

Department of Social Systems and Management
Discussion Paper Series

No.1289

Estimating the Markov-Switching
Almost Ideal Demand Systems:a Bayesian ApproachbySatoshi KABE and Yuichiro KANAZAWA

March 2012

UNIVERSITY OF TSUKUBA
Tsukuba, Ibaraki 305-8573
JAPAN

Estimating the Markov-Switching Almost
Ideal Demand Systems: A Bayesian approach
Satoshi KABE ∗Yuichiro KANAZAWA †Abstract
Allais and Nichèle (2007) proposed a Markov-switching almost
ideal demand system (MS-AIDS) model by extending the idea ofHamilton (1989). This model enables us to determine when the regime
shifts occurred and to estimate parameters characterized across thedifferent regimes. Moreover, degree of belongingness to each of theregimes and transitions between regimes are quantified by the probabilities. In this paper, we propose a Bayesian estimation for MSAIDS model and illustrate applicability of our proposed method. The
Bayesian estimation has some important advantages. First, it enables us to avoid the singularity problem suggested by Hamilton (1990,
1991). Second, our proposed Bayesian estimation ensures that transition probabilities lie between zero and one. Third, Bayesian estimation is able to avoid the messy calculations entailed in the scorefunctions of log-likelihood. We then run a simulation study to confirm the validity of the proposed method. In the empirical study on
the Japanese meat market, we found that our Bayesian estimation
improves the mean squared errors for all meat products over the maximum likelihood estimation, while successfully capturing the regime
shifts of meat demand coinciding with the timing of Bovine Spongiform Encephalopathy (BSE) cases in Japan and U.S.
Keywords: Bayesian estimation, Gibbs sampler, Markov-switching model,
Almost ideal demand system, Japanese meat market
JEL codes: C11, C51, D12, Q11∗

Doctoral Student, Graduate School of Systems and Information, University ofTsukuba. E-mail: k0420214@sk.tsukuba.ac.jp.
†
Professor of Statistics, Department of Social Systems and Management, University ofTsukuba. Email: kanazawa@sk.tsukuba.ac.jp.

1

Introduction

In empirical demand analysis, the almost ideal demand system (AIDS) model
proposed by Deaton and Muellbauer (1980) has been widely used. This model
assumes that market demand can be thought of as if it were the outcomeof decisions by a rational representative consumer. The AIDS model is able
to test the homogeneity and symmetry restrictions of demand theory as the
Rotterdam model (Theil, 1965) is able to and applies the second-order approximation to any arbitrary functions in the PIGLOG expenditure function
such as the Translog model (Christensen et al., 1975). Several researchers
made modifications to the original AIDS model. For instance, Cooper and
McLaren (1992) modified the model so that the range restrictions placed on
the budget share can be reflected on the model. Banks et al. (1997) also
made modification to the linear relationship between the expenditure andthe demand. These models have also been employed extensively.
In the previous studies on the structural shifts in demand, Moosa andBaxter (2002) developed the time-varying coefficient AIDS model. They
introduced the stochastic trend and seasonality terms into the linear approximated AIDS (LA-AIDS) model so that it can be applied to unstable demand
structure in the alcoholic beverage market in the U.K. Ishida et al. (2006,2010) employed the gradual switching AIDS model. They set the transition
functions into the AIDS model to capture the gradual shifts following Bovine
Spongiform Encephalopathy (henceforth BSE) and bird flu outbreaks in theJapanese meat market. The latter model assumes that researchers know thestructural change points in demand in advance.
Obviously modeling abrupt changes in demand caused by the unique exogenous events and estimating these change points are the next step. Allais
and Nichèle (2007) seem to be the first to propose a Markov-switching almost ideal demand system (MS-AIDS) model extending the idea of Hamilton(1989) 1 . This model enables us to determine when the regime shifts occurred
and to estimate parameters characterized across the different regimes. Moreover, degree of belongingness to each of the regimes and transitions between
regimes are quantified by the probabilities. They analyzed the French meatand fish demands over the period 1991 - 2002 and detected the abrupt changes
due to the two BSE crises in France. Kabe and Kanazawa (2012) also assessed the structural change points in the Japanese meat market during 1998
- 2006 via MS-AIDS model. They found the structural change point coincid1

Hamilton (1989) proposed the Markov-switching model to date the timing of recessions
and booms with real gross national product (GNP) data in U.S. He found that the regime
shift from positive to negative growth rate has a recurrent feature of the U.S. businesscycle.

1

ing with the timing of first reported case of BSE, but not of bird flu. In bothof these instances, MS-AIDS model is found to be quite effective in detecting
abrupt changes in demand using monthly aggregate data.
In Allais and Nichèle (2007), they estimate the parameters including transition probabilities via maximum likelihood (ML) estimation. However, whenthe variance-covariance matrices differ between regimes, a singularity problem arises when the determinant of variance-covariance matrix is close to zero,sending log-likelihood of MS-AIDS model to infinity, and making numerical
optimization methods (e.g., Newton-Raphson method) break down. This
problem is well-known in the literature on estimation of mixture of normal
distributions. They also estimate the transition probabilities via ML estimation without any constraints, although the transition probabilities have to lie
between zero and one inclusive.
To avoid the singularity problem, subjective judgment is required in deciding what constitutes a suitable region for plausible value of the variancecovariance matrices, so that Hamilton (1990, 1991) suggests the Bayesianestimation as a simple solution of the singularity problem. Hamilton (1991,
p.37) stated that “the [Bayesian] approach is intuitively appealing and trivial
to implement. Monte Carlo analysis suggests that this approach can consistently improve the MSE’s for a wide variety of underlying models.”
Bayesian estimation enables us to incorporate the prior information on
the variance-covariance matrices to the conjugate prior distributions. Moreover, Bayesian estimation can provide us with the posterior distributions oftransition probabilities in the unit intervals, avoiding problems associatedwith unconstrained ML estimation. Finally, Bayesian estimation is able toreplace the messy calculations entailed in the score functions of log-likelihoodfor MS-AIDS model with computationally simple Gibbs sampler. To the bestof our knowledge, no Bayesian estimation method is proposed to solve theses
problems associated with the MS-AIDS model.
In this paper, we propose a Bayesian method to estimate parametersin MS-AIDS model along with the transition probabilities. We then run asimulation study to confirm the validity of the proposed Bayesian method.
To illustrate its applicability, we take the proposed method to the Japanesemeat market data and examine the regime shifts caused by the food safetyconcerns such as BSE and bird flu.
The rest of this paper is organized as follows. Section 2 briefly describes
the Markov-switching AIDS model and introduces the necessary notations.
Then section 3 proposes the Bayesian estimation, and we illustrate our proposed method by simulation in section 4. Section 5 presents the empirical
study on the Japanese meat market via the proposed Bayesian estimation
method. Finally, section 6 discusses the merits of the proposed Bayesian2method relative to the ML estimation we employed in Kabe and Kanazawa(2012), and then we point to future directions of the research.

2

Markov-Switching AIDS model

Suppose that st is an unobserved random variable that takes an integer value
in 1, 2, . . . , K to express “regime” or “state” at time t, then budget share of ith product at time t, w̄it which is defined!
as pit qit /m0t with price pit , quantity
qit and expenditure (or budget) m0t (= i pit qit ) takes the following form:w̄it = αi,st +N"γij,st log pjt + βi,st log

j=1

#

m0t
Pt

$

(2.1)

where Pt is a price index which is defined by
N
"

N

N

1 ""
log Pt = α0,st +αk,st log pkt +γkj,st log pkt log pjt
2
j=1
k=1
k=1

(2.2)

and α0,st , αi,st , γij,st and βi,st (i, j = 1, 2, . . . , N ) are regime-dependent parameters.
The parameters in (2.1) and (2.2) have the theoretical constraints 2 asfollows[Adding up]

N
"
i=1

[Homogeneity]

αi,st = 1,N"γij,st = 0,

i=1

N
"

N
"

βi,st = 0,

(2.3a)

i=1

γij,st = 0,

(2.3b)

γij,st = γji,st .

(2.3c)

j=1[Symmetry]

Following the previous studies (Rickertsen, 1996; Allais and Nichèle, 2007;
Ishida et al., 2010), we include a trend effect, seasonal effect and habit effectinto the intercept term αi,st as
αi,st = ᾱi,st + νi,st t + δ1,i d1,t + δ2,i d2,t +N"

φij w̄j,t−1

(2.4)

j=1

2

“Adding up” guarantees that the total expenditure is equal to the sum of expenditureson the category of products under consideration. “Homogeneity” guarantees that if prices
of products increase to τ p1t , . . . , τ pN t for a scalar τ > 0, representative consumer has toincrease his expenditure from m0t to τ m0t to keep his utility level. “Symmetry” guarantees
that the substitution effect in the Slutsky equation is symmetric.

3

where d1,t and d2,t are dummy variables%%
1 if t is August1d1,t =
d2,t =
0 otherwise0

if t is December
otherwise.

As for seasonal effect, we set the dummy variables to adjust the seasonality
in budget shares. The budget shares for meat and fish are considered to shift
due to the seasonal habits (e.g., summer camp, gift-giving tradition, yearend party and so forth) in August and December. Furthermore, we include
a habit effect which is defined as a linear function of one-lagged budget
shares (Rickertsen, 1996; Allais and Nichèle, 2007).
to satisfy the!N In order !
adding up condition, we impose the restriction i=1 ᾱi,st = 1, N
i=1 νi,st = 0,!N!N!N
δ1,i = i=1 δ2,i = 0 and i=1 φij = 0. We also impose the restriction!i=1N
j=1 φij = 0 to avoid the identification problem.
Using the theoretical constraints in (2.3a), (2.3b) and (2.3c), the MSAIDS model (2.1) can be rewritten as
#
$
#
$
N
−1
"
pjt
m0t
w̄it = αi,st +γij,st log+ βi,st log
(2.5)
pN t
Pt
j=1
where i = 1, 2, . . . , N − 1. Imposing the restriction
term αi,st in (2.5) is expressed asαi,st = ᾱi,st + νi,st t + δ1,i d1,t + δ2,i d2,t +

N
−1
"
j=1

!N

j=1 φij = 0, intercept

φij (w̄j,t−1 − w̄N,t−1 ).

The MS-AIDS model employs the Markov switching mechanism which
is developed by Hamilton (1989). The Markov switching mechanism can
express switching of regimes by using the unobserved random variables that
follow the Markov process. To apply the Markov switching mechanism, we
assume that transitions between regimes are governed by a K-state Markov
chain with transition probabilities:
Pr (st = j|st−1 = i) = πij ,i, j = 1, 2, . . . , K

and the transition matrix is defined as
π11 π21 . . . πK1
 π12 π22 . . . πK2 Π =  ..
..
.. 
..
 .
.
.
. 
π1K π2K . . . πKK
where πi1 + πi2 + · · · + πiK = 1, i = 1, 2, . . . , K.
4

(2.6)

(2.7)

3

Bayesian Estimation

Let wt be a (N −1)×1 vector of budget shares at time t, w̄it (i = 1, 2, . . . , N −1) and we define the matrix of explanatory variables for regime-dependent(1)
parameters ᾱi,st , γi1,st , γi2,st , . . . , γiN −1,st , βi,st , νi,st as X t and for regime(0)independent parameters δ1,i , δ2,i , φi1 , φi2 , . . . , φi,N −1 as X t .
Given the value of price index (2.2), the MS-AIDS model (2.5) can be firstrewritten by separating the parts that depend on regimes and by including
the error term εit asw̄it = ᾱi,st +

N
−1
"
j=1

γij,st log

#

pjt
pN t

+ δ1,i d1,t + δ2,i d2,t +

$

N
−1
"
j=1

+ βi,st log

#

m0t
Pt

$

+ νi,st t

φij (w̄j,t−1 − w̄N,t−1 ) + εit(3.1)

and thus can further be rewritten as the matrix form:
(1)

(0)

wt = X t θ st + X t θ 0 + εt

(3.2)

where εt ∼ N (0, Σst ) and Σst is also regime-dependent parameter such that(1)
Σst = Σj if time t belongs to regime j. The size of the matrices X t and(0)
X t are (N − 1) × [ 3(N − 1) + N (N − 1)/2 ] and (N − 1) × (N − 1)(N + 1).
ExampleLet us consider the case that the number of products N is four. Then 3 × 15
(1)
matrix X t is defined as,(1)
Xt ≡ I3 P t M t T twhere I 3 is a 3 × 3 identity matrix,log( mP0tt )0,log( mP0tt )Mt ≡ 
m0t0log( Pt )
and





Tt ≡ 

t

0
t

0

t



,


log( pp1t) log( pp2t
) log( pp3t
)
0
0
0
4t
4t
4t
.
0
log( pp1t)0log( pp2t) log( pp3t)
0
Pt ≡ 
4t
4t
4t
p1t
p2t
p3t
0
0
log( p4t )0log( p4t ) log( p4t )5The 15 element parameter vector θ st is defined asᾱst γs t 
θ st ≡  βs tν stwhere ᾱst ≡ [ ᾱ1,st ᾱ2,st ᾱ3,st ]" , γ st ≡ [ γ11,st γ12,st γ13,st γ22,st γ23,st γ33,st ]" ,β st ≡ [ β1,st β2,st β3,st ]" and ν st ≡ [ ν1,st ν2,st ν3,st ]" .
(0)
The 3 × 15 matrix X t is defined as,(0)
X t ≡ D 1t D 2t W 1t W 2t W 3t
where

and



D 1t ≡ 

W jt ≡ 

d1,t

0
d1,t

0

d1,t

w̄j,t−1 − w̄4,t−1
0





,

D 2t ≡ 

d2,t

0
d2,t

0

d2,t



,

0
w̄j,t−1 − w̄4,t−1

The 15 element parameter vector θ 0 is defined as δ1
 δ2  θ0 ≡ 
 φ1 
 φ2 
φ3w̄j,t−1 − w̄4,t−1.

where δ 1 ≡ [ δ11 δ12 δ13 ]" , δ 2 ≡ [ δ21 δ22 δ23 ]" and φj ≡ [ φ1j φ2j φ3j ]" .
Since the MS-AIDS model is parameterized nonlinear due to the price index (2.2), the estimate cannot be written as closed form. Instead, we assume
that value of price index is already known and draw samples of parameters
via Gibbs sampler. Afterward, these samples are used to update the valueof price index and then samples of parameters are generated with the newprice index. We repeat this process until convergence.
To obtain the likelihood function of MS-AIDS model, we denote the setof variables obtained from t = 1 through time t as
Yt ≡ {w1 , w2 , . . . , wt }, St ≡ {s1 , s2 , . . . , st },Xt ≡ {x1 , x2 , . . . , xt }6

where xt is a 1 × (2N + 3) vector of explanatory variables at time t in(3.1): xt ≡ [ 1 log(p1t /pN t ) · · · log(pN −1t /pN t ) log(m0t /Pt ) t d1,t d2,tw̄1,t−1 − w̄N,t−1 · · · w̄N −1,t−1 − w̄N,t−1 ].
Then likelihood function L(·|·) is defined as
L(θ, π|YT , ST , XT ) = L(π|ST )L(θ|YT , ST , XT )(3.3)

where θ ≡ {θ 0 , θ 1 , θ 2 , . . . , θ K , Σ1 , Σ2 , . . . , ΣK } and π ≡ {πij : i, j = 1, 2, . . . , K}.
Given a prior distribution p(θ, π) = p(θ)p(π) 3 , we obtain the posterior distributions with respect to θ and to π asp(θ, π|YT , ST , XT ) ∝ L(θ, π|YT , ST , XT )p(θ, π)= L(π|ST )p(π) × L(θ|YT , ST , XT )p(θ)∝ p(π|ST ) × p(θ|YT , ST , XT ).

(3.4)

Now we compute each of the terms on the right hand side of (3.4). To be able
to do this, we first need to generate discrete latent variables s1 , s2 , . . . , sT torepresent regimes.

3.1

Sampling of latent variables s1 , s2 , . . . , sT

Since ST ≡ {s1 , s2 , . . . , sT } is a sequence of unobservable finite discrete random variables, we need to generate samples s1 , s2 , . . . , sT to compute theposterior distributions in (3.4). To generate samples of latent variables
s1 , s2 , . . . , sT , we apply the multi-move sampler (e.g., Carter and Kohn, 1994;
Chib, 1996): Given the data obtained through time t, Ωt ≡ {Yt , Xt } and setof parameters Θ ≡ {θ, π}, we consider a joint distribution f (ST |ΩT , Θ) asf (ST |ΩT , Θ) = f (s1 , s2 , . . . , sT |ΩT , Θ)
= Pr(sT |ΩT , Θ) Pr(sT −1 |sT , ΩT −1 , Θ) · · · Pr(s1 |s2 , Ω1 , Θ)
= Pr(sT |ΩT , Θ)

T
−1
.
t=1

Pr(st |st+1 , Ωt , Θ)

(3.5)

if ST ≡ {s1 , s2 , . . . , sT } is assumed to follow Markov process. This usage ofMarkovian in reverse order is justified by virtue of Bayes theorem as we see
below
Pr(st |st+1 , Ωt , Θ) =3

Pr(st+1 |st ) Pr(st |Ωt , Θ)Pr(st+1 |Ωt , Θ)That is, the prior of θ and the prior of π are independent.

7

Pr(st+1 |st ) Pr(st |Ωt , Θ)
= !K
st =1 Pr(st+1 |st ) Pr(st |Ωt , Θ)

(3.6)

where Pr(st+1 |st ) is a transition probability. Notice that Pr(st |st+1 , Ωt , Θ) in(3.5) can be computed from (3.6). The quantity Pr(st |Ωt , Θ) can be derived
by using the Hamilton filter (Hamilton, 1989).
ExampleSuppose that the number of regimes K is two, we generate the samplesof s1 , s2 , . . . , sT as follows: Drawing a random number from the uniform
distribution between 0 and 1. If the generated number is less than or equal
to the Pr(sT −1 = 1|sT , ΩT −1 , Θ) in (3.6) for t = T − 1, we set sT −1 = 1,otherwise, set sT −1 = 2. We can generate the sample of st for t = T − 1, T −2, . . . , 1 backwards this way.
In order to calculate Pr(sT −1 = 1|sT , ΩT −1 , Θ), we use sT ∼ Pr(sT |ΩT , Θ)from the Hamilton filter as stated before in both the numerator and thedenominator of (3.6). We also need π11 = Pr(sT = 1|sT −1 = 1) and π22 =Pr(sT = 2|sT −1 = 2) for st = 1 and 2. Generate π11 and π22 from (3.10) and(3.11) respectively and calculate π12 = 1 − π11 , π21 = 1 − π22 from (3.8) and(3.9) respectively.

3.2

Sampling of transition probabilities πij

Given the samples of latent variables s1 , s2 , . . . , sT , the likelihood function
L(π|ST ) appears on the right hand side of (3.4) is defined as
L(π|ST ) =

K .
K
.

n

πijij

i=1 j=1

where nij is the total number of transitions from i to j from t = 1 to t = T .
Suppose that the i-th column vector of transition matrix (2.7) is denotedby π i = [πi1 πi2 · · · πiK ]" and let the prior distribution of π i , independentlyof π j (j '= i) be a K-dimensional Dirichlet distribution 4 :π i ∼ Dir(ui1 , ui2 , . . . , uiK ),4

The Dirichlet distribution for π i is defined asp(π i |ui1 , ui2 , . . . , uiK ) =where 0 ≤ πij ≤ 1,

!K

Γ(ui0 )
uiK −1π ui1 −1 · · · πiK
Γ(ui1 ) · · · Γ(uiK ) i1

j=1 πij = 1, uij > 0, and ui0 =8!K

j=1 uij .

then posterior distribution of π i is given asp(π i |ST ) ∝ L(π i |ST )p(π i )
∝

K
.
j=1

n

uiK −1ui1 −1πijij × (πi1
· · · πiK)

niK +uiK −1
ni1 +ui1 −1 ni2 +ui2 −1
= πi1
πi2
· · · πiK.

(3.7)

Therefore we have
π i |ST ∼ Dir(ni1 + ui1 , ni2 + ui2 , . . . , niK + uiK ),i = 1, 2, . . . , K.

This corresponds to the first term on the right hand side of (3.4).
ExampleSuppose that the number of regimes K is two, likelihood function is definedasL(π|ST ) =

2 .
2
.

n

πijij

i=1 j=1
n11
n22
= π11
(1 − π11 )n12 × π22(1 − π22 )n21

= L(π11 , π22 |ST )becauseπ11 + π12 = 1,(3.8)

π21 + π22 = 1.

(3.9)

and
Then we can use beta distribution as the conjugate prior distributions withrespect to π11 and to π22 5 :π11 ∼ Beta(u11 , u12 ),π22 ∼ Beta(u22 , u21 ).

Since posterior distributions for π11 and π22 are given asp(π11 , π22 |ST ) ∝ L(π11 , π22 |ST )p(π11 )p(π22 ),5

The beta distribution for π11 is defined asp(π11 |u11 , u12 ) =Γ(u11 + u12 ) u11 −1π(1 − π11 )u12 −1
Γ(u11 )Γ(u12 ) 11

where 0 ≤ π11 ≤ 1 and u11 , u12 > 0.

9

n11 +u11 −1
n22 +u22 −1∝ π11(1 − π11 )n12 +u12 −1 × π22(1 − π22 )n21 +u21 −1 ,we haveand

π11 |ST ∼ Beta(n11 + u11 − 1, n12 + u12 − 1),(3.10)

π22 |ST ∼ Beta(n22 + u22 − 1, n21 + u21 − 1).

(3.11)

Here n11 , n12 , n21 , and n22 are the numbers of transitions from regime 1 to
1, 1 to 2, 2 to 1, and 2 to 2 observed over the course of s1 , . . . , sT , and theses1 , . . . , sT are generated in Example in section 3.1.

3.3

Sampling of parameters θ 0 , θ j , j = 1, 2, . . . , K

To evaluate the posterior distribution p(θ|YT , ST , XT ) in (3.4) via Gibbs
Ksampler, we need conditional distributions of {θ j }K
j=0 given {Σj }j=1 and ofKK{Σj }j=1 given {θ j }j=0 . To make the formulation clear, we assume that priordistribution p(θ) 6 can be written as
Kp(θ) ≡ p({θ j }K
j=0 , {Σj }j=1 )K= p({θ j }Kj=0 )p({Σj }j=1 ).

Since posterior distribution p(θ|YT , ST , XT ) in (3.4) can be rewritten as
Kp(θ|YT , ST , XT ) ≡ p({θ j }K
j=0 , {Σj }j=1 |YT , ST , XT ),we haveKp({θ j }K
j=0 , {Σj }j=1 |YT , ST , XT )K
KK∝ L({θ j }K
j=0 , {Σj }j=1 |YT , ST , XT )p({θ j }j=0 )p({Σj }j=1 ).

(3.12)K
Given {Σj }K
j=1 , YT , ST , and XT , conditional posterior distributions for {θ j }j=0
K
is expressed by dividing both sides of (3.12) by p({Σj }j=1 )K
p({θ j }K
j=0 |YT , ST , XT , {Σj }j=1 )K
K∝ L({θ j }K
j=0 , {Σj }j=1 |YT , ST , XT )p({θ j }j=0 ).

(3.13)

Given the samples of latent variables s1 , s2 , . . . , sT obtained in section
3.1 and variance-covariance matrices Σj (j = 1, 2, . . . , K) to be described in(3.17), we re-express the MS-AIDS model (3.2) as
w t = X t θ ∗ + εt6

(3.14)

In other words, we assume that the prior distribution’s location parameters {θ j }K
j=0
and scale-like parameters {Σj }Kcanbefreely
moved
and
form
a
K-dimensional
rectanj=1
gular parameter space.

10where εt ∼ N (0, Σst ). The matrix X t in (3.14) is defined as
/
0
(1)
(1)
(0)
X t = 1{st = 1}X (1)
1{s
=
2}X
·
·
·
1{s
=
K}X
X
t
t
t
t
t
t

with indicator function 1{st = j} taking scalar value 1 if st = j, or 0 otherwise 7 , and parameter vector θ ∗ is defined asθ1
 θ2 
 .. ∗θ ≡  . .
 θK 
θ0
To generate samples of θ ∗ , we derive the posterior distribution of θ ∗ conditional on Σ1 , Σ2 , . . . , ΣK from (3.13). Applying the multivariate normal
distribution N (µ, V ) as a conjugate prior p(θ ∗ ), conditional posterior distribution of θ ∗ in (3.13) is derived asp(θ ∗ |YT , ST , XT , {Σj }K
j=1 )
1
2
34
T
.
1
∗ " −1∗− N 2−1− 12∝(2π)|Σst | exp − (wt − X t θ ) Σst (wt − X t θ )
2
t=1
2
3
1 ∗
∗
− 12
" −1
× |V | exp − (θ − µ) V (θ − µ)
2
5 ∗
6
" −1
∝ exp (θ − b) B (θ ∗ − b)

where

b=B

7 T
"
t=1

−1
X "t Σ−1µst w t + V

8

, B

−1

=

T
"

−1
X "t Σ−1.
st X t + V

t=1

Then the conditional posterior distribution of θ ∗ is
θ ∗ |YT , ST , XT , {Σj }K
j=1 ∼ N (b, B) .

7

(3.15)

That is, when st = k, the matrix X t consists of k − 1 of matrices of size (N − 1) ×
(1)
[3(N − 1) + N (N − 1)/2] whose elements are all zero, X t , and K − k of matrices of size(0)(N − 1) × [3(N − 1) + N (N − 1)/2] whose elements are all zero, and X t , all aligned fromleft to right.

11

3.4

Sampling of parameters Σj , j = 1, 2, . . . , K

We assume that Σ1 , Σ2 , . . . , ΣK are independent, then conditional posterior
distribution for {Σj }K
j=1 is expressed asKp({Σj }K
j=1 |YT , ST , XT , {θ j }j=0 ) =

∝

K
.
j=1

K
.
j=1

p(Σj |YT , ST , XT , {θ j }K
j=0 )L(Σj , {θ j }K
j=0 |YT , ST , XT )p(Σj ).
(3.16)

To generate samples of Σj , j = 1, 2, . . . , K, we derive the conditional
posterior distribution of Σj from (3.16). Applying the inverse Wishart distribution IW (νj , Λj ) as a conjugate prior p(Σj ), conditional posterior distribution of Σj is derived asp(Σj |YT , ST , XT , θ ∗ )#$4. 1
1 " −1− N 2−1− 12∝(2π)|Σj | exp − εt Σj εt
2t∈{t:st =j}#$ν +(N −1)+1
1 5 −1 6− j
2
× |Σj |
exp − tr Σj Λj
2
7
%
7 T
898
"
ν +(N −1)+1+nj
1
−1
− j
"
2
∝ |Σj |
exp − tr Σjεt εt 1{st = j} + Λj2
t=1
where nj is the total number of time t belonging to regime j. Then conditional
posterior distribution of Σj is
7
8
T
"
∗
"
Σj |YT , ST , XT , θ ∼ IW νj + nj ,εt εt 1{st = j} + Λj
(3.17)
t=1

where j = 1, 2, . . . , K.
From (3.15) and (3.17), we are able to construct Gibbs sampler algorithm
by generating θ ∗ and substituting these into (3.17) and then generating Σjwith the generated θ ∗ and substituting those back into (3.15).

4

Simulation study

In this section, we illustrate our proposed method by simulation. To generate
simulation data, we consider the case that the number of products N =12

4, the number of regimes K = 2, and the number of observations T =800. The price data of four products p1t , p2t , p3t , and p4t generate fromthe uniform distributions such that p1t ∼ U (240, 380), p2t ∼ U (130, 150),
p3t ∼ U (90, 100), and p4t ∼ U (120, 200). Total expenditure (or budget) on
four products, m0t , generates from U (7500, 16000). We specify unobserved
latent variables ST = {s1 , s2 , . . . , sT } such that st = 1 if 1 ≤ t ≤ 400, andst = 2 if 401 ≤ t ≤ 800 8 .
To simplify the MS-AIDS model (2.1), parameter αi,st in (2.4) is set toαi,st = ᾱi,st and we estimate the parameters via the Gibbs sampling algorithmwith (3.6), (3.10), (3.11), (3.15) and (3.17). The prior distributions areparameterized by setting µ = 0, V = 104 I 24 , νj = 10, Λj = 10−3 I 3 (j =1, 2), u11 = u22 = 5, and u12 = u21 = 2. When we generate the samples ofparameters via the Gibbs sampling algorithm, we restrict a priori that, in
our simulated data, at least 40% of observations lie in each regime in order
to avoid identification problem.
The Gibbs sampling algorithm is run so that the first 40, 000 samples
are discarded as burn-in and then the next 40, 000 samples are recorded.
To test the convergence of samples to the posterior distributions, we apply
the Geweke (1992)’s convergence diagnostic. The first 10% and last 50% of
the recorded simulated data are used to conduct this test, as suggested by
Geweke (1992).
Tables 1 and 2 show the posterior means, posterior standard deviations(SD), 95% credible intervals (sometimes referred as 95% Bayesian confidence
intervals as well), and Geweke’s convergence diagnostic statistics (CD) forall parameters in regimes 1 and 2. We confirm that Geweke’s convergence
diagnostic statistics for all parameters are within ±1.96 range. For α’s, γ’s,and β’s, their posterior distributions are all normal, so that the means and
50% percentiles in principle coincide for the same estimated, while this is
not the case for σ’s because the posteriors for σ’s are inverse Wishart andthey are not symmetric. The Geweke’s convergence diagnostic statistics inTables 1 and 2 show that we do not reject the null hypothesis for equality ofthe means of the first 10% and the last 50% of the recorded simulated dataat the 5% significant level. Tables 1 and 2 also show that posterior meansof all parameters are similar to those true values and 95% credible intervals
include true values.
In Table 3, transition probabilities π11 and π22 pass the Geweke’s convergence diagnostic and posterior means of transition probabilities indicate
8

These number are set to mimic Japanese meat (beef, pork, chicken and fish) market
from 1998 to 2006 to be analyzed in this paper, so that the prices and the expenditures
are in Japanese yen. In the actual analysis, the number of observations are only available
monthly with 108 observations.

13

Table 1: Estimated Parameters in Regime 1

α1
α2
α3
γ11
γ12
γ13
γ22
γ23
γ33
β1
β2
β3
2
σ11
σ12
σ13
2
σ22
σ23
2
σ33

Mean
0.41010.49550.29370.0188-0.0177-0.0437
0.0437-0.02900.0973-0.0626-0.0595-0.01820.0001080.000007
0.000006
0.000099
0.000001
0.000091

SD
0.01140.01100.01090.00350.00330.00320.00710.00630.00710.00250.00240.00230.000008
0.000005
0.000005
0.000007
0.000005
0.000006

2.5%
0.38790.47400.27250.0119-0.0241-0.0500
0.0298-0.04140.0835-0.0674-0.0642-0.02270.000094-0.000003-0.000003
0.000086-0.000008
0.0000791450%
0.41010.49550.29370.0188-0.0177-0.0437
0.0437-0.0291
0.0973-0.0626-0.0595-0.01820.000107
0.000007
0.000006
0.000099
0.000001
0.000090

97.5%
CD
0.4322 1.3850
0.5170 -1.3797
0.3151 -0.2398
0.0256 -1.2725-0.0112 1.2060-0.0375 0.0054
0.0577 -0.2960-0.0168 0.2734
0.1112 -0.7818-0.0577 -1.1847-0.0549 1.2374-0.0138 -0.0367
0.000124 1.8725
0.000017 -1.0644
0.000016 0.2284
0.000114 -1.2798
0.000010 0.3664
0.000104 0.4843

True value
0.40
0.50
0.30
0.02
-0.02
-0.04
0.05
-0.03
0.10
-0.06
-0.06
-0.02
0.000100.00000
0.00000
0.000100.00000
0.00010

Table 2: Estimated Parameters in Regime 2

α1
α2
α3
γ11
γ12
γ13
γ22
γ23
γ33
β1
β2
β3
2
σ11
σ12
σ13
2
σ22
σ23
2
σ33

Mean
0.27970.40650.50980.0217-0.0279-0.05420.0531-0.0223
0.1092-0.0459-0.0719-0.0305
0.000137
0.000003
0.000000
0.000162-0.000002
0.000150

SD
2.5%
50%
0.01300.25400.27970.01400.37920.40640.01400.48210.50980.00400.01400.02170.0039-0.0356-0.0279
0.0039-0.0619-0.05420.00930.03480.05310.0085-0.0391-0.02220.00940.09070.10920.0027-0.0511-0.0459
0.0029-0.0777-0.07190.0028-0.0360-0.0306
0.000010 0.000119 0.000136
0.000007 -0.000012 0.000003
0.000007 -0.000014 0.000000
0.000011 0.000141 0.000161
0.000008 -0.000017 -0.000002
0.000011 0.000131 0.000150

97.5%
0.30510.43400.53700.0296-0.0202-0.0467
0.0715-0.0056
0.1279-0.0406-0.0663-0.02500.000157
0.000017
0.000014
0.000186
0.000013
0.000172CD
0.8683-1.7004-0.9716-1.4201
0.48810.26470.6860-0.93420.7255-0.5835
1.52171.1880-1.5203-1.5000
0.19451.1823-0.1355-0.7952

True value
0.30
0.40
0.50
0.02
-0.03
-0.05
0.05
-0.02
0.10
-0.05
-0.07
-0.03
0.000150.00000
0.00000
0.000150.00000
0.00015

that simulated budget share data tends to stay for a long time in the same
regime. This result obviously reflects the specification of latent variables
ST = {s1 , s2 , . . . , sT } such that st = 1 if 1 ≤ t ≤ 400, and st = 2 if
401 ≤ t ≤ 800. We also calculate the probability of being regime j, (j = 1, 2)
at time t asn1"(i)Pr{st = j} =1{st = j}
(4.1)
n i=1

where n is the number of recorded samples to summarize the posterior dis(i)tributions, and st is a i-th sample of latent variable st . Figure 1 shows that

π11
π22

Table 3: Estimated Transition Probability
Mean SD
2.5%
50%
97.5%
0.9951 0.0035 0.9863 0.9958 0.9994
0.9975 0.0025 0.9908 0.9983 0.9999

15CD-0.8778-0.5904

1.0
0.8
0.6
0.4
0.2
0.0

0.0

0.2

0.4

0.6

0.8

1.0

regime shift from st = 1 to st = 2 is observed at time t = 400.

0

200

400

600

800

0

Time

200

400

600

800

Time

Figure 1: The left panel shows probability of st = 1, while the right panel
shows the probability of st = 2.

5

Empirical study on Japanese meat market

In the previous studies on structural changes of demands in the Japanesemeat market, for example, Jin and Koo (2003) identified a structural change
point coinciding with the first BSE case in Japan by using the non-parametrictests. Peterson and Chen (2005) focused on the type and origin of beef products (i.e., wagyu 9 , dairy, U.S., and Australian beef) and showed the difference of impacts on the beef products due to the first BSE case in Japan.
Ishida et al. (2006, 2010) examined not only the structural change due tothe BSE but also the bird flu in Japan. Previous studies such as Peterson
and Chen (2005) and Ishida et al. (2006, 2010) applied the gradual switchingmodel proposed by Ohtani and Katayama (1986) to examine the gradualshifts in demands of meat products. This model utilizes a transition function to express a gradual shift pattern in demand under the assumption thatstarting-points of structural shift are already known. In this study, we examine the structural change points without any prior information about change
points in the Japanese meat market via the proposed Bayesian MS-AIDS
model. Also, we compute the elasticities of price and expenditure in eachregime to examine the change of consumers’ purchasing behavior pattern ofmeat products in Japan.
9Japanese native beef cattle

16

0.20
0.15
0.10
0.05

Budget Share

0.25

0.30

The Ministry of Internal Affairs and Communications in Japan provides
us with the household expenditure survey data (i.e., Family Income andExpenditure Survey). The household expenditure survey data includes themonthly time series data about average expenditure and price of meat andfish products along with others. In this study, we used the average expenditure and price data of beef, pork, chicken and fish over January 1998 to
December 2006 (108 months). Figure 2 plots the budget shares of meatproducts from January 1998 through December 2006.

Beef
Pork

0.00

Chicken

98

99

00

01

02

03

04

05

06

07

Year

Figure 2: Plot of budget share data
In the Japanese meat market, there exists serious food safety concerns
regarding to the BSE and bird flu during the period under this study. The
BSE was initially recognized in Japan in September 2001. The budget shareof beef dramatically declined after the first reported case of BSE in Japan,
while those of pork and chicken increased (see Figure 2). After the first
BSE case, budget share of beef gradually recovered, however it was still relatively small as compared with that of pork. During the study period, thefirst BSE case in U.S. was confirmed on December 2003. Although the U.S.
was one of the largest exporters to Japan, Japanese government announced
a ban on import of American beef immediately. On December 2005 theJapanese government resumed importing American beef under stricter conditions. Nevertheless banned specified-risk materials of beef products were
found from the imported beef from the U.S. on January 2006. Afterward,
17

Japanese government imposed a ban on import of U.S. beef until July 2006.
The bird flu is an infectious disease caused by avian influenza viruses(e.g., H5N1 virus). The H5N1 virus is the most highly pathogenic strain.
Many people infected with the H5N1 virus have died in Vietnam, Indonesia,
Thailand and other Southeast Asian countries. On January 2004, the first
H5N1-infected poultry were discovered in Japan. The budget share of chicken
declined slightly and then it recovered within a short period of time.

5.1

Estimation Results

We estimate parameters of MS-AIDS model (2.1) with the intercept parameters in (2.4). The Gibbs sampling algorithm is run so that the first 5, 000 samples are discarded as burn-in and then the next 25, 000 samples are recorded.
The prior distributions are parameterized by setting µ = 0, V = 104 I 45 ,
νj = 10, Λj = 10−3 I 3 (j = 1, 2), u11 = u22 = 5, and u12 = u21 = 2. We
also restrict a priori that, in our observed data, at least 40% of observations
lie in each regime in order to avoid identification problem within the Gibbs
sampling algorithm.
In this study, we examine the following four models: model 1 only includes
intercept parameter ᾱi,st . Model 2 includes seasonal effects on August andDecember into model 1, and model 3 adds a habit effect into model 2. Finally,model 4 further incorporates a trend effect into model 3.
Model 1 αi,st = ᾱi,stModel 2 αi,st = ᾱi,st + δ1,i d1,t + δ2,i d2,tModel 3 αi,st = ᾱi,st + δ1,i d1,t + δ2,i d2,t +!N

j=1 φij w̄j,t−1

Model 4 αi,st = ᾱi,st + νi,st t + δ1,i d1,t + δ2,i d2,t +!N

j=1 φij w̄j,t−1

Table 4: Log-Marginal Likelihood and Log-Bayes Factor
Model1Model2Model3Model4Model1 1098.416———
Model218.927 1117.343——
Model362.15743.230 1160.573—
Model463.45244.5251.295 1161.868
Table 4 shows the logarithmic marginal likelihood of model i, log-MLi (i =1, 2, 3, 4) as diagonal elements and logarithmic Bayes factors (see Appendix
18C), log-BFij for model i against model j as off-diagonal elements. To obtainthe marginal likelihoods for candidate models, we use the method proposed
by Newton and Raftery (1994). Although models 3 and 4 have large logarithmic marginal likelihoods relative to the other models, logarithmic Bayes
factor for model 4 against model 3, log-BF43 (= 1.295) indicates “positive”(Kass and Raftery, 1995, p.777) evidence in favor of model 4. Therefore weconclude that model 4 fits the data best.
Table 5: Estimated Parameters of MS-AIDS model in Regime1
Mean
SD
2.5%
50%
97.5%
CD
ᾱ1
0.28660.12620.04160.28500.5373 0.6055
ᾱ20.6046
0.09250.41180.60740.7780 -0.3348
ᾱ30.32850.07940.16940.32900.4824 -0.7486
γ110.0245
0.0613-0.11310.03500.1229 -0.0015
γ12-0.00390.0437-0.0701-0.0129
0.0951 0.3383
γ13-0.0549
0.0335-0.1315-0.0534
0.0060 -0.4092
γ22
0.03350.0473-0.0693
0.03680.1174 -0.1208
γ23-0.01100.0403-0.0947-0.0096
0.0653 -0.7934
γ33
0.0891
0.0480-0.00200.08790.1892 0.7943
β1-0.0301
0.0265-0.0817-0.0301
0.0226 -0.6585
β2-0.09360.0193-0.1320-0.0936-0.0553 -0.0383
β3-0.0195
0.0170-0.0524-0.0198
0.0147 0.7026
2
σ11 0.000061 0.000013 0.000041 0.000060 0.000091 1.3920
σ12 -0.000003 0.000008 -0.000018 -0.000003 0.000013 -0.5299
σ13 -0.000008 0.000007 -0.000022 -0.000008 0.000006 -0.10792
σ220.000035 0.000008 0.000024 0.000034 0.000053 0.4690
σ23 0.000006 0.000005 -0.000004 0.000006 0.000016 0.29852
σ33
0.000029 0.000006 0.000019 0.000028 0.000044 1.1468

Tables 5 and 6 show the results of parameters for beef, pork and chicken.
The parameters for fish are estimated from the adding-up condition (2.3a).
Tables 5 and 6 show the the posterior means, posterior standard deviations(SD), 95% credible intervals, and Geweke’s convergence diagnostic statistics(CD) for all parameters in MS-AIDS model (2.1) and variance-covariancematrices in regimes 1 and 2. To carry out the Geweke’s convergence diagnostic, we used the first 10% and last 50% of the recorded simulated data
and Tables 5 and 6 show that all parameters pass the Geweke’s convergence
diagnostic at 5% significant level. In the Bayesian framework, if a 95% credible interval does not include zero, estimated parameters are interpreted as
19

Table 6: Estimated Parameters of MS-AIDS model in Regime2
Mean
SD
2.5%
50%
97.5%
CD
ᾱ1
0.19260.1858-0.14180.18170.5981 -0.4185
ᾱ20.65420.12550.39930.65930.8891 -0.4349
ᾱ30.28380.10170.08120.28430.4830 -0.6659
γ11
0.10320.0682-0.0267
0.10560.2270 -0.1384
γ12-0.0484
0.0355-0.1164-0.04890.0185 0.4018
γ13-0.0560
0.0299-0.1154-0.0555
0.0011 0.9316
γ22
0.04320.0417-0.0368
0.04250.1272 0.0669
γ23
0.0236
0.0343-0.0436
0.02340.0923 -0.3294
γ33
0.06280.0399-0.0146
0.06200.1429 -1.1898
β1-0.03290.0385-0.1131-0.03180.0392 0.8659
β2-0.0949
0.0246-0.1421-0.0954-0.0454 0.5194
β3-0.01120.0210-0.0529-0.01110.0301 0.0316
2σ110.000161 0.000045 0.000096 0.000154 0.000267 -0.7579
σ12 -0.000033 0.000018 -0.000074 -0.000031 -0.000004 0.2701
σ13 -0.000029 0.000015 -0.000062 -0.000027 -0.000003 0.3091
2
σ220.000045 0.000010 0.000030 0.000044 0.000069 -1.0053
σ23 0.000004 0.000008 -0.000011 0.000004 0.000020 -0.0595
2
σ33
0.000042 0.000010 0.000027 0.000041 0.000065 -1.3308

22
2
the significant parameters. Thus ᾱ1 , ᾱ2 , ᾱ3 , β2 , σ11, σ22, and σ33
in Table 5222and ᾱ2 , ᾱ3 , β2 , σ11 , σ12 , σ13 , σ22 , and σ33 in Table 6 are regarded significantly
different from zero. These parameters in MS-AIDS model (2.1) are used to
calculate the price and expenditure elasticities.
Next we show the results of estimated parameters in the intercept term
(2.4). In Table 7, only trend effect of beef in regime 1 has a significant negative effect, while the other parameters include zero within the 95% credibleintervals. As for seasonal effects in Table 8, beef has a significant positiveeffect, while chicken has a significant negative effect both in August. On
the other hand, seasonal effects on December are not significant for all meatproducts. Finally, Table 9 shows that the budget shares of beef and pork forprevious period have positive impacts on the current budget shares.
To compare our proposed Bayesian estimation with the ML estimation
proposed in Allais and Nichèle (2007) and employed in Kabe and Kanazawa(2012), we calculate the mean squared errors (MSEs) for estimated budget
shares. The MSEs of Bayesian estimation are evaluated by the posteriormeans of estimates of budget shares generated within the Gibbs sampler. The

20

Table 7: Trend Effect in MS-AIDS model
Trend (Regime 1)
ν11 (beef)
ν21 (pork)
ν31 (chicken)

Mean
SD
2.5%
50%
97.5%
-0.00038 0.00014 -0.00064 -0.00040 -0.00007
0.00003 0.00010 -0.00018 0.00004 0.00021
0.00002 0.00008 -0.00016 0.00002 0.00017

CD
0.2165-0.9640-0.6504

Mean
SD
0.00016 0.00022
0.00019 0.00012
0.00015 0.00012

CD0.9870-0.0507-1.7246

Trend (Regime 2)
ν12 (beef)ν22 (pork)
ν32 (chicken)

2.5%-0.00029-0.00005-0.00009

50%
0.000170.000190.0001597.5%
0.00058
0.000440.00038

Table 8: Seasonal Effect in MS-AIDS model
Seasonal (Aug)
Mean
SD
2.5%
50%
97.5% CD
δ11 (beef)
0.0108 0.0040 0.0031 0.0108 0.0189 -0.1119
δ12 (pork)-0.0040 0.0026 -0.0093 -0.0040 0.0010 0.8173
δ13 (chicken)-0.0054 0.0024 -0.0101 -0.0054 -0.0006 0.7323
Seasonal (Dec)
δ21 (beef)δ22 (pork)
δ23 (chicken)

Mean
SD
2.5%
50%
97.5% CD
0.0119 0.0124 -0.0111 0.0114 0.0375 0.0939-0.0142 0.0104 -0.0355 -0.0137 0.0045 -0.3565
0.0128 0.0079 -0.0028 0.0128 0.0281 -0.6845

Table 9: Habit Effect in MS-AIDS model

φ11 (beef)
φ22 (pork)
φ33 (chicken)

Mean
0.31830.05100.4454SD
2.5%
50%
97.5% CD
0.0772 0.1636 0.3197 0.4696 -0.0619
0.0776 -0.1010 0.0513 0.2030 0.0906
0.1158 0.2136 0.4478 0.6683 -1.1871

results of MSEs with respect to our proposed Bayesian estimation (Bayes)and ML estimation (MLE) are given in Table 10. Our Bayesian estimation21

improves the MSEs for all products over ML estimation. This result reflects
the goodness of fit to the budget share data of our Bayesian estimation.
Table 10: Mean squared errors (MSEs)
Beef
Pork
Chicken
Fish
−4
−4
−4
Bayes 0.805 × 10
0.172 × 10
0.152 × 10
0.603 × 10−4
MLE 1.177 × 10−4 0.264 × 10−4 0.158 × 10−4 0.657 × 10−4
Figure 3 plots the probability of being regime 2 and budget share data ofbeef and pork from January 1998 through December 2006 under the proposed
Bayesian method. We calculate the probability Pr{st = 2} from (4.1). In
Figure 3, regime shift from st = 1 to st = 2 is observed at the timing of first
BSE case in Japan in September 2001 and then the probability Pr{st = 2}
gradually declines until the end of 2003 along with increase in budget shareof beef. With the timing of first BSE case in U.S. in December 2003, we
observe a high probability of being regime 2 once again. Since then, structureof budget share tends to stay in regime 2.
The first regime shift in Figure 3 reflects the switching of consumers’
preference from beef to pork triggered by the BSE scare in Japan in September 2001. The second regime shift after the first BSE discovery in U.S. in
December 2003 might have arisen due to the ban on import of American
beef. Since the ban on importing U.S. beef led to the shortage of beef supply
in the domestic meat market, consumers may have been forced to purchasemore pork instead of beef. In addition, Table 11 shows that posterior meansof transition probabilities π11 and π22 are relatively high. This implies that
there is little chance for switching from regime 1 to regime 2 and from regime
2 to regime 1.
Figure 4 shows the results of probability of being regime 2, Pr(st =: under the ML estimation. We estimated the probability from the2|Ωt , Θ)
Hamilton filter using data set obtained through time t, Ωt , and ML esti: in model 4 (see Kabe and Kanazawa, 2012). The regime shift at
mates Θthe timing of first BSE case in Japan in September 2001 is observed in Figure
4. Nevertheless, when compared with the result of regime shift in Figure 3,
the probability’s gradual decline due to the recovery of beef budget sharefollowing the first BSE case in Japan observed in Figure 3 no longer can beobserved in Figure 4. Unlike Figure 3, we cannot identify the regime shift at
the timing of first BSE case in U.S. in December 2003 in Figure 4.
We calculate the average budget share of i-th product at regime st = j

22

0.25
0.15
0.10
0.05

Budget share

0.20

1.00
0.80
0.60
0.40

Probability of being Regime2

0.20

Probability of being Regime2
Budget share of Beef

0.00

0.00

Budget share of Pork

98

99

00

01

02

03

04

05

06

07

Year

Figure 3: Probability of being regime 2, Pr{st = 2} and budget share data ofbeef and pork under the proposed Bayesian estimation. Two vertical dashed lines
indicate the first BSE case in Japan on September 2001 and the first BSE case in
U.S. on December 2003.

Table 11: Estimated Transition Probabilities

π11
π22

as

Mean SD
0.9520 0.0282
0.9619 0.0300

!T

2.5%
0.88300.8868t=1
w̄i,st =j = !
T50%
0.95720.96891{st = j}w̄itt=1 1{st = j}

97.5%
0.9911
0.9979CD-0.1400
1.0742.

Table 12 shows that regime 1 is characterized by a higher beef budget sharerelative to that of pork, while regime 2 is characterized by the reversal ofthese two budget shares.
Since substitution occurs mostly between beef and pork in regimes 1 and
2 (see Table 12), we focus on the price and expenditure elasticities for beef
and pork. Using the estimated parameters in Tables 5, 6, 7, 8 and 9, wePE
calculate the Marshallian price elasticity ηij,sand expenditure elasticity ηi,s
t
t
23

0.25
0.15
0.10
0.05

Budget share

0.20

1.00
0.80
0.60
0.40

Probability of being Regime2

0.20

Probability of being Regime2
Budget share of Beef

0.00

0.00

Budget share of Pork

98

99

00

01

02

03

04

05

06

07

Year

: and budget share data
Figure 4: Probability of being regime 2, Pr(st = 2|Ωt , Θ)of beef and pork under ML estimation.

Table 12: Posterior mean of average budget share
Regime 1 Regime 2
Beef
0.20750.1799Pork
0.18250.2071Chicken0.08780.0961
Fish
0.52220.5170at regime st for each 25, 000 samples generated via Gibbs sampler as
;
<
N
"
γ
β
ij,st
i,st
P
ηij,s
= −κij +−αj,st +
γkj,st log p̄k,st ,tw̄i,stw̄i,st
k=1
E
ηi,s
=
t

βi,st
+ 1,
w̄i,st

(5.1)
(5.2)

where κij = 1 for i = j and κij = 0 for i '= j, and p̄k,st is an average price atregime st .
In Table 13, we show the posterior means and 95% credible intervals of
price and expenditure elasticities for beef and pork. Although own-price elas24

ticities of pork have significant negative impacts in both regimes, own-priceelasticity of beef in regime 2 includes zero within the 95% credible interval.
Since American beef was banned and in short supply then, beef prices tended
to increase in regime 2. Hence this price inelastic beef purchasing behavior
in regime 2 in Table 13 implies that those who had kept purchasing beef in
regime 2 did so regardless of its price.
Moreover substitution between beef and pork due to the change of theirrelative prices is rejected statistically because cross-price elasticities are notsignificant in both regimes. This consumer behavior represented in Table 13
shows that this insignificance stayed almost unchanged over the two regimes.
Finally, expenditure elasticities of beef and pork are as expected significantly positive in both regimes and they do not seem to have changed
significantly between these regimes. In Japan, since beef unit price is almost
as twice as high as pork unit price, it is not surprising that beef expenditureelasticity is much higher than pork expenditure elasticity.
Table 13: Price elasticities and Expenditure elasticities
Regime1
Price ( ηijP )Expenditure ( ηijE )
Beef
Pork
Beef
-0.81710.06000.8552(-1.4682, -0.3440) (-0.2839, 0.5459)(0.6088, 1.1092)
Pork

0.1483(-0.2299, 0.7208)

Regime2

Price ( ηijP )Beef-0.3357(-1.0458, 0.3696)

Beef-0.5142(-1.0636, -0.0365)

Expenditure ( ηijE )Pork-0.1679(-0.5573, 0.2385)

Pork-0.0961-0.5050(-0.4100, 0.2209) (-0.9362, -0.0620)
1) 95% credible interval in parentheses

6

0.4874(0.2783, 0.6939)

0.8168(0.3694, 1.2182)
0.5421(0.3172, 0.7797)

Conclusion

In this paper, we proposed the Bayesian estimation for MS-AIDS model proposed by Allais and Nichèle (2007) and illustrated the applicability of our25

proposed method via simulated and real data. The proposed Bayesian estimation has some important advantages. First, it enables us to avoid thesingularity problem suggested in Hamilton (1990, 1991). In the Bayesianframework, we can use conjugate prior distributions to incorporate the priorinformation about variance-covariance matrices in advance. On the other
hand, ML estimation via numerical optimization methods (e.g., NewtonRaphson method) has to depend on sensible selection of initial values ofparameters to avoid singularity points on the parameter space. Second, ourproposed Bayesian estimation by design ensures that transition probabilities
be located between zero and one by generating the samples from the beta
distributions within the Gibbs sampler. Third, there is no need to calculate
the score functions of log-likelihood, unlike ML estimation which employedin Allais and Nichèle (2007) and Kabe and Kanazawa (2012), is computationally very intensive. In our Bayesian estimation, posterior distributions ofparameters are expressed as the standard formula (e.g., multivariate normal
and inverse Wishart distributions). Thus each of parameters can be easily
simulated via Gibbs sampler.
In the empirical study on the Japanese meat market, we found that ourBayesian estimation improves the mean squared errors for all meat products
compared with the ML estimation. Moreover we found the regime shift inthe budget shares of meat products in Figures 3 depicts much more sophisticated and realistic picture of regime transition than Figure 4. Specifically, in
Figure 3, probability of being regime 2, Pr{st = 2}, estimated via Bayesianestimation shows the regime shifts at the timing of first reported cases of
BSE both in Japan in September 2001 and also in U.S. in December 2003.
On the other hand, Figure 4 shows a single regime shift at the timing offirst BSE case in Japan. Since ML parameter estimates are given as the: in Figpoint solutions, the probability of being regime 2, Pr(st = 2|Ωt , Θ)
ure 4 ignores the uncertainty about parameters and making the probability: in Figure 4 closer to zero or one. Perhaps as Scott (2002,
Pr(st = 2|Ωt , Θ)
p.345) observed, ignoring uncertainly about the parameter may have contributed to such result.
Finally, we discuss the further extension of MS-AIDS model. Several
studies have extended the Hamilton (1989)’s Markov-switching model. In
particular, they focused on a useful modification of transition probabilities.
For example, Diebold et al. (1994) introduced the time-varying transition
probabilities into the Markov-switching model, and also they allowed thetransition probabilities to evolve as logistic functions of economic variables.
Alternatively, Markov-switching model assumes that latent variables controlling regime shifts are exogenous. Kim et al. (2008) relaxed this exogenous
regime-switching assumption and proposed a Markov-switching regression
26

model with endogenous regime switching. The extensions to MS-AIDS modelin these directions will be interesting for future research.

Conditional posterior distributions of θ ∗and ΣstA

To generate samples of θ ∗ , we derive the conditional posterior distributionsof θ ∗ from (3.13). Since Σ−1
st is symmetric and positive definite, there exists−1/2
a nonsingular matrix Σst such that
−1/2" −1/2
Σ−1
Σst .
st = ΣstMultiplying Σ−1/2
on the both sides of (3.14), we havest
Σ−1/2
wt = Σ−1/2
X t θ ∗ + Σ−1/2
εt .
st
st
st

(A.1)

Let us denote w̃t ≡ Σ−1/2
wt , X̃ t ≡ Σ−1/2
X t , and ε̃t ≡ Σ−1/2
εt . Applyingststst
the multivariate normal distribution N (µ, V ) as a conjugate prior p(θ ∗ ),conditional posterior distribution of θ ∗ is derived as
∗

34
1
∗ "
∗
exp − (w̃t − X̃ t θ ) (w̃t − X̃ t θ )
2
t=1
2
3
1 ∗
∗
− 12
" −1
× |V | exp − (θ − µ) V (θ − µ)
2
%
9
T
1"
= exp −(w̃t − X̃ t θ ∗ )" (w̃t − X̃ t θ ∗ )
2 t=1
2
3
1 ∗
∗
− 12
" −1
× |V | exp − (θ − µ) V (θ − µ) .
2
(A.2)

p(θ |YT , ST , XT , {Σj }Kj=1 ) ∝

T 1
.

2

!
!
"
"
Let us consider the OLS estimator θ:∗ = ( Tt=1 X̃ t X̃ t )−1 ( Tt=1 X̃ t w̃t ),then (A.2) is written asp(θ ∗ |YT , ST , XT , {Σj }K
j=1 )
%
9
T
>" =
>
1 "=
∝ exp −w̃t − X̃ t θ:∗ + X̃ t θ:∗ − X̃ t θ ∗w̃t − X̃ t θ:∗ + X̃ t θ:∗ − X̃ t θ ∗
2 t=1
2
3
1 ∗
∗
− 12
" −1
× |V | exp − (θ − µ) V (θ − µ) .
(A.3)
2
27

Since the orthogonality condition for OLS estimator θ:∗ implies that
T
"
t=1

=
>
"
X̃ t w̃t − X̃ t θ:∗ = 0,

the first term in (A.3) is expressed as
%
9
T =
T =
>
=
>
>
=
>
"
"
"
"
1
1
exp −
w̃t − X̃ t θ:∗w̃t − X̃ t θ:∗ −X̃ t θ:∗ − X̃ t θ ∗X̃ t θ:∗ − X̃ t θ ∗
2 t=1
2 t=1
%
9
T
T =
>" =
> 1"
>" "
=
>
1 "=
∗
∗
∗
∗
∗
∗
= exp −w̃t − X̃ t θ:w̃t − X̃ t θ: −θ: − θ X̃ t X̃ t θ: − θ
2 t=1
2 t=1
%
9
T
>" "
=
>
1 " = :∗
∝ exp −
θ − θ ∗ X̃ t X̃ t θ:∗ − θ ∗
2 t=1
%
7 T
8
9
>" "
=
>
1 = :∗"= exp − θ − θ ∗X̃ t X̃ tθ:∗ − θ ∗
.
(A.4)
2
t=1
Substituting (A.4) into (A.3), we havep(θ ∗ |YT , ST , XT , {Σj }K
j=1 )
%
7 T
8
9
>" "
=
>
1 = :∗"∝ exp − θ − θ ∗X̃ t X̃ tθ:∗ − θ ∗
2
t=1
2
3
1
1
∗
∗
−2
" −1
× |V | exp − (θ − µ) V (θ − µ)
2
%
7 T
8
9
>" "
=
> 1
1 = :∗
"
∗
∗
∗
∗
∗
−1
"
∝ exp − θ − θX̃ t X̃ t
θ: − θ − (θ − µ) V (θ − µ)
2
2
t=1
%
;7 T
8
<9
=
>=
>"
" "
1
∗
∗
∗
∗
∗
∗
−1
= exp − trX̃ t X̃ t
θ: − θ
θ: − θ + V (θ − µ)(θ − µ)"
2
t=1
(A.5)
where “tr” denotes the trace of matrix. We notice that
;7 T
8
<
=
>=
>"
" "
tr
X̃ X̃ tθ:∗ − θ ∗ θ:∗ − θ ∗ + V −1 (θ ∗ − µ)(θ ∗ − µ)"
t

t=1

= tr

;7 T
"
t=1

"
X̃ t X̃ t

8

=

"
"
)
)
θ:∗ θ:∗ − θ:∗ θ ∗ − θ ∗ θ:∗ + θ ∗ θ ∗

28

>

=

∗)

∗ ∗)

+ V −1 θ θ − θ ∗ µ" − µθ + µµ"

= tr

;7 T
"

"
X̃ t X̃ t

t=1

=

8

=

+ V −1 θ θ − 2µθ + µµ"

= tr

;7 T
"

"
X̃ t X̃ t + V −1

t=1

+

7 T
"

"

X̃ t X̃ t

t=1

= tr

;7 T
"

8

8

>

<

∗ ∗)

θ θ −2"X̃ t X̃ t + V −1

where
C0 =

8

∗ ∗)

θ θ −27 T""
X̃ t X̃ t

t=1

<

−2

= tr



;7 T
"

t=1

"

X̃ t X̃ t

87 T
"

"
X̃ t X̃ t + V −1"

X̃ t X̃ t

t=1

8

∗ ∗)

"
X̃ t X̃ t

8"
X̃ t X̃ t

8

t=1

%7 T
"
t=1

8

Substituting the OLS estimator θ:∗ =(A.6), we have
;7 T
8
" "
)
tr
X̃ t X̃ t + V −1 θ ∗ θ ∗

7
T
 "

>

%7 T
"

"
θ:∗ θ:∗ + V −1 µµ"

t=1

t=1

<

"
)
)
θ:∗ θ:∗ − 2θ:∗ θ ∗ + θ ∗ θ ∗

∗)

∗ ∗)

>

)
θ:∗ + V −1 µ θ ∗9∗)θ:∗ + V −1 µ θ + C 0

<

(A.6)

"
θ:∗ θ:∗ + V −1 µµ" .

=!

>−1 =!
>
"
"
T
T
into
t=1 X̃ t X̃ t
t=1 X̃ t w̃ t

8−1 7 T"θ θ −2

9

7 T
"

t=1

"

X̃ t w̃t8+ V −1 µ"X̃ t w̃t + V −1 µ

8

∗)







)

θ∗ + C 0 

θ + C0<t=1
7 t=17 T
8−1 7 T
8 
8
T


"
"
" "
)
)
"
"
= tr 
X̃ t X̃ t + V −1θ∗θ∗ − 2X̃ t X̃ t + V −1
X̃ t w̃t + V −1 µ θ ∗


t=1
t=1
t=1
<

+ C0 .

(A.7)

29

Let us denote
b=B

7 T
"

"

X̃ t w̃t + V −1 µ ,

t=1

B

−1

=

T
"

8

"

X̃ t X̃ t + V −1 ,

(A.8)
(A.9)

t=1

then (A.7) can be rewritten as
>
0
/
=
)
)
tr B −1 θ ∗ θ ∗ − 2bθ ∗ + C 0,= tr B −1 (θ ∗ − b) (θ ∗ − b)" − B −1 bb" + C 0 .

(A.10)

Substituting (A.10) into (A.5), we havep(θ ∗ |YT , ST , XT , {Σj }K
j=1 )
2
3
1 , −1 ∗"∗"−1∝ exp − tr B (θ − b) (θ − b) − B bb + C 0
2
2
3
1 , −1 ∗"∗∝ exp − tr B (θ − b) (θ − b)
2
2
3
1 ∗
"
∗
−1
= exp − (θ − b) B (θ − b) .
2

(A.11)

Since w̃t = Σ−1/2
wt and X̃ t = Σ−1/2
X t , (A.8) and (A.9) can be rewritten
st
st
as
7 T
8
"
−1
b=B
X "t Σ−1µ ,(A.12)
st w t + V
t=1

B −1 =T"−1
X "t Σ−1.
st X t + V(A.13)

t=1

From (A.11), (A.12) and (A.13), conditional posterior distribution of θ ∗ is
θ ∗ |YT , ST , XT , {Σj }K
j=1 ∼ N (b, B) .

(A.14)

To generate samples of Σj , j = 1, 2, . . . , K, we derive the conditional
posterior distribution of Σj from (3.16). Applying the inverse Wishart distribution IW (νj , Λj ) as a conjugate prior p(Σj ), conditional posterior distribution of Σj is derived as#$4. 1
1 " −1∗− N 2−1− 12
p(Σj |YT , ST , XT , θ ) ∝(2π)|Σj | exp − εt Σj εt
2t∈{t:st =j}30

$
1 5 −1 6
× |Σj |
exp − tr Σj Λj
2
#
$4
. 1
1 5 −1 " 6− N 2−1− 12=(2π)|Σj | exp − tr Σj εt εt2t∈{t:st =j}#$ν +(N −1)+1
1 5 −1 6− j
2
× |Σj |
exp − tr Σj Λj
2
7
%
98
T
"
nj
1
∝ |Σj |− 2 exp − tr Σ−1εt ε"t 1{st = j}
j
2
# t=1
$
νj +(N −1)+1
1 5 −1 6−
2
× |Σj |
exp − tr Σj Λj
2
7
%
7 T
898
"
νj +(N −1)+1+nj
1
2= |Σj |−
exp − tr Σ−1εt ε"t 1{st = j} + Λjj2
t=1
(A.15)
ν +(N −1)+1
2

− j

#

where nj is the total number of time t belonging to regime j, and 1{st = j}
is indicator function which takes 1 if st = j, otherwise, takes 0. From (A.15),
conditional posterior distribution of Σj is
7
8
T"
Σj |YT , ST , XT , θ ∗ ∼ IW νj + nj ,εt ε"t 1{st = j} + Λj
(A.16)
t=1

where j = 1, 2, . . . , K.

B

Geweke (1992)’s convergence diagnostic

Geweke (1992) proposed a convergence diagnostic for Markov chains basedon a test for equality of the means of the first and last part of a Markov
chain after a burn-in period. Given the samples of parameter {θ(j) }nj=1 , weconsider the two sample averages θ̄A and θ̄B such that
n

A
1 "
θ̄A =
θ(j)
nA j=1n
1 " (j)and θ̄B =θnB j=n∗(B.1)

where n∗ = n − nB + 1. From these values in (B.1), Geweke (1992) proposes
the following statistic called convergence diagnostic (CD):CD = Eθ̄A − θ̄B

Var(θ̄A ) + Var(θ̄B )
31

(B.2)

where Var(θ̄A ) and Var(θ̄B ) are variances of θ̄A and θ̄B . If the sequence of
θ(j) is weak stationary, (B.2) converges in distribution to the standard normal
from the central limit theorem (see Hamilton, 1994). Hence if absolute valueof CD in (B.2) is greater than 1.96, null hypothesis H0 : θ̄A = θ̄B is rejected
at 5% significant level and the convergence of Markov chain is judged not
yet achieved.
To carry out the convergence diagnostic test proposed by Geweke (1992),
we need to evaluate the variances Var(θ̄A ) and Var(θ̄B ) in (B.2). Let us
define the autocovarivance γt−s such thatFG
γt−s = Cov θ(s) , θ(t)(B.3)
then Var(θ̄A ) in (B.2) is

Var(θ̄A ) = Var

7

n

A
1 "
θ(j)
nA j=1

8

nA "
nA
F
G
1 "
= 2
Cov θ(s) , θ(t)nA s=1 t=1
n

=

n

A "
A
1 "
γt−s .
n2A s=1 t=1(B.4)

Here we can assume (B.3) because we assume the sequence of θ(j) is weakstationary. Let r = t − s, (1 − nA ≤ r ≤ nA − 1), (B.4) can be rewritten as
n

n"
A −1

n

A "
A
1 "
1
γt−s = 22nA s=1 t=1
nA

1
=
nA

r=−(nA −1)n"
A −1

r=−(nA −1)(nA − |r|)γr
#

|r|
1−
nA

$

γr .

(B.5)

We define the population spectrum at frequency ω to be∞

1 "fA (ω) =γr exp(−i∗ ωr)
2π r=−∞(B.6)√
where i∗ = −1. To evaluate (B.5), we construct the sample analog of (B.6),
which is known as the sample periodogram:
1fˆA (ω) =
2π

n"
A −1

r=−(nA −1)32γ̂r exp(−i∗ ωr).

(B.7)

If the sample size nA is large enough, then γ̂r is close enough to γr and ω = 0,
then
#
$
n"
A −1
|r|
1−
γr ≈ 2π fˆA (0).
(B.8)
nA
r=−(nA −1)

Therefore Var(θ̄A ) is estimated by 2π fˆA (0)/nA . In a similar fashion, Var(θ̄B )is also estimated by 2π fˆB (0)/nB .

C

Estimation of marginal likelihood

Given the parameter θ ∈ Θ, marginal likelihood for data y ≡ {y1 , y2 , . . . , yT }conditional on model M is defined as
Hp(y|M ) =L(θ|y, M )p(θ|M )dθ
(C.1)
Θ

where
L(θ|y, M ) =

T
.
t=1

p(yt |θ, M )and p(θ|M ) is a prior distribution for model M . To evaluate the marginal
likelihood p(y|M ) in (C.1), we suppose that p(θ|M ) is a proper density. Then
we notice thatH1=p(θ|M )dθΘ4
H 1p(y|M )p(θ|y, M )=p(θ|M )dθ
Θ L(θ|y, M )p(θ|M )H
1= p(y|M )p(θ|y, M )dθ(C.2)
Θ L(θ|y, M )wherep(y|M )p(θ|y, M )p(θ|y, M )p(θ|y, M )= L(θ|y ,M )p(θ|M ) == 1.
L(θ|y, M )p(θ|M )p(θ|y, M )p(y |M )Given the samples {θ(j) }nj=1 from the posterior distribution p(θ|y, M ), Newton and Raftery (1994) estimates the marginal likelihood p(y|M ) in (C.2)
as
; n
<−1
1"
1
p̂(y|M ) =(C.3)
n j=1 L(θ(j) |y, M )33and the Bayes factor for model i against model j is obtained as
BFij =

p̂(y|Mi ).
p̂(y|Mj )(C.4)

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