ˆˆˆ

)ˆˆ(ˆˆ

1

11

1

gt

ut

vt

t

t

t

t

t

t

g

u

v

g

u

v

1

1

1

333231

232221

131211

333231

232221

131211

1

1

1

Propagation

Add lags to match dynamics of data (Del Negro-Schorfeide, Smets-Wouters)

ttxtt vxi ˆˆˆ Taylor rule

tttp

tp

pt

ttttttt

xE

EixEh

xh

hx

ˆˆ1

ˆ1

ˆ

)ˆˆ(ˆ1

1ˆ

1ˆ

11

11

11

29.01

35.01

p

p

h

h

Solution of complex models

11

10101

101

10110

tttt

tttt

tttt

BvAXXE

vBAXAAXE

vBXAXEA

A B

Blanchard-Kahn technique relies on invertibility of A0 in state-space form.

QZ decomposition

For models where A0 is not invertible

10110 tttt vBXAXEA

uppertriangular

QZ decomposition: s.t. ,,, ZQ

1

0

''

''

AZQ

AZQ

Recursive equations

101

22

1211

1

1

22

1211 )'(~

~

0~

~

0

tt

t

tt

t vBQy

w

yE

w

111211112111~~~~

tttttt vRywyEw

1222122~~

tttt vRyyE

stable

unstable

Recursive structure means unstable equation can be solved first

Solution strategy

Solve unstable transformed equation ty

~

Translate back into original problem

tw~

t

t

y

w

Substitute into stable transformed equation

Simulation possibilities

Stylised facts

Impulse response functions

Forecast error variance decomposition

Optimised Taylor rule

What are best values for parameters in Taylor rule ?ttxtt vxi ˆˆˆ

Introduce an (ad hoc) objective function for policy

)ˆˆˆ(min 222

0titxt

i

i ix

Brute force approach

Try all possible combinations of Taylor rule parameters

Check whether Blanchard-Kahn conditions are satisfied for each combination

For each combination satisfying B-K condition, simulate and calculate variances

Brute force method

Calculate simulated loss for each combination

Best (optimal) coefficients are those satisfying B-K conditions and leading to smallest simulated loss

Grid search

x

0 1 2

2

1

For each point check B-K conditions

Find lowest loss amongst points satisfying B-K

condition

Next steps

Ex 14: Analysis of model with 3 shocks

Ex 15: Analysis of model with lags

Ex 16: Optimisation of Taylor rule coefficients