数理解析研究所講究録
第 1804 巻 2012 年 127-143

127

Estimating the Markov-Switching Almost Ideal
Demand Systems: A Bayesian approach
Satoshi KABE 1
Department of Social Systems and Management,
Graduate School of Systems and Information,
University of Tsukuba.
Yuichiro KANAZAWA 2
Graduate School of Systems and Information,
University of Tsukuba.

1

Introduction

In empirical demand analysis, the almost ideal demand system (AIDS) model proposed
by Deaton and Muellbauer (1980) has been widely used. This $mo$ del assumes that market
demand can be thought of as if it were the outcome of decisions by a rational representative
consumer. In the previous studies on the structural shifts in demand, Moosa and Baxter(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 thatit 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 thetransition functions into the AIDS model to capture the gradual shifts following Bovine
Spongiform Encephalopathy (henceforth BSE) and bird flu outbreaks in the Japanesemeat market. The latter model assumes that researchers know the structural change
points in demand in advance.
Obviously modeling abrupt changes in demand caused by the unique exogenous eventsand estimating these change points are the next step. Allais and Nich\‘ele (2007) seem to be
the first to propose a Markov-switching almost ideal demand system ( $MS$ -AIDS) model
extending the idea of Hamilton (1989) 3. 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 meat and 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 undthe structural change point coinciding with the timing of first reported case of BSE, but
not of bird flu. In both of these instances, $MS$ -AIDS model is found to be quite effective
in detecting abrupt changes in demand using monthly aggregate data.
In Allais and Nich\‘ele (2007), they estimate the parameters including transition probabilities via maximum likelihood ( $ML$ ) estimation. However, when the variance-covariance
$fo$

$1E$

-mail: k0420214@sk.tsukuba.ae.jp.

2Professor of Statistics, Email: kanazawa@sk.tsukuba.ac.jp.
3Hamilton (1989) proposed the Markov-switching model to date the tilning of recessions and booms

with real gross national product (GNP) data in U.S. He found that the regime shift from positive tonegative growth rate has a recurrent feature of the U. . business cycle.
$S$

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matrices differ between regimes, a singularity problem arises when the determinant ofvariance-covariance matrix is close to zero, sending -likelihood of $MS$ -AIDS model toinfinity, 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 zeroand one inclusive.
To avoid the singularity problem, subjective judgment is required in deciding what
constitutes a suitable region for plausible value of the variance-covariance matrices, sothat Hamilton (1990, 1991) suggests the Bayesian estimation 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 thisapproach can consistently improve the MSE’s for a wide variety of underlying models.”
Bayesian estimation enables us to incorporate the prior information on the variancecovariance matrices to the conjugate prior distributions. Moreover, Bayesian estimation
can provide us with the posterior distributions of transition probabilities in the unit intervals, avoiding problems associated with unconstrained $ML$ estimation. Finally, Bayesianestimation is able to replace the messy calculations entailed in the score functions of loglikelihood for $MS$ -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 toestimate parameters in $MS$-AIDS model along with the transition probabilities. To illustrate its applicability, we take the proposed method to the Japanese meat market dataand examine the regime shifts caused by the food safety concems such as BSE and birdflu.
The rest of this paper is organized as follows. Section 2 briefly describes the Markovswitching AIDS model and introduces the necessary notations. Then section 3 proposes
the Bayesian estimation, and section 4 presents the empirical study on the Japanese meat
market via the proposed Bayesian estimation method. Finally, section 5 discusses the
merits of the proposed Bayesian method relative to the $ML$ estimation we employed in
Kabe and Kanazawa (2012), and then we point to future directions of the research.
$\log$

2

Markov-Switching AIDS model

Suppose that is an unobserved random variable that takes an integer value in 1, 2, . . . , $K$
to express “regime” or “state” at time , then budget share of i-th product at time
which is defined aswith price , quantityand expenditure (or budget)
takes the following form:$s_{t}$$t$$p_{\’{i} t}q_{it}/m_{0t}$$t,\overline{w}_{it}$$p_{it}$$q_{it}$$m_{0t}$$(= \sum_{i}p_{it}q_{it})$$\overline{w}_{it}=\alpha_{i,s_{t}}+\sum_{j=1}^{N}\gamma_{ij,s_{t}}\log p_{jt}+\beta_{i,s_{t}}\log(\frac{m_{0t}}{P_{t}})$where$P_{t}$(2. 1)

is a price index which is defined by$\log P_{t}=\alpha_{0,s_{t}}+\sum_{k=1}^{N}\alpha_{k_{St}},\log p_{kt}+\frac{1}{2}\sum_{k=1}^{N}\sum_{j=1}^{N}\gamma_{kj,s_{t}}\log p_{kt}\log p_{jt}$

(2.2)

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and
and $\beta_{i,s_{t}}(i,j=1,2, \ldots, N)$ are regime-dependent parameters.
The parameters in (2.1) and (2.2) have the theoretical constraints 4 as follows$\alpha_{0,s_{t}},$$\alpha_{i,s_{t}},$$\gamma_{ij_{)}s_{t}}$[Adding up]$\sum_{i=1}^{N}\alpha_{i,s_{t}}=1,$[Homogeneity]$\sum_{i=1}^{N}\gamma_{ij,s_{t}}=0,$$\sum_{i=1}^{N}\beta_{i,s_{t}}=0$,(2.3a)(2.3b)$\sum_{j=1}^{N}\gamma_{ij,s_{t}}=0,$[Symmetry]$\gamma_{ij,s_{t}}=\gamma_{ji,s_{t}}$.

(2.3c)

Following the previous studies (Rickertsen, 1996; Allais and Nich\‘ele, 2007; Ishida et al.,2010), we include a trend effect, seasonal effect and habit effect into the intercept term$\alpha_{i,s_{t}}$

as

(2.4)

$\alpha_{i,s_{t}}=\overline{\alpha}_{i,s_{t}}+v_{i,s_{t}}t+\delta_{1,i}d_{1,t}+\delta_{2,i}d_{2,t}+\sum_{j=1}^{N}\phi_{ij}\overline{w}_{j,t-1}$

where$d_{1,t}$and$d_{2,t}$are dummy variables$d_{1,t}=\{\begin{array}{l}1 if t is August\{\end{array}$$d_{2,t}=$$0$otherwise1

if is December$0$

otherwise.

$t$

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, year-end party and so forth) in Augustand December. Furthermore, we include a habit effect which is defined as alinear functionof one-lagged budget shares (Rickertsen, 1996; Allais and Nich\‘ele, 2007). In order tosatisfy the adding up condition, we impose the restriction
and. We also impose the restriction
to avoid the identification problem.
Using the theoretical constraints in (2.3a), (2.3b) and (2.3c), the $MS$ -AIDS model(2.1) can be rewritten as$\sum_{i=1}^{N}\overline{\alpha}_{i,St}=1,$$\sum_{i=1}^{N}\delta_{1,i}=\sum_{i=1}^{N}\delta_{2,i}=0$$\sum_{i=1}^{N}\nu_{i,s_{t}}=0,$$\sum_{i=1}^{N}\phi_{ij}=0$$\sum_{j=1}^{N}\phi_{ij}=0$(2.5)$\overline{w}_{it}=\alpha_{i,s_{t}}+\sum_{j=1}^{N-1}\gamma_{ij,s_{t}}\log(\frac{p_{jt}}{p_{Nt}})+\beta_{i,s_{t}}\log(\frac{m_{0t}}{P_{t}})$where $i=1,2,$$N-1$ . Imposing the restriction(2.5) is expressed as$\ldots,$$\sum_{j=1}^{N}\phi_{ij}=0$, intercept term$\alpha_{i,s_{t}}=\overline{\alpha}_{i,s_{t}}+\nu_{\’{i},s_{t}}t+\delta_{1,i}d_{1,t}+\delta_{2,i}d_{2,t}+\sum_{\prime,J^{=1}}^{N-1}\phi_{ij}(\overline{w}_{j,t-1}-\overline{w}_{N,t-1})$$\alpha_{i,s_{t}}$in.

4‘Adding up” guarantees that the total expenditure is equal to the sum of expenditures on the
category of products under consideration. “Homogeneity” guarantees that if prices of products increasetofor a scalar $\tau>0$ , representative consumer has to increase his expenditure from
to
keep
to
his utility level. “Symmetry” guarantees that the substitution effect in the Slutsky equation
is symmetric.
$\tau p_{1t},$$\tau m_{0t}$$\ldots,$$\tau p_{Nt}$$m_{0t}$130
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 theMarkov switching mechanism, we assume that transitions between regimes are govemedby a -state Markov chain with transition probabilities:
$K$

(2.6)

$Pr(s_{t}=j|s_{t-1}=i)=\pi_{ij}, i,j=1,2, \ldots, K$

and the transition matrix is defined as(2.7)$\Pi=\{\begin{array}{llll}\pi_{11} \pi_{21} \cdots \pi_{K1}\pi_{12} \pi_{22} \cdots \pi_{K2}\vdots \vdots \ddots \vdots\pi_{1K} \pi_{2K} \cdots \pi_{KK}\end{array}\}$where $\pi_{i1}+\pi_{i2}+\cdots+\pi_{iK}=1,$ $i=1,2,$3$\ldots,$$K.$

Bayesian EstimationLet
be $a(N-1)\cross 1$ vector of budget shares at time $t,\overline{w}_{it}(i=1,2, \ldots, N-1)$ and we
define the matrix of explanatory variables for regime-dependent parametersasand for regime-independent parameters$w_{t}$$\overline{\alpha}_{i,s_{t}},$$\gamma_{i2,s_{t}},$$\ldots,$$\phi_{i2},$$\ldots,$$\gamma_{iN-1,s_{t}},$$\phi_{i,N-1}$

as$\beta_{i,s_{t}},$$X_{t}^{(1)}$$\delta_{1,i},$$\nu_{i,s_{t}}$$\gamma_{i1,s_{t}},$$\delta_{2,i},$$\phi_{i1},$$X_{t}^{(0)}.$Given the value of price index (2.2), the $MS$ -AIDS model (2.5) can be first rewritten
by separating the parts that depend on regimes and by including the error termas$\epsilon_{it}$$\overline{w}_{it}=\overline{\alpha}_{i,s_{t}}+\sum_{j=1}^{N-1}\gamma_{ij,s_{t}}\log(\frac{p_{jt}}{p_{Nt}})+\beta_{i,s_{t}}\log(\frac{m_{0t}}{P_{t}})+v_{i,s_{t}}t$(3. 1)$+ \delta_{1,i}d_{1,t}+\delta_{2,i}d_{2,t}+\sum_{j=1}^{N-1}\phi_{ij}(\overline{w}_{j,t-1}-\overline{w}_{N,t-1})+\epsilon_{it}$and thus can further be rewritten as the matrix form:(3.2)$w_{t}=X_{t}^{(1)}\theta_{s_{t}}+X_{t}^{(0)}\theta_{0}+\epsilon_{t}$

where
and
is also regime-dependent parameter such that
iftime belongs to regime . The size of the matrices
andare $(N-1)\cross[3(N-$$1)+N(N-1)/2]$ and $(N-1)\cross(N-1)(N+1)$ .
$\epsilon_{t}\sim \mathcal{N}(0, \Sigma_{s_{t}})$$t$$\Sigma_{s_{t}}=\Sigma_{j}$$\Sigma_{s_{t}}$$j$$X_{t}^{(1)}$$X_{t}^{(0)}$ExampleLet us consider the case that the number of products $N$ is four. Thenis defined as$X_{t}^{(1)}\equiv[I_{3} P_{t} M_{t} T_{t}]$$3\cross 15$

matrix$X_{t}^{(1)}$

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where

$I_{3}$

is a$3\cross 3$

identity matrix,$M_{t}\equiv\{log(^{m_{P_{t}}}-\Delta L)0 log(\mathscr{X}) 0log(^{m_{P_{t}}}-\Delta t)\}, T_{t}\equiv\{t0 t 0t\},$and$P_{t} \equiv[00 \log_{0}\log(\frac{}{})(\frac{p_{1t}p_{2t}p^{4t}}{p_{4t}}) \log(\frac{p_{1t}}{p_{4t}})\log(\frac{p_{3t}}{p_{4t}})0 \log(_{p}\frac{2t}{4t})00_{R} \log(\frac{}{})\log(\frac{p_{3t}}{pp_{4t}^{4t}p_{2t}})0 \log(\frac{p_{3t}}{p_{4t}})00]$

The 15 element parameter vector$\theta_{s\iota}$

is defined as$\theta_{St}\equiv\{\begin{array}{l}\overline{\alpha}_{s_{t}}\sqrt{}\gamma_{s_{t}}s_{t}v_{s_{t}}\end{array}\}$where$\overline{\alpha}_{s_{t}}\equiv[\overline{\alpha}_{1,s_{t}}\overline{\alpha}_{2,s_{t}}\overline{\alpha}_{3,s_{t}}]’,$$\beta_{2,8t}\beta_{3,s_{t}}|’$

The

and
matrix$3\cross 15$$\gamma_{s_{t}}\equiv[\gamma_{11,s_{t}}\gamma_{12,s_{t}}\gamma_{13,s_{t}}\gamma_{22,s_{t}}\gamma_{23,s_{t}}\gamma_{33,s_{t}}]’,$$\beta_{s_{t}}\equiv[\beta_{1,s_{t}}$$\nu_{s_{t}}\equiv[v_{1,s_{t}}v_{2,s_{t}}\nu_{3,s_{t}}]’.$$X_{t}^{(0)}$is defined as$X_{t}^{(0)}\equiv[D_{1t} D_{2t} W_{1t} W_{2t} W_{3t}]$where$D_{1t}\equiv\{d_{1,t}0 d_{1,t} 0d_{1,t}\}, D_{2t}\equiv\{d_{2,t}0 d_{2,t} 0d_{2,t}\},$

and$W_{jt}\equiv\{\overline{w}_{j,t-1}-\overline{w}_{4,t-1}0 \overline{w}_{j,t-1}-\overline{w}_{4,t-1} 0\overline{w}_{j,t-1}-\overline{w}_{4,t-1}\}$

The 15 element parameter vector$\theta_{0}$is defined as$\theta_{0}\equiv\{\begin{array}{l}\delta_{1}\delta_{2}\phi_{1}\phi_{2}\phi_{3}\end{array}\}$where$\delta_{1}\equiv[\delta_{11}\delta_{12}\delta_{13}]’,$$\delta_{2}\equiv[\delta_{21}\delta_{22}\delta_{23}]’$and$\phi_{j}\equiv[\phi_{1j}\phi_{2j}\phi_{3j}]’.$

Since the $MS$ -AIDS model is parameterized nonlinear due to the price index (2.2), theestimate 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, thesesamples are used to update the value of price index and then samples of parameters are
generated with the new price index. We repeat this process until convergence.

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To obtain the likelihood function of $MS$ -AIDS model, we denote the set of variables
obtained from $t=1$ through time as$t$$\mathcal{Y}_{t}\equiv\{w_{1}, w_{2}, \ldots, w_{t}\}, S_{t}\equiv\{s_{1}, s_{2}, \ldots, s_{t}\},$$\mathcal{X}_{t}\equiv\{x_{1}, x_{2}, \ldots, x_{t}\}$where$x_{t}$

is a $1\cross(2N+3)$ vector of explanatory variables at time$t$in (3.1): $x_{t}\equiv[1$$\log(p_{1t}/p_{Nt})\cdots\log(p_{N-1t}/p_{Nt})\log(m_{0t}/P_{t})td_{1,t}d_{2,t}\overline{w}_{1,t-1}-\overline{w}_{N,t-1}\cdots\overline{w}_{N-1,t-1}-$$\overline{w}_{N,t-1}].$

Then likelihood function$\mathcal{L}(\cdot|\cdot)$is defined as(3.3)$\mathcal{L}(\theta, \pi|\mathcal{Y}\tau, \mathcal{S}_{T}, \mathcal{X}_{T})=\mathcal{L}(\pi|\mathcal{S}_{T})\mathcal{L}(\theta|\mathcal{Y}\tau, \mathcal{S}_{T}, \mathcal{X}_{T})$where
and $\pi\equiv\{\pi_{ij} : i,j=1,2, \ldots, K\}$ . Givena prior distribution $p(\theta, \pi)=p(\theta)p(\pi)5$ , we obtain the posterior distributions withrespect to and to as$\theta\equiv\{\theta_{0}, \theta_{1}, \theta_{2}, \ldots, \theta_{K}, \Sigma_{1}, \Sigma_{2}, \ldots, \Sigma_{K}\}$$\theta$$\pi$$p(\theta, \pi|\mathcal{Y}_{T}, S_{T}, \mathcal{X}_{T})\propto \mathcal{L}(\theta, \pi|\mathcal{Y}\tau, S_{T}, \mathcal{X}_{T})p(\theta, \pi)$$=\mathcal{L}(\pi|S_{T})p(\pi)\cross \mathcal{L}(\theta|\mathcal{Y}\tau, S_{T}, \mathcal{X}_{T})p(\theta)$$\propto p(\pi|\mathcal{S}_{T})\cross p(\theta|\mathcal{Y}\tau,S_{T}, \mathcal{X}_{T})$

.

(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
to represent regimes.
$s_{1},$

3.1

Sampling of latent variables$s_{1},$$s_{2},$$s_{2},$$\ldots,$$\ldots,$$s_{T}$$s_{T}$

Since
is a sequence of unobservable finite discrete random variables,
we need to generate samples
to compute the posterior distributions in (3.4).
To generate samples of latent variables, we apply the multi-move sampler(e.g., Carter and Kohn, 1994; Chib, 1996): Given the data obtained through time
, we consider ajoint distribution
and set of parameters$S_{T}\equiv\{s_{1}, s_{2}, \ldots, s_{T}\}$$s_{1},$$s_{2},$$\ldots,$$s_{T}$$s_{1},$$s_{2},$$\ldots,$$s_{T}$$t,$$\Theta\equiv\{\theta, \pi\}$$\Omega_{t}\equiv$$f(\mathcal{S}_{T}|\Omega_{T}, \Theta)$$as\{\mathcal{Y}_{t}, \mathcal{X}_{t}\}$$f(\mathcal{S}_{T}|\Omega_{T}, \Theta)=f(s_{1}, s_{2}, \ldots, s_{T}|\Omega_{T},\Theta)$$=Pr(s_{T}|\Omega_{T}, \Theta)Pr(s_{T-1}|s_{T}, \Omega_{T-1}, \Theta)\cdots Pr(s_{1}|s_{2}, \Omega_{1}, \Theta)$$= Pr(s_{T}|\Omega_{T}, \Theta)\prod_{t=1}^{T-1}Pr(\mathcal{S}_{t}|s_{t+1}, \Omega_{t}, \Theta)$(3.5)

is assumed to follow Markov process. This usage of Markovian inreverse order is justified by virtue of Bayes theorem as we see below
if$S_{T}\equiv\{s_{1}, s_{2}, \ldots, s_{T}\}$$Pr(s_{t}|s_{t+1}, \Omega_{t}, \Theta)=\frac{Pr(s_{t+1}|s_{t})Pr(s_{t}|\Omega_{t},\Theta)}{Pr(s_{t+1}|\Omega_{t},\Theta)}$$= \frac{Pr(s_{t+1}|s_{t})Pr(s_{t}|\Omega_{t},\Theta)}{\sum_{s_{t}=1}^{K}Pr(s_{t+1}|s_{t})Pr(s_{t}|\Omega_{t},\Theta)}$

(3.6)

where
is a transition probability. Notice thatin (3.5) can becomputed from (3.6). The quantity
can be derived by using the Hamilton
filter (Hamilton, 1989).
5That is, the prior of and the prior of are independent.
$Pr(\mathcal{S}_{t+1}|s_{t})$$Pr(s_{t}|\mathcal{S}_{t+1}, \Omega_{t}, \Theta)$$Pr(s_{t}|\Omega_{t}, \Theta)$$\theta$

$\pi$

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3.2

Sampling of transition probabilities$\pi_{ij}$Given the samples of latent variables, the likelihood function
pears on the right hand side of (3.4) is defined as$s_{1},$$s_{2},$$\ldots,$$s_{T}$$\mathcal{L}(\pi|S_{T})$ap-$\mathcal{L}(\pi|\mathcal{S}_{T})=\prod_{\iota’=1}^{K}\prod_{j=1}^{K}\pi_{ij}^{n_{ij}}$

whereis the total number of transitions from to from $t=1$ to $t=T.$
Suppose that the i-th column vector of transition matrix (2.7) is denoted byand let the prior distribution of , independently of $\pi_{j}(j\neq i)$ be a$K$-dimensional Dirichlet distribution 6 :$i$$n_{ij}$$j$$\pi_{i}=$$[\pi_{i1}\pi_{i2}\cdots\pi_{iK}]’$$\pi_{i}$$\pi_{i}\sim Dir(u_{i1}, u_{i2}, \ldots, u_{iK})$then posterior distribution of$\pi_{i}$,is given as$p(\pi_{i}|\mathcal{S}_{T})\propto \mathcal{L}(\pi_{i}|\mathcal{S}_{T})p(\pi_{i})$$\propto\prod_{j=1}^{K}\pi_{ij}^{n_{ij}}\cross(\prime K$(3.7)$=\pi_{i1}^{n_{i1}+u_{i1}-1}\pi_{i2}^{n_{i2}+u_{i2}-1}\cdots\pi_{iK}^{n_{\iota K}+u_{iK}-1}$

Therefore we have$\pi_{i}|S_{T}\sim Dir(n_{i1}+u_{i1}, n_{i2}+u_{i2}, \ldots, n_{iK}+u_{iK}) , i=1,2, \ldots, K.$

This corresponds to the first term on the right hand side of (3.4).

3.3

Sampling of parameters

To evaluate the posterior distribution$\theta_{0},$$\theta_{j},$$j=1,2,$$p(\theta|\mathcal{Y}_{T}, \mathcal{S}_{T}, \mathcal{X}_{T})$distribut,$\ldots,$$K$in (3.4) via Gibbs sampler, we needmake$theformu1ationc1eai^{onsof\{\theta_{j}\}_{j=0}^{K}given\{\Sigma_{j}\}_{j=1}^{K}andof\{\Sigma_{j}\}_{\theta^{=1}}^{K}given\{\theta_{j}\}_{j=0}^{K}.To}weassumethatpriordistributionp(\theta)canbewrittenasConditiona1$$p(\theta)\equiv p(\{\theta_{j}\}_{j=0}^{K}, \{\Sigma_{j}\}_{j=1}^{K})$$=p(\{\theta_{j}\}_{j=0}^{K})p(\{\Sigma_{j}\}_{j=1}^{K})$Since posterior distribution$p(\theta|\mathcal{Y}_{T}, S_{T}, \mathcal{X}_{T})$.

in (3.4) can be rewritten as,$p( \theta|\mathcal{Y}_{T}, \mathcal{S}_{T}, \mathcal{X}_{T})\equiv p(\{\theta_{j}\}_{j=0}^{K}, \{\sum_{j}\}_{j=1}^{K}|\mathcal{Y}_{T}, \mathcal{S}_{T}, \mathcal{X}_{T})$6The Dirichlet distribution for$\pi_{i}$is defined as$p( \pi_{i}|u_{i1},u_{i2}, \ldots,u_{iK})=\frac{\Gamma(u_{i0})}{\Gamma(u_{i1})\cdots\Gamma(u_{iK})}\pi_{i1}^{u-1}\cdots\pi_{iK}^{u-1}i1:K$

where $0\leq\pi_{ij}\leq 1,$$u_{ij}>0$ , and7In other words, we assume that the prior distribution’s location parametersand scale-likeparameters
can be freely moved and form a $K$ -dimensional rectangular parameter space.
$\sum_{j=1}^{K}\pi_{ij}=1,$$u_{i0}= \sum_{j=1}^{K}u_{ij}.$$\{\theta_{j}\}_{j=0}^{K}$$\{\Sigma_{j}\}_{j=1}^{K}$134
we have$p(\{\theta_{j}\}_{j=0}^{K}, \{\Sigma_{j}\}_{j=1}^{K}|\mathcal{Y}\tau, S_{T}, \mathcal{X}_{T})$$\propto \mathcal{L}(\{\theta_{j}\}_{j=0}^{K}, \{\Sigma_{j}\}_{j=1}^{K}|\mathcal{Y}\tau, \mathcal{S}_{T}, \mathcal{X}_{T})p(\{\theta_{j}\}_{j=0}^{K})p(\{\Sigma_{j}\}_{j=1}^{K})$, conditional posterior distributions for, andpressed by dividing both sides of (3.8) by

Given$\{\Sigma_{j}\}_{j=1}^{K},$$S_{T}$$\mathcal{Y}\tau,$$\mathcal{X}_{T}$

(3.8)

.
$\{\theta_{j}\}_{j=0}^{K}$

is ex-$p(\{\Sigma_{j}\}_{j=1}^{K})$$p(\{\theta_{j}\}_{j=0}^{K}|\mathcal{Y}_{T}, \mathcal{S}_{T}, \mathcal{X}_{T}, \{\Sigma_{j}\}_{j=1}^{K})$

.

(3.9)

$\propto \mathcal{L}(\{\theta_{j}\}_{j=0}^{K}, \{\Sigma_{j}\}_{j=1}^{K}|\mathcal{Y}_{T}, S_{T}, \mathcal{X}_{T})p(\{\theta_{j}\}_{j=0}^{K})$Given the samples of latent variables$s_{1},$

covariance matrices$MS$ -AIDS model (3.2) as$\Sigma_{j}(j=1,2, \ldots, K)$

obtained in section 3.1 and varianceto be described in (3.13), we re-express the$s_{2},$$\ldots,$$s_{T}$(3.10)$w_{t}=X_{t}\theta^{*}+\epsilon_{t}$where$\epsilon_{t}\sim \mathcal{N}(0, \Sigma_{s_{t}})$. The matrix$X_{t}=[1\{s_{t}=1\}X_{t}^{(1)}$$X_{t}$in (3.10) is defined as$1\{s_{t}=2\}X_{t}^{(1)}$...

$1\{s_{t}=K\}X_{t}^{(1)}$with indicator function $1\{s_{t}=j\}$ taking scalar value 1 if $s_{t}=j$ , orparameter vector
is defined as$0$$X_{t}^{(0)}]$otherwise 8, and$\theta^{*}$$\theta^{*}\equiv\{\begin{array}{l}\theta_{1}\theta_{2}\vdots\theta_{K}\theta_{0}\end{array}\}$conditional on
To generate samples of , we derive the posterior distribution ofasfrom (3.9). Applying the multivariate normal distributionin (3.9) is derived as, conditional posterior distribution ofa conjugate prior$\theta^{*}$$\theta^{*}$$\Sigma_{1},$$\mathcal{N}(\mu, V)$$\Sigma_{K}$$\Sigma_{2},$$\ldots,$$\theta^{*}$$p(\theta^{*})$$p(\theta^{*}|\mathcal{Y}\tau, \mathcal{S}_{T}, \mathcal{X}_{T}, \{\Sigma_{j}\}_{j=1}^{K})$$\propto\prod_{t=1}^{T}[(2\pi)^{-\frac{N-1}{2}}|\Sigma_{s_{t}}|^{-\frac{1}{2}}\exp\{-\frac{1}{2}(w_{t}-X_{t}\theta^{*})’\Sigma_{s_{t}}^{-1}(w_{t}-X_{t}\theta^{*})\}]$$\cross|V|^{-\frac{1}{2}}\exp\{-\frac{1}{2}(\theta^{*}-\mu)’V^{-1}(\theta^{*}-\mu)\}$$\propto\exp\{(\theta^{*}-b)’B^{-1}(\theta^{*}-b)\}$where$b=B( \sum_{t=1}^{T}X_{t}’\Sigma_{s_{t}}^{-1}w_{t}+V^{-1}\mu), B^{-1}=\sum_{t=1}^{T}X_{t}’\Sigma_{s_{t}}^{-1}X_{t}+V^{-1}$

Then the conditional posterior distribution of$\theta^{*}$

is$\theta^{*}|\mathcal{Y}_{T}, \mathcal{S}_{T}, \mathcal{X}_{T}, \{\Sigma_{j}\}_{j=1}^{K}\sim \mathcal{N}(b, B)$.

(3.11)

8That is, when $s_{t}=k$ , the matrix consists of $k-1$ of matrices of size $(N-1)\cross[3(N-1)+N(N-1)/2]$, and $K-k$ of matrices of size $(N-1)\cross[3(N-1)+N(N-1)/2]$ whose
whose elements are all zero,, all aligned from left to right.
elements are all zero, and$X_{t}$$X_{t}^{(1)}$$X_{t}^{(0)}$

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3.4

Sampling of parameters

We assume thatforis expressed as$\Sigma_{1},$$\Sigma_{K}$$\Sigma_{2},$$\ldots,$$\Sigma_{j},$$j=1,2,$$\ldots,$$K$

are independent, then conditional posterior distribution$\{\Sigma_{j}\}_{j=1}^{K}$$p( \{\Sigma_{j}\}_{j=1}^{K}|\mathcal{Y}_{T}, S_{T}, \mathcal{X}_{T}, \{\theta_{j}\}_{j=0}^{K})=\prod_{j=1}^{K}p(\Sigma_{j}|\mathcal{Y}\tau, \mathcal{S}_{T}, \mathcal{X}_{T}, \{\theta_{j}\}_{j=0}^{K})$$\propto\prod_{j=1}^{K}\mathcal{L}(\Sigma_{j}, \{\theta_{j}\}_{j=0}^{K}|\mathcal{Y}\tau, \mathcal{S}_{T}, \mathcal{X}_{T})p(\Sigma_{j})$.

(3.12)$K$ , we derive the conditional posterior
To generate samples of$j=1,2,$
distribution of
from (3.12). Applying the inverse Wishart distribution
as aconjugate prior, conditional posterior distribution ofis derived as$\Sigma_{j},$$\ldots,$$\Sigma_{j}$$\mathcal{I}\mathcal{W}(v_{j}, \Lambda_{j})$$p(\Sigma_{j})$$\Sigma_{j}$$p(\Sigma_{j}|\mathcal{Y}\tau, \mathcal{S}_{T}, \mathcal{X}_{T}, \theta^{*})$$\propto\prod_{tt\in\{t:s=j\}}[(2\pi)^{-\frac{N-1}{2}}|\Sigma_{j}|^{-\frac{1}{2}}\exp(-\frac{1}{2}\epsilon_{t}’\Sigma_{J}^{-1}\prime\epsilon_{t})]$$\cross|\Sigma_{j}|^{-\frac{\nu_{j}+(N-1)+1}{2}}\exp(-\frac{1}{2}$trtr$\propto|\Sigma_{j}|^{-\frac{\nu_{J}\prime+(N-1)+1+n_{7}}{2}}\exp(-\frac{1}{2}$whereis the total number of time
distribution ofis$n_{j}$$t$$\{\Sigma_{j}^{-1}\Lambda_{j}\})$$\{\Sigma_{j}^{-1}(\sum_{t=1}^{T}\epsilon_{t}\epsilon_{t}’1\{s_{t}=j\}+\Lambda_{j})\})$

belonging to regime . Then conditional posterior$j$$\Sigma_{j}$$\Sigma_{j}|\mathcal{Y}_{T}, \mathcal{S}_{T}, \mathcal{X}_{T}, \theta^{*}\sim \mathcal{I}\mathcal{W}(\nu_{j}+n_{j}, \sum_{t=1}^{T}\epsilon_{t}\epsilon_{t}’1\{s_{t}=j\}+\Lambda_{j})$(3.13)$K.$
where $j=1,2,$
From (3.11) and (3.13), we are able to construct Gibbs sampler algorithm by generating
and substituting these into (3.13) and then generatingwith the generated
and
substituting those back into (3.11).
$\ldots,$$\theta^{*}$$\Sigma_{j}$4$\theta^{*}$

Empirical study on Japanese meat market

The Ministry of Internal Affairs and Communications in Japan provides us with thehousehold expenditure survey data (i.e., Family Income and Expenditure Survey). The
household expenditure survey data includes the monthly time series data about average
expenditure and price of meat and fish 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 1 plots the budget shares of meat products fromJanuary 1998 through December 2006.

136$98 99 00 01 02 03 04 05$

06

Year

Figure 1: Plot of budget share data

4.1

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 asburn-in and then the next 25, 000 samples are recorded. The prior distributions areparameterized by setting $\mu=0,$ $V=10^{4}I_{45},$ $\nu_{j}=10,$ $\Lambda_{j}=10^{-3}I_{3}(j=1,2),$ $u_{11}=$$u_{22}=5$ , and $u_{12}=u_{21}=2$ . 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: $mo$ del 1 only includes intercept
parameter. Mode12 includes seasonal effects on August and December into model
1, and mode13 adds a habit effect into mode12. Finally, mode14 further incorporates atrend effect into mode13.
$\overline{\alpha}_{i,s_{t}}$Modell$\alpha_{i,s_{t}}=\overline{\alpha}_{i,s_{t}}$Mode12$\alpha_{i,s_{t}}=\overline{\alpha}_{i,s_{1}}+\delta_{1,i}d_{1,t}+\delta_{2,i}d_{2,t}$Mode13$\alpha_{i,s_{t}}=\overline{\alpha}_{i,s_{t}}+\delta_{1,i}d_{1,t}+\delta_{2,i}d_{2,t}+\sum_{j=1}^{N}\phi_{ij}\overline{w}_{j,t-1}$

Mode14$\alpha_{i,s_{t}}=\overline{\alpha}_{i,s_{t}}+\nu_{i,s_{t}}t+\delta_{1,i}d_{1,t}+\delta_{2,i}d_{2,t}+\sum_{j=1}^{N}\phi_{ij}\overline{w}_{j,t-1}$

Table 1 shows the logarithmic marginal likelihood of model $\log-ML_{i}(i=1,2,3,4)$ as
diagonal elements and logarithmic Bayes factors, $\log-BF_{ij}$ for model against model as
off-diagonal elements. To obtain the marginal likelihoods for candidate models, we use themethod 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 mode14 against $mo$ de13, $\log-BF_{43}(=1.295)$ indicates “positive” (Kass and Raftery,
$i,$

$i$

$j$

137

Table 2: Estimated Parameters of $MS$ -AIDS model in Regimel

1995, p.777) evidence in favor of mode14. Therefore we conclude that mode14 fits thedata best.
Tables 2 and 3 show the results of parameters for beef, pork and chicken. The parameters for fish are estimated from the adding-up condition (2.3a). Tables 2 and 3 show
the the posterior means, posterior standard deviations ( $SD$ ), 95% credible intervals, andGeweke’s convergence diagnostic statistics ( $CD$ ) for all parameters in $MS$-AIDS model(2.1) and variance-covariance matrices 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 2 and 3 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 the significant parameters.
Thus
, and
in Table 2 and, and
in Table 3 are regarded significantly different from zero. These parameters in $MS$ -AIDS
model (2.1) are used to calculate the price and expenditure elasticities.
To compare our proposed Bayesian estimation with the $ML$ estimation proposed in$\overline{\alpha}_{1},\overline{\alpha}_{2},\overline{\alpha}_{3},$$\beta_{2},$$\sigma_{11}^{2},$$\sigma_{22}^{2}$$\sigma_{33}^{2}$$\overline{\alpha}_{2},\overline{\alpha}_{3},$$\beta_{2},$$\sigma_{11}^{2},$$\sigma_{12},$$\sigma_{13},$$\sigma_{22}^{2}$$\sigma_{33}^{2}$

138
Table 3: Estimated Parameters of $MS$-AIDS model in Regime2

Allais and Nich\‘ele (2007) and employed in Kabe and Kanazawa (2012), we calculate themean squared errors (MSEs) for estimated budget shares. The MSEs of Bayesian estimation are evaluated by the posterior means of estimates of budget shares generated within
the Gibbs sampler. The results of MSEs with respect to our proposed Bayesian estimation (Bayes) and $ML$ estimation (MLE) are given in Table 4. Our Bayesian estimation
improves the MSEs for all products over $ML$ estimation. This result reflects the goodnessof fit to the budget share data of our Bayesian estimation.
Table 4: Mean squared errors (MSEs)

Figure 2 plots the probability of being regime 2 and budget share data of beef andpork from January 1998 through December 2006 under the proposed Bayesian method.
We calculate the probability $Pr\{s_{t}=2\}$ . In Figure 2, regime shift from $s_{t}=1$ to $s_{t}=2$
is observed at the timing of first BSE case in Japan in September 2001 and then theprobability $Pr\{s_{t}=2\}$ gradually declines until the end of 2003 along with increase inbudget share of beef. With the timing of first BSE case in U. . in December 2003, we
observe a high probability of being regime 2 once again. Since then, structure of budget
share tends to stay in regime 2.
The first regime shift in Figure 2 reflects the switching of consumers’ preference from
$S$

139
beef to pork triggered by the BSE scare in Japan in September 2001. The second regime
shift after the first BSE discovery in U. . in December 2003 might have arisen due to theban on import of American beef. Since the ban on importing U. . beef led to the shortageof beef supply in the domestic meat market, consumers may have been forced to purchasemore pork instead of beef.
Figure 3 shows the results of probability of being regime 2,
under
the $ML$ estimation. We estimated the probability from the Hamilton filter using data set
obtained through time
, and $ML$ estimatesin mode14 (see Kabe and Kanazawa,2012). The regime shift at the timing of first BSE case in Japan in September 2001
is observed in Figure 3. Nevertheless, when compared with the result of regime shift
in Figure 2, the probability’s gradual decline due to the recovery of beef budget sharefollowing the first BSE case in Japan observed in Figure 2 no longer can be observed in
Figure 3. Unlike Figure 2, we cannot identify the regime shift at the timing of first BSE
case in U. . in December 2003 in Figure 3.
$S$

$S$

$Pr(s_{t}=2|\Omega_{t},\hat{\Theta})$$t,$$\hat{\Theta}$$\Omega_{t}$$S$$98 99 00 01 02 03 04 05 06 07$
Year

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

We calculate the average budget share of i-th product at regime $s_{t}=j$ as$\overline{w}_{i,s_{t}=j}=\frac{\sum_{t--1}^{T}1\{s_{t}=j\}\overline{w}_{it}}{\sum_{t=1}^{T}1\{s_{t}=j\}}.$

Table 5 shows that regime 1 is characterized by a higher beef budget share relative tothat of pork, while regime 2 is characterized by the reversal of these two budget shares.
Since substitution occurs mostly between beef and pork in regimes 1 and 2 (see Table
5 , we focus on the price and expenditure elasticities for beef and pork. We calculate
$)$

140

98

99

01

00

02

03

04

05

06

Year

Figure 3: Probability of being regime 2,
pork under$ML$$Pr(s_{t}=2|\Omega_{t},\hat{\Theta})$and budget share data of beef andestimation.

Table 5: Posterior mean of average budget share

the Marshallian price elasticity
and expenditure elasticity
25, 000 samples generated via Gibbs sampler as$\eta_{ij,s_{t}}^{P}$$\eta_{i,s_{t}}^{E}$at regime$\eta_{ij,s_{t}}^{P}=-\kappa_{ij}+\frac{\gamma_{ij,s_{l}}}{\overline{w}_{i,s_{t}}}-\frac{\beta_{i,s_{t}}}{\overline{w}_{i,s_{t}}}[\alpha_{j,s_{t}}+\sum_{k=1}^{N}\gamma_{kj,s_{t}}\log\overline{p}_{k,s_{t}}]$$\eta_{i_{St}}^{E}=\frac{\beta_{i,s_{t}}}{\overline{w}_{i,s_{t}}}+1$,$s_{t}$

for each(4. 1)
(4.2)

,

where
for $i=j$ andfor $i\neq j$ , and , is an average price at regime
In Table 6, we show the posterior means and 95% credible intervals of price andexpenditure elasticities for beef and pork. Although own-price elasticities of pork havesignificant negative impacts in both regimes, own-price elasticity 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 beefpurchasing behavior in regime 2 in Table 6 implies that those who had kept purchasing
beef in regime 2 did so regardless of its price.
$\kappa_{ij}=1$$\kappa_{ij}=0$$\overline{p}_{k_{8_{t}}}$$s_{t}.$

141

RegimelRegime2

Table 6: Price elasticities and Expenditure elasticities
Price
Expenditure$(\eta_{i\dot{\eta}}^{P})$Price

Expenditure$(\eta_{ir}^{P})$$(\eta_{i\gamma}^{E})$$(\eta_{i\gamma}^{E})$

1 95% credible interval in parentheses$)$

5

Conclusion

In this paper, we proposed the Bayesian estimation for $MS$ -AIDS model proposed by Allais and Nich\‘ele (2007) and illustrated the applicability of our proposed method via realdata. The proposed Bayesian estimation has some important advantages. First, it enables
us to avoid the singularity problem suggested in Hamilton (1990, 1991). In the Bayesianframework, we can use conjugate prior distributions to incorporate the prior information
about variance-covariance matrices in advance. On the other hand, $ML$ estimation via numerical optimization methods (e.g., Newton-Raphson method) has to depend on sensible
selection of initial values of parameters to avoid singularity points on the parameter space.
Second, our proposed 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 employed in Allais and Nich\‘ele (2007) andKabe and Kanazawa (2012), is computationally very intensive. In our Bayesian estimation, posterior distributions of parameters are expressed as the standard formula (e.g.,multivariate normal and inverse Wishart distributions). Thus each of parameters can beeasily simulated via Gibbs sampler.
In the empirical study on the Japanese meat market, we found that our Bayesianestimation improves the mean squared errors for all meat products compared with the$ML$ estimation. Moreover we found the regime shift in the budget shares of meat products
in Figures 2 depicts much more sophisticated and realistic picture of regime transition thanFigure 3. Specifically, in Figure 2, probability of being regime 2, $Pr\{s_{t}=2\}$ , estimatedvia Bayesian estimation shows the regime shifts at the timing of first reported cases of
BSE both in Japan in September 2001 and also in U. . in December 2003. On the other
hand, Figure 3 shows a single regime shift at the timing of first BSE case in Japan. Since
$S$

142
parameter $estimates\wedge$ are given as the point solutions, the probability of being regime
2,
in Figure 3 ignores the uncertainty about parameters and making theprobability
in Figure 3 closer to zero or one. Perhaps as Scott (2002,
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 haveextended the Hamilton $(1989)$ ’s Markov-switching model. In particular, they focusedon a useful $mo$ dification of transition probabilities. For example, Diebold et al. (1994)
introduced the time-varying transition probabilities into the Markov-switching model, andalso they allowed the transition probabilities to evolve as logistic functions of economic
variables. Altematively, Markov-switching model assumes that latent variables controlling
regime shifts are exogenous. Kim et al. (2008) relaxed this exogenous regime-switchingassumption and proposed a Markov-switching regression model with endogenous regime
switching. The extensions to $MS$-AIDS model in these directions will be interesting for
future research.
$ML$$Pr(s_{t}=2|\Omega_{t}, \Theta)$$Pr(s_{t}=2|\Omega_{t},\hat{\Theta})$

References
Allais, O., and V. Nich\‘ele (2007) ‘Capturing structural changes in french meat and fish demand over the period 1991-2002.’ European review of agricultural economics 34(4), 517-

538

Banks, J., R. Blundell, and A. Lewbel (1997) ‘Quadratic engel curves and consumer
demand.’ Review of Economics and Statistics 79(4), 527-539

Carter, C., and R. Kohn (1994) ‘On gibbs sampling for state space models.’ Biometrika
81(3), 541-553
Chib, S. (1996) ‘Calculating posterior distributions and modal estimates in markov mixture models.’ Journal of Econometrics 75(1), 79-97

Christensen, L. ., D. W. Jorgenson, and L. J. Lau (1975) ‘Transcendental logarithmic
utihty functions.’ American Economic Review 65(3), 367-383$R$

Cooper, R. ., and K. R. McLaren (1992) ‘An empirically oriented demand system withimproved regularity properties.’ Canadian Journal of Economics 25(3), 652-668$J$

Deaton, A., and J. Muellbauer (1980) ‘An almost ideal demand system.’ The American
Economic Review 70(3), 312-326
Diebold, F. ., J. H. Lee, and G. Weinbach (1994) ‘Regime switching with time-varying
transition probabilities.’ in C. Hargreaves (ed.), Nonstationary Time Series Analysis
and Cointegration.(Advanced Texts in Econometrics, CWJ Granger and G. Mizon,eds.), 283-302. Oxford: Oxford University Press.
$X$

Hamilton, J. . (1989)
new approach to the economic analysis of nonstationary
time series and the business cycle.‘ Econometrica: Journal of the Econometric Society 57(2), 357-384
$D$

‘ $A$

143
-(1990) ‘Analysis of time series subject to changes in regime.’ Journal of Econometrics$45(1-2),$ $39-70$-(1991) ‘$A$quasi-bayesian approach to estimating parameters for mixtures of normal
Economic Statistics pp. 27-39

distributions.’ Joumal of Business$\mathcal{E}i$

Ishida, T., N. Ishikawa, and M. Fukushige (2006) ‘Impact of bse and bird flu on consumers’
meat demand in japan.’ DISCUSSION PAPER SERIES in Osaka university
Ishida, T., N. Ishikawa, and M. Fukushige (2010) ‘Impact of bse and bird flu on consumers’
meat demand in japan.’ Applied Economics 42(1), $49-56$

Jin, H. ., and W. W. Koo (2003) ‘The effect of the bse outbreak in japan on consumers’
preferences.’ European Review of Agricultural Economics 30(2), 173-192$J$

Kabe, S., and Y. Kanazawa (2012) ‘Another view of impact of bse crisis in japanese meat
market through the almost ideal demand system model with markov switching.’ Department of Social Systems and Management Discussion Paper Series No.1288 (Universityof Tsukuba), forthcoming in Applied Economics Letters (Accepted on December 2011).
URL: http://www.sk.tsukuba.ac.jp/SSM/libraries/list1276.php
Kass, R.

., and A. E. Raftery (1995) ‘Bayes factors.’ The Journal of the American
Statistical Association 90(430), 773-795$E$

Kim, C. ., J. Piger, and R. Startz (2008) ‘Estimation of markov regime-switching regression models with endogenous switching.’ Joumal of Econometrics 143(2), 263-273$J$Moosa, I. ., and J. L. Baxter (2002) ‘Modeling the trend and seasonal within an aids
model of demand for alcoholic beverage in the united kingdom.‘ Joumal of Applied
Econometrics 17(2), 95-106$A$

Newton, M. ., and A. E. Raftery (1994) ‘Approximate bayesian inference with theweighted likelihood bootstrap.’ Joumal of the Royal Statistical Society, Series
56(1), 3-48
$A$

$B$

Ohtani, K., and S. Katayama (1986) ‘ gradual switching regression $mo$ del with autocorrelated errors.’ Economics Letters 21 (2), 169-172$A$

Peterson, H. ., and Y. J. K. Chen (2005) ‘The impact of bse on japanese retail meat$313-327$
demand.’ Agribusin$H$$es\mathcal{S}21(3),$Rickertsen, K. (1996) ‘Structural change and the demand for meat and fish in norway.’
European Review of Agricultuml Economics 23(3), 316-330

Scott, S. . (2002) ‘Bayesian methods for hidden markov models.’ Journal of the American
Statistical Association 97(457), 337-351$L$

Theil, H. (1965) ‘The information approach to demand analysis.’ Econometnca 33(1), 67-87