HIAS-E-128

Posterior Inferences on Incomplete
Structural Models:The Minimal Econometric Interpretation

Takashi Kano
Hitotsubashi University

March 2023

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
tel:+81 42 580 8668 http://hias.hit-u.ac.jp/
HIAS discussion papers can be downloaded without charge from:https://hdl.handle.net/10086/27202
https://ideas.repec.org/s/hit/hiasdp.html

Posterior Inferences on Incomplete Structural Models:The Minimal Econometric Interpretation
Takashi Kano
Graduate School of Economics
Hitotsubashi University
Naka 2-1, Kunitachi, Tokyo
186-8601, JAPAN
Email: tkano@econ.hit-u.ac.jp
Current Draft: March 14, 2023
Abstract
The minimal econometric interpretation (MEI) of DSGE models provides a formal model evaluation and comparison of misspeciﬁed nonlinear dynamic stochastic general equilibrium (DSGE)
models based on atheoretical reference models. The MEI approach recognizes DSGE models asincomplete econometric tools that provide only prior distributions of targeted population moments
but have no implications for actual data and sample moments. This study, based on the MEI approach, develops a Bayesian posterior inference method. Prior distributions of targeted population
moments simulated by the DSGE model restrict the hyperparameters of Dirichlet distributions.
These are natural conjugate priors for multinomial distributions followed by corresponding posterior distributions estimated by the reference model. The Pólya marginal likelihood of the resulting
restricted Dirichlet-multinomial model has a tractive approximated log-linear representation of theJensen-Shannon divergence, which the proposed distribution-matching posterior inference uses as
the limited information likelihood function. Monte Carlo experiments indicate that the MEI posterior sampler correctly infers calibrated structural parameters of an equilibrium asset pricing modeland detects the true model with posterior odds ratios.
Key Words : Bayesian posterior inference, Minimum econometric interpretation, Nonlinear DSGE
model, Model misspeciﬁcation, Equilibrium asset pricing model
JEL Classiﬁcation Number : C11, C52, E37†

I wish to thank Gianni Amisano, Toru Kitagawa, Jim Nason, and Toshi Watanabe and conference participants fromthe 4th Hitotsubashi Summer Institute (HSI 2019) and 16th International Conference on Computational and Financial
Econometrics (CFE 2022) for their insightful discussions and helpful comments. I wish to express my gratitude for the
grant-in-aid for scientiﬁc research from the Japan Society for the Promotion of Science (Nos. 24330060, 17H02542,
17H00985, and 20H00073) and ﬁnancial support from the Hitotsubashi Institute for Advanced Study. I am solely
responsible for any errors and misinterpretations in this study.
‡
Contact information: Graduate School of Economics, Hitotsubashi University, Naka 2-1, Kunitachi-city, Tokyo, 1868601, Japan. Emails: tkano@econ.hit-u.ac.jp.

“And now here is my secret, a very simple secret: It is only with the heart that one
can see rightly; what is essential is invisible to the eye.” (Saint-Exupéry, The Little
Prince).
“Assuming the population moment is equal to the sample moment can be treacherous.”(Geweke 2010).

1. Introduction
In his seminal study, Geweke (2010) develops the minimal econometric interpretation (MEI)of dynamic stochastic general equilibrium (DSGE) models, generalizing the Bayesian calibration
approach of DeJong et al. (1996) as a crucial predecessor. What makes the MEI approach sharply
distinct from the conventional likelihood-based inferences and standard moment-matching calibration exercises is its central premise. In the MEI approach, DSGE models only need to provide priordistributions of unobservable population moments. These are dimensions of the macroeconomic
reality that DSGE modelers intend to mimic. The MEI approach treats DSGE models as incomplete
econometric tools, assuming that they have no direct implications on either actual data or sample
moments. This perspective of MEI on incomplete structural models enables formal implementation of Bayesian evaluation and comparison of nonlinear DSGE models even under potential model
misspeciﬁcation (e.g., stochastic singularity).1
However, population moments are unobservable. Hence, the MEI approach requires a denselyparameterized atheoretical econometric model as a statistical reference to establish the posterior
distributions of targeted population moments from observed data. Essentially, the reference modelacts as a hypothetical bridge between DSGE models and actual data through posterior distributionsof targeted population moments.
The MEI approach evaluates a DSGE model by measuring the degree of overlap between
prior distributions of the targeted population moments implied by the DSGE model and the corresponding posterior distributions estimated by the reference model. The higher the degree of overlap,
the better the ﬁt of the DSGE model to the macroeconomic reality of interest. Hence, when conducting formal model evaluation and comparison of the DSGE model, the MEI approach utilizes
1

Canova (2007), DeJong and Dave (2011), Del Negro and Schorfheide (2011), and Fernández-Villaverde et al.
(2016) review the MEI approach as a Bayesian model comparison method of DSGE models subject to model misspecifciation.

1

the sampling variation of the targeted population moments implied by the statistical reference modeland places much weaker weight on the corresponding sample moments and the underlying observed
data.2 This characteristic of the MEI approach formalizes the critical point made by Eichenbaum(1991), and reemphasized by Geweke (2010), with the epigram for empirical DSGE research thatassuming that the population moment is equal to the sample moment can be treacherous; observed
data and macroeconomic reality of interest are distinct objects.3
This study aims to develop a posterior inference method based on the MEI approach. The
key idea behind the proposed method is to update the prior distributions of the structural parameters by matching the synthetic distributions of the targeted population moments implied by the
DSGE model (hereafter, theoretical moment distribution) with the corresponding posterior distributions estimated using the statistical reference model (hereafter, empirical moment distribution) as
closely as possible. Essentially, this study formalizes distribution-matching, not moment-matching
posterior inference of the DSGE model.
While the MEI approach provides a formal, explicit, simple, and easy model evaluation,comparison, and development method, a practical concern for the intrinsic nature of MEI is thatthe resulting decision-making could be aﬀected by implausibly assumed prior distributions of thestructural parameters of the DSGE model. For instance, when facing a signiﬁcant deviation in
the synthetic distribution of a targeted population moment simulated by the DSGE model fromthe corresponding posterior distribution estimated with the reference model, an MEI user cannot
distinguish a priori whether this failure stems from the misspeciﬁcation of the proposed DSGE
model or implausible prior distributions of its structural parameters. As a formal Bayesian decisionmaking scheme, the MEI approach requires a parameter-updating process to avoid this ambiguity.
The proposed distribution-matching posterior inference method starts by discretizing the em2

Close cousins of the MEI approach are the prior-predictive analysis (Box 1980; Canova 1994; Lancaster 2004;
Geweke 2005) and posterior-predictive analysis (Gelman et al. 2003; Faust and Gupta 2012). These speciﬁcation
tests measure how well prior and posterior distributions of the targeted sample moments (i.e., the checking functions)
implied by the DSGE models contain their observed values with data, calculating prior and posterior predictive pvalues. Geweke (2010), however, argues that the prior predictive check is Bayesian but the posterior predictive checkis not (see also Lancaster 2004). Faust and Gupta (2012) discusses the practical advantage of the posterior predictive
check for DSGE modelers and policy makers.
3
The MEI approach is applied by Nason and Rogers (2006) and Kano and Nason (2014) to model evaluation andcomparisons among small open-economy models and new Keynesian monetary models, respectively. Loria et al.
(2022) also exploit the MEI approach to construct from calibrated DSGE models mixture priors for population parameters of statistical unobserved component models.

2pirical and theoretical moment distributions over ﬁnite grid points. The discretized empirical distribution of a targeted population moment is assumed to follow a multinomial distribution conditionalon a mass probability vector attached to ﬁnite grid points. This study assumes that the sole roleof the DSGE model is to provide the prior distribution of the mass probability vector. In Bayesian
literature, a Dirichlet distribution with hyper-parameters considered the concentration parameters
acts as a natural conjugate prior for the multinomial distribution’s mass probability vector. The
DSGE model then imposes restrictions reﬂected in the discretized theoretical moment distribution
on Dirichlet concentration parameters. Hence, based on the MEI approach, this study emphasizes
that the DSGE model has no direct implications for the actual data and sample moments.
The resulting restricted Dirichlet distribution and empirical multinomial distribution jointly
constructs the Dirichlet-multinomial (DM) model. Importantly, after integrating out the mass probability vector, the DM model yields the marginal likelihood of the empirical moment distribution,
conditional on the theoretical counterpart with a Pólya distribution. This DM marginal likelihood
plays a crucial role as a likelihood function in the proposed MEI posterior inference method. Moreover, as a mathematical proposition, this study shows that the logarithmic kernel of the resulting
DM marginal likelihood has a tractable approximated representation of the Jensen-Shannon (JS)divergence between the empirical and theoretical moment distributions.
The shape of the JS divergence depends on the relative size of the simulated draws for thetheoretical moment distribution compared to that of the empirical one. When the number of draws
for the theoretical moment distribution is suﬃciently large relative to that of the empirical one, JS
divergence converges to the Kullback-Leibler (KL) divergence of the empirical moment distribution
from the theoretical one. This establishes a quasi-likelihood function calculated from the multinomial distribution with mass probabilities restricted by theoretical moment distribution. However,when the number of draws for the theoretical distribution is minimum and equal to one, JS divergence converges to the predictive density function of the DM model, which closely traces out theempirical moment distribution. Hence, JS divergence oﬀers a density kernel for a minimum distance (MD) estimator in this extreme case. This novel ﬁnding extends the central propositions Del
Negro and Schorfheide (2004) established in their DSGE-VAR framework toward more ﬂexibly
selected population moments that only nonlinear DSGE models can target.

3

This study proposes a two-step Markov chain Monte Carlo (MCMC) procedure as the MEI
posterior inference method. This treats JS divergence as the logarithm of the likelihood function(hereafter, JS likelihood). The ﬁrst step applies a conventional MCMC sampler for the reduced-form
parameters of the statistical reference model and constructs posterior distributions of the targeted
population moments as empirical moment distributions. Given the resulting empirical moment distributions, the second step implements a posterior sampler induced by the JS likelihood to jointly
simulate the theoretical moment distributions and posterior distributions of the structural parameters of the underlying DSGE model. The proposed MEI posterior sampler draws targeted population
moments from the DSGE model and calculates the JS likelihood. Premultiplied by suitable priordistributions of structural parameters, the constructed JS likelihood then yields the kernel of theposterior probability density for a random-walk Metropolis-Hastings (RW-MH) algorithm for thetheoretical moment distributions and posterior distributions of structural parameters. Moreover,
the MEI posterior sampler enables the estimation of the marginal likelihood of the underlying data,
conditional on DSGE and reference models, leading to formal Bayesian posterior model evaluation,comparison, and criticism.
To investigate how the proposed MEI posterior sampler works, this study conducts Monte
Carlo experiments with Labadie’s (1989) equilibrium consumption-based asset-pricing model, which
Geweke (2010) exercises with the MEI approach. Labadie’s (1989) model is notable for two reasons. First, this single-shocked model is subjected to the stochastic singularity problem for conventional full-information likelihood inference with multiple time-series data. Second, the model
is nonlinear in deriving the equilibrium price of the risky asset. Hence, the linear inference method(e.g., the DSGE-VAR method) cannot be applied.
Given the calibrated values of structural parameters, Labadie’s (1989) model simulates synthetic time-series data of the risk-free rate, equity premium, and consumption growth rate with arealistic sample length. Following Geweke (2010), this study adopts the ﬁrst-order vector autoregression (VAR) as the statistical reference model. In the ﬁrst step of the MCMC procedure, sixtargeted population moments from the standard normal-inverted Wishart model are simulated andempirical moment distributions are constructed. Under correctly speciﬁed informative prior distributions of structural parameters, the MEI posterior sampler in the second step of the MCMC
procedure generates theoretical moment distributions tightly overlapping their empirical counter4

parts. Hence, resulting posterior distributions of structural parameters were close to corresponding
calibrated values. Moreover, the Monte Carlo investigation shows that this successful performanceof the MEI posterior sampler is robust against two diﬀerent settings of prior distributions of structural parameters: uniform and misspeciﬁed prior distributions. The estimated posterior odds ratios
indicate that the model with correctly speciﬁed informative prior distributions dominates models
with the other two prior distributions.
This study emphasizes two novel aspects of the proposed MEI posterior sampler for literatureon the Bayesian inference of DSGE models. First, based on the MEI approach, the MEI posterior
sampler considers the sampling variations of the targeted unobservable population moments but
weakens dependence on actual data and sample moments. Essentially, the MEI posterior sampler
can be interpreted as a data augmentation technique using empirical moment distributions to avoid
overﬁtting actual data and enhance the generalization ability of the inferred DSGE model.
Second, the MEI posterior sampler is a limited-information Bayesian inference method that
does not rely on the full-information likelihood function of the underlying DSGE model.4 In
the extreme case with the minimum number of draws for theoretical moment distributions, the
MEI posterior sampler nests the approximated Bayesian computation with the MCMC sampler(MCMC-ABC) of Marjoram et al. (2003) and Forneron and Ng (2018). Hence, it oﬀers a Bayesian
distribution-matching indirect inference method, which can easily be implemented for invisible essential population characteristics of macroeconomic reality using potentially misspeciﬁed nonlinear
DSGE models.
The remainder of this paper is organized as follows. Section 2 introduces the proposed MEI
posterior inference method. Section 3 discusses the construction and implementation of the Monte
Carlo experiments. Section 4 reports the results of the Monte Carlo experiments. Finally, Section
5 concludes the study.

4

Kim (2002) developed a Bayesian posterior inference method with a limited information likelihood from the objective function of an extremum estimator. This resembles the Laplace type estimator of Chernozhukov and Hong(2003). Inoue and Shintani (2018) showed that the resulting simulated quasi marginal likelihood can be used for formal Bayesian model evaluation and comparison. Fernández-Villaverde et al. (2016) provided a compact survey ofBayesian limited-information inference methods.

5

2. The MEI posterior inference
Let A and E denote the DSGE and empirical models characterized by the parameter vectors
θA and θE , respectively. These two models can simulate the vector of unobservable population
moments M which a researcher explains as I distinct macroeconomic facts: M = E(Z), where Z
is the vector of the corresponding sample moments, and E is the mathematical expectations operator. Let mi denote the i-th element of the population moment vector M = [m1 , m2 , · · · , mI ].
The DSGE model A is parsimonious and can be nonlinear when simulating the population moment
vector. Let mA,i denote the i-th element of the population moment vector MA , which is simulatedby the DSGE model A equipped with the structural parameter vector θA (hereafter the “theoretical” moment). The theoretical moment mA,i is a deterministic nonlinear function of the structuralparameter vector θA , mA,i (θA ). The empirical model E is atheoretical with a higher degree of freedom to simulate the reduced-form posterior distribution of the population moment vector. Hence,
let mE,i represent the i-th element of the population moment vector ME , which is estimated by theempirical model E equipped with the parameter vector θE (hereafter the “empirical” moment).
2.1. The Dirichlet-multinomial model for discretized population moment distributions
The most important device this study relies on is the Dirichlet-multinomial (DM) model,
which characterizes the probability distributions of the discretized population moment distributions.5 Assume that the population moment mi has the ﬁnite support Si = [si , s̄i ]. Suppose that
support Si is decomposed into K mutually exclusive subintervals sk,i for k = 1, · · · K. Let pk,i ≥ 0denote the mass probability of the event wherein the population moment mi drops to the k-th subinterval sk,i . Let pi ≡ [p1,i , p2,i , · · · , pK,i ] denote a vector comprising pk,i that satisﬁes the regularity
P
condition K
k=1 pk,i = 1.
Suppose that the two models A and E generate collections of theoretical and empirical moj
Nments, mA,i ≡ {mjA,i }M
j=1 and mE,i ≡ {mE,i }j=1 , where M ≥ 1 and N ≥ 1 are numbers ofelements in the theoretical and empirical moment collections, respectively. The former and latter collections are equivalent to the theoretical and empirical moment distributions. Let ΘA dej M
note the collection of M structural parameter vectors, ΘA ≡ {θA}j=1 . Moreover, I deﬁne col-lections MA and ME by horizontally stacking I theoretical and empirical moment distributions,
5

The Dirichlet-multinomial model is introduced by Gelman et al. (2003) and Lancaster (2004).

6

MA ≡ [mA,1 , · · · , mA,I ], and ME ≡ [mE,1 , · · · , mE,I ], respectively.
Consider the following discretization of the empirical moment distribution mE,i over massprobability vector pi for i = 1, · · · , I. Let nk,i ≥ 0 for k = 1, · · · , K denote the number ofPdraws of mE,i that drop into the k-th subinterval sk,i . The condition K
k=1 nk,i = N is satisﬁed.
The probability of mE,i conditional on pi is then characterized by a multinomial distribution withparameter ni = [n1,i , n2,i , · · · , nK,i ]:KY

Γ(N + 1)p(mE,i |pi ) = QK

k=1 Γ(nk,i + 1) k=1(pk,i )nk,i ,(1)where Γ(.) is the Gamma function.
Let αk,i ≥ 1 represent one plus the number of draws of the theoretical moment mA,i , whichP
drops into the k-th subinterval sk,i . Condition K
k=1 αk,i = M + K is then satisﬁed. Consider ajoint event wherein αk,i −1 draws of mA,i drop into the k-th subinterval sk,i with probability pk,i fork = 1, · · · , K simultaneously. The joint probability of pi conditional on mA,i is then characterized
by the Dirichlet distribution with concentration parameter vector αi = [α1,i , α2,i , · · · , αK,i ]:Γ(M + K) Y
(pk,i )αk,i −1 .
p(pi |mA,i ) = QK
k=1 Γ(αk,i ) k=1
K(2)

The DSGE model A imposes theoretical restrictions on the concentration parameter vector αi .
The restricted Dirichlet distribution (2) acts as a natural conjugate prior distribution for multinomial distribution (1). The Dirichlet-multinomial (DM) model consisting of eqs.(1) and (2) imply
that the conditional distribution of pi on mA,i and mE,i is characterized by the Dirichlet distribution
with parameter ni + αi :KΓ(N + M + K) Yp(pi |mA,i , mE,i ) ∝ p(pi |mA,i )p(mE,i |pi ) = QK
(pk,i )nk,i +αk,i −1 .
Γ(n
+
α
)
k,i
k,i k=1
k=1

where nk,i + αk,i − 1 ≥ 1 for k = 1, · · · , K and

(3)

PK

k=1 (nk,i + αk,i ) = N + M + K.

The DM model’s important property is its analytical representation of the marginal likelihood function. Integrating out pi from the DM model provides the analytical form of the marginal
likelihood function with Pólya distribution

7

Z

K
Γ(N + 1)Γ(M + K) Y Γ(nk,i + αk,i )p(mE,i |mA,i ) = p(mE,i |pi )p(pi |mA,i )dpi =.
Γ(N + M + K) k=1 Γ(nk,i + 1)Γ(αk,i )(4)
The posterior inference of the DSGE model A stems from the DM marginal likelihood function (4).

The diﬃculty is that the DM marginal likelihood (4) becomes inﬁnite for large values of N
and M as Gamma functions explode. This study shows that the above DM marginal likelihood(4) has a tractable approximation with the density kernel of the Jensen-Shannon (JS) divergence
between empirical and theoretical distributions.6 To establish the main proposition, the ratio of thetotal number of draws in the theoretical moment distribution to that of the empirical one is deﬁned
as λ(≡ (M + K)/N ). For k = 1, · · · , K, let ζk,i and qk,i denote frequencies wherein the empiricaland theoretical moments drop into the k-th subinterval, ζk,i ≡ nk,i /N and qk,i ≡ αk,i /(M + K),
respectively. Particularly, ζk,i is equivalent to the maximum likelihood (ML) estimate of pk,i in
multinomial distribution (1). Let ζ i and qi denote empirical and theoretical probability mass vectors
ζ i ≡ [ζ1,i , ζ2,i , · · · , ζk,i ] and qi ≡ [q1,i , q2,i , · · · , qK,i ], respectively. Appendix A establishes thefollowing proposition:
Proposition 1. The logarithm of the marginal likelihood function of the DM model (4) is approximated as follows:ln pλ (mE,i |mA,i ) ≈ ln N − (1 + λ)N DJS (ζ i || qi ),(5)

where DJS (ζ i || qi ) denotes the JS divergence between the empirical and theoretical distributions.



K
1 X
1
λ
DJS (ζ i || qi ) =ζk,i ln ζk,i − lnζk,i +qk,i
1 + λ k=1
1+λ
1+λ



K
λ X
λ
1
ζk,i +
qk,i+
qk,i ln qk,i − ln
1 + λ k=11+λ1+λwith the standard regularity condition 0 × ln 0 = 0.
The approximated DM marginal likelihood (i.e., the JS likelihood function (5)) has three important
properties. First, as the JS divergence is nonnegative, the JS likelihood (5) is maximized when
6

The JS divergence is a generalized form of the Kullback-Leibler (KL) divergence with symmetry. Lin (1991)
discusses the properties of JS divergence.

8

the empirical and theoretical distributions match exactly: limζ i →qi ln pλ (mE,i |mA,i ) → ln N .7
Second, when the number of draws in the theoretical moment distribution, M , becomes suﬃciently
large relative to that of the empirical one, N (i.e., λ → ∞), it is the case that
lim ln pλ (mE,i |mA,i ) → ln N − N

λ→∞

K
X

ζk,i (ln ζk,i − ln qk,i ) ∝ −N DKL (ζ i || qi ).

(6)

k=1

Hence, the JS likelihood becomes proportional to the negative of the Kullback-Leibler (KL) divergence of ζ i from qi , denoted by DKL (ζ i || qi ). As the logarithm of multinomial distribution (1) isrepresented by the KL divergence,ln p(mE,i |pi ) ≈ −N

KX

ζk,i (ln ζk,i − ln pk,i ) = −N DKL (ζ i || pi ),k=1

Function (6) is obtained by replacing the unrestricted probability mass vector pi with its theoreticalcounterpart qi . Hence, when λ → ∞, the JS likelihood (5) approaches the quasi-likelihood function
constructed by the multinomial distribution imposing the theoretical restriction of the DSGE model
A. Notably, this second property corresponds to the ﬁrst proposition established by Del Negro andSchorfheide (2004, Proposition 1).
Third, as Appendix B shows, when the number of draws in the theoretical moment distribution M takes the minimum value of 1, or equivalently, λ → (K + 1)/N , the JS likelihood (5)becomes

lim ln pλ (mE,i |mA,i ) →λ→ K+1NK
XI[mA,i ∈ sk,i ] ln

k=1

nk,i + 2
N +K +1,(7)where I[mA,i ∈ sk,i ] is the indicator function that takes the value of 1 if mA,i drops into the subinnk,i +2
terval sk,i and 0 otherwise. The term N +K+1 is the predictive density of the DM model (3),at which the theoretical moment that draws mA,i drops into the k-th subinterval sk,i .8 For a suﬃciently large N , predictive density is approximated well by the frequency of the empirical moment
distribution ζk,i . Hence, the theoretical moment that draws mA,i from the JS likelihood (7) closely
7

If the empirical and theoretical distributions are uniform, ζk = qk = 1/K, the JS divergence becomes minimized
and the JS likelihood (5) is maximized to ln N .
8
The DM model (3) implies that the predictive density of a new draw mA,i conditional on mE,i and mA,i ispλ (mA,i ∈ sk,i |mE,i , mA,i ) =1
λζk,i +
qk,i .
1+λ1+λThe predictive probability is the weighted average between the ML estimate ζk,i of the multinomial distribution (1)and the expectation of the Dirichlet distribution (2) with relative weights 1/(1 + λ) and λ/(1 + λ). As qk,i > 0, theJS likelihood function (7) is well-deﬁned even when the ML estimate ζk,i is zero.

9

traces the shape of the empirical moment distribution mE,i . This third property echoes the second proposition of Del Negro and Schorfheide (2004, Proposition 2). In this case, the posterior
estimate of the structural parameter θA can be interpreted as a minimum distance (MD) estimate
with KL divergence. This is obtained by ﬁtting the theoretical moment distribution to the empiricalcounterpart as closely as possible.9
2.2. The posterior joint distributions of ΘA , mA,i , mE,i , and pi
This subsection introduces the joint posterior distribution of ΘA , mA,i , mE.i , and pi conditional on data y and the empirical and DSGE models E and A. In doing so, this subsection borrows
Condition 4.1 and Proposition 4.2 of Geweke (2010), which are cornerstones of the MEI approach:Condition 4.1 (Geweke 2010). Conditional on the empirical and DSGE models, E and A,
p(mi , θE , y|A, E) = p(mi |A)p(θE |mi , E)p(y|θE , mi , E).
Condition 4.1 of Geweke (2010) implies that the sole function of the DSGE model A is to provide
a prior distribution of population moment mi . Empirical model E is incomplete as it provides noproper prior distribution of mi : p(mi |E) ∝ const. Then, the following proposition holds:
Proposition 4.2 (Geweke 2010). Under Condition 4.1,
Rp(mi , θE , y|A, E)dθE,
p(mi |A, E)
R
p(mi |A)p(θE |mi , E)p(y|θE , mi , E)dθE=,
p(mi |A)Z= p(θE |mi , E)p(y|θE , mi , E)dθE ,p(y|mi , A, E) == p(y|mi , E).
This proposition guarantees the core argument of the MEI approach; the DSGE model A has nodirect implication on the data probability y.
The joint distribution of ΘA , mA,i , mE,i , pi , and y conditional on the two models E and A
then are characterized as follows:9

λ1
ζ i + 1+λqi ).
Appendix B also shows that function (7) is proportional to the KL divergence DKL (qi || 1+λ10

p(ΘA , mA,i , mE,i , pi , y|A, E) = p(ΘA |A) p(mA,i |ΘA , A) p(mE,i , pi , y|mA,i , A, E)= p(ΘA |A) p(pi |mA,i (ΘA )) p(mE,i , y|pi , A, E)= p(ΘA |A) p(pi |mA,i (ΘA )) p(mE,i |pi ) p(y|mE,i , E).

(8)Particularly, the ﬁrst equality stems from Condition 4.1, and the sole role of DSGE model A is toR
generate the prior distribution of the population moment p(mA,i |A) = p(mA,i |ΘA , A)p(ΘA |A)dΘA .
As the theoretical moment mA,i is provided as a deterministic nonlinear function of the structuralparameter θA , the distribution p(mA,i |ΘA , A) degenerates to a mass point. The second term of thesecond equality results from the distribution of pi conditional on mA,i and is given by the Dirichlet distribution (2). The third term of the third equality stems from multinomial distribution (1).
Finally, the fourth term of the third equality reﬂects Proposition 4.2, which guarantees equalityp(y|mE,i , A, E) = p(y|mE,i , E).10
From Bayes’ law, p(y|mE,i , E) =p(mE,i |y,E)p(y|E), by substituting this result into eq.(8) andp(mE,i |E)

dividing the result by the marginal data density p(y|A, E) yields the posterior joint distribution of
ΘA , mA,i , mE,i , and pi on the data y and the two models A and E:
p(ΘA , mA,i , mE,i , pi |y, A, E) =p(ΘA , mA,i , mE,i , pi , y|A, E)
p(y|A, E)= p(ΘA |A) p(pi |mA,i (ΘA )) p(mE,i |pi )p(mE,i |y, E) p(y|E)
p(mE,i |E)p(y|A, E)∝ p(ΘA |A) p(pi |mA,i (ΘA )) p(mE,i |pi ) p(mE,i |y, E),

10
To show that the joint distribution (8) indeed leads to the MEI approach, conﬁrm that marginalizing out ΘA , mA,i ,mE,i , and pi from the joint distribution (8) yields the marginal data density of y conditional on the two models A andE:p(y|A, E)ZZ
Z Z=p(ΘA |A)p(mA,i |ΘA , A)p(pi |mA,i )p(mE,i |pi )p(y|mE,i , E)dΘA dpi dmA,i dmE,i ,

Z

mE,i

Z

mE,i

Z

mE,i

Z

mA,i

Z i

p

Z

mA,i

pi

ΘA

p(mA,i |A)p(pi |mA,i )p(mE,i |pi )p(y|mE,i , E)dpi dmA,i dmE,i ,=p(mA,i |A)p(mE,i |mA,i )p(y|mE,i , E)dmA,i dmE,i ,=mA,ip(mE,i |A)p(y|mE,i , E)dmE,i .

=mE,i

As established in Proposition 4.3 by Geweke (2010), marginal data density p(y|A, E) is given by the convolution of
two densities. This is the same marginal data density that the MEI approach uses for formal model comparison.

11where the proportionality in the third line results from p(mE,i |E) ∝ const. If pi is marginalized
out, then the joint distribution becomes the following:p(ΘA , mA,i , mE,i |y, A, E) ∝ p(ΘA |A) pλ (mE,i |mA,i (ΘA )) p(mE,i |y, E),(9)where the second term of the RHS results from the DM marginal likelihood (4) and JS likelihood(5) in Proposition 1.
2.3. An MCMC procedure for the posterior joint distribution
This study proposes the posterior inference of the MEI approach using the following two-step
MCMC procedure:Two-step MCMC procedureStep 1. Given y and the empirical model E, draw mE,i from p(mE,i |y, E) for i = 1, · · · , I.
Step 2. Given ME , draw ΘA and MA, from p(ΘA |A)ΠIi=1 pλ (mE,i |mA,i (ΘA )).
Resulting draws of ΘA and MA construct the posterior draws from p(ΘA , MA |y, A, E).



The posterior draws of mE,i in Step 1 are implemented using the Gibbs sampling procedure withan atheoretical empirical model. Let p(y|θE , E) denote the likelihood function of empirical model
E with parameter θE given data y. Given prior distribution p(θE |E), parameter θE is drawn fromthe posterior kernelp(θE |y, E) ∝ p(y|θE , E)p(θE |E).
Given a draw of θE , the empirical moment mE,i is simulated as a deterministic nonlinear function, mE,i (θE ). This process is repeated N times to construct a draw of mE,i from the conditional
distribution p(mE,i |y, E).
To implement Step 2, this study focuses on the case with M = 1 because it is computationally infeasible to simultaneously draw a large number of theoretical moments mA,i by solving
the underlying nonlinear DSGE model for diﬀerent values of the structural parameters ΘA . WithM = 1, collection ΘA degenerates to a single vector θA . Hence, the conditional density becomes:pλ (θA |ME ) = p(θA |A)ΠIi=1 pλ (mE,i |mA,i (θA )).
Because there is no analytical representation of pλ (θA |ME ), Step 2 adopts the following RW-MH

12

algorithm.

The RW-MH algorithm in Step 2



old
old
and corresponding conditional probability p(θA|ME ),
Given the initial draw θA
new2(a). Draw a new candidate of the structural parameter θA
from
new
old
θA
= θA
+ v,

v ∼ i.i.d.(0, τ Ω)

where, v is a random variate drawn from an i.i.d. with the mean of zero and diagonalvariance-covariance matrix Ω accompanied by the conﬁguration parameter τ .
new2(b). Calculate the conditional probability pλ (θA|ME ). Computenewpλ (θA|ME )new oldr(θA |θA ) = min 1,.
oldpλ (θA|ME )newnew old2(c). Draw a uniform random variate u ∼ U [0, 1]. Accept θA = θA
if r(θA|θA ) ≥ u.
old
Keep θA = θAotherwise.
old= θA . Repeat 2(a)-(d) many times.
2(d). Set θA

The proposed RW-MH procedure can be interpreted as a generalization of the approximatedBayesian computation with the MCMC sampler (MCMC-ABC) developed by Marjoram et al.
(2003) and extended by Forneron and Ng (2018). To understand this interpretation, suppose thatthe empirical moment distribution mE,i degenerates to a single scaler value of the sample momentmE,i . The JS likelihood (7) then becomes:3
lim ln pλ (mE,i |mA,i ) →I[mA,i ∩ mE,i ∈ sk,i ] ln
λ→K+1
K +2
k=1

 

Si
3
= I |mA,i − mE,i | ≤
ln
.
K
K +2
K
X

This function takes the value of ln

3
K+2



if the simulated moment mA,i drops to the same subin-

terval sk,i , where the sample moment mE,i resides, and 0 otherwise. Essentially, only when theabsolute distance between mA,i and mE,i is less than the width of the subinterval Si /K does the3
JS likelihood take a positive value of K+2. Given this particular distance function, the proposed

RW-MH procedure yields the same Markov kernel as the MCMC-ABC.

13

2.4. The marginal likelihood estimation and model comparison
In the MEI approach, the marginal likelihood of the DSGE model A is evaluated relative tothat of the empirical model E. This relative marginal likelihood ψλ (y|A, E) ≡ p(y|A, E)/p(y|E)
is approximated by the modiﬁed harmonic mean estimator by Geweke (1999): for J posterior draws,"j
1 X f (θA)ψ̂λ (y|A, E) =j
J j=1 pλ (θA |ME )
J

#−1
,

j
is drawn from the posterior joint distribution (9) using the MCMC procedure. Following
where θA

the recommendation of Geweke (1999), this study uses, for the function f (θA ), a truncated normal
approximation of the posterior distribution for θA .
The formal model comparison between the two nonlinear DSGE models A1 and A2 is implemented with the estimated relative marginal likelihoods ψ̂λ (y|A1 , E) and ψ̂λ (y|A2 , E). Given
the prior model probabilities p(A1 ) and p(A2 ), the posterior odds ratio of A1 versus A2 conditionalon the reference model E isp(A1 )ψ̂λ (y|A1 , E)p(A2 )ψ̂λ (y|A2 , E)=p(A1 )p(y|A1 , E).
p(A2 )p(y|A2 , E)

When greater than one, the posterior odds ratio is in favor of the model A1 . The nonlinear DSGE
model A1 can then replicate the empirical moment distributions estimated by the reference model
E better than the other nonlinear DSGE model A2 .

3. Monte Carlo experiments on the MEI posterior sampler with anequilibrium asset-pricing model
To show how the proposed MEI posterior inference method works practically, this study
conducts Monte Carlo experiments with an equilibrium consumption-based asset-pricing modelby Labadie (1989) as a simple example. Generally, an equilibrium asset-pricing model prices arisky asset based on the conditional covariance between the stochastic discount factor and futurerisky returns. The implied multiplicity makes it infeasible for a linearized model to price a risky
asset in equilibrium. As a result, drawing a Bayesian posterior inference of an equilibrium assetpricing model conditional on time-series data of asset prices becomes quite diﬃcult. This section
shows that the proposed MEI posterior sampler easily draws a posterior inference of a nonlinear14

equilibrium asset-pricing model, as in Labadie (1989).
The population moment vector M investigated in this section is a six-dimensional column
vector. This includes the population means of the risk-free rate E(rtf ), equity premium E(ept ), andconsumption growth rate E(ln κt ), the population variances of the equity premium V(ept ) and consumption growth rate V(ln κt ), and the population ﬁrst-order autocorrelation of the consumption
growth rate Cor(ln κt );M ≡ [mi ] = [E(rtf ), E(ept ), V(ept ), E(ln κt ), V(ln κt ), Cor(ln κt )]′ .
Geweke (2010) focuses on the population means of the risk-free rate and equity premium. Thisis because the standard equilibrium asset pricing models, as in Mehra and Prescott (1985) andLabadie (1989), are not intended to explain higher-order population moments of two asset prices.
Additionally, this study includes the population variance of the equity premium and the population
ﬁrst and second moments related to the consumption growth rate to more precisely identify the
consumption growth rate process.
3.1. Labadie’s (1989) model and the true distributions of population moments
Labadie’s (1989) model is a continuous state-space generalization of Mehra and Prescott’s(1985) equilibrium consumption-based asset-pricing model. While Labadie (1989) investigates
the role of inﬂation tax in the equity premium puzzle, the Monte Carlo exercises focus only on theimplications of the model for the real rates of return, following Geweke (2010).11
A representative investor maximizes the lifetime utility function
Et

∞
X
i=0

β

1−γ
i ct+i − 11−γ,

0 < β < 1,

γ > 0,(10)

where β, γ, and Et are the subjective discount factor, degree of relative risk aversion, and mathematical conditional expectation operator, respectively. The objective function (10) is maximized,
subject to the budget constraint as follows:
qt zt+1 + pt bt+1 + ct ≤ (qt + et )zt + bt ,where ct , et , qt , pt , zt , and bt are consumption, stochastic endowment, price of the risky equity,

11

Geweke (2010) investigates four distinct consumption-based equilibrium asset-pricing models for the real rates ofreturn of risk-free and risky assets.

15

price of the risk-free bond, amount of risky equity, and amount of risk-free bonds, respectively.
The growth rate of endowment ln κt follows the AR(1) process:ln κt ≡ ln et /et−1 = δ0 + δ1 ln κt−1 + υt ,

υt ∼ i.i.d.N (0, σe2 ),(11)where δ0 , δ1 , and υt are the unconditional mean and AR root of the endowment growth rate andendowment growth shock, respectively. Endowment growth rate shock υt is assumed to follow anormal distribution with a mean of zero and variance of σe2 . Thus, κt follows a lognormal process.
Hence, the model contains ﬁve structural parameters: θA = [β, γ, µ, δ0 , δ1 , σe2 ].
In equilibrium, consumption should be equal to the endowment ct = et . Imposing thismarket-clearing condition on the ﬁrst-order necessary conditions (FONCs) for equity holdings andrisk-free bond holdings gives the following:andqt e−γ= βEt e−γtt+1 (qt+1 + et+1 ),(12)

pt e−γ= βEt e−γ
t
t+1 .

(13)

If a new variable ht is deﬁned by qt e−γ
t , then the FONC for equity holdings (12) can be rewritten
as follows:

ht = βEt (ht+1 + e1−γt+1 ).

(14)

Eq.(14) is a linear rational expectation model with respect to ht . Solving eq.(14) for ht by forward
iterations yields the fundamental solution:
ht =

∞
X

β i Et eρt+i ,

(15)

i=1

where ρ ≡ 1 − γ.
Under the assumption of log-normality of the endowment growth rate, Labadie (1989) provesthat if the long-run average growth rate of the marginal utility times the real dividend eρt is ofexponential order less than β −1 , the variable ht in eq.(15) can be solved with respect to the equity
price qt as follows:
qt = e t∞
Xa

Aj κt j ,

(16)

j=1

where A1 = β exp[ρ(δ0 + 0.5ρσ 2 )] and a1 = ρδ1 , and for j > 1,
Aj+1 = Aj β exp[(aj + ρ)(δ0 + 0.5(aj + ρ)σ 2 )],

16

and

aj+1 = (aj + ρ)δ1 .

∗Labadie (1989) provides an algorithm to ﬁnd ﬁxed points A∗ , a∗ , and Ht = A∗ κat that satisfy
qt = et Ht and solves the nonlinear equation (16). The real rate of equity return rtq is then given by
the following:∗
Ht + 1A∗ κat + 1
qt + e tq
1 + rt =
= κt
= κt
.
∗
qt−1
Ht−1
A∗ κat−1(17)

From eq.(13), the risk-free rate rtf is as follows:
1
1 + rtf

2 2 −γδ1.
= pt = βEt κ−γ
t+1 = β exp(−γδ0 + 0.5γ σ )κt(18)

Equity premium ept is deﬁned by the diﬀerence between the two rates of returns rtq and rtf : ept ≡rtq −rtf . The rate of equity return (17) and risk-free rate (18) are nonlinear functions of the structuralparameters θA and the state variable κt .
To construct the true model denoted by R, this study follows Labadie’s (1989) calibration.
The true value of β is set to 0.980, that of γ is 2.000, that of δ0 is 0.017, that of δ1 is 0.180, andthat of σe2 is 0.003. Four true moments E(rtf |R), E(ln κt |R), V(ln κt |R), and Cor(ln κt |R) can becalculated using analytical values.
γδ0γ 2 σe2 2f−1
E(rt |R) = β exp+(δ − 0.5) − 1,
1 − δ1 1 − δ12 1
σe2
δ0, V(ln κt |R) =, and Cor(ln κt |R) = δ1 .
E(ln κt |R) =1 − δ11 − δ12
The true population mean and variance of the equity premium, E(ept |R) and V(ept |R), have noanalytical representations. Given the corresponding initial state variable ln κ0 = δ0 /(1 − δ1 ),eqs.(11), (17), and (18) generate a synthetic time series of ept for Ttrue = 1, 000 quarter-periods.
Then, the true population moments E(ept |R) and V(ept |R) are approximated as the time-series
PTtrue−1mean and variance of the synthetic data of ept : E(ept |R) = Ttrue
t=1 ept and V(ept |R) =
P
Ttrue
−1
2
Ttrue
t=1 (ept − E(ept |R)) . The true moment vector MR is then constructed as follows:MR ≡ 100 × [E(rtf |R), E(ept |R), V(ept |R), E(ln κt |R), V(ln κt |R), Cor(ln κt |R)].
The second column of Table 1 displays the true population moments MR . For example, thetrue model yields 5.725 % of a mean risk-free rate E(rtf |R), 0.493 % of the mean equity premiumE(ept |R), and 27.5% of the variance in the equity premium V(ept |R). Particularly, the small meanequity premium of 0.493 % implies that Labadie’s (1989) model suﬀers from an equity premium17

puzzle.
3.2. Posterior inference of the empirical model E: Step 1
Step 1 of the MEI posterior inference method employs empirical model E to infer the empirical moment distributions from the actual data. Following Geweke (2010), this study utilizes
ﬁrst-order vector autoregression (VAR) as the empirical model E to simulate the posterior distribution of the empirical moment vector mE . Information set yt = [rtf , ept , ln κt ]′ follows a trivariateﬁrst-order VAR(1)yt − µ = F (yt−1 − µ) + ut ,ut ∼ i.i.d.N (0, Σ),(19)

where µ is the 3 × 1 column vector of the unconditional mean of yt , F is the 3 × 3 coeﬃcientmatrix, and ut is the 3 × 1 reduced-form disturbance vector that follows the multivariate normal
distribution with a mean of 0 and the variance-covariance matrix Σ. The empirical moment vector
ME is a linear mapping of VAR parameters µ, F , and Σ;
ME = 100 × [E(rtf |E), E(ept |E), V(ept |E), E(ln κt |E), V(ln κt |E), Cor(ln κt |E)]2
2
= 100 × [µ1 , µ2 , σep, µ3 , σlnκ , ρln κ ],22and σlnwhere µi for i = 1, 2, 3 is the i-th element of the constant vector µ, and σep
κ are the (2,2)thand (3,3)th elements of the variance-covariance matrix Σy , which is given as the solution of thediscrete Lyapunov equation Σy = F Σy F ′ + Σ. Finally, ρln κ is the (3,3)th element of the ﬁrst-order
autocorrelation matrix from the VAR (19).
This study implements the draws of the VAR parameters from the posterior distributionp(θE |y, E) using the Gibbs sampling procedure for the standard Normal-inverted Wishart model.
The prior joint distributions of µ, F , and Σ are set to be improper. The resulting posterior distribution of Σ follows an inverted Wishart distribution. Conditional on a draw of Σ, the posterior joint
distribution of µ and F is a multivariate normal. The realistic sample length is set to TE = 200
at a quarterly frequency. From synthetic data with 10,000 sample lengths simulated from the truemodel, the last 200 data points are extracted to construct data y. The Gibbs sampling procedure for
the Normal-inverted Wishart model simulates posterior distributions of θE and mE conditional on
the synthetic data y. The number of Gibbs sampling draws N is set to 30,000.
Figure 1 shows the empirical moment distribution ME . Each window plots the kernelsmoothed densities of the posterior distribution of an empirical moment and the corresponding true18

moment as solid gray and vertical dashed blue lines, respectively. The upper-left, upper-middle,
upper-right, lower-left, lower-middle, and lower-right windows correspond to E(rtf |E), E(ept |E),
V(ept |E), V(ln κt |E), V(ln κt |E), and Cor(ln κt |E), respectively. The third and fourth columnsof Table 2 report the posterior means and standard deviations of the empirical population moments,
respectively.
Figure 1 shows that the empirical model E suﬀers from small-sample bias in inferring thetrue population moments from the synthetic data y. The upper-left window of Figure 1 shows thatthe posterior mean of E(rtf |E) was slightly larger than the true value E(rtf |R). The third columnof Table 2 reports that the posterior mean of E(rtf |E) is 5.813, which overestimates the true valueof 5.725. However, the degree of the small-sample bias is not so severe: the posterior standard
deviation of E(rtf |E) is 0.173. This implies that the 90 % Bayesian credible interval suﬃciently
includes the true value. Regarding the ﬁve other empirical moments, the upper-middle, upper-right,
lower-left, lower-middle, and lower-right windows of Figure 1 show that the posterior distributionsof E(ept |E), E(ln κt |E), V(ept |E), V(ln κt |E), and Cor(ln κt |E) contain corresponding true
values close to their modal values within the 90% Bayesian credible intervals. Thus, even with asmall sample bias and posterior uncertainty, owing to the short length of data y, empirical model(19) infers the true values of the targeted population moments with high precision. This property ofthe empirical model E is necessary for the successful inference of the true values of the structuralparameters through the MEI posterior sampler.
3.3. Posterior inference of the structural model A: Step 2
Step 2 implements the RW-MH algorithm to infer the DSGE model A. Given a draw of θAfrom the candidate distribution in Step 2(a), the four theoretical moments E(rtf |A), E(ln κt |A),
V(ln κt |A), and Cor(ln κt |A) are calculated using their analytical values.
γδ0γ 2 σe2 2f−1
E(rt |A) = β exp+(δ − 0.5) − 1,
1 − δ1 1 − δ12 1
δ0
σe2
E(ln κt |A) =, V(ln κt |A) =, and Cor(ln κt |A) = δ1 .
1 − δ11 − δ12
The theoretical population mean and variance of the equity premium, E(ept |A) and V(ept |A), haveno analytical representations. Given the corresponding initial state variable ln κ0 = δ0 /(1 − δ1 ),eqs.(11), (17), and (18) generate a synthetic time series of ept for TA = 1, 000 sample periods.

19

Theoretical moments E(ept |A) and V(ept |A) are then approximated as the time-series mean andP Avariance, respectively, of the synthetic data of ept ; E(ept |A) = TA−1 Tt=1
ept and V(ept |A) =P
TA
TA−1 t=1(ept − E(ept |A))2 . The theoretical moment vector MA (θA ) is given by the following:
MA (θA ) ≡ 100 × [E(rtf |A), E(ept |A), V(ept |A), E(ln κt |A), V(ln κt |A), Cor(ln κt |A)].
Eq.(7) provides the JS likelihood ΣIi ln pλ (mE,i |mA,i (θA )), given the empirical moment distribution
ME .
To construct prior distributions of the structural parameters p(θA |A), this study adopts threestrategies. The ﬁrst strategy (the third-ﬁfth columns of Table 2), denoted “Prior I,” assumes thecorrectly speciﬁed informative prior distributions for the structural parameters θA with the priormeans matching the true values; the prior distribution of β is the Beta distribution with a mean of
0.980 and standard deviation of 0.001; that of γ is the Gamma distribution with a mean of 2.000
and standard deviation of 1.500; that of δ0 is the normal distribution with a mean of 0.017 andstandard deviation of 0.005; that of δ1 is the normal distribution with a mean of 0.180 and standard
deviation of 0.100; and that of σe2 is the inverted Gamma distribution with a mean of 0.003 andstandard deviation of 0.001.
The second strategy assumes the uniform distributions with suitable supports for the ﬁve
structural parameters (the sixth and seventh columns of Table 2, denoted “Prior II”). The second
prior strategy checks how well the MEI posterior sampler recovers true values of the structuralparameters θA without any prior information. Finally, the third strategy (the eighth-tenth columnsof Table 2 denoted “Prior III”), sets the prior mean of the relative risk aversion parameter γ to anincorrectly high value of 5.000 and keeps prior distributions of the other structural parameters to
the same as in Prior I. Hence, the third prior strategy scrutinizes how the MEI posterior sampler
updates the misspeciﬁed prior position of γ toward the true value of 2.000 exploiting the posterior
information of the empirical moment distributions ME .

4. Monte Carlo experiment results
This section reports the results of the Monte Carlo experiments with the MEI posterior sampler. Given the empirical posterior moments ME (reported as solid gray lines in Figure 1), the
RW-MH algorithm in Step 2 simulates the posterior distributions of the structural parameters θAthrough the theoretical moments MA .
20

Initial values of θA,0 are set to be far from the true values for initiating the RW-MH algo2
rithms: β0 = 0.95, γ0 = 1.5, δ0,0 = 0.001, δ1,0 = 0.001, and σe,0
= 0.001. Consequently, initialvalues of the theoretical population moments mA,0 are distinct from their true values. Subsequent
MCMC draws are monitored to check how the theoretical moment distributions MA converge tostationary distributions overlapping the empirical moment distributions ME and how closely simulated posterior distributions of the structural parameters θA recover the corresponding true values.
This study conﬁgures the MEI posterior sampler as follows. The ﬁnite support of E(rtf ) is[0.0, 20.0], that of E(ept ) is [−1.0, 3.0], that of V(ept ) is [0.0, 80.0], that of E(ln κt ) is [0.0, 5.0],
that of V(ln κt ) is [0.0, 80.0], and that of Cor(ln κt ) is [−0.5, 0.5]. Grid number K for discretizing
ﬁnite supports is set to 300. Number N of posterior draws for the empirical moment distribution is
set to 30,000. Subsection 2.1 shows that this study sets the number M for the theoretical moment
distribution to a minimum value of 1. In this case, weight λ becomes 0.01, and JS likelihood(5) degenerates to eq.(7). The length of the MCMC sampling is set to 200,000. The ﬁrst 20,000
MCMC samples are burned in to guarantee convergence to stationary distributions. The RW-MH
draws maintain a 40 % rejection rate by suitably setting the conﬁguration parameter τ .12
4.1. MEI posterior inferences of θA and mA under Prior I
Figure 2 plots the kernel-smoothed densities of the empirical moment distributions ME as
solid grey lines and simulated theoretical moment distributions MA under Prior I as solid blacklines. The upper-left window corresponds to the population mean of the risk-free rate, E(rtf );the upper-middle window corresponds to the population mean of the equity premium, E(ept ); theupper-right window corresponds to the population variance of the equity premium, V(ept ); thelower-left window corresponds to the population mean of the consumption growth rate, E(ln κt );the lower-middle window corresponds to the population variance of the consumption growth rate,
V(ln κt ); and the lower-right window corresponds to the population ﬁrst-order autocorrelation of
the consumption growth rate, Cor(ln κt ). The ﬁfth and sixth columns of Table 1 report the posteriormeans and standard deviations of the theoretical moment distributions under Prior I.
The upper-left, upper-right, and lower-middle windows in Figure 2 clearly show that the the12This study executes the entire procedure of the proposed MEI posterior inference by Python 3.9 codes. A MacBook
Air with the M2 chip and 24GB RAM provides suﬃciently quick implementation taking 149 minutes. Program codes
are available upon request.

21

oretical posterior distributions of E(rtf |A), V(ept |A), and V(ln κt |A) overlap with their empiricalcounterparts of E(rtf |E), V(ept |E), and V(ln κt |E). The third and ﬁfth columns of Table 1 showthat the posterior means of these theoretical population moments, 5.805, 24.86, and 28.04, arequite close to those of their empirical counterparts, 5.813, 24.86, and 29.12, respectively. Figure 2
also shows that the theoretical posterior distributions of the other population moments E(ept |A),
E(ln κt |A), and Cor(ln κt |A) overlap with their empirical counterparts E(ept |E), E(ln κt |E), andCor(ln κt |E). However, the cross-equation restrictions imposed by Labadie’s (1989) model lead to
fewer overlapping degrees. Note that in Table 1, for all theoretical moments, their posterior meansare close to not only those of the empirical moments but also true values; indeed, one posterior
standard deviation interval of the theoretical moment distributions contain the corresponding true
values.
Figure 3 depicts posterior distributions of the structural parameters θA . The upper-left, uppermiddle, upper-right, lower-left, and lower-middle windows correspond to β, γ, σe2 , δ0 , and δ1 , respectively. Each window plots the kernel-smoothed density of the posterior distribution of a structural parameter as a solid black line, the calibrated true value as the vertical dashed blue line, andthe prior distribution as the dashed red line. The third and fourth columns of Table 3 report theposterior means and standard deviations of the structural parameters under Prior I, respectively,
with the corresponding calibrated true values in the second column.
The windows in Figure 3 show that the MEI posterior sampler successfully detects the calibrated true values of the structural parameters. All posterior distributions of the structural parameters θA contain the corresponding true values close to their posterior means. Particularly, posterior
modal values of β, γ, and δ0 almost perfectly match the true values. The third column of Table 3
conﬁrms that the posterior means of the structural parameters are almost identical to the true values. Moreover, posterior distributions of the structural parameters tend to be more symmetric andnarrower than their corresponding prior distributions. Hence, the MEI posterior sampler correctly
updates prior belief on the positions of the structural parameters toward the true values by exploiting
the empirical moment distributions ME eﬀectively.
4.2. Convergence and eﬃciency of the MEI posterior sampler
To check the convergence and eﬃciency of the MEI posterior sampler, Figures 4 and 5 plot

22the time series of the simulated draws of the theoretical moments MA and structural parametersθA , respectively. The upper-left, upper-middle, upper-right, and lower-middle windows in Figure
4 show that the posterior draws of E(rtf |A), E(ept |A), V(ept |A), and V(ln κt |A), represented by
the gray-solid lines, quickly converge to stationary distributions around empirical means (blacksolid lines) and true values (dotted black lines) with negligible autocorrelations. The lower-leftand lower-right windows of Figure 4 show that the posterior draws of E(ln κt |A) and Cor(ln κt |A)converge to stationary distributions, although they are less eﬃcient with higher autocorrelations.
Simulated posterior draws of the structural parameters β, γ, σe2 , and δ0 reported in Figure 5 quickly
converged to stationary distributions around the calibrated true values with small autocorrelations,
except for the posterior draws of δ1 subject to high autocorrelation.13 Therefore, based on this visualevidence, the MEI posterior sampler achieves stable convergence to stationary posterior distributions. However, it yields diﬀerent degrees of eﬃciency over structural parameters and population
moments.
To check the convergence and eﬃciency of the MEI posterior sampler more formally, thisstudy calculates the convergence criterion |Z| proposed by Geweke (1992) and the ineﬃciency
factor proposed by Chib (2001) for each of the structural parameters as well as the population
moments.14 The second and third columns in Table 4 report the former and latter statistics, respectively. For all structural parameters and population moments, convergence criteria are less than
1.959. Hence, the null of a convergence cannot be rejected at the conventional 5 % signiﬁcancelevel. However, as shown in the fourth column, ineﬃciency factors are large for most structural parameters and population moments. The minimum and maximum values of the ineﬃciency factors
are 91 and 14,754, respectively, for E(rtf |A) and Cor(ln κt |A). Particularly, ineﬃciency factors
are greater than 2,000 for structural parameters, except σe2 . Therefore, the proposed MEI posterior sampler suﬀers from a substantial ineﬃciency problem; correct posterior inferences require
suﬃciently long posterior simulations.
4.3. Robustness check with Priors II: Role of prior information
This subsection examines the extent to which successful posterior inferences of the MEI apBy construction, simulated draws of δ1 are equivalent to those of Cor(ln κt |A).
Particularly, in this study, Geweke’s convergence criterion uses the ﬁrst 20 % and last 40% posterior draws for theﬁrst and second distributions, respectively. The ineﬃcient factor exploits the estimated autocorrelation coeﬃcients up
to 20,000 lags.
13

14

23

proach under Prior I depend on the correctly speciﬁed prior belief about the true values of structuralparameters. The sixth and seventh columns of Table 2 show that Prior II adopts uniform prior distributions for all structural parameters. Hence, the results of the MEI posterior sampler under Prior II
reﬂect only information of the empirical moment distributions ME but are independent of any priorbelief in the positions of the structural parameters. Figure 6 depicts the kernel-smoothed densitiesof the empirical moment distributions ME as solid gray lines and simulated theoretical moment
distributions MA under Prior II as solid black lines. The seventh and eighth columns of Table 1
report the posterior means and standard deviations of the theoretical moment distributions underPrior II.
Notably, each window in Figure 6 is visually indistinguishable from the corresponding window in Figure 2. Particularly, the simulated posterior distributions of E(rtf |A), V(ept |A), andV(ln κt |A) overlap with their empirical counterparts, E(rtf |E), V(ept |E), and V(ln κt |E), as
shown in Figure 2. The seventh and eighth columns of Table 1 also conﬁrm this visual impression; the posterior means and standard deviations of the theoretical moment distributions underPrior II are close and quite similar to those under Prior I.
Results from Prior II imply that the MEI posterior sampler successfully traces the empiricalmoment distributions and, as a result, recovers the true positions of the structural parameters even
without any prior information of the structural parameters. Figure 7 shows simulated posterior
distributions of the structural parameters θA under Prior II. Construction of Figure 7 is identicalto that of Figure 3 under Prior I. The MEI posterior sampler under Prior II implies the kernelsmoothed densities of the posterior distributions of the structural parameters (solid black lines) that
contain the corresponding true values (vertical blue dashed lines) close to their posterior means,even with ﬂat prior distributions (red dashed lines). However, several crucial diﬀerences in theposterior inferences of the structural parameters between Priors I and II are shown. The ﬁfth andsixth columns of Table 3 show the posterior means and standard deviations of structural parameters
under Prior II. First, the posterior mean of γ under Prior II is 2.415, which is much larger than 2.076
under Prior I and the calibrated true value of 2.000. Second, posterior standard deviations underPrior II were larger than those under Prior I.
These diﬀerences in the posterior inferences of the structural parameters are correctly re-24

ﬂected in the lower marginal likelihood under Prior II than under Prior I. The last two rows of Table
3 show the estimated log relative marginal likelihood ln ψ̂λ (y|A, E) and the posterior odds ratio(relative to Prior I) for each speciﬁcation. The estimated log relative marginal likelihood underPrior II is -719,685.2, which is much lower than that under Prior I (-719,673.8). The resulting posterior odds ratio of Prior II relative to Prior I is 0.000. The result of the formal model comparison
validates the general conjecture that correct prior information leads to a higher posterior model
probability.
4.4. Robustness check with Priors III: Updating misspeciﬁed prior information
A major concern of the MEI approach originally developed by Geweke (2010) is that themodel evaluation and comparison depend on the prior information of the structural parameters because the original MEI approach is absent from the updating process of θA through data. Hence, ifprior information is crucially misspeciﬁed, the resulting model evaluation, comparison, and criticism with the MEI approach are heavily distorted. The last subsection shows that the MEI posterior
sampler proposed in this study updates misspeciﬁed prior information correctly using informationof the empirical moment distributions. Simultaneously, the misspeciﬁed prior information of a particular structural parameter leads to biased posterior inferences of all other structural parameters
through the cross-equation restrictions of the DSGE model. The posterior odds ratio identiﬁes the
distortion caused by the misspeciﬁcation of prior information of the structural parameters.
Recall that Prior III sets the prior mean of γ to an incorrectly higher value of 5.000. Figure
8 plots the kernel-smoothed densities of the empirical and theoretical moment distributions, ME
and MA under Prior III, as shown in Figures 2 and 6 for Priors I and II. As observed in the lowerright window, the MEI posterior sampler under Prior III results in the worst overlap of the theoretical
moment distribution with the empirical one with respect to E(ln κt ). The last two columns of Table
1 also numerically conﬁrm this property of Prior III; the posterior mean of E(ln κt |A) under Prior
III is 1.706, which is much lower than those of 2.119 and 2.007 under Priors I and II, respectively,
and far from its empirical counterpart of 2.256 as well.
Figure 9 plots the kernel-smoothed densities of the posterior distributions of the structuralparameters, as in Figures 3 and 7 for Priors I and II. The upper-middle window of Figure 9 conﬁrms
that the prior distribution of γ is misspeciﬁed with a larger prior mean of 5.000. Two important25

results are worth mentioning. First, although the posterior distribution of γ remains biased to theright, the MEI posterior sampler updates the misspeciﬁed prior distribution of γ correctly, toward
the true value of 2.000. The seventh column of Table 3 shows that the posterior mean of γ underPrior III is 3.672, which is close to the true value of 2.000. Second, the misspeciﬁcation of theprior mean of γ results in biased posterior inferences on not only γ itself but also on other structuralparameters. Particularly, the posterior means of the consumption growth rate parameters δ0 and δ1are biased downward, as shown in the last two columns of Table 3. Hence, the misspeciﬁed priorbelief about the position of a single structural parameter tends to distort the posterior inferences ofthe entire parameter space of the DSGE model through cross-equation restrictions. The estimatedlog relative marginal likelihood under Prior III is -719,675.3, which is signiﬁcantly lower than that
under Prior I. The resulting posterior odds ratio of Prior III relative to Prior I was 0.223. Thus, theformal Bayesian model comparison prefers Prior I to Prior III and correctly identiﬁes the distorting
eﬀect of the misspeciﬁed prior belief of the structural parameters.

5. Concluding remarks
This study developed a distribution-matching Bayesian indirect inference method extending
the MEI approach by Geweke (2010). Following the central premise of the MEI approach, theproposed Bayesian strategy assumes that DSGE models only provide prior distributions of unobservable population moments. Based on a statistical reference model to estimate empirical moment
distributions, the developed MEI posterior sampler simulates the theoretical moment distributions
to overlap empirical moments as tightly as possible and, as a result, infers the posterior distributionsof the structural parameters of the DSGE model. Because deriving a full-information likelihood
is unnecessary, this simulation-based method makes drawing posterior inferences of potentially
misspeciﬁed nonlinear DSGE models with ﬂexible exogenous impulses possible.
This study formalizes the MEI posterior sampler utilizing the restricted DM model and its approximated representation of JS divergence as a parametric probability model of mixing discretizedempirical and theoretical moment distributions. As an inheritance of the original MEI approach,the proposed posterior inference method reﬂects sampling variations of the targeted population moments in the posterior inference of the DSGE model under investigation. Essentially, it acts as a data26

augmentation technique that avoids overﬁtting actual data and enhances the generalization abilityof the inferred DSGE model.
Monte Carlo experiments based on Labadie’s (1989) nonlinear equilibrium asset-pricing
model provide solid evidence that the proposed MEI posterior sampler successfully detects calibrated values of the structural parameters. Moreover, the estimated posterior odds ratios distinguishthe model with correctly speciﬁed prior distributions of the structural parameters from those withuniform and misspeciﬁed prior distributions.
The MEI posterior inference scheme is not necessarily speciﬁc to DSGE models. Moreover,
it is applicable to the estimation and inference problems of structural models in broad ﬁelds of
economics. Particularly, its main characteristic of distribution matching enables researchers to draw
posterior inferences of structural models with heterogeneous economic agents, which have stringent
implications on distributions of household income and ﬁrm productivity.
Finally, the proposed MEI posterior inference scheme had several limitations. First, simultaneous draws of the theoretical population moments from a nonlinear DSGE model are computationally costly for a large number M. Hence, the MEI posterior sampler is practically infeasible
over continuous variations in the weight λ in the JS likelihood. This limitation implies that an MEI
user cannot infer the degree of misspeciﬁcation of the DSGE model through the posterior odds
ratios calculated for diﬀerent values of λ, as exercised in the DSGE-VAR method by Del Negroand Schorfheide (2004). Second, more eﬃcient candidate densities in the RW-MH algorithm areneeded to improve the ineﬃciency of the MEI posterior sampler (Section 4.2). These are the main
challenges that this study’s Bayesian strategy needs to address in the future.

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Canova, F., 2007, Methods for Applied Macroeconomic Research, Princeton University Press,
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Chernozhukov, V., Hong, Han, 2003, An MCMC approach to classical estimation, Journal of
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Chib, S., 2001, Markov chain Monte Carlo methods: computation and inference , in Heckman, J.
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DeJong, D. N., Ingram, B. F., Whiteman, C. H., 1996, A Bayesian approach to calibration, Journalof Business & Economic Statistics, 14, 1-9.
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Dijk, H., eds, The Oxford Handbook of Bayesian Econometrics, 293-389, Oxford University
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Eichenbaum, M., 1991, Real business-cycle theory: wisdom or whimsy?, Journal of Economic
Dynamics and Control, 15, 607-626.
Faust, J., and Gupta, A.., 2012, Posterior predictive analysis for evaluating DSGE models, NBER
Working Paper, 17906.
Fernández-Villaverde, J., Rubio-Ramirez, J. F., Schorfheide, F., 2016, Solution and estimation
methods for DSGE models, in Taylor, J. B., Uhlig, H. eds., Handbook of Macroeconomics,
vol.2A, 527-724.
Forneron, J. J., Ng, S., 2018, The ABC of simulation estimation with auxiliary statistics, Journalof Econometric, 205, 112-139.
Gelman, A., Carlin, J. B., Stern, H. S., Rubin, D. B., 2003, Bayesian Data Analysis, second edition,
Chapman & Hall/CRC, Boca Raton, FL.
Geweke, J., 1992, Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, in Bernardo et al. eds Bayesian Statistics, vol.4, 169-194, Clarendon Press, Oxford, U.K.
Geweke, J., 1999, Using simulation methods for Bayesian econometric models: inference, development and communication, Econometric Reviews, 18, 1-126.
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Inoue, A., Shintani, M., 2018, Quasi Bayesian model selection, Quantitative Economics, 9, 12651297.
Kano, T., Nason, J. M., 2014, Business cycle implications of internal consumption habit in new
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28

Kim, J. Y., 2002, Limited information likelihood and Bayesian analysis, Journal of Econometrics,
107, 175-193.
Labadie, P., 1989, Stochastic inﬂation and the equity premium, Journal of Monetary Economics,
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Lancaster, T., 2004, An Introduction to Modern Bayesian Econometrics, Blackwell, Malden, MA.
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29

Appendix A: Proof of Proposition 1
In this appendix, I omit subscript i without loss of generality. To prove Proposition 1, ﬁrst note that the Gamma
function Γ(x) is equal to the factorial (x − 1)! for variable x. Stirling’s approximation implies that for a large numberof x, ln x! ≈ x ln x − x + 12 ln(2πx) = (x + 1) ln x − x + 12 ln(2π/x). As the last term is dominated for large x, it is
the case thatln Γ(x + 1) = ln x! ≈ (x + 1) ln x − x.
Note that Pólya distribution (4) can be rewritten as
N !(M + K − 1)! Y (nk + αk − 1)!
(N + M + K − 1)!
nk !(αk − 1)!
Kp(mE |mA ) =k=1

N !(M + K)! N + M + K Y (nk + αk )! αk(N + M + K)! M + K
nk !αk ! nk + αk
K

=

k=1

K
N !(M + K)! 1 + λ Y (nk + αk )! αk=(N + M + K)! λnk !αk ! nk + αk
k=1

Taking the logarithm of both sides of the above result and applying Stirling’s approximation of the factorial yields:ln p(mE |mA ) = ln

1+λ+ ln N ! + ln(M + K)! − ln(N + M + K)!
λ
K
K
K
K
K
X
X
X
X
X
+
ln(nk + αk )! −ln nk ! −ln αk ! +ln αk −ln(nk + αk )
k=1

k=1

k=1

k=1

k=1

1+λ
≈ ln
+ (N + 1) ln N + (M + K + 1) ln(M + K) − (N + M + K + 1) ln(N + M + K)
λ
K
K
K
X
X
X
+
(nk + αk ) ln(nk + αk ) −nk ln nk −αk ln αkk=1

= ln N +KX(nk + αk ) ln

k=1

k=1

nk + α k
N +M +K

k=1






αk
−
nk ln
−
αk ln
N
M +K
k=1
k=1

 X



  
 
K
K
X
nk + α knk + α kαknk− ln= ln N −
nk ln
−
αk ln
− ln
N
N +M +K
M +K
N +M +K
K
X

n 
k

K
X

k=1

k=1

(A.1)
Note that the term (nk + αk )/(N + M + K) in the third and fourth terms on the RHS of the last equality in eq.(A.1)
can be rewritten as the weighted average between ζk and qk with weights 1/(1 + λ) and λ/(1 + λ):
nk + α k
1
λ
=
ζk +
qk .
N +M +K1+λ1+λSubstituting this relation into eq.(A.1) establishes Proposition 1:



1
1
λ
ln p(mE |mA ) ≈ ln N − (1 + λ)N
ζk ln (ζk ) − ln
ζk +
qk
1+λ
1+λ
1+λ
k=1



K
X
λ
1
λ
− (1 + λ)N
qk ln (qk ) − ln
ζk +
qk .
1+λ
1+λ
1+λ
K
X

k=1

30

Appendix B: Approximated marginal density when λ → (K + 1)/N
Suppose that a single theoretical draw mA drops into the k ′ -th subinterval sk′ . For k ̸= k ′ , αk = 1 and
qk = 1/(K + 1). For k ′ , αk′ = 2 and qk′ = 2/(K + 1). When λ → (K + 1)/N , eq.(A.1) becomes:
 



 n  X   n + 1 X
nk + 11kk′
lim ln p(mE |mA ∈ sk ) ≈ ln N +
nk ln
− ln
+
ln
− ln
N +K +1
N
N +K +1
K +1
λ→ K+1
N
k̸=k′
k̸=k′







 
n ′ 
nk ′ + 2
nk ′ + 22k+ nk′ ln
− ln
+ 2 ln− ln
N +K +1
N
N +K +1
K +1
 



K
 n  X   n + 1 X
nk + 11kk= ln N +
nk ln
− ln
+
ln
− ln
N +K +1
N
N +K +1
K +1
k=1
k̸=k′
 



 



nk ′ + 2
2
nk ′ + 2
nk ′ + 1+ 2 ln− ln+ nk′ ln
− ln
N +K +1
K +1
N +K +1
N +K +1
The ﬁrst two terms on the RHS of the last equality are constant. The last term is approximately zero when N is
suﬃciently high. Then it is the case that


 



X   nk + 1 
1
nk ′ + 22′
ln
lim ln p(mE |mA ∈ sk ) ∝− ln+ 2 ln− ln
N +K +1
K +1
N +K +1
K +1
λ→ K+1
′
N
k̸=k


λ
1
ζk +
qk
= −(K + 1)
qk ln (qk ) − ln
1+λ
1+λ
k=1


λ
1
ζ+
q
∝ −DKL q
1+λ
1+λ


nk ′ + 2∝ ln
N +K +1λ1
λ1
1+λ ζ + 1+λ q denotes KL divergence from 1+λ ζ + 1+λ q to q. This implies


K
X


where DKL q

lim ln p(mE |mA ) ∝ −λ→ K+1NKX

I[mA ∈ sk ]DKL q

k=1

∝

K
X



I[mA ∈ sk ] ln

k=1

1
λ
ζ+
q
1+λ
1+λ




nk + 2
N +K +1

where I[mA ∈ sk ] is the indicator function that takes the value of 1 if mA drops into the k-th subinterval sk and 0
otherwise. Note that when λ → (K + 1)/N ≈ 0 the marginal likelihood is approximately proportional to the negative
KL divergence from the empirical distribution ζ to q.
Appendix C: Stationarity of the MEI sampler
The proposed MEI sampler satisﬁes the reversibility (detailed balance) condition:oldpλ (θA|ME )new
old newnewpλ (θA |ME )r(θA |θA ) = pλ (θA |ME ) min 1,
new |M )pλ (θAEnewold= min pλ (θA |ME ), pλ (θA |ME )newpλ (θA|ME )old= pλ (θA |ME ) min 1,
old |M )pλ (θAE
old
new old= pλ (θA|ME )r(θA|θA ).

31(C.1)

old new
For the Markov kernel K(θA|θA ) deﬁned byZ
old new
old newK(θA|θA ) = r(θA|θA ) +

old newoldoldnew[1 − r(θA|θA )]dθA× I[θA= θA],

old newnewoldnew= r(θA|θA ) + α(θA
)I[θA
= θA
]

the posterior distribution satisﬁesZ
old new
new
new
K(θA
|θA )pλ (θA|ME )dθAZZ
old new
new
new
new
old
new
new
new
= r(θA|θA )pλ (θA|ME )dθA+ α(θA)I[θA= θA]pλ (θA|ME )dθAZ
old new
new
new
old
old
= r(θA|θA )pλ (θA|ME )dθA+ α(θA)pλ (θA
|ME )
Z
old
old
new oldoldnew= r(θA|θA )pλ (θA|ME )dθA+ α(θA)pλ (θA|ME ) from eq. (C.1)
Z
Z
new oldoldnew
new old
newold= r(θA|θA )pλ (θA|ME )dθA+ [1 − r(θA|θA )]dθApλ (θA
|ME )
Z
Z
old
new old
newold
new old
newold= pλ (θA|ME ) + r(θA|θA )dθApλ (θA|ME ) − r(θA|θA )dθApλ (θA|ME )old= pλ (θA|ME ).

32

Table 1: True, Empirical, and Theoretical Distributions of Population Moments m

E(rtf )E(ept )V(ept )
E(ln κt )V(ln κt )Cor(ln κt )True mR

Empirical mE

5.725
0.493
27.50
2.075
31.03
0.180

Mean
5.813
0.619
24.86
2.256
29.12
0.129

S.D.
0.173
0.313
2.603
0.456
3.143
0.077

Theoretical mA
Prior I
Mean
S.D.
5.805 0.196
0.471 0.130
24.86 2.027
2.119 0.312
28.04 2.269
0.178 0.047

Theoretical mA
Prior II
Mean
S.D.
5.821 0.330
0.510 0.181
27.17 2.197
2.007 0.436
28.27 2.326
0.174 0.052

Theoretical mA
Prior III
Mean
S.D.
5.836 0.203
0.647 0.174
24.93 2.198
1.706 0.213
28.03 2.252
0.135 0.039

Table 2: True Values and Prior Distributions of Structural Parameters
True
β
γ
δ0
δ1
σe2

0.980
2.000
0.017
0.180
0.003

Prior I
Dist.
Mean
Beta
0.980
Gamma 2.000
Normal 0.017
Normal 0.180
InvGam 0.003

S.D.
0.001
1.500
0.005
0.100
0.001

Prior IIDist.
Support
Uniform [0.001, 0.999]
Uniform [0.001, 10.00]
Uniform [0.001, 0.500]
Uniform [0.001, 0.500]
Uniform [0.001, 0.100]

Prior III
Dist.
Mean
Beta
0.980
Gamma 5.000
Normal 0.017
Normal 0.180
InvGam 0.003

Table 3: Posterior Distributions of Structural Parameters
True
β
γ
δ0
δ1
σe2
ln ψ̂λ (y|A, E)
Odds Ratio vs. Prior I

0.980
2.000
0.017
0.180
0.003

Prior I
Mean S.D.
0.981 0.007
2.076 0.533
0.017 0.002
0.178 0.047
0.003 0.000-719673.8
1

33Prior II
Mean S.D.
0.981 0.011
2.415 1.044
0.016 0.003
0.174 0.052
0.003 0.000-719685.2
0.000

Prior III
Mean S.D.
0.985 0.006
3.672 0.856
0.014 0.002
0.137 0.040
0.003 0.001-719675.3
0.223

S.D.
0.001
1.500
0.005
0.100
0.001

Table 4: Convergence and Efficiency of MEI Posterior Simulator

β
γ
δ0
δ1
σe2
E(rtf |A)E(ept |A)
V(ept |A)
E(ln κt |A)V(ln κt |A)Cor(ln κt |A)

Prior I|Z|
Ineﬃciency
0.089
2,058
0.017
2,139
0.093
5,574
0.091
14,478
0.189
634
0.076
91
0.197
3,851
0.572
172
0.198
3,073
0.205
238
0.132
14,754

Prior II|Z|
Ineﬃciency
0.752
6,201
0.581
3,371
0.578
6,277
0.889
12,906
1.075
673
0.739
638
0.880
5,850
1.466
284
0.364
5,940
1.026
142
0.986
13,28534

Prior II|Z|
Ineﬃciency
0.396
1,854
0.158
1,769
0.070
3,162
0.718
5,288
0.874
847
0.010
139
1.050
3,889
1.261
376
0.763
2,020
1.167
452
0.989
5,409

35
Figure 1: Empirical Distributions of Population Moments

36
Figure 2: Empirical and Theoretical Distributions of Population Moments: Prior I

37
Figure 3: Posterior Distributions of Structural Parameters: Prior I

38
Figure 4: Convergence of Theoretical Moments mA

39
Figure 5: Convergence of Structural Parameters θA

40
Figure 6: Empirical and Theoretical Distributions of Population Moments: Prior II

41
Figure 7: Posterior Distributions of Structural Parameters: Prior II

42
Figure 8: Empirical and Theoretical Distributions of Population Moments: Prior III

43
Figure 9: Posterior Distributions of Structural Parameters: Prior III