Department of Social Systems and Management
Discussion Paper Series

No.1288

Another View of Impact of BSE Crisis
in Japanese Meat Market throughthe Almost Ideal Demand System Modelwith Markov SwitchingbySatoshi KABE and Yuichiro KANAZAWA

February 2012

UNIVERSITY OF TSUKUBA
Tsukuba, Ibaraki 305-8573
JAPANAnother view of impact of BSE crisis inJapanese meat market through the almost
ideal demand system model with Markov
switching
Satoshi KABE ∗Yuichiro KANAZAWA †Abstract
During the period in this study, there have been serious food safetyconcerns regarding the Bovine Spongiform Encephalopathy (henceforth BSE) and bird flu in Japan. Previous studies such as Peterson
and Chen (2005) and Ishida et al. (2006, 2010) assumed that structural change points in demands have already known and they specified transition functions at the change points. On the other hand,
we detected the structural change points without any prior information about change points. To assess the structural change points in
the Japanese meat market, we employed the Markov switching almost
ideal demand system (MS-AIDS) model (Allais and Nichèle, 2007). In
this study, we found the structural change point coinciding with thetiming of first reported case of BSE, but not of bird flu.

∗

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

1

Introduction

Food safety issues (e.g., virus, bacteria, and toxic chemicals ) are said to influence the consumers’ choices and preferences for food. Thus it is important
to examine the change of consumption pattern as a results of reported foodsafety concerns.
In the Japanese meat market, there have been serious food safety concerns
regarding to the Bovine Spongiform Encephalopathy (henceforth BSE) andbird flu. The BSE was initially recognized in cattle in the UK in 1986. The
UK government announced that consumption of beef infected by BSE is suspected to cause new variant Creutzfeldt Jacob Disease (vCJD). Within the
period from 1987 to 2002, the total number of infected cattle grew to 183,000in the UK (Jin and Koo, 2003). The BSE crisis in the UK resulted in an
immediate and significant decline in beef consumption throughout Europe
including Ireland, Switzerland, France, Portugal, Belgium, and the Netherlands in the 1990s.
On September 2001, the Japanese government announced the first BSE
case in the country. It was the first reported BSE case outside Europe. The
cow believed to have the BSE was a five-year-old Holstein and was located in
Chiba prefecture. The Japanese government expended more than one billion
dollar over the subsequent half year to restore food safety (Peterson andChen, 2005) and began BSE testing for all cattle in October 2001. It alsobanned imports of meat and bone meal to reduce the possibility of outbreak.
However thirty six cases of BSE was confirmed during 2001 - 2009.
The bird flu is an infectious disease of birds (e.g., chicken, duck, andturkey) caused by avian influenza viruses. There are many subtypes ofavian influenza viruses, but only some strains of four types have been highly
pathogenic in human (e.g., H5N1 virus). The H5N1 virus is the most highly1pathogenic strain. Many people infected with the H5N1 virus have died in
Vietnam, Indonesia, Thailand and other Southeast Asia countries.
On January 2004, the first infection case with H5N1 virus in Japan was
confirmed in Yamaguchi prefecture. Since then the chicken infected with
H5N1 virus are periodically found in various regions in Japan. With the
spread of bird flu in Japan, consumers are worried about the safety of chicken
despite the fact that the Ministry of Health, Labor and Welfare in Japanannounced that there is virtually no risk H5N1 infection from eating theinfected chicken itself or the products made out of the infected chicken.
In the previous studies about structural change of demand 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, dairy, U.S., and Australian beef) and showed the differenceof 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 to the BSE
but also the bird flu in Japan and empirically investigated the difference ofstructural changes caused by BSE and bird flu.
Previous studies such as Peterson and Chen (2005) and Ishida et al. (2006,2010) applied the gradual switching model (Ohtani and Katayama, 1986) toexamine the nature of structural change in demands. This model utilizes
a transition function to express a gradual shift pattern in demands under
the assumption that starting-points of structural shift have already known.
For example, the starting-points of structural shift in Ishida et al. (2006,2010) were specified at the timing of first reported cases of BSE and bird flu
in Japan. The authors assumed the four types of transition functions andselected a transition function which the likelihood is maximized.

2

In this paper, we detect the structural change points without any priorinformation about change points in the market and at the same time examine
when and how often the structure of demands changed. Also we computethe elasticities for price and expenditure to examine the change of consumers’
behavior for meat products in Japan.
For these purposes, we employ the Markov switching almost ideal demand system (MS-AIDS) model proposed by Allais and Nichèle (2007). Thismodel employs the Markov switching mechanism to capture the dynamic shift
patterns. The Markov switching mechanism allows the frequent structuralchanges at random points, and it is suitable to uncover distinct dynamic
patterns in different periods.
This paper is organized as follows. In section 2, we introduce the Markovswitching AIDS (MS-AIDS) model. Details of the model are in Appendix. In
the next section, the estimation procedure is described. The data and results
are presented in section 4, and the last section contains the conclusion of thisstudy.

2

Markov-Switching AIDS Model

Deaton and Muellbauer (1980) developed the almost ideal demand system(AIDS) model which is widely used in the empirical demand analysis evennow, and several researchers extended the AIDS models (e.g., Cooper and
McLaren, 1992; Banks et al., 1997; Moosa and Baxter, 2002; Ishida et al.,2006, 2010; Allais and Nichèle, 2007). The AIDS model is able to carry out
the statistical test of theoretical constraints in demand theory, like the Rotterdam model (Theil, 1965) and applies the second-order Taylor expansion
to the unknown functions, like the Translog model (Christensen et al., 1975).

3

In the MS-AIDS model, representative consumer’s demand for a categoryof products under consideration during the given time-period consists of apredetermined number of “regimes” or “states” and this model can estimate
the degree of “belongingness” to these “regimes” by probability.
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 k=1 j=1
k=1

(2.2)

where 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 1 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]1“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.

4

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

where d1,t and d2,t are dummy variables1 if t is August
1d1,t =
d2,t =

0 otherwise
0

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). In order to satisfy the!
!Nadding up condition, we impose the restriction N
i=1 ᾱi,st = 1,i=1 νi,st = 0,!N!N!Ni=1 δ1,i =i=1 δ2,i = 0 and
i=1 φij = 0. We also impose the restriction!N
j=1 φij = 0 to avoid the identification problem.

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 ,5i, j = 1, 2, . . . , K(2.5)

and the transition matrix is defined asπ
π21 . . . πK1
 11



 π12 π22 . . . πK2 

Π=
 ..
..
.. 
...
 .
.
. 


π1K π2K . . . πKK(2.6)

where πi1 + πi2 + · · · + πiK = 1, i = 1, 2, . . . , K.

3

Estimation

The parameters of MS-AIDS model are estimated by iterating the following
steps:
Step 0. Set the initial values of parameters and set g = 0.
Step 1. Given the parameters at gth iteration, calculate the conditional
probabilities about the value of st from the Hamilton filter (Hamilton,1989).
Step 2. Calculate the score functions with respect to parameters ( see detailin Appendix ).
Step 3. Find the maximum likelihood estimates of parameters via BHHH
algorithm (Berndt et al., 1974) and set g = g + 1.
Step 4. Repeat steps 1 - 3 until the log-likelihood does not change.
In Allais and Nichèle (2007), authors estimate the transition probabilities by maximum likelihood estimation, but if transition probability has aboundary solution such as πij = 0 or 1, asymptotic normality of transition
probabilities does not hold. To avoid this problem, we reparameterize the6transition probability πij , 0 ≤ πij ≤ 1 as λij = log (πij /πiK ), natural sufficient statistics for the multinomial model, −∞ < λij < ∞ and calculatescore function with respect to λij . Once, we estimate the parameter λij , thenwe calculate the transition probability πij back using the standard formulaπiK =1
1 + exp(λi1 ) + exp(λi2 ) + · · · + exp(λiK−1 )πij =exp(λij ).
1 + exp(λi1 ) + exp(λi2 ) + · · · + exp(λiK−1 )andAssuming that budget share equations follow a multivariate normal distribution, Allais and Nichèle (2007) estimates variance-covariance matrix Σstalong with other parameters. However maximum log-likelihood becomes infinite if determinant of variance-covariance matrix |Σst | goes to zero andnumerical maximization algorithm (e.g., Newton-Raphson method) breaks/ st after estidown. Therefore we derive the maximum likelihood estimator Σmating all parameters other than Σst (see detail in Appendix).

4

Data and Results

The Ministry of Internal Affairs and Communications in Japan provides us
with the household 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 andprice 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. Fish, having consistently been thedominant source of protein for the Japanese for many years, has budget share7

of almost 50% of meat products (beef, pork, chicken, and fish). Therefore weomit the budget share of fish to reveal the budget share movement of meat

0.20
0.15
0.10
0.05

Budget Share

0.25

0.30

products other than fish when we plot Figure 1.

Beef
Pork

0.00

Chicken

98

99

00

01

02

03

04

05

06

07

Year

Figure 1: Plot of budget share data
During this period (see Figure 1), there exist the first reported case of
BSE and of bird flu in Japan. The first BSE case was confirmed on September
2001 and beef consumption dwindled dramatically and then has gradually
recovered. However the budget share of beef has not recovered at the samelevel as in the pre-BSE period. On the other hand, the change of budget
share of chicken due to the first reported case of bird flu on January 2004
was relatively small compared with the change of beef demand due to the first
BSE case. The budget share of chicken has recovered completely and evenincreased afterwards. Finally budget share of pork has gradually increased
after the first BSE case on September 2001 but has experienced significant
seasonal declines at the end of every year.
8

Next we consider the seasonality in budget shares for meat products. To
examine the seasonal variations in the budget shares, we calculate the averagebudget shares of meat products per month for January 1998 - December 2000
and for January 2003 - December 2006 respectively. This way we can avoid
the influence of BSE. Figures 2 and 3 display the average budget shares ofmeat products per month for the two periods. During these periods, theaverage monthly budget shares for meat products do not significantly varythroughout the year except for the pork’s decline in Decembers. Figures 2
and 3 also show the average monthly budget share for beef peaked in August.
In this study, therefore, we examine the following four models: model 1
only includes intercept parameter ᾱi,st . Model 2 includes seasonal effects on
August and December into model 1, and model 3 adds a habit effect intomodel 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

In Table 1, we computed the Akaike’s information criterion (AIC) andlikelihood ratio (LR) statistic 2 for each of the candidate models when thenumber of regimes is two. The LR statistic shows that log-likelihood of model
4 is significantly higher than those of other candidate models, and also model4 has a smallest AIC. Therefore we conclude that model 4 fits the data best.
2

We assumed the asymptotic normality of ML estimator (see detail in Bickel et al.

(1998)) and applied the model selection criteria (e.g., AIC) and LR test. However, if
nuisance parameters are not identified under the null hypothesis, LR test is not valid
because the LR statistic does not have a standard asymptotic distribution (Garcia, 1998).

9

0.25
0.20
0.15
0.10

Budget Share

0.05

Beef
Pork

0.00

Chicken

Jan

Feb

Mar

Apr

May

Jun

July

Aug

Sep

Oct

Nov

Dec

month

Figure 2: Average monthly budget shares of meat products in January 1998

0.15
0.10
0.05

Budget Share

0.20

0.25

- December 2000

Beef
Pork

0.00

Chicken

Jan

Feb

Mar

Apr

May

Jun

July

Aug

Sep

Oct

Nov

Dec

month

Figure 3: Average monthly budget shares of meat products in January 2003
- December 200610

Table 1: Model selection
LR statics

d.f .

χ20.05 (d.f.)-2244.24

211.95

21

32.67

44

-2288.55

155.6415

25.00

1243.3953-2380.77

45.41

13

12.59

1266.0959-2414.19

——

6

——

Model

Log-likelihood# of parameters

AIC

11160.1238

21188.273
41) AIC : -2 log-likelihood + 2 ( # of parameters )2) LR statics : 2 ( log-likelihood(null model) - log-likelihood(alternative model) )3) d.f. : degree of freedom

Next, we examine the structural change points in the Japanese meat
market. Using the Hamilton filter, we calculate the conditional probability/ (j = 1, 2) based onof each time point being at regime st , Pr(st = j|Ωt ; Θ)/the data obtained through time t, Ωt and set of estimated parameters Θin MS-AIDS model. In Figure 4, we find that structure of budget shares is

estimated to change almost instantaneously from regime 1 to regime 2 at thetiming of first reported case of BSE on September 2001. However, we did
not find any structural change points at the timing of bird flu, even when thenumber of regimes is increased from two to three. This result implies that
there is no apparent effect on the consumers’ behavior due to the bird flu in
the Japanese meat market.
/ we calculate the avUsing the conditional probability Pr(st = j|Ωt ; Θ),erage budget share of ith product at regime st = j as!T/
t=1 Pr(st = j|Ωt ; Θ)w̄itw̄i,st =j = !
.
T/Pr(st = j|Ωt ; Θ)
t=1

Table 2 shows that beef and pork have significant changes between regime 1
and 2. Regime shift between regime 1 and regime 2 is considered due to theconsumers’ preference shift from beef to pork after the first BSE case.
11

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 4: Probability of being Regime 2 and Budget share data of Beef and Pork

Table 2: Estimated average budget share
Regime1

Regime2Beef

0.21550.1797Pork

0.17810.2049Chicken0.0862

0.0955

Fish

0.52020.519912

We show the estimates of parameters in Tables 3 and 4. Table 3 showsthat all parameters for prices (γij ) except for the budget share of chicken
relative to its own price are not significant. On the other hand, budget
share of beef has a significant parameter for expenditure in regime 1 and theparameter for pork is significant in both regimes. These figures are used to
calculate price and expenditure elasticities in Table 5.
Table 3: Estimated Parameters of MS-AIDS Model
Regime1

Regime2

Estimate

Sd. Error

t-value

Estimate

Sd. Error

t-valueᾱ10.43590.15072.8920∗∗∗ᾱ10.24430.31140.7845ᾱ20.48400.0754

6.4184∗∗∗ᾱ20.54290.14213.8199∗∗∗ᾱ30.28730.09443.0430∗∗∗ᾱ30.22990.10742.1408∗∗γ110.00680.06570.1035γ110.05520.08020.6879γ12-0.0174

0.0485-0.3596γ12-0.0197

0.0347-0.5677γ13-0.0446

0.0384-1.1592

γ13-0.0372

0.0277-1.3400

γ22

0.05230.0364

1.4362

γ220.03780.0301

1.2575γ23-0.0373

0.0348-1.0715

γ23

0.00650.02640.2467γ33

0.10420.04042.5807∗∗

γ33

0.06390.0297

2.1539β1-0.0672

0.0325-2.0659∗∗

β1-0.0404

0.0675-0.5980β2-0.0687

0.0167-4.1228∗∗∗

β2-0.07610.0303-2.5097β3-0.0126

0.0180-0.6970β3-0.00430.0231-0.1875∗∗∗∗1) Significant level : ∗∗ 5%, ∗ ∗ ∗ 1%

In Table 4, significantly negative trend effect in the budget share of beef
is observed in regime 1. We find that each of meat products has a different
seasonal variation for August and December. The seasonal effects for Augustdummy variables are estimated to be significant for beef, while those for
December dummy variables are not significant except pork. Furthermore,
habit effects in the budget shares are estimated to be significant for beef and13

Table 4: Estimated Parameters in Intercept term
Trend (Regime 1)

Trend (Regime 2)
Estimate

Sd. Error

t-valueν1-0.00039

0.00016-2.4463ν2-0.00001

0.00009-0.1515

ν3-0.00006

0.00006-1.1065

Estimate

Sd. Error

t-valueδ11

0.01150.00552.0894δ12-0.00480.0028δ13-0.0049

0.0029∗∗Seasonal (Aug)

Estimate Sd. Error

t-valueν1

0.000220.00032

0.6876ν20.000170.000161.0501ν30.000150.000111.4211

Estimate Sd. Error

t-value

Seasonal (Dec)

∗∗

δ21

0.01860.0196

0.9480-1.6942δ22-0.02540.0120-2.1287-1.6673δ23

0.00790.0124

0.6329∗∗Transition
Habit effectprobability
Estimate

Sd. Error

t-value

φ110.18870.07092.6619φ220.04400.0831

0.5294

φ33

0.39390.07475.2704∗∗

Estimate Sd. Error

t-value

λ11

4.32970.142730.3483∗∗∗

λ21-5.1701

1.1361-4.5509∗∗∗∗∗∗1) Significant level : ∗∗ 5%, ∗ ∗ ∗ 1%chicken. This result means that the budget shares of beef and chicken forprevious period have positive impacts on the current budget shares. Finally,
Table 4 shows that estimated reparameterized transition probabilities λ11
and λ21 are significant, and transition probabilities π11 and π22 (= 1 − π21 )are estimated to be 0.987 and 0.994. Therefore, structure of demand in theJapanese meat market tends to stay for a long time within the same regime.
Next, we consider the change of elasticities for beef and pork after the
first BSE case. We do not compute elasticities for chicken primary because
the price of chicken has changed very little over the study period, but also
because we already found that substitution occurs mostly between beef andpork. Using the estimated parameters in Tables 3 and 4, we can calculatePE
the Marshallian price elasticity ηij,sand expenditure elasticity ηi,s
at regimett

14

st as
0
1
N
"
γ
β
ij,s
i,s
t
t
P
ηij,s
= −κij +−αj,st +
γkj,st log p̄k,st ,tw̄i,stw̄i,st
k=1

(4.1)

E
ηi,s
=
t

(4.2)

βi,st
+ 1,
w̄i,stwhere κij = 1 for i = j and κij = 0 for i %= j, and p̄k,st is an average priceat regime st . We also compute their associated standard errors via deltamethod. Table 5 shows the results of price and expenditure elasticities in
regimes 1 and 2, respectively. The own-price elasticities of beef and pork
Table 5: Price elasticities and Expenditure elasticities
Price (ηijP )Regime 1
Beef
Beef

Pork
∗∗

-0.815(-2.311)

Pork

0.688

(0.227)(3.801)

0.091-0.529(-3.143)

∗∗∗

0.615

∗∗∗
∗∗∗

(7.563)

Price (ηijP )
Beef

Pork

0.062

(0.384)

Regime2

Beef

Expenditure (ηiE )Expenditure (ηiE )Pork-0.618

0.006

0.147

(-1.582)(0.025)(5.277)0.032-0.624(0.181)(-3.429)

∗∗∗

0.145

∗∗∗
∗∗∗

(4.345)1) t-value in parentheses
2) Significant level : ∗∗ 5%, ∗ ∗ ∗ 1%

in regime 1 are significantly negative, but beef price elasticity in regime 2
is not significant. From the estimated cross-price elasticities, no statisticallysignificant price-driven substitution occurred between beef and pork in bothregimes. Based on the estimated expenditure elasticities, we find that consumptions of beef and pork are responsive to the change in total expenditure
on meat products in both regimes.
15

5

Conclusion and Discussion

Previous studies such as Peterson and Chen (2005) and Ishida et al. (2006,2010) assumed that starting-points of structural shift in demands are known.
For example, Ishida et al. (2006, 2010) assumed that there are two structuralshifts, the first at the timing of the first reported cases of BSE in September
2001 and the second at the first reported case of bird flu in January 2004,
both in Japan. However such structural shifts, if any, ought to be reflectedin the demand data itself. Thus, in this study, we applied the Markovswitching almost ideal demand system (MS-AIDS) model proposed by Allais
and Nichèle (2007) to uncover the structural change points in the Japanesemeat market.
First, we illustrate that MS-AIDS model is able to identify the timing of
the first reported case of BSE in September 2001 as the structural change
point, but the same model does not recognize the timing of the first bird
flu. This result shows that there is no apparent impact of bird flu on theJapanese consumers’ meat consumption. Regime 1 before the first BSE ischaracterized by higher beef budget share relative to the pork budget share,
while regime 2 is characterized by the reversal of budget shares between beef
and pork (see Table 2). The first BSE changed consumers’ preference awayfrom beef to pork.
Next, we find that own-price elasticity of beef in regime 1 is significantly
negative, but that in regime 2 is no longer significant. This shift in regime 2
is the reflection of the fact that food safety on beef becomes more important
than its price to Japanese consumers. There were two streams of events thatwere likely to have contributed to this heightened awareness of safety of beef
products, one domestic, and the other international.
Domestically, in October 18, 2001, a month after the first reported BSE
16case in Japan, the Japanese government started complete survey of domestic
cattle. At the same time the government instituted buyback program of domestic beef processed before that date. A well known food importer/producer
Snow Brand Food Company disguised imported beef as domestic beef to illegally take advantage of the government buyback program. It was sincerevealed that many other major meat companies includes Nippon Ham committed the same subsidy fraud. As a result consumers have become weary ofbeef in general.
Internationally, the first BSE case in the U.S., the largest or second largest
beef exporter to Japan during 2001-2003, was confirmed on December 2003.
The Japanese government immediately announced a ban on import of American beef. On December 2005 the Japanese government resumed importing
American beef under stricter conditions. Nevertheless banned specified-riskmaterials of beef products were found from the imported beef from the U.S.
since January 2006.
Finally, we note some issues which are left for future work. This study
does not establish direct relationship between the regime shift of meat demands and the consumers’ awareness of risk of vCJD and bird flu, though
the regime shift was found to coincide with the first BSE case timewise.
This awareness itself could be affected by consumers demographics such asage, composition of the family, household income (Adda, 2007). In addition,
consumers’ choice of meat products could be influenced by the country-oforigin information (Peterson and Chen, 2005) and by the mass media such as
news papers and TV (Verbeke and Ward, 2001). Therefore, future researchis needed for integration of theses effects into the Markov-switching almost
ideal demand system model.

17

References
Adda, J. (2007) ‘Behavior towards health risks: An empirical study using
the “mad cow” crisis as an experiment.’ Journal of Risk and Uncertainty
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Verbeke, W., and R. W. Ward (2001) ‘A fresh meat almost ideal demand system incorporating negative tv press and advertising impact.’ Agricultural
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A

Log-likelihood function of MS-AIDS model

Let wt be a (N −1)×1 vector of budget shares at time t, w̄it (i = 1, 2, . . . , N −1) and xt be a vector of explanatory variables. We define a parameter vector
in regime st as θ st and error term of MS-AIDS model as εt = [ε1t , ε2t , . . . , εN −1,t ]" ∼N (0, Σst ) with a (N − 1) × (N − 1) variance-covariance matrix Σst .
Suppose that distribution of wt conditional on xt , st and θ = [θ 1 , . . . , θ K ,Σ1 , . . . , ΣK ]" is defined as p(wt |xt , st ; θ) and let π be a vector of transition
probabilities πij , i = 1, 2, . . . , K, j = 1, 2, . . . , K − 1. Then the conditional
log-likelihood function with respect to parameter set Θ ≡ {θ, π} under all
the observations (wt , xt ), t = 1, 2, . . . , T is
0 K
1
T
"
"
-(Θ) =log
p(wt |xt , st ; θ) Pr(st |Ωt−1 ; Θ)
t=1

=

T
"
t=1

(A.1)

st =1

log p(wt |Z t ; Θ)

(A.2)

where Ωt is an information set containing all observations obtained throughtime t: Ωt ≡ {wt , wt−1 , . . . , w1 , xt , xt−1 , . . . , x1 } and Z t is an information
set of lags of wt and observable explanatory variables obtained through time
t: Z t ≡ {wt−1 , wt−2 , . . . , w1 , xt , xt−1 , . . . , x1 }. The conditional probabilityPr(st |Ωt ; Θ) represents the probability about the value of st based on dataobtained through time t and based on parameter set Θ.
The maximum likelihood estimator of variance-covariance matrix Σst is
derived as follows: From (A.1) and (A.2), a first derivative of log-likelihood
20

function with respect to inverse variance-covariance matrix Σ−1
st of regimest = j is
∂-(Θ)
∂Σ−1
st =j
"
!
=
t

=

"
t

!

∂
1
st p(w t |xt , st ; θ) Pr(st |Ωt−1 ; θ, π)!

st p(w t |xt , st ; θ) Pr(st |Ωt−1 ; θ, π)
∂Σ−1
st =j

1
∂p(wt |xt , st = j; θ) Pr(st = j|Ωt−1 ; θ, π)∂Σ−1st p(w t |xt , st ; θ) Pr(st |Ωt−1 ; θ, π)
st =j" p(wt |xt , st = j; θ) Pr(st = j|Ωt−1 ; θ, π) ∂ log [p(wt |xt , st = j; θ) Pr(st = j|Ωt−1 ; θ, π)]!
=∂Σ−1st p(w t |xt , st ; θ) Pr(st |Ωt−1 ; θ, π)
st =j
t
"
∂ log p(wt |xt , st = j; θ)=Pr(st = j|Ωt ; Θ)
.
−1
∂Σ
s
=j
t
t
Assuming thatp(wt |xt , st ; θ) = (2π)

− N 2−1|Σst |− 12#$
1 " −1
exp − εt Σst εt ,2

we have2
3∂ log p(wt |xt , st = j; θ)∂N −1
1
1 " −1−1=−
log(2π) + log |Σst | − εt Σst εt
2
2
2
∂Σ−1
∂Σ−1
st =j
st =j
4 " −1 5
1 ∂ log |Σst =j |−1 1 ∂ εt Σst =j εt
=
−
2
2
∂Σ−1
∂Σ−1
st =j
st =j
6
7
−1
−1
1 ∂ log |Σst =j | 1 ∂tr Σst =j εt ε"t=−2 ∂Σ−1
2
∂Σ−1
st =j
st =j
1
1 "
= Σst =j − εt εt
2
2
and

2
3
T
"
∂-(Θ)
1
1 "
=
Pr(st = j|Ωt ; Θ) Σst =j − εt εt = 0.
2
2
∂Σ−1
st =j
t=1

Therefore we have/ st =j =
Σ

!T

t=1
!
T

Pr(st = j|Ωt ; Θ)ε̂t ε̂"tt=1 Pr(st = j|Ωt ; Θ)

21

.

B

Calculation of score function

Score function with respect to θ
The first derivative of log-likelihood with respect to θ is
∂ log p(wt |Z t ; Θ)∂θK
" ∂ log p(wt |xt , st ; θ)=Pr(st |Ωt ; Θ)
∂θ
s =1
t

+

t−1 "
K
"

∂ log p(wτ |xτ , sτ ; θ){Pr(sτ |Ωt ; Θ) − Pr(sτ |Ωt−1 ; Θ)}∂θτ =1 s =1

(B.1)

τ

for t = 2, 3, . . . , T andK∂ log p(w1 |Z 1 ; Θ) " ∂ log p(w1 |x1 , s1 ; θ)=Pr(s1 |Ω1 ; Θ).
∂θ
∂θ
s =1

(B.2)

1

Score function with respect to πij
The first derivative of log-likelihood with respect to the transition probabilities πij , i = 1, 2, . . . , K, j = 1, 2, . . . , K − 1 is
∂ log p(wt |Z t ; Θ)∂πij
−1
= πij−1 Pr(st = j, st−1 = i|Ωt ; Θ) − πiK
Pr(st = K, st−1 = i|Ωt ; Θ)+ πij−1
−1
− πiK

+

t−1
"

τ =2
t−1
"
τ =2

[Pr(sτ = j, sτ −1 = i|Ωt ; Θ) − Pr(sτ = j, sτ −1 = i|Ωt−1 ; Θ)][Pr(sτ = K, sτ −1 = i|Ωt ; Θ) − Pr(sτ = K, sτ −1 = i|Ωt−1 ; Θ)]K"∂ log Pr(s1 ; π)

s1 =1

∂πij

[Pr(s1 |Ωt ; Θ) − Pr(s1 |Ωt−1 ; Θ)](B.3)

for i = 1, 2, . . . , K, j = 1, 2, . . . , K − 1 and t = 2, 3, . . . , T , and when t = 1,
K
∂ log p(w1 |Z 1 ; Θ) " ∂ log Pr(s1 ; π)=Pr(s1 |Ω1 ; Θ).
∂πij
∂πij
s =1
1

22

(B.4)

To satisfy 0 ≤ πij ≤ 1 andprobability πij as follows!K

λij = log (πij /πiK ) ,j=1 πij = 1, we reparameterize the transitioni = 1, 2, . . . , K, j = 1, 2, . . . , K − 1and estimate the parameter λij instead of the transition probability πij .
To calculate the score function with respect to the parameter λij , weapply the chain rule as follows∂ log p(wt |Z t ; Θ)
∂ log p(wt |Z t ; Θ) ∂πij
=
×
∂λij
∂πij
∂λij
and the partial derivative of πij with respect to λij is obtained from∂λij∂=log (πij /πiK )
∂πij
∂πij
∂ log(πij ) ∂ log(πiK )
=
−
∂πij
∂πij
∂ log(πij ) ∂ log(πiK ) ∂πiK
=
−
×
∂πij
∂πiK
∂πij
1
1
=
+
,
πij πiK
where i = 1, 2, . . . , K, j = 1, 2, . . . , K − 1.

C

Calculation of joint probabilities

We need to assess the changes in the probabilities Pr(sτ |Ωt ; Θ)−Pr(sτ |Ωt−1 ; Θ)in (B.1) and Pr(sτ , sτ −1 |Ωt ; Θ) − Pr(sτ , sτ −1 |Ωt−1 ; Θ) in (B.3). With joint
probabilities Pr(st−1 , sτ , sτ −1 |Ωt−1 ; Θ) and Pr(st , sτ , sτ −1 |Ωt ; Θ), the changes
in the probabilities Pr(sτ |Ωt ; Θ) − Pr(sτ |Ωt−1 ; Θ) and Pr(sτ , sτ −1 |Ωt ; Θ) −Pr(sτ , sτ −1 |Ωt−1 ; Θ) are calculated asPr(sτ , sτ −1 |Ωt ; Θ) − Pr(sτ , sτ −1 |Ωt−1 ; Θ)
23

=

"

"

Pr(st−1 , sτ , sτ −1 |Ωt−1 ; Θ)

(C.1)

Pr(sτ |Ωt ; Θ) − Pr(sτ |Ωt−1 ; Θ)""=Pr(sτ , sτ −1 |Ωt ; Θ) −Pr(sτ , sτ −1 |Ωt−1 ; Θ).

(C.2)

st

Pr(st , sτ , sτ −1 |Ωt ; Θ) −

st−1

and

sτ −1

sτ −1

The derivation of joint probabilities Pr(st−1 , sτ , sτ −1 |Ωt−1 ; Θ) and Pr(st , sτ , sτ −1 |Ωt ; Θ)is as shown below.
Suppose that we have a joint probability for regimes st−1 , sτ and sτ −1 :Pr(st−1 , sτ , sτ −1 |Ωt−1 ; Θ)

(C.3)

for τ < t, a joint probability Pr(st , sτ , sτ −1 |Ωt ; Θ) is calculated as follows
Pr(st , sτ , sτ −1 |Ωt ; Θ) = Pr(st , sτ , sτ −1 |wt , xt , Ωt−1 ; Θ)!
g(st , sτ , sτ −1 , wt |xt , Ωt−1 ; Θ)st−1 g(st−1 , st , sτ , sτ −1 , w t |xt , Ωt−1 ; Θ)==p(wt |xt , Ωt−1 ; Θ)p(wt |xt , Ωt−1 ; Θ)!
st−1 g(st−1 , st , sτ , sτ −1 , w t |xt , Ωt−1 ; Θ)! !
=! !
g(st−1 , st , sτ , sτ −1 , wt |xt , Ωt−1 ; Θ)stst−1
! sτ sτ −1st−1 Pr(st−1 , st , sτ , sτ −1 |Ωt−1 ; Θ)p(w t |xt , st−1 , st , sτ , sτ −1 ; Θ)
! !
=! !
.
st
st−1
sτ
sτ −1 Pr(st−1 , st , sτ , sτ −1 |Ωt−1 ; Θ)p(w t |xt , st−1 , st , sτ , sτ −1 ; Θ)(C.4)

Since the transition is Markovian
Pr(st−1 , st , sτ , sτ −1 |Ωt−1 ; Θ) = Pr(st |st−1 , sτ , sτ −1 ) Pr(st−1 , sτ , sτ −1 |Ωt−1 ; Θ)
= Pr(st |st−1 ) Pr(st−1 , sτ , sτ −1 |Ωt−1 ; Θ)and we havep(wt |xt , st−1 , st , sτ , sτ −1 ; Θ) = p(wt |xt , st ; Θ),24

(C.4) can be rewritten as
Pr(st , sτ , sτ −1 |Ωt ; Θ)!
st−1 Pr(st |st−1 ) Pr(st−1 , sτ , sτ −1 |Ωt−1 ; Θ)p(w t |xt , st ; Θ)
! !
=! !
st
st−1
sτ
sτ −1 Pr(st |st−1 ) Pr(st−1 , sτ , sτ −1 |Ωt−1 ; Θ)p(w t |xt , st ; Θ)

(C.5)

where Pr(st |st−1 ) is the transition probability and p(wt |xt , st ; Θ) is the conditional density at time t.

25