sequential monte carlo methods (for dsge models) · bayesian estimation of dsge models, with e....
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Sequential Monte Carlo Methods (for DSGEModels)
Frank Schorfheide∗
∗University of Pennsylvania, PIER, CEPR, and NBER
October 23, 2017
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Some References
• “Solution and Estimation of DSGE Models,” with J.Fernandez-Villaverde and Juan Rubio-Ramirez, 2016, in: Handbookof Macroeconomics, vol. 2A
• Bayesian Estimation of DSGE Models, with E. Herbst, 2015,Princeton University Press
• “Sequential Monte Carlo Sampling for DSGE Models,” with E.Herbst, 2014, Journal of Econometrics
• See syllabus for additional references.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Some Background
• DSGE model: dynamic model of the macroeconomy, indexed by θ –vector of preference and technology parameters. Used forforecasting, policy experiments, interpreting past events.
• Bayesian analysis of DSGE models:
p(θ|Y ) =p(Y |θ)p(θ)
p(Y )∝ p(Y |θ)p(θ).
• Computational hurdles: numerical solution of model leads tostate-space representation =⇒ likelihood approximation =⇒posterior sampler.
• “Standard” approach for (linearized) models (Schorfheide, 2000;Otrok, 2001):
• Model solution: log-linearize and use linear rational expectationssystem solver.
• Evaluation of p(Y |θ): Kalman filter• Posterior draws θi : MCMC
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Sequential Monte Carlo (SMC) Methods
SMC can help to
Lecture 1
• approximate the posterior of θ: Chopin (2002) ... Durham andGeweke (2013) ... Creal (2007), Herbst and Schorfheide (2014)
Lecture 2
• approximate the likelihood function (particle filtering): Gordon,Salmond, and Smith (1993) ... Fernandez-Villaverde andRubio-Ramirez (2007)
• or both: SMC 2: Chopin, Jacob, and Papaspiliopoulos (2012) ...Herbst and Schorfheide (2015)
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Lecture 1
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
A Loglinearized New Keynesian DSGE Model
• Consumption Euler equation:
yt = Et [yt+1]− 1
τ
(Rt − Et [πt+1]− Et [zt+1]
)+ gt − Et [gt+1]
• New Keynesian Phillips curve:
πt = βEt [πt+1] + κ(yt − gt),
where
κ = τ1− ννπ2φ
• Monetary policy rule:
Rt = ρR Rt−1 + (1− ρR )ψ1πt + (1− ρR )ψ2 (yt − gt) + εR,t
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Exogenous Processes
• Technology:
lnAt = ln γ + lnAt−1 + ln zt , ln zt = ρz ln zt−1 + εz,t .
• Government spending / aggregate demand: define gt = 1/(1− ζt);assume
ln gt = (1− ρg ) ln g + ρg ln gt−1 + εg ,t .
• Monetary policy shock εR,t is assumed to be serially uncorrelated.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Solving A Linear Rational Expectations System – A SimpleExample
• Consider
yt =1
θEt [yt+1] + εt , (1)
where εt ∼ iid(0, 1) and θ ∈ Θ = [0, 2].
• Introduce conditional expectation ξt = Et [yt+1] and forecast errorηt = yt − ξt−1.
• Thus,
ξt = θξt−1 − θεt + θηt . (2)
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
A Simple Example
• Determinacy: θ > 1. Then only stable solution:
ξt = 0, ηt = εt , yt = εt (3)
• Indeterminacy: θ ≤ 1 the stability requirement imposes norestrictions on forecast error:
ηt = Mεt + ζt . (4)
• For simplicity assume now ζt = 0. Then
yt − θyt−1 = Mεt − θεt−1. (5)
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
At the End of the Day...
• We obtain a transition equation for the vector st :
st = Φ1(θ)st−1 + Φε(θ)εt .
• The coefficient matrices Φ1(θ) and Φε(θ) are functions of theparameters of the DSGE model.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Measurement Equation
• Connect model variables st with observables yt .
• In NK model:
YGRt = γ(Q) + 100(yt − yt−1 + zt)
INFLt = π(A) + 400πt
INTt = π(A) + r (A) + 4γ(Q) + 400Rt .
where
γ = 1 +γ(Q)
100, β =
1
1 + r (A)/400, π = 1 +
π(A)
400.
• More generically:
yt = Ψ0(θ) + Ψ1(θ)t + Ψ2(θ)st +ut︸︷︷︸optional
.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
State-Space Representation and Likelihood
• Measurement:
yt = Ψ0(θ) + Ψ1(θ)t + Ψ2(θ)st +ut︸︷︷︸optional
• State transition:
st = Φ1(θ)st−1 + Φε(θ)εt
• Joint density for the observations and latent states:
p(Y1:T ,S1:T |θ) =T∏
t=1
p(yt , st |Y1:t−1,S1:t−1, θ)
=T∏
t=1
p(yt |st , θ)p(st |st−1, θ).
• Compute the marginal p(Y1:T |θ). We will discuss how in Lecture 2.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Where Are We Headed? ... Bayesian Inference
• Ingredients of Bayesian Analysis:
• Likelihood function p(Y |θ)
• Prior density p(θ)
• Marginal data density p(Y ) =∫p(Y |θ)p(θ)dφ
• Bayes Theorem:
p(θ|Y ) =p(Y |θ)p(θ)
p(Y )∝ p(Y |θ)p(θ)
• Implementation: usually by generating a sequence of draws (notnecessarily iid) from posterior
θi ∼ p(θ|Y ), i = 1, . . . ,N
• Algorithms: direct sampling, accept/reject sampling, importancesampling, Markov chain Monte Carlo sampling, sequential MonteCarlo sampling...
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Bayesian Inference – Posterior
• Focus on SMC algorithm that generates draws θiNi=1 from
posterior distributions of parameters.
• Draws can then be transformed into objects of interest, h(θi ), andunder suitable conditions a Monte Carlo average of the form
hN =1
N
N∑i=1
h(θi ) ≈ Eπ[h]
∫h(θ)p(θ|Y )dθ.
• Strong law of large numbers (SLLN), central limit theorem (CLT)...
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Bayesian Inference – Decision Making
• The posterior expected loss of decision δ(·):
ρ(δ(·)|Y
)=
∫Θ
L(θ, δ(Y )
)p(θ|Y )dθ.
• Bayes decision minimizes the posterior expected loss:
δ∗(Y ) = argmind ρ(δ(·)|Y
).
• Approximate ρ(δ(·)|Y
)by a Monte Carlo average
ρN
(δ(·)|Y
)=
1
N
N∑i=1
L(θi , δ(·)
).
• Then compute
δ∗N (Y ) = argmind ρN
(δ(·)|Y
).
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Bayesian Inference
• Point estimation:
• Quadratic loss: posterior mean
• Absolute error loss: posterior median
• Interval/Set estimation Pπθ ∈ C (Y ) = 1− α:
• highest posterior density sets
• equal-tail-probability intervals
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Implementation: Sampling from Posterior
• DSGE model posteriors are often non-elliptical, e.g., multimodalposteriors may arise because it is difficult to
• disentangle internaland externalpropagationmechanisms;
• disentangle therelative importance ofshocks.
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0θ2
θ1
• Economic Example: is wage growth persistent because
1 wage setters find it very costly to adjust wages?
2 exogenous shocks affect the substitutability of labor inputs andhence markups?
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Sampling from Posterior
• If posterior distributions are irregular, standard MCMC methods canbe inaccurate (examples will follow).
• SMC samplers often generate more precise approximations ofposteriors in the same amount of time.
• SMC can be parallelized.
• SMC = importance sampling on steroids =⇒ We will first reviewimportance sampling.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Importance Sampling
• Approximate π(·) by using a different, tractable density g(θ) that iseasy to sample from.
• For more general problems, posterior density may be unnormalized.So we write
π(θ) =p(Y |θ)p(θ)
p(Y )=
f (θ)∫f (θ)dθ
.
• Importance sampling is based on the identity
Eπ[h(θ)] =
∫h(θ)π(θ)dθ =
∫Θh(θ) f (θ)
g(θ)g(θ)dθ∫Θ
f (θ)g(θ)g(θ)dθ
.
• (Unnormalized) importance weight:
w(θ) =f (θ)
g(θ).
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Importance Sampling
1 For i = 1 to N, draw θi iid∼ g(θ) and compute the unnormalizedimportance weights
w i = w(θi ) =f (θi )
g(θi ).
2 Compute the normalized importance weights
W i =w i
1N
∑Ni=1 w
i.
An approximation of Eπ[h(θ)] is given by
hN =1
N
N∑i=1
W ih(θi ).
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Illustration
If θi ’s are draws from g(·) then
Eπ[h] ≈1N
∑Ni=1 h(θi )w(θi )
1N
∑Ni=1 w(θi )
, w(θ) =f (θ)
g(θ).
6 4 2 0 2 4 60.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
f
g1
g2
6 4 2 0 2 4 6
weights
f/g1
f/g2
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Accuracy
• Since we are generating iid draws from g(θ), it’s fairlystraightforward to derive a CLT:
√N(hN−Eπ[h]) =⇒ N
(0,Ω(h)
), where Ω(h) = Vg [(π/g)(h−Eπ[h])].
• Using a crude approximation (see, e.g., Liu (2008)), we can factorizeΩ(h) as follows:
Ω(h) ≈ Vπ[h](Vg [π/g ] + 1
).
The approximation highlights that the larger the variance of theimportance weights, the less accurate the Monte Carloapproximation relative to the accuracy that could be achieved withan iid sample from the posterior.
• Users often monitor
ESS = NVπ[h]
Ω(h)≈ N
1 + Vg [π/g ].
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
From Importance Sampling to Sequential ImportanceSampling
• In general, it’s hard to construct a good proposal density g(θ),
• especially if the posterior has several peaks and valleys.
• Idea - Part 1: it might be easier to find a proposal density for
πn(θ) =[p(Y |θ)]φnp(θ)∫[p(Y |θ)]φnp(θ)dθ
=fn(θ)
Zn.
at least if φn is close to zero.
• Idea - Part 2: We can try to turn a proposal density for πn into aproposal density for πn+1 and iterate, letting φn −→ φN = 1.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Illustration: Tempered Posteriors of θ1
θ1
0.00.2
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n10
2030
4050
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1
2
3
4
5
πn(θ) =[p(Y |θ)]φnp(θ)∫[p(Y |θ)]φnp(θ)dθ
=fn(θ)
Zn, φn =
(n
Nφ
)λ
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
SMC Algorithm: A Graphical Illustration
C S M C S M C S M
−10
−5
0
5
10
φ0 φ1 φ2 φ3
• πn(θ) is represented by a swarm of particles θin,W
inN
i=1:
hn,N =1
N
N∑i=1
W inh(θi
n)a.s.−→ Eπn [h(θn)].
• C is Correction; S is Selection; and M is Mutation.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
SMC Algorithm
1 Initialization. (φ0 = 0). Draw the initial particles from the prior:
θi1
iid∼ p(θ) and W i1 = 1, i = 1, . . . ,N.
2 Recursion. For n = 1, . . . ,Nφ,
1 Correction. Reweight the particles from stage n − 1 by defining theincremental weights
w in = [p(Y |θi
n−1)]φn−φn−1 (6)
and the normalized weights
W in =
w inW
in−1
1N
∑Ni=1 w
inW i
n−1
, i = 1, . . . ,N. (7)
An approximation of Eπn [h(θ)] is given by
hn,N =1
N
N∑i=1
W inh(θi
n−1). (8)
2 Selection.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
SMC Algorithm
1 Initialization.
2 Recursion. For n = 1, . . . ,Nφ,
1 Correction.2 Selection. (Optional Resampling) Let θN
i=1 denote N iid drawsfrom a multinomial distribution characterized by support points andweights θi
n−1, WinN
i=1 and set W in = 1.
An approximation of Eπn [h(θ)] is given by
hn,N =1
N
N∑i=1
W inh(θi
n). (9)
3 Mutation. Propagate the particles θi ,Win via NMH steps of a MH
algorithm with transition density θin ∼ Kn(θn|θi
n; ζn) and stationarydistribution πn(θ). An approximation of Eπn [h(θ)] is given by
hn,N =1
N
N∑i=1
h(θin)W i
n . (10)
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Remarks
• Correction Step:• reweight particles from iteration n− 1 to create importance sampling
approximation of Eπn [h(θ)]
• Selection Step: the resampling of the particles• (good) equalizes the particle weights and thereby increases accuracy
of subsequent importance sampling approximations;• (not good) adds a bit of noise to the MC approximation.
• Mutation Step: changes particle values• adapts particles to posterior πn(θ);• imagine we don’t do it: then we would be using draws from prior
p(θ) to approximate posterior π(θ), which can’t be good!
θ1
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F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
More on Transition Kernel in Mutation Step
• Transition kernel Kn(θ|θn−1; ζn): generated by running M steps of aMetropolis-Hastings algorithm.
• Lessons from DSGE model MCMC:• blocking of parameters can reduces persistence of Markov chain;• mixture proposal density avoids “getting stuck.”
• Blocking: Partition the parameter vector θn into Nblocks equally sizedblocks, denoted by θn,b, b = 1, . . . ,Nblocks . (We generate the blocksfor n = 1, . . . ,Nφ randomly prior to running the SMC algorithm.)
• Example: random walk proposal density:
ϑb|(θin,b,m−1, θ
in,−b,m,Σ
∗n,b) ∼ N
(θi
n,b,m−1, c2n Σ∗
n,b
).
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Adaptive Choice of ζn = (Σ∗n, cn)
• Infeasible adaption:• Let Σ∗
n = Vπn [θ].• Adjust scaling factor according to
cn = cn−1f(1− Rn−1(ζn−1)
),
where Rn−1(·) is population rejection rate from iteration n − 1 and
f (x) = 0.95 + 0.10e16(x−0.25)
1 + e16(x−0.25).
• Feasible adaption – use output from stage n− 1 to replace ζn by ζn:
• Use particle approximations of Eπn [θ] and Vπn [θ] based onθi
n−1, WinN
i=1.
• Use actual rejection rate from stage n − 1 to calculatecn = cn−1f
(Rn−1(ζn−1)
).
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
More on Resampling
• So far, we have used multinomial resampling. It’s fairly intuitive andit is straightforward to obtain a CLT.
• But: multinominal resampling is not particularly efficient.
• The Herbst-Schorfheide book contains a section on alternativeresampling schemes (stratified resampling, residual resampling...)
• These alternative techniques are designed to achieve a variancereduction.
• Most resampling algorithms are not parallelizable because they relyon the normalized particle weights.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Application 1: Small Scale New Keynesian Model
• We will take a look at the effect of various tuning choices onaccuracy:
• Tempering schedule λ: λ = 1 is linear, λ > 1 is convex.
• Number of stages Nφ versus number of particles N.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Effect of λ on Inefficiency Factors InEffN [θ]
1 2 3 4 5 6 7 8100
101
102
103
104
λ
Notes: The figure depicts hairs of InEffN [θ] as function of λ. Theinefficiency factors are computed based on Nrun = 50 runs of the SMCalgorithm. Each hair corresponds to a DSGE model parameter.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Number of Stages Nφ vs Number of Particles N
ρg σr ρr ρz σg κ σz π(A) ψ1 γ(Q) ψ2 r (A) τ10−3
10−2
10−1
100
101
Nφ = 400, N = 250Nφ = 200, N = 500Nφ = 100, N = 1000
Nφ = 50, N = 2000Nφ = 25, N = 4000
Notes: Plot of V[θ]/Vπ[θ] for a specific configuration of the SMCalgorithm. The inefficiency factors are computed based on Nrun = 50runs of the SMC algorithm. Nblocks = 1, λ = 2, NMH = 1.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
A Few Words on Posterior Model Probabilities
• Posterior model probabilities
πi,T =πi,0p(Y1:T |Mi )∑M
j=1 πj,0p(Y1:T |Mj )
where
p(Y1:T |Mi ) =
∫p(Y1:T |θ(i),Mi )p(θ(i)|Mi )dθ(i)
• For any model:
ln p(Y1:T |Mi ) =T∑
t=1
ln
∫p(yt |θ(i),Y1:t−1,Mi )p(θ(i)|Y1:t−1,Mi )dθ(i)
• Marginal data density p(Y1:T |Mi ) arises as a by-product of SMC.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Marginal Likelihood Approximation
• Recall w in = [p(Y |θi
n−1)]φn−φn−1 .
• Then
1
N
N∑i=1
w inW
in−1 ≈
∫[p(Y |θ)]φn−φn−1
pφn−1 (Y |θ)p(θ)∫pφn−1 (Y |θ)p(θ)dθ
dθ
=
∫p(Y |θ)φnp(θ)dθ∫p(Y |θ)φn−1p(θ)dθ
• Thus,
Nφ∏n=1
(1
N
N∑i=1
w inW
in−1
)≈∫
p(Y |θ)p(θ)dθ.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
SMC Marginal Data Density Estimates
Nφ = 100 Nφ = 400N Mean(ln p(Y )) SD(ln p(Y )) Mean(ln p(Y )) SD(ln p(Y ))500 -352.19 (3.18) -346.12 (0.20)1,000 -349.19 (1.98) -346.17 (0.14)2,000 -348.57 (1.65) -346.16 (0.12)4,000 -347.74 (0.92) -346.16 (0.07)
Notes: Table shows mean and standard deviation of log marginal datadensity estimates as a function of the number of particles N computedover Nrun = 50 runs of the SMC sampler with Nblocks = 4, λ = 2, andNMH = 1.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Application 2: Estimation of Smets and Wouters (2007)Model
• Benchmark macro model, has been estimated many (many) times.
• “Core” of many larger-scale models.
• 36 estimated parameters.
• RWMH: 10 million draws (5 million discarded); SMC: 500 stageswith 12,000 particles.
• We run the RWM (using a particular version of a parallelizedMCMC) and the SMC algorithm on 24 processors for the sameamount of time.
• We estimate the SW model twenty times using RWM and SMC andget essentially identical results.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Application 2: Estimation of Smets and Wouters (2007)Model
• More interesting question: how does quality of posterior simulatorschange as one makes the priors more diffuse?
• Replace Beta by Uniform distributions; increase variances ofparameters with Gamma and Normal prior by factor of 3.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
SW Model with DIFFUSE Prior: Estimation stability RWH(black) versus SMC (red)
l ιw µp µw ρw ξw rπ−4
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4
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
A Measure of Effective Number of Draws
• Suppose we could generate iid Neff draws from posterior, then
Eπ[θ]approx∼ N
(Eπ[θ],
1
NeffVπ[θ]
).
• We can measure the variance of Eπ[θ] by running SMC and RWMalgorithm repeatedly.
• Then,
Neff ≈Vπ[θ]
V[Eπ[θ]
]
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Effective Number of Draws
SMC RWMHParameter Mean STD(Mean) Neff Mean STD(Mean) Neff
σl 3.06 0.04 1058 3.04 0.15 60l -0.06 0.07 732 -0.01 0.16 177ιp 0.11 0.00 637 0.12 0.02 19h 0.70 0.00 522 0.69 0.03 5Φ 1.71 0.01 514 1.69 0.04 10rπ 2.78 0.02 507 2.76 0.03 159ρb 0.19 0.01 440 0.21 0.08 3ϕ 8.12 0.16 266 7.98 1.03 6σp 0.14 0.00 126 0.15 0.04 1ξp 0.72 0.01 91 0.73 0.03 5ιw 0.73 0.02 87 0.72 0.03 36µp 0.77 0.02 77 0.80 0.10 3ρw 0.69 0.04 49 0.69 0.09 11µw 0.63 0.05 49 0.63 0.09 11ξw 0.93 0.01 43 0.93 0.02 8
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
A Closer Look at the Posterior: Two Modes
Parameter Mode 1 Mode 2ξw 0.844 0.962ιw 0.812 0.918ρw 0.997 0.394µw 0.978 0.267Log Posterior -804.14 -803.51
• Mode 1 implies that wage persistence is driven by extremelyexogenous persistent wage markup shocks.
• Mode 2 implies that wage persistence is driven by endogenousamplification of shocks through the wage Calvo and indexationparameter.
• SMC is able to capture the two modes.
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
A Closer Look at the Posterior: Internal ξw versus Externalρw Propagation
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10
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1
ξw
ρw
F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models
Stability of Posterior Computations: RWH (black) versusSMC (red)
P (ξw > ρw) P (ρw > µw) P (ξw > µw) P (ξp > ρp) P (ρp > µp) P (ξp > µp)0
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F. Schorfheide Econ 556a (Part 2, Guest Lecture): SMC for DSGE Models