Estimation of moment-based models with latent
variables
work in progress
Ra¤aella Giacomini and Giuseppe Ragusa
UCL/Cemmap and UCI/Luiss
UPenn, 5/4/2010
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
UPenn, 5/4/2010
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Dynamic latent variables in macroeconomic models
E.g., time-varying parameters, structural shocks, stochastic
volatility etc.
Typical parametric setting: X T = (X1 , ..., XT ) = (Y T , Z T ),
Y T observable, Z T latent
Joint density p (X , θ 0 ) = p (Y T jZ T , θ 0 )p (Z T , θ 0 ) =)
estimation of θ 0 based on integrated likelihood
b
θ = arg max
θ
Z
p (Y T jZ T , θ )p (Z T , θ )dZ T
Integrated likelihood computed by state-space methods
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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Existing state-space methods
State equation ! p (Z T , θ )
known in closed form
Observation equation ! p (Y T jZ T , θ ) "…ltering" density
known in closed form (e.g. Kalman …lter) or easy to simulate
Integral can be computed by MCMC methods
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Moments and latent variables
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State-space methods for limited information
models?
We consider the following scenario:
p (Z T , θ ) known ! state equation same as before
p (Y T jZ T , θ ) unknown. Only information about θ is in the form
of (non-linear) moment conditions
Et
1
[g (Yt , Zt , θ )] = 0
! substitute observation equation with moment conditions
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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Applications. GMM with time-varying parameters
Example #1. Time-varying "structural" parameters:
E [g (Yt , βt )] = 0
βt = Φβt
1
+ εt , εt
iidN (0, Σ)
E [ ] de…ned with respect to joint distribution of Yt and βt
Want to estimate θ = (Φ, Σ) and sequence of "smoothed" βt
Application: Cogley and Sbordone’s (2005) analysis of stability
of structural parameters in a Calvo model of in‡ation
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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Applications. "Robust" stochastic volatility
estimation
Example #2.
Yt = σt εt
log σ2t = α + β log σ2t
1
+ vt , vt
iidN (0, 1)
Existing estimation methods require distributional assumption on
εt (typically N (0, 1))
Problem: does not capture "fat tails" of …nancial data =)
include jumps or use fat-tailed distribution for εt (not as
straightforward as in GARCH case)
Our method is robust to misspeci…cation in distribution of εt
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
UPenn, 5/4/2010
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Applications. Nonlinear DSGE models
Example #3. Prototypical DSGE model. Optimality conditions:
Et
1
[m(Yt , St , Zt , β)] = 0
St = f (St
Zt = ΦZt
1 , Yt , Zt , β )
1
+ εt , εt
iidN (0, Σ)
Want to estimate θ = ( β, Φ, Σ)
Yt = observable variables
St = endogenous latent variables
Zt = exogenous latent variables
m ( ) and f ( ) known
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Moments and latent variables
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An and Schorfheide (2007) DSGE model
In AS model, the endogenous latent variable equation has a
simple form:
St = f (Yt , Zt , β)
(1)
Can substitute St and rewrite the equilibrium conditions as
Et
Zt = ΦZt
1
[g (Yt , Zt , β)] = 0
1
+ εt , εt
iid N (0, Σ)
Warning: not all DSGEs …t this framework
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
UPenn, 5/4/2010
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Existing approaches to estimation of DSGE models
1
2
Theory does not provide likelihood ! must use approximation
methods
Linearize around steady state (Smets and Wouters, 2003;
Woodford, 2003)
Solve the model to …nd policy functions Yt = h(St , Zt )
Construct likelihood by Kalman …lter
3
Nonlinear approximations (Fernandez-Villaverde and
Rubio-Ramirez, 2005)
Solve the model (numerically or analytically in the case of
second order approximations around steady state) to …nd policy
functions
Construct likelihood by nonlinear state-space methods (e.g.,
particle …lter)
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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Drawbacks of existing likelihood-based approaches
1
2
Linearization = possible loss of information
(Fernandez-Villaverde and Rubio-Ramirez, 2005)
Must impose structure to solve the model
1
2
3
Add "shocks"/measurement error to avoid stochastic singularity
Restrict parameters to rule out indeterminacy (multiple rational
expectations solutions)
Nonlinear state-space methods computationally intensive (must
solve the model for each parameter draw) =) so far mostly
applied to simple models
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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10 / 35
Relationship with simulation-based method of
moments
GMM, SMM, EMM, Indirect inference (eg, Ruge-Murcia, 2010)
Di¤erence: requires knowledge of p (Y T jZ T ) or focuses on
moments of the type
EY [g (Y , β)] = 0,
(2)
where g (Y , β) can be computed by simulation
In our case, the model gives
EY,Z [m (Y , Z , β)] = 0
=) can be written as (2) only if p (Z jY ) known
Unlike these methods, we directly obtain estimates of the
smoothed latent variables
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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The idea
Propose methods for estimating non-linear moment-based
models that "exploit" the information contained in the moment
conditions
Methods are:
1
2
Computationally convenient
Classical or Bayesian
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
UPenn, 5/4/2010
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Key elements of methodology
Recall problem we want to solve (e.g., classical framework)
max
θ
Z
p (Y T jZ T , θ )
p (Z T , θ )dZ T
" unknown
" known
Two steps:
1
2
3
Approximate the unknown likelihood p (Y T jZ T , θ )
Integrate out the latent variables using classical or Bayesian
methods
For DSGEs: from an exact likelihood of the approximate
model.... to an approximate likelihood of the exact model
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
UPenn, 5/4/2010
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Approximate likelihoods
We consider two di¤erent approximation strategies
Both use projection theory (for no latent variables, Kim (2002),
Chernozhukov and Hong (2003), Ragusa (2009)): out of all
probability measures satisfying the moment conditions, choose
the one that minimizes the Kullback-Leibler information distance
Method 1 does not require solving the model (but not applicable
to models with dynamic latent endogenous variables)
Method 2 applicable to all models but requires solution of
(approximate) model
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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Approximate likelihoods - Method 1
Find density that satis…es moment conditions and minimizes
distance from the true density: gives approximate likelihood
exp
e
p (Y T jZ T , θ ) ∝
1 0
g Y T , Z T , θ VT 1 Y T , Z T , θ gT Y T , Z T , θ
2 T
gT Z T , θ
VT Y T , Z T , θ
1
= p
T
T
∑ g (Yt , Zt , θ )
wt
1
t =1
= Var (gT Y T , Z T , θ ), wt
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
1
instruments
UPenn, 5/4/2010
15 / 35
Approximate likelihoods - Method 1
exp
e
p (Y T jZ T , θ ) ∝
1 0
g Y T , Z T , θ VT 1 Y T , Z T , θ gT Y T , Z T , θ
2 T
e
p (Y T jZ T , θ ) is a simple transformation of the GMM objective
function.
Intuition:
When (Z T , θ ) is consistent with the model gT Y T , Z T , θ
0
T
T
=) pe(Y jZ , θ ) close to max value of 1.
When (Z T , θ ) is inconsistent with the moment conditions =)
large values of
gT0 Y T , Z T , θ VT 1 Y T , Z T , θ gT Y T , Z T , θ =)
p
e(Y T jZ T , θ ) 0.
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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Approximate likelihoods - Method 2
t
t 1)
Write p (Y T jZ T ) = ΠT
t =1 p (Yt jZ , Y
Choose approximate density b
p (Yt jZ t , Y t 1 , θ ) (does not need
to satisfy moment condition but easy to calculate) For DSGEs, e.g., linearize model around steady state and apply
Kalman …lter =) b
p (Yt jZ t , Y t 1 , θ ) are the …ltered densities
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
UPenn, 5/4/2010
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Approximate likelihoods - Method 2
"Tilt" b
p (Yt jZ t , Y t 1 , θ ) towards moment condition
Et 1 [g (Yt , Zt , θ )] = 0 , new density e
p ( ) satis…es moment
condition and minimizes Kullback Leibler distance from b
p( ) :
Solve problem:
min
h 2H
Z Z
s.t.
log
Z Z
h(Yt jZ t , Y t 1 )
b
p (Yt jZ t , Y t 1 , θ )
b
p Yt jZ t , Y t
g (Yt , Zt , θ )h(Yt jZ t , Y t
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
1
1
, θ dYt dF Z t ,
)dYt dF (Zt ) = 0
UPenn, 5/4/2010
18 / 35
Approximate likelihoods - Method 2
Under regularity conditions the solution is
where
e
p (Yt jZ t , Y t 1 , θ )
= exp fη t + λt g (Yt , Zt , θ )g b
p (Yt jZ t , Y t
(η t , λt ) = arg min
η,λ
Z
1
, θ)
exp fη + λg (Yt , Zt , θ )g b
p (Yt jZ t , Y t
1
, θ )dY
λt = "weights for each moment condition"; η t = integration
constant
(η t , λt ) are functions of Z t , Y t 1 , θ
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
UPenn, 5/4/2010
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Approximate likelihoods - Method 2
e
p (Yt jZ t , Y t
1
, θ) =
exp fη t + λt g (Yt , Zt , θ )g b
p (Yt jZ t , Y t
1
,θ
In practice, approximate integral and compute (η t , λt ) by
simulating N times from b
p (Yt jZ t , Y t 1 , θ ) =)
(η t , λt ) = arg min
η,λ
1
N
N
∑ exp
i =1
n
(i )
η + λg Yt , Zt , θ
o
Well-behaved objective function =) for DSGEs, small additional
computational cost relative to Kalman …lter (cf. particle …lter?)
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
UPenn, 5/4/2010
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The two methods in a simple case
No latent variables, Y T = (Y1 , ..., YT ) mean µ0 , variance σ20
Moment condition identifying parameters are
g1 (Yt , µ, σ2 ) = Yt
g2 (Yt , µ, σ2 ) = Yt2
µ
σ2
Method 1:
b2
b, σ
µ
= arg max exp
θ =(µ,σ2 )
1 0
g Y T , θ VT 1 Y T , θ gT Y T , θ
2 T
=) our estimator is same as GMM (Chernozhukov and Hong
(2003))
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Moments and latent variables
UPenn, 5/4/2010
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The two methods in a simple case
2
Method 2: Start from
n pdf of N (µ, σo) :
1
µ)2 and "tilt it" towards
b
p (Yt ) = p 1 exp
2σ (Yt
2πσ
moment conditions
1
e
p (Yt ) = exp η + λ1 (Yt µ) + λ2 Yt2 σ2 p
e
2πσ
µ0 µ
λ1 =
;
σ0
σ
1
1
λ2 =
2σ 2σ0
1
2 (Y t
No tilting if µ = µ0 , σ2 = σ20
In this case e
p (Yt ) N (µ0 , σ20 ) =) our estimator is the same
as (Q)MLE
Normality here is a special result - e
p ( ) no longer normal if e.g.,
g ( ) non-linear
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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Step 2. Integrate out latent variables
Classical estimation approach: solve
b
θ = max
θ
Z
e
p (Y T jZ T , θ )p (Z T , θ )dZ T
using Jacquier, Johannes and Polson (2007) to compute
integral here works well in our limited experience
Bayesian estimation approach: assume prior for θ (and Z0 ),
π (θ ) and calculate the approximate posterior
e
p (θ, Z T jY T ) ∝ e
p (Y T jZ T , θ )p (Z T j θ ) π ( θ )
Integration of latent variables step is the same as previous
literature
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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23 / 35
Econometric properties
For method 2 (tilted density), can show that MLE based on
approximate integrated likelihood e
p (Y T , θ ) is consistent for
θ = arg min
θ
Z
log
e
p (Y T , θ )
p (Y T )
p (Y T )dY T
θ = parameter that sets the approximate density that is
consistent with the moment conditions as close as possible to
true density
In particular if moment condition uniquely identi…es parameter
θ 0 , by construction θ = θ 0
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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Econometric properties
Back to simple example: Yt iid (µ0 , σ20 ),
g (Yt , θ ) = (Yt µ, Yt2 σ2 ), initial density b
p N (µ, σ2 )
If tilt towards both moments, approximate density
e
p N (µ0 , σ20 ) =) our estimator (=QMLE) consistent for true
parameters
What if tilt towards only one moment condition?
E.g., only use g2 (Yt , θ ) = Yt2 σ2 =) p
e N ( σ σ0 , σ20 )
Variance estimated consistently; mean not estimated
consistently
Suggests that not using moments can cause distortions =)
need to understand tradeo¤s between too many/too few
moments
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
µ
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25 / 35
Econometric properties
Hypothesis testing, model selection relatively straightforward for
method 2
E.g., could test whether λ (or individual components) = 0 ,
understand importance of non-linearities in DSGE models
Open issue: identi…cation (here assumed but challenging
because of presence of latent variables + nonlinearity of moment
conditions)
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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Method 1 in a simple example
Data-generating process
Yt = .9Zt + vt iid N (0, 1)
Zt = .9Zt 1 + εt iid N (0, 1)
Moment condition
E [Zt (Yt
βZt )] = 0
Zt = ρZt
1
+ εt
g (Yt , Zt , β) = Zt (Yt βZt )
Priors: β U (0, 2), ρ U (0, 1), Z0
iid N (0, 1)
N (0, 1
1
),
ρ2
T = 100
Use Jacquier et al. (2007)
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Moments and latent variables
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1.5
1.0
0.5
0.0
Density
2.0
2.5
3.0
Distribution of β
−3
−2
−1
0
c(−2, −2)
1
2
3
6
4
2
0
Density
8
10
Distribution of ρ
−1.0
−0.5
0.0
ρ
0.5
1.0
−6
Smoothed Probabilities
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Smoothed p(z|x)
Actual z
●
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4
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0
20
40
60
Time
80
100
Simulation: AS New Keynesian model
1 = βEt e
1 ν τĉt
(e
νφπ 2
1) = (e π̂t
τ ĉt +1 +τ ĉt +R̂t ẑt +1 π̂ t +1
1)
1
1 π̂t
1
e +
2ν
2ν
βE (e π̂t +1
e ĉt
ŷt
= e
ĝt
(3)
φπ 2 π̂t
(e
2
1) e
(4)
τ ĉt +1 +τ ĉt +ŷt +1 yˆt +π̂ t +1
1) 2
R̂t = ρr R̂t 1 + (1 ρr )ψ1 π̂ t + (1
ẑt = ρz ẑt 1 + σz εz,t
ĝt = ρg ĝt 1 + σg εg ,t
ρr )ψ2 (ŷt
(5)
ĝt ) + σR εR ,t(6)
ε’s independent N (0, 1)
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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AS New Keynesian model
Observable variables: Yt = (Xt , π t , Rt )0 (output, in‡ation and
interest rate), where
Xt = γ(Q ) + 100(ŷt
ŷt
1
+ ẑt )
π t = π (A) + 400π̂ t
Rt = π (A) + r (A) + 4γ(Q ) + 400R̂t .
ŷt , R̂t , π̂ t = deviation from steady state
Endogenous latent variable: St = b
ct = deviation from steady
state of consumption
Exogenous latent variables: Zt = (b
zt , gbt )0 = technology and
government spending
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
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AS model in compact form
(4) implies expression for St as a function of Yt and Zt =)
substitute into moment conditions
Write policy rule as moment conditions
Choose instruments to transform Et [ ] into E [ ]
Write model as
E [g (Yt +1 , Yt , Zt +1 , Zt , θ )] = 0
Zt =
ρz 0
0 ρg
Zt
1 + εt , εt
iidN
0
0
,
σ2z 0
0 σ2g
g ( ) is 11 1, θ =
(τ, ν, φ, 1/g, ψ1 , ψ2 , ρR , σR , π (A) , γ(Q ) , r (A) , ρz , ρg , σz , σg )
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Moments and latent variables
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AS model posterior
Approximate posterior
e
p (θ, Z T jY T ) ∝ exp
1 0
gT Y T , Z T , θ VT 1 Y T , Z T , θ gT Y
2
T
∏ p(zt jzt
t =1
T
1, θ)
∏ p(gt jgt
t =1
1 , γ )p (z0 , g0 j γ )
z0 and g0 drawn from their stationary distributions
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Moments and latent variables
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Simulation exercise
Same DGP as AS:
Generate a time series (T = 80) from a second order
approximation to the model
Parameters and priors as in AS
Compare posteriors for θ obtained by our method to those in AS
(both linear and nonlinear solution methods)
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Moments and latent variables
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AS estimation results
Draws from priors and posteriors for parameters
π (A ) , γ (Q ) , r (A ) , ρ z , ρ g , σ z , σ g
Red lines = true parameter values
Estimation time: 100,000 MCMC draws 6 days
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Moments and latent variables
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72
Figure 17: Posterior Draws: Linear versus Quadratic Approximation II
Linear/Kalman Posterior
1
0.8
0.8
0.8
0.6
γ(Q)
1
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
2
4
0
6
2
4
0
6
6
6
6
5
5
5
π(A)
7
(A)
7
4
4
3
3
2
2
2
1
1
2
0
1
r(A)
1
2
1.4
1.2
1.2
1.2
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
1.5
0.2
0.5
1
ρg
0.2
0.5
6
6
5
4
4
4
σz
5
3
3
2
2
1
1
1
0.005
σg
Notes: See Figure 16.
0.01
0
0
0.005
σg
0.01
x 10
3
2
0
1.5
−3
x 10
5
0
1
ρg
−3
x 10
σz
σz
1.5
ρg
−3
6
2
1
ρz
ρz
1
z
1
1
1
r(A)
1.4
0.2
0.5
0
r(A)
1.4
0.8
6
4
3
1
4
π(A)
7
0
2
π(A)
π
π(A)
π(A)
ρ
Quadratic/Particle Posterior
1
γ(Q)
γ
(Q)
Prior
0
0
0.005
σg
0.01
72
Figure 17: Posterior Draws: Linear versus Quadratic Approximation II
Linear/Kalman Posterior
1
0.8
0.8
0.8
0.6
γ(Q)
1
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
2
4
0
6
2
4
0
6
6
6
6
5
5
5
π(A)
7
(A)
7
4
4
3
3
2
2
2
1
1
2
0
1
r(A)
1
2
1.4
1.2
1.2
1.2
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
1.5
0.2
0.5
1
ρg
0.2
0.5
6
6
5
4
4
4
σz
5
3
3
2
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1
1
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0.005
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Notes: See Figure 16.
0.01
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0.01
x 10
3
2
0
1.5
−3
x 10
5
0
1
ρg
−3
x 10
σz
σz
1.5
ρg
−3
6
2
1
ρz
ρz
1
z
1
1
1
r(A)
1.4
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0.5
0
r(A)
1.4
0.8
6
4
3
1
4
π(A)
7
0
2
π(A)
π
π(A)
π(A)
ρ
Quadratic/Particle Posterior
1
γ(Q)
γ
(Q)
Prior
0
0
0.005
σg
0.01
Our estimation results
Draws from priors and posteriors for parameters
π (A ) , γ (Q ) , r (A ) , ρ z , ρ g , σ z , σ g
Red lines = true parameter values
Estimation time: 2 million MCMC draws 4-5 hours
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
UPenn, 5/4/2010
34 / 35
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σz
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0.010
0.000
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sigma_g
0.008
0.010
Conclusion
Two new methods for estimating structural parameters in
moment-based models that depend on dynamic latent variables
Projection-based approximate likelihoods that satisfy the
moment conditions
Marries the computational convenience of MCMC in
high-dimensional problems with the ability of GMM to handle
nonlinear moment conditions
Directly delivers "smoothed" latent variables
Potential for estimating realistic models and understanding
importance of non-linearities
Giacomini and Ragusa (UCL/Cemmap and UCI/Luiss)
Moments and latent variables
UPenn, 5/4/2010
35 / 35