advanced dynamic models martin ellison university of warwick and cepr bank of england, december 2005
TRANSCRIPT
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