Learning about the parameters and the dynamics of
DSGE models: identification and estimation
Fabio Canova
IGIER, Universitat Pompeu Fabra and CEPR
Luca Sala
IGIER, Universitá Bocconi
Oslo, May 2005
1
DSGE models have quickly become the benchmark for:
• Understanding business cycles
• Policy analysis
Everybody wants one!!
2
How are DSGE typically estimated?
Full Information
1. Maximum Likelihood or ”Bayesian Maximum Likelihood”:
”weighted average” of the prior and the likelihood
Limited information
1. GMM
2. Indirect Inference approach:
”minimum distance” estimation → matching impulse responses
3
Bayesian maximum likelihood and indirect inference
Get the unique stationary rational expectations solution
→ restricted VAR (state-space)
GMM
Work with the first order conditions (expectations still there)
4
Stationary RE solution of the model:
Yt = A(θ0)Yt−1 + B(θ0)ut
Maximum Likelihood
θ̂ = argmax L(Y, θ)
θ
Bayesian: θ̂ = argmax L(Y, θ)πθ
θ
Matching impulse responses (conditional on some shock j):
j
model impulse responses: YtM (θ) = C(θ)(L)ut
j
data impulse responses: Yt = Ŵ (L)ut
θ̂ = argmax − D(θ)
θ
where: D(θ) = ||Yt − YtM (θ)||Ω̂
5
Under what conditions can we recover structural parameters?
Identifiability
Mapping from the likelihood/distance to the parameters
• Likelihood: L(y, θ1) 6= L(y, θ2) for any y
• Distance:
- −D(θ) has a unique maximum 0 at θ = θ0
- Hessian is positive definite and has full rank
- D(θ) is globally concave
Additional issue
Curvature of D(θ) is ”sufficient” (”Weak identification” in GMM)
6
Different objective functions may have different ”identification
power” (similar to the choice of instruments in GMM)
In DSGE, the shape of the functions L(.) and D(.) is non-linear
and too complicated to be worked out analitically
↓
Identifiability is far from clear
7
Results
- Many (model-loss function) combinations display some identification problems
- Warning on the interpretation of the estimated parameters
- Warning on the structural interpretation of the model
- Proposals for applied researchers
8
Example 1
yt = a1Etyt+1 + a2(it − Etπt+1) + v1t
πt = a3Etπt+1 + a4yt + v2t
it = a5Etπt+1 + v3t
(1)
(2)
(3)
Solution (log-lin from SS):





ŷt
1 0 a2
v1t





 π̂t  =  a4 1 a2a4   v2t 
ît
0 0
1
v3t
• Some parameters disappear from the solution
• Different shocks identify different parameters
9
Example 2: a Neo-Keynesian model
yt =
1
1
h
yt−1 +
Et yt+1 + (it − Etπt+1 ) + v1t
1+h
1+h
ϕ
πt =
ω
β
(ϕ + ϑ)(1 − ζβ)(1 − ζ)
πt−1 +
πt+1 +
yt + v2t
1 + ωβ
1 + ωβ
(1 + ωβ)ζ
it = φr it−1 + (1 − φr )(φπ πt−1 + φy yt−1 ) + v3t
h: degree of habit persistence (.85)
ϑ: inverse elasticity of labor supply (3)
ϕ: relative risk aversion (2)
β: discount factor (.985)
ω: degree of price indexation (.25)
ζ: degree of price stickiness (.68)
φr , φπ , φy : policy parameters (.2, 1.55, 1.1)
v1t : AR(ρ1 ) (.65)
v2t : AR(ρ2 ) (.65)
v3t : i.i.d.
10
−4
β = 0.985
φ=2
4
4
4
2
2
2
2
0
0
0
0.98
ϑ=3
ζ = 0.68
φr = 0.2
0.99
1
2
0
10
8
6
4
2
3
0
5
10
15
20
1
0.99
5
2
0
3
0.4
0.6
0.8
1
φπ = 1.55
0.1
0.2
5
0
10
0.4
0
0.3
15
20
0
0.99
2
0.6
0.8
1
0.1
0.2
1
0.5
0
5
3
10
15
20
1.5
0
2
1
0
0.4
0.6
0.8
1
0.99
2
3
5
10
15
20
0
1
0.4
0.6
0.8
1
0.5
0.1
0.2
0
0.3
0.1
0.2
0.3
0.02
10
0.01
1
0.985
200
0.1
0.05
0
0.3
0.98
20
10
0
10
8
6
4
2
0
10
2
0.05
0.985
1
0.4
200
4
0
0.98
0.2
0
5
0
0.5
0.985
10
3
2
1
10
0.98
0.2
1
1.5
0 x 10−3
1
2
0.2
1.5
0
2
5
1
1.5
2
0.2
0.1
0
1
ρ1 = 0.65
φy = 1.1
0.985
5
0
0.9
1
1.1
1.2
1.3
0
1
0
1
0.6
0.65
0.7
−1
1
0
0.5
−1
0
0.6
0.65
0.7
0.05
0
0.02
0.01
0
0.9
1
1.1
1.2
1.3
0
0.5
ρ2 = 0.65
ω = 0.7
x 10
x 10
x 10
4
0
h = 0.85
−3
−5
−3
x 10
0.5
0
1
0.9
1
1.1
1.2
1.3
0
0.6
0.65
0.7
−1
1
0.65
0.7
0.9
1
1.1
1.2
1.3
0.5
0.6
0.65
0.7
0
0.6
0
1
0
1
0.6
0.65
0.7
0.6
0.65
0.7
0.5
−1 x 10−3
0.6
0.65
0.7
0
0.5
2
0.5
0.6
0.7
0.8
0.9
0
0.04
0.5
0.6
0.7
0.8
0.9
0.8
0.9
IS shock
1
0
0.6
0.7
0.8
0.9
0.7
0.8
0.9
Cost push shock
1
0
0
0.5
0.6
0.7
0.8
0.9
0.05
2
0.02
0.7
0 x 10−3
0.5
0.7
0.8
0.9
1
Monetary policy shock
0
0.7
0.8
0.9
All shocks
11
1
Minimizing the distance
Monetary shock
Monetary shock
12
−0.4
10
8
6
2
labor supply elasticity
1
01
0.
2
0.75
4
4
00
−0.3
01
0
0.
6
0.
−0.2
8
1
−0.1
0.0
labor supply elasticity
10
0.1
0.7
0.65
0.66
price stickiness
0.68
Monetary shock
0.7
0.72
0.74
price stickiness
0.76
Monetary shock
12
10
8
6
labor supply elasticity
4
2
0.24
0.26
0.28
0.3
price indexation
4
0.1
−0.2
0.01
−0.15
6
0.001
−0.1
8
0.001
−0.05
0.01
labor supply elasticity
10
2
0.24
0.26
0.28
price indexation
0.3
12
0.78
Along the ridge
−7
−5
Monetary shock
x 10
Monetary shock
x 10
8
9
7
8
7
6
6
5
5
4
4
3
3
2
2
1
0
1
5
10
15
20
25
0
20
40
60
80
100
120
13
Minimizing the distance
Cost push shock
Cost push shock
12
0.90.7
10
8
6
2
labor supply elasticity
0.7
0.9
4
2
0.75
4
1
0. .01 1
0 .0
0
6
3
−20
0
1
−15
8
0.
−10
01
.0
10
0
0
.
0.
−5
0.5
0.3
labor supply elasticity
10
5
0.
0.7
0.65
0.66
price stickiness
0.68
Cost push shock
0.7
0.72
0.74
price stickiness
0.76
Cost push shock
12
−4
10
8
6
labor supply elasticity
4
2
0.24
0.26
0.28
0.3
price indexation
0.7
0.5
0.3
0.7
0.5
4
2
0.24
0.9
−3
6
0.1
−2
8
0.3
0.1
0.01
0.
001
0.01
−1
0.9
labor supply elasticity
10
0.26
0.28
price indexation
0.3
14
0.78
Histograms − Monetary shock
beta = 0.985
risk aversion = 2
10
inverse labor elasticity = 3
60
1.5
40
1
20
0.5
5
0
0
0.5
1
price stickiness = 0.68
1.5
0
1.8
1.9
2
2.1
phi r = 0.2
2.2
2.3
0
20
10
1
10
5
0.5
0
0.6
0.7
0.8
phi x = 1.1
0.9
1
1
0
−0.5
0
0.5
rho1 = 0.65
1
0
0
−2
6
6
4
4
2
2
1
0
2
phi π = 1.55
2
4
rho2 = 0.65
3
4
6
8
0.8
0.9
0.5
0
−5
0
5
10
omega = 0.25
15
30
0
0.4
0.5
0.6
0.7
habit = 0.85
0.8
0.8
1.1
0
0.4
0.5
0.6
0.7
20
20
10
10
0
−0.2
0
0.2
0.4
0.6
0
0.6
0.7
0.9
1
15
IRFs − Monetary shock
Gap
0.2
0
−0.2
−0.4
−0.6
0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
π
0.1
0
−0.1
−0.2
−0.3
0
2
4
6
8
10
interest rate
1
0.5
0
−0.5
−1
0
2
4
6
8
10
20
16
Histograms − Cost push shock
beta = 0.985
risk aversion = 2
inverse labor elasticity = 3
1
100
10
0.5
50
5
0
0.8
0.85 0.9
0.95
price stickiness = 0.68
1
15
0
1.94
1.96
1.98
phi r = 0.2
0
2
100
40
50
20
5
10
15
phi π = 1.55
20
1.7
1.8
rho2 = 0.65
2
10
5
0
0.6
0.65
0.7
0.75
phi x = 1.1
0.8
40
20
0
1.2
1.4
omega = 0.25
1.6
0
0.2
0.25
0.3
rho1 = 0.65
0.35
0
1.5
4
2000
2
1000
0
0.5
0
0.6
0.7
habit = 0.85
1.6
0.65
0.655
1.9
0.66
200
15
10
100
5
0
0.05
0.1
0.15
0.2
0.25
0
0.85
0.9
0.95
1
17
IRFs − Cost push shock
interest rate
0
−0.2
−0.4
−0.6
−0.8
−1
0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
π
1.5
1
0.5
0
0
2
4
6
8
10
Gap
2
1.5
1
0.5
0
0
2
4
6
8
10
20
18
Example 3: a state-of-the-art DSGE model
0
0
=
=
0
=
0
0
0
=
=
=
0
=
0
=
0
=
0
=
0
=
−kt+1 + (1 − δ)kt + δxt
−ut + ψrt
ηδ
ηδ
xt + (1 − )ct − ηkt − (1 − η)Nt − ηut − ezt
r̄
r̄
−Rt + φr Rt−1 + (1 − φr )(φπ πt + φy yt ) + ert
−yt + ηkt + (1 − η)Nt + ηut + ezt
−Nt + kt − wt + (1 + ψ)rt
h
1−h
h
ct+1 − ct +
ct−1 −
(Rt − πt+1 )]
Et [
1+h
1+h
(1 + h)ϕ
1
χ−1
β
1
β
xt+1 − xt +
xt−1 +
qt +
ext+1 −
ext]
Et [
1+β
1+β
1+β
1+β
1+β
Et [πt+1 − Rt − qt + β(1 − δ)qt+1 + βr̄rt+1 ]
β
γp
Et [
πt+1 − πt +
πt−1 + Tp (ηrt + (1 − η)wt − ezt + ept )]
1 + βγp
1 + βγp
1
β
β
wt+1 − wt +
wt−1 +
πt+1 −
Et [
1 + βγp
1+β
1+β
1 + βγw
ϕ
γw
(wt − σNt −
πt +
(ct − hct−1 ) − ewt )]
1+β
1 + βγw t−1
1−h
19
δ
ψ
η
ϕ
β
ζp
γp
φy
φr
Tp ≡
Tw ≡
depreciation rate (.0182)
parameter (.564)
share of capital (.209)
risk aversion coefficient (3.014)
discount factor (.991)
price stickiness (.887)
price indexation (.862)
response to y (.234)
int. rate smoothing (.779)
λw
π̄
h
σl
χ−1
ζw
γw
φπ
wage markup (1.2)
steady state π (1.016)
habit persistence (.448)
inverse elasticity of labor supply (2.145)
investment’s elasticity to Tobin’s q (.15)
wage stickiness (.62)
wage indexation (.221)
response to π (1.454)
(1−βζp )(1−ζp )
(1+βγp )ζp
(1−βζw )(1−ζw )
(1+β)(1+(1+λw )σl λ−1
w )ζw
20
−7
−7
x 10
delta = 0.018 x 10eta = 0.209
2
1.5
1.5
1
0.5
0
−7
−8
4
−5
−5
x 10sigc = 3.014
habit = 0.448 x 10 chi = 6.3
x 10
beta = 0.991 x 10
4
3
3
1.5
1
2
2
1
0.5
1
1
0.5
1
0.5
0
0
0
0
0
0.015 0.02
0.2 0.25 0.988
0.99
0.992
0.994
0.4 0.45 0.5
5 6 7
2.5 3 3.5
−7
−3
−6
−5
−8
−7
zeta
=
0.62
zeta
=
0.887
gamma
= 0.221
gammap = 0.862
x 10sigl = 2.145 x 10psi = 0.564 x 10 p
x 10
x 10
x 10
w
w
3
3
1.5
4
2
2
1
1.5
3
1
2
0.5
1
2
1
1
0.5
0
0
0
0
0
0
2
3
0.5
0.6
0.85 0.9
0.8
0.9
0.6
0.7 0.15 0.2 0.25
−6
−7
−7
−4
lambda
= 0.234
x 10
x 10phiy = 0.275 x 10phipi = 1.454 x 10phir = 0.779 rhoz = 0.997
w
1
1
2
4
3
0.5
1.5
3
2
0
0.5
1
2
1
−0.5
0.5
1
0
0.2 0.25 0.3
0
0.2
0.3
0
1.45
1.5
0
0.75
0.8
−1
0.980.99 1
Monetary shocks
21
−6
−5
6
−6
−8
x 10
delta = 0.018 x 10eta = 0.209
8
4
4
−5
x 10sigc = 3.014
−5
x 10beta = 0.991 x 10
habit = 0.448 x 10chi = 6.3
8
3
1
6
6
2
4
2
2
4
0.5
1
2
0
0.015 0.02
−5
x 10sigl = 2.145
0
0
0
0
0
0.2 0.25 0.988
0.99
0.992
0.994
0.4 0.45 0.5
5 6 7
2.5 3 3.5
−6
−4
−3
−7
−7
zeta
=
0.62
zeta
=
0.887
gamma
=
0.862
gamma
= 0.221
psi = 0.564
x 10
x 10
x 10 p
x 10
x 10
w
p
w
2
8
2
4
3
1
1.5
6
1
4
0.5
2
1
2
2
2
0.5
1
0
0
0
0
0
0
2
3
0.5
0.6
0.85 0.9
0.8
0.9
0.6
0.7 0.15 0.2 0.25
−7
−4
−5
−4
−6
lambdaw = 0.234
x 10
x 10phiy = 0.275 x 10phipi = 1.454 x 10phir = 0.779 x 10rhoz = 0.997
1
12
1.5
1.5
10
4
1
2
0.5
0
0.5
0.2 0.25 0.3
0
8
3
1
0.5
6
4
1
0.2
0.3
0
1.45
Both shocks
2
1.5
0
0.75
0.8
0.980.99 1
22
Table 3: Estimates of various models, matching monetary policy shocks
Baseline
Estimates
Case 1
Estimates
Case 2
Estimates
Case 3
Estimates
Case 4
Estimates
Case 5
Estimates
Case 6
Estimates
ζp
0.887
0.833
0.000
0.397
0.000
0.395
0.000
0.442
0.887
0.901
0.887
0.928
0.887
0.895
γp
0.862
0.549
0.862
0.010
0.000
0.010
0.862
0.001
0.000
0.280
0.000
0.302
0.000
0.321
ζw
0.620
0.604
0.620
0.654
0.620
0.653
0.620
0.673
0.000
0.011
0.620
0.586
0.000
0.071
γw
0.221
0.379
0.221
0.419
0.221
0.411
0.000
0.407
0.221
0.010
0.801
0.155
0.221
0.010
Obj.Fun.
1.46 e-06
4.51 e-07
4.50 e-07
6.54 e-07
1.94 e-07
3.50 e-07
7.35 e-06
23
Inflation
Interest rate
0.1
0.5
0
0
−0.1
0
5
Real10
wage
15
20
0
−0.5
0
5
10
Investment
15
20
0
5
Hours10
worked
15
20
0
5 Capacity10
utilisation15
20
0
5
10
15
quarters after shock
20
0
−0.5
−0.5
−1
−1.5
0
5
10
Consumption
15
20
−1
0
0.2
−0.1
0
−0.2
0
5
10
output
15
20
−0.2
0
0.5
−0.2
0
−0.4
0
5
10
15
quarters after shock
20
−0.5
No wage stickyness, no price indexation
24