Discussion Paper No. 1

Modeling Preference Change
through Brand Satiation
Nobuhiko Terui and Shohei Hasegawa
Graduate School of Economics and Management
Tohoku University
Sendai, Japan
April, 2013

Data Science and Service Research
Discussion Paper

Center for Data Science and Service Research

Graduate School of Economic and Management
Tohoku University27-1 Kawauchi, Aobaku
Sendai 980-8576, JAPANModeling Preference Change through Brand Satiation

byNobuhiko Terui
Graduate School of Economics and Management
Tohoku University
Sendai, Japanterui@econ.tohoku.ac.jp
andShohei Hasegawa
Graduate School of Economics and Management
Tohoku University
Sendai, Japanshohei.hasegawa@gmail.com

April, 20131

Modeling Preference Change through Brand Satiation

Abstract
In this study, we develop structural models of preference change due to consumer state
dependence through satiation by purchase experience.

A dynamic factor model with switching

structure is proposed to explain consumer preference changes.

Two types of dynamic factor models

are separately applied to baseline and satiation parameters in a direct utility model that accommodates
multiple discreteness data.

The first dynamic factor model has a switching structure for consumer

preference, and decomposes brand baselines into time-invariant factor loadings for the coordinates ofbrand positions and time-varying factor scores for consumer preference directions.

The second

dynamic factor model applied to satiation parameters extracts the consumer level of satiation in a
product category, and this is used as a causal variable in a switching equation to show when and how
preferences change over time according to the level of brand satiation.

The brand positions andtemporal changes of heterogeneous preferences are jointly depicted in a dynamic joint space map.
The empirical analysis of a panel dataset shows that our proposed dynamic model, implyingthat consumers change their preferences when previous brand satiation exceeds the admissible leveland preference directions are determined by the previous level of satiation, performs better thanalternative specifications, such as a static model with no preference change and a dynamic model
without structures which imply that preference changes whenever a consumer purchases a product.

Key words and phrases:
Structural Modeling, Brand Positioning, Consumption Experience, Joint Space Map, Dynamic
Factor Model, Multiple Discreteness Data, Satiation, Switching Structure2Modeling Preference Change through Brand Satiation

1. Introduction
Satiation through consumer experience has been discussed in several ways by using different
definitions and for different purposes.

The first stream of research on satiation is related toconsumer brand-switching behavior in a category that has been discussed in the literature and labeled
as variety seeking, e.g., Bawa (1990), Jeuland (1978), Lattin (1987), Johnson, Herrmann, and Gutsche(1995), McAlister (1982), and Lattin and McAlister (1985).

These studies assume thatvariety-seeking behavior is caused by the satiation dynamics of the desired attributes at a given
time and that consumers switch brands when the inferential attribute, such as caffeine in the drink
category in McAlister (1982), is satiated.

They investigated their hypotheses on the basis ofexperimental data.
The second stream of research on satiation is related to intertemporal choice.
defined as the factor of carryover effect of consumption from one period to the next.

Satiation is
Brand satiation

decreases the utility incorporating satiation due to previous consumption, which is called thediscounted utility model in Baucells and Sarin (2007).

This is extended to a model that captures theeffect of past consumption by the habituation model, e.g., Wathieu (1997, 2004) and Baucells andSarin (2010).

These analyses are conducted on the basis of a normative approach by using analyticalmodels.
The third stream of research on satiation is related to an economic model with diminishingreturn of marginal utility proposed by Kim et al. (2002, 2009), where the satiation parameter means
the curvature of an direct utility function to explain consumer multiple-choice behavior.
therein is related to the effect of broadening the width of choice.

Satiation

In particular, Hasegawa et al.

(2012) proposed the model extracting dynamic satiation score for individual consumer by using
dynamic factor model in a choice model which accommodates multiple discreteness data with direct3

utility.

They proposed a model in which dynamics allow the factor scores to evolve over time,reflecting variation in household satiation sensitivity. The analysis of a panel dataset of corn chips
purchases indicates that respondent satiation is better explained by a low-dimensional factor structure,
leading to implications for product line assortment in the face of quickening satiation.
In this study, we investigate change in consumer preference from the point of view of satiation
due to purchase experience.

We propose a model to decompose baseline parameters of a direct

utility function into time-invariant factor loadings representing brand positions, and a time-varying
factor score, the weight of the lower-dimensional axis, and therefore, the preference direction in alower-dimensional attribute space.
This is an extension of the concept and models for analyzing market structure in the literature
called a joint space map, and originally proposed by Hauser and Shugan (1983).
are found in the following studies.

Several extensions

Elrod (1988) proposed a perceptual map model based on

consumer-choice behavior by employing factor analytic decomposition of a brand-specific intercept in
the utility function.

Chintagunta (1994) extended the model to heterogeneous consumers grouped

into a finite number of segments.

Erdem (1996) incorporated a higher level of heterogeneity in themodel to give continuous mixture models, where heterogeneous parameters were integrated by thesimulated maximum likelihood method, leading only to homogeneous model parameter estimates.
Moreover, the time-varying intercept connecting to last purchase behavior is incorporated such thatthe intercept value increases if the same brand is chosen as the last purchase and vice versa; andbrand-loyal consumers or variety seekers are identified by reviewing this data, although the brandpositioning as well as preference vectors are time invariant.

Recently, focusing on the effect of new

product entry on market structure, Rutz and Sonnier (2011) proposed a choice model to represent the
dynamic change of brand positions, where consumer preference is kept constant.
Most previous studies assume that consumer preference is independent of previous behavior,and that choice is driven temporally by marketing strategies such as pricing and promotions.
4

In thisstudy, we relax this assumption in a way that preference is state dependent similar to the case in
Erdem (1996), and propose a model to test the hypothesis that preference changes in the process ofpurchase experience, and explore the structure when it occurs.
Our model extends those in previous studies in several ways.

First, our model is an extension

to the dynamic model describing market structure in terms of product and consumers.

Second, ourmodel accommodates individual consumer heterogeneity, and then we depict the dynamic joint space
map for an individual consumer.

Third, we incorporate a mechanism to switch the preference

direction in a model, i.e., we develop structural modeling of preference change in a testable way
empirically by the data.
As a contribution of structured choice modeling, we incorporate two types of dynamic factor
models in a choice model with a direct utility function that accommodates multiple discreteness data.
Then, the factor score extracted by the dynamic factor model applied to satiation parameters drives
the change of preference direction, which is defined by factor score vector from the second dynamic
factor model applied to baseline parameters.

Preference changes when the satiation level in theprevious state exceeds the admissible range, and it does not change otherwise.

We compare themodel with alternative models, which are categorized by “dynamics” and “structure.”

This includes

the model without state dependence, i.e., the steady preference model, the dynamic model without
structure which implies that preference changes whenever a consumer purchases a product.
The remainder of this article is organized as follows.

In section 2, we present the models bydefining the utility function and distributional assumptions of the model variables as well as relatedcomparative models.

Section 3 describes the empirical results of an application of the proposedmodel to a panel dataset of corn chips purchases.
concludes this study with a summary.

In section 4, we discuss implications, and section 5

The algorithm for model estimation and parameter

identification issues are provided in the appendix.

5

2. Model
Utility Function and Likelihood
We employ the utility function proposed by Bhat (2005) and used in Hasegawa et al. (2012) byincorporating product attributes and dynamic effects in the baseline utility and satiation parameters.
Consumer h’s utility over j

U

x

where

,…,x

1, … , m varieties at time t are defined as follows:

0.

ψ

ln γ

x

1

ln z

,

(1)

is the vector of quantity demanded by consumer h at t, z

represents the outside good, and ψ
γ

∑

,z

, γ

j

1, … , m are parameters restricted as ψis the baseline value of marginal utility for a product j when x

0 and0, and γ

is asatiation parameter that affects the rate at which marginal utility diminishes.
The stochastic model is obtained by assuming that the baseline utility parameter has an error, orthat ψ

exp ψrespectively.
constraintEεwhere ψand ε

are unrestricted and independent errors,

Then, the likelihood function is obtained by maximizing (1) subject to the budget′z

E , whereis the total expenditure.
Qand, respectively, mean price and quantity vector, andThis is accomplished by creating the auxiliary equation as follows:

U

,z

λ

′

z

E

.

(2)

By employing the Kuhn–Tucker conditions of constrained utility maximization, we obtain an
expression that relates the observed demand to the error terms as follows:

ε

ψ

ln γ

x

1

p

ln

′

E
ε

ψ

ln γ

x

1

6

ln

if x

0

(3)

if x

0.

(4)

p x

E

′

Then, the likelihood function is composed of a combination of density and mass, arising from theinterior and corner solutions, respectively. We assume that it follows independently normal
distribution ε ~N 0, 1 , as developed in Hasegawa et al. (2012).

Baseline and Preference Dynamics with Switching Structure
We assume that the baseline parameters are well projected into a lower-dimensional attributed
space, as is done in the choice map:;

N 0, V

Each row vector of factor loadings matrix
corresponding factor score vector

diag v , … , v.

(5)defines the coordinate of brand position and, indicating consumer h’s preference direction at time t.

Weassume that the preference direction will change when consumer satiation level exceeds theadmissible level r , but does not change otherwise.

Then, the first dynamic factor model is

described as follows:
f

where

r

β

,β

′

;

N 0, Iand we set the hierarchical model as β

0 for identification in the empirical application.

behavior theory.

if f

r

if f

r ,

N β ,1 k

(6)
1, 2.

We setThis formulation is grounded in consumer

That is, the existence of threshold r as a reference point to evaluate current statusis based on the adaptation-level theory of Helson (1964), and consumer asymmetric response between
regimes is supported by the prospect theory of Kahneman and Tversky (1979).

Satiation Dynamics
We relate m satiation parameters to p brand characteristics pusing information provided to

us by the product manufacturer in a linear mapping, similar to that found in conjoint analysis.
are used as γTheyc α , and are organized in a matrix form by
,
7

(7)

where γ

exp γ, c is a vector of characteristics for the jis the matrix constituted by a row vector.

product of dimension p, andOur previous study, Hasegawa et al. (2012), shows that thebrand characteristics data are well connected to satiation and one factor is appropriate for our dataset
used in the empirical application.

In addition, it was shown that this information is also effective inallowing us to know how these characteristics affect the demand for a product and whether a subset ofcharacteristics is responsible for satiation for a firm’s offerings.
We next decompose the part worth

regarding satiation into a time-invariant factor loading

and one-dimensional factor f

to define the second dynamic factor model as follows:

matrixff;

f

N 0, Σν ;νdiag σ , … , σ(8)

N 0,1 .

(9)

Specification (9) defines a non-parametric model for temporal dynamics, and it accommodates atrend component locally over time in the non-stationary part worth and satiation parameters.

(See

the literatures in time series analysis, e,g, Harvey, A.,1989, Kitagawa and Gersh,1984 and West andHarrison ,1997) The factor score moves more smoothly when the variance of factor score is smaller
than the part worth’s variance, as is employed and discussed in Terui et al. (2010) and Hasegawa et al.
(2012).

Alternative Models
We compare our model with six alternative models.

The first model is static on the preference

direction defined by the ordinary factor model, and is denoted as (Static).
;

N 0, V

diag v , … , v

.

(10)

The second alternative is the dynamic model as follows:;

N 0, I .

This is non-parametric modeling of a stochastic trend in time series
prior in state space modeling.

(11), and is called smoothness

We call this model as “non-structured” in the sense that there are no8

causal variables or structural parameters in the equation.
(11) a non-parametric dynamic factor model (NDF).

We call the model represented by equation

This specification was successfully employedin Hasegawa et al. (2012) to capture the locally linear stochastic trend for a non-stationary series.

The third model describes that the preference direction has a structural equation that is determined bysatiation at a previous period as a causal variable:;fh1 ht 1

N 0, I .

(12)

We call the model represented by (12) as the structured dynamic factor model (SDF).

Both the NDF

and SDF have a common property that preference changes whenever a consumer purchases a product.
The next class of models has a switching mechanism regarding the timing of preference change.
We assume that satiation drives the change when the satiation level exceeds the admissible range, andit does not drive the change otherwise.

This class has some variations in types.

The first model is

an SDF model with switching structure, called a switching non-parametric dynamic model (SSDF1):;

N 0, I

if f

r

if f

r

(13)

The second model is our proposed dynamic factor model (SSDF2) shown in (6). The third model iscomposed of these models, called a hybrid dynamic factor model (SSDF3):;fh1 ht 1

N 0, I

if f
if f

r
r

(14)

The fourth model is an autoregressive model (SSDF4):fh1 ht 1;h2

N 0, I

if f
if f

r
r

(15)

These models provide a comprehensive set for assessing the benefit of the proposed dynamic
model for describing preference change.
93. Empirical Application
Data and Variables
Chips Data
Data were obtained through an experiment involving undergraduate students at a large
university. Students were included in the experiment if they frequently purchased salty snacks for
personal consumption.

Students were allocated a $2.00 weekly budget and asked to purchase among

eight varieties of corn chips.
six packets each week.

The offerings were priced at $0.33, allowing the students to select up to

The regular price of a corn chips packet was $0.99.

The students were toldthat any unused budget allocation would be paid in cash at the end of the experiment.

By offering

the chips at reduced prices, we hoped to induce higher levels of consumption, which might provide
useful information about satiation.

Students were instructed to purchase the chips for their own

consumption, not for the consumption of others.

These data were previously analyzed by Kim et al.

(2009) using a subset of product characteristics and a stationary demand model.
Table 1 lists the offerings and associated product characteristics that were provided by the cornchips manufacturer.

The chip varieties and characteristics are disguised for proprietary purposes, butreflect summary taste characteristics such as “citrus,” “red pepper,” and “treated corn” that aremeaningful to the manufacturer.

The experiment was conducted over a seven-week period, resulting

in a total of 634 observations for 101 subjects. The data for each purchase occasion is composed ofa vector of purchase quantities of each of the eight corn chip varieties, and the quantity of the outside
good that was set equal to the unspent budget allocation.

Previous analysis of a portion of thecharacteristics reported in Table 1 indicated that product characteristics could successfully be related
to baseline utility in a static model of a choice model.

== Table 1 ==10

Summary statistics of the data are reported in Table 2.

Very few of the purchase occasions

resulted in a corner solution where just one of the varieties were selected.

Purchase incidence of thevarieties ranged from 168 to 244, indicating that no variety was dominant in the data.

The

prevalence of interior solutions points to the need for a demand model that can accommodate interior
solutions.

== Table 2 ==Model Comparison
We employ Bayesian MCMC methods to evaluate the joint posterior density for these models.
Algorithms for model estimation are provided in the appendix. Models converged relatively quickly
and were estimated on the basis of 20,000 iterations of the Markov chain after 10,000 burn-in
samples.The interpretation of satiation parameters and the number of factors are robust throughout themodels.
Table 3 reports two measures of model plausibility for each model: the log marginal density(ML) and Bayesian deviance information criterion (DIC).

The DIC is a measure of modelcomparison that explicitly penalizes a model for its number of parameters (Spiegelhalter et al., 2002).
We use the DIC instead of calculating model performance on a holdout set of data because our student
panel experienced a fair amount of attrition toward the end of the study, particularly in the sixth andseventh weeks.

The loss of data toward the end of the panel makes it difficult to compare theout-of-sample predictions, particularly with dynamic models.
differ greatly in their fit to the data.

== Table 3 ==11

The results indicate that the models

First, we find that incorporating dynamics into the baseline parameters ψleads to adramatic improvement in model fit in terms of criteria by observing dramatic improvement between
the static and dynamic models.

The static model fit shows DIC = 11067.6, and log ML = −5088.8.

On the other hand, the dynamic models have approximately 80% lesser DIC and 60% greater log ML
values.
Second, the comparison among the dynamic models shows that the switching models dominate
the steady changing models in both criteria.

This means that preference changes occasionally andoccurs relative to the level of satiation, which is constituted in the previous period through purchaseexperience.

In addition to the fact that NDF performs slightly better than the causal model (SDF) in

steadily changing models, the switching structure with the level of satiation is intrinsic to capture
consumer behavior.
Finally, the best model is our proposed model (SSDF2).

This means, in addition to the switchingmechanism above, that preference change is caused by the previous level of satiation.
model is the structured dynamic factor autoregressive model (SSDF4).

The next best

This means that theparametric structure works better than non-parametric local trend models (SSDF1 and SSDF3).

Parameter Estimates
Table 4 summarizes the posterior distributions of parameters for the model SSDF2, thebest-fitting model.

== Table 4 ==First, the bottom portion, Table 4(c), reports estimates of parameters in the second dynamic
factor model, i.e., product attribute part-worth on satiation, factor loadings, and variances.
12

Theand have opposite signs, implying that f

estimates forrepresents an excitement(anti-satiation) factor score, the same as that shown in Hasegawa et al. (2012).
worth

The estimates of partmean the importance weights of product characteristics c1-c12 on the satiation.

The

characteristics (c4, c5, c10, c12) have positive large numbers of estimates, implying that thesecharacteristics contribute to brand satiation.

In contract, (c1, c8, c11) with negative large values arecharacteristics which make consumers excitement (anti-satiation).

Finally, (c2, c3, c6, c7, c9) havealmost zero impact on satiation.
Next, the top portion, Table 4(a), reports estimates of parameters in the first dynamic factor
model with switching structure, and the means across households and over time.

Since the posterior

distributions of some of these quantities are skewed, we also report the posterior median when needed.
We observe that the estimated baselinepurchase records.

for products are almost proportional to their shares in

The factor loading matrix

significantly estimated.

, as well as the variance matrix V, are almost

Finally, the middle portion, Table 4(b), reports the posterior mean andmedian for coefficient parameters in the switching equation.

The satiation level f

direction of preference in the first dimension as E β

1.323 standard deviation S. D.

0.885 .

On the other hand, it does not affect the second dimension as E β

explains the

0.112(S.D.:0.854).

== Figure 1 ==Figure 1 depicts a histogram of individual consumer propensity score k to change theirpreference.

The score is defined bykwhere k

∑R k

T

k

T

(20)

⁄R is the posterior probability of consumer h’s changing preference at periodt across R iterations of MCMC, and kis the indicator taking binary according to the switchingmechanism
13

k

1

if f

0

0

if f

0

(21)

The figure shows that many consumers change their preference, as E k
consumers with k

0.8 is 71.3%.

= 0.791 and the share of

On the other hand, it is true that some consumer preferences

do not change much.

== Figure 2 ==Figure 2 shows histograms of estimated coefficients on the switching equation.
figures are the estimates of ββ

for the first dimension of the preference direction and those offor the second dimension, respectively.

The heterogeneous distribution across consumers is

relatively stable, although slightly skewed to the right for β
0.112.

The left and rightgConsidering the relation g

β
β

ω
ω

f

.

Eβ

1.323 and E β, this means that the satiation level

affects the first dimension more than the second dimension.

4. Discussion
In this section, we investigate the dynamics of preference change by considering three panelists
with three patterns that exhibit frequent change of preference over time (panelist #97), moderately
frequent change (panelist #35), and not changing (panelist #15), and then consider how estimates forindividual consumers are related to observed purchase behavior and switching structure.

The

dynamic joint space maps are depicted.
Figure 3(a) shows that the preference direction moves every time for panelist #97.

The purchaserecord indicates multiple purchases with a broad range, a satiation score (minus f value) that remains
at a high level, and then the switching mechanism works such that the coordinates in the attributed
space move all the time.

This consumer can be characterized as a variety seeker.

14

The satiation

(excitement) level affects both coordinates positively, and this impact is much greater for the first
dimension.

== Figure 3 ==Figure 3(b) provides the map and tables for panelist #35, showing moderately frequent change.
Preference changes up to the third time purchase, but does not change anymore after this period.

Wenote that the preference direction is not heading to any product during the first period, and four
varieties of corn chips of a single quantity were purchased at this time.

This could imply that she

was unfamiliar with this product category and getting excited as she purchased them.

The satiation(excitement) level negatively and positively affects the first and the second coordinates to change,
respectively.
Figure 3(c) provides the map and tables for panelist #15, showing no preference changes.

The

record for purchasing D is consistent with the preference direction.
The satiation (excitement) level negatively and positively affects the first and second coordinates tochange, respectively, and it is much greater for the first coordinate.

5. Concluding Remarks
In this study, we propose a dynamic model of consumer preference change through purchaseexperience in a direct utility model that accommodates multiple discrete choices.
structural modeling of preference in two ways.

We develop

The first is a cause–effect model of preference

directions based on the satiation level in a product category, and the second is a switching structure
indicating when the change occurs relative to the level of satiation.
From a modeling perspective, two dynamic factor models are applied to baseline and satiation
parameters to extract dynamic factor scores.

The first dynamic factor model has a switching15

structure based on the factor score according to the level of factor score derived from a second
dynamic factor model, where the switching occurs when the second score exceeds some admissiblelevel, and it does not occur otherwise.

This is motivated by the adaption level theory and theprospect theory for our switching structure.

We could furthermore attribute it to the assumption thata consumer has latitude of acceptance for satiation, as satiation and variety-seeking behavior ingeneral, which has been studied extensively in marketing, e.g., McAlister and Pessemier (1982), withtaxonomies proposed for explaining variation in consumer choice.
We extensively compare the models, including a static model implying that preference does notchange at all, a dynamic model without a switching structure on preference change, and dynamic
models with switching structures.

The models in the last category are composed of non-parametriclocal linear trend, parametric regression, and their hybrid models.

The measures of model fit, log of

marginal likelihood and DIC, support the model with a switching structure and parametric regression.
This means that preference will change occasionally after a consumer is satiated enough, and that it
stays the same until the critical level.
The empirical application shows that the mode of switching is heterogeneous over consumers.
70 % or more of consumers change their preference by 80% of their purchase opportunities.

On theother hand, another portion of consumers is almost uniformly distributed over other levels of change.
The investigations of individual consumers are consistent with their choice behavior.

That is, theconsumer with frequent change of preference has a variety-seeking purchase record with broader
range of choices, the consumer not changing preference has a narrow range of choices, and theconsumer with moderate preference change behaves in between.

Furthermore, the regression

coefficients provide useful information on satiation to the coordinates of preference dimensions.
We have some limitations in this research. The empirical findings above are only obtained byapplying the model to the dataset we used.

The investigation regarding empirical generalizationneeds more datasets to be analyzed in a variety of categories.
16

When this is done, the causal variable

on the structural equation to determine preference as well as switching structure could be different
depending on the characteristics of the category. Another limitation is on the modeling.

That is,contrary to our assumption of continuous quantity of purchase for analytical equilibrium solutions, weobserve a discrete number of purchase quantities.

We leave these problems for future research.

17

Appendix – Identification Condition and MCMC Algorithm
We explain the identification condition for the dynamic factor model, and summarize the prior andconditional posterior distribution used for our proposed one-factor model below.

1. Identification Condition on and Priors for Factor Models
For a two-factor model applied to baseline parameters, we restrict the loadings to achieve statistical
identification:
1
a
a

0
1
a

a

a

(A.1)

This restriction due to the factor model being applied to parameters of a latent utility is stronger than
Geweke and Zhou’s (1996) condition for the conventional factor model.
We then define prior distribution of factor model as:σ
N a ,A ;a

IG n /2, s /2
a ,a

(A.2)

′

N

for 3

,

k

p,

(A.3)

as suggested in Lee (2007).
The first column of (A.1) is set on factor loadings
parameter.

for a one-factor model applied to satiation

The following same prior distributions are employedvb

IG n /2, s /2

N b ,B

for 2

(A.4)
k

p,

(A.5)

2. Prior Distributions on Hyper ParametersPriora

N a ,AN

βv

Setting
a

0,

,

A

,
β

N β ,ν

IG n ⁄2 , s ⁄2

n

18

100
100

0,

νβ0

10

2,

s

2

3. Conditional Posterior Distributions for MCMC
We run 20,000 MCMC iterations for all models, and we used last 10,000 iterations to calculate
posterior distribution of model parameters.
|

(1)

,

, ,

,
|

p

,

, ,

,
⁄

det| |

′

exp

⁄2

(A.6)

L
The term Ltis the likelihood function for consumer h

1, … , H at purchase time

1, … , T , where the likelihood function is composed of a combination of density and mass,

arising from the interior and corner solutions, respectively, and is defined as
||

,…,

,…,

See the details in Hasegawa et al. (2012).
Setting r

1, … , R to MCMC iterations, we use Metropolis–Hastings with a random walkalgorithm, each h

1, … , H and t

1, … , T .
ψ;

(A.7)

N 0, 0.5

ψThe acceptance probability is
min
|

(2)

,

p

,

p

, ,
,

,

, ,

(A.8)

,

, ,f ,
p|,, ,f ,
det| |

⁄

exp

′

f

f

⁄2

(A.9)

L
As fort, we use Metropolis–Hastings with a random walk algorithm, each h

1, … , H and1, … , T .
α;αN 0, 0.01(A.10)

The acceptance probability is
min

p

,

p

, ,
,

19

, ,

,
,

(A.11)

(3)

|

,f ,

Under the assumption of uncorrelated a ’s, we define′

then make downward stacking over h,,ααdefine,…,α T′: T

f ,f ,
′

; H′

, ′,

: ∏H
′

1, and

T
′

,

′

,f T

,

,

: T

1 matrix, and1 matrix.

Similarly, we

′

′
H

T

1.

and explanatory matrix .

Then, we have the regression equation with coefficient parameter vectora: ∏H

N a ,A ,(A.12)

where
A

A

′

σ

,

a

The identification condition is considered when k
(4)

,

|

σ

′

1.

,

In the same way as
′

A A a

′

,
′

,

;
′

,

,

, we define
′ ′
H

,

: ∏H

,

H

′ ′

T

,

,

′
T

2 matrix and

: ∏H

T

: T

ψ

2 matrix,,ψ,…,ψ T

′

: T

1,

1.
N

,

,

(A.13)

where
′

v

,

a

The identification condition is considered when j(5) f ,

|

,

′

v
2.

, ,

We reformulate measurement equation (Equations (8) and (13)) and system equation (Equations (9)and (14)).
Measurement equation:
0

f

;

0

N 0,

0

(A.14)

0

System equation:
1

f

where

0
1 K

K
β

,β

′

ν

f

and

20

;

ν

N 0,

1
0

0
K

,

(A.15)

K

1, if fr
(A.16)

K
where we define f

f0, if fr ,by interpreting the factor of anti-satiation for f.

We use Carter and Kohn (1994) for a time-varying coefficient in state space model expressed as
Equation (A.14) and Equation (A.15)

(6)

′

,β

β

|f ,

,
N β

β

,

′

1

k

1,2

(A.17)

where
′

β

′

1

.

and

are the row vector collected in case of regime f

and

0 at all t), posterior is β

If

(K

(7)

β ,β

′

νβ

or K

1 .

N β , 1 by the homogeneity.

N β , H

νβ

where
H

r

|
β

β

β

∑H

β

νβ β

21

k

1,2

(A.18)

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23

Table 1 Product Varieties and Characteristics
Varieties
Variety A
Variery B
Variety C
Variety D
Variety E
Variety F
Variety G
Variety H

c1
1.00
0.00
1.00
0.00
1.00
0.00
0.00
0.00

c2
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2.17

c3
0.00
0.00
3.17
0.00
0.00
0.00
3.17
0.67

c4
0.00
0.00
0.00
0.00
0.50
0.00
0.00
0.00

c5
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.50

c6
0.00
1.00
0.00
1.33
1.83
0.00
0.00
0.00

c7
4.00
3.00
0.00
0.88
4.25
4.00
0.00
3.38

c8
1.50
0.67
1.33
3.67
0.00
1.50
1.33
0.00

c9
4.00
3.83
4.17
3.33
4.17
4.00
4.17
3.83

c10
2.00
4.25
4.00
5.50
5.00
2.00
4.00
0.00

Table 2 Purchase Summary
Varieties

Purchaseincidence

Variety A
Variery B
Variety C
Variety D
Variety E
Variety F
Variety G
Variety H
Total

168
177
188
180
190
244
235
218
1600

Total
purchasequantity
224
262
231
235
295
446
338
277
2308

CornersolutionInterior
solution

4 (.02)2 (.01)
6 (.02)12 (0.01)

168 (1.00)
173 (0.98)
188 (1.00)
180 (1.00)
188 (0.99)
238 (0.98)
235 (1.00)
218 (1.00)
1588 (0.99)

Kim et al. (2007)
Table 3 Model Comparison

Model

DIC
11067.6
NDF (No structure) 8649.0
SteadySDF (Strcture)8756.7SSDF18435.0
Dynamic
SSDF2
8170.0Switching
SSDF38425.9SSDF48283.1
Static24ML-5088.8-3124.5-3171.8-2829.4-2754.1-2829.5-2758.1

c11
0.00
2.00
0.00
2.25
1.00
0.00
0.00
0.00

c12
1.75
0.00
0.00
0.00
0.00
1.75
0.00
0.00

Table 4 Parameter Estimate(a) Baseline Parameters
V
A
B
C
D
E
F
G
H

Mean
-1.763-1.668-1.782-1.552-2.127-0.841-0.836-1.074

Median-2.307-2.291-1.870-1.900-2.316-0.685-0.385-0.846

S.D(3.026)(3.339)(3.146)(3.065)(3.782)(3.007)(3.129)(2.716)

Mean
1.000
0.000
1.110
1.000
1.066
0.008
1.045
0.772
1.184
-0.353
0.683
1.108
0.690
1.395
0.806
0.863

S.D
(0.107)(0.084)(0.098)(0.085)(0.110)(0.129)(0.093)(0.118)(0.105)(0.129)(0.092)(0.144)(0.093)

Mean
3.140
4.528
3.003
3.304
3.773
4.722
2.479
3.183

S.D
(0.567)(0.673)(0.545)(0.576)(0.642)(0.653)(0.478)(0.534)

These numbers show the grand mean and median of panelist’s estimates over time across panel members.
Number of b and V show the posterior mean and the posterior standard deviation in parentheses.

(b) Switching Equation: Baseline ParametersMean-1.3230.112

ββMedian-1.4280.162

S.D(0.885)(0.854)Mean-1.322
0.110

S.D(0.186)(0.189)These numbers show the grand mean of panelist’s estimates across panel members.

(c) Satiation Parameters: Characteristic Level
Σ
c1
c2
c3
c4
c5
c6
c7
c8
c9
c10
c11
c12

Mean
-1.230
0.007
0.081
0.547
0.421
0.052
0.012
-0.224-0.0330.220-0.1290.283

Median-1.396
0.006
0.094
0.603
0.469
0.054
0.016
-0.246-0.0330.253-0.143
0.325

S.D(1.455)(0.033)(0.096)(0.658)(0.498)(0.076)(0.035)(0.262)(0.056)(0.253)(0.151)(0.332)Mean
1.000-0.003-0.064-0.441-0.336-0.042-0.0100.180
0.029-0.1740.102-0.226

S.D(0.029)(0.043)(0.049)(0.042)(0.035)(0.026)(0.033)(0.020)(0.037)(0.037)(0.053)

Mean
0.187
0.105
0.063
0.461
0.322
0.136
0.040
0.069
0.034
0.034
0.094
0.120

S.D
(0.058)(0.026)(0.012)(0.107)(0.056)(0.024)(0.006)(0.010)(0.004)(0.004)(0.019)(0.028)

These numbers show the grand mean and median of panelist’s estimates over time across panel members.
Number ofand Σ show the posterior mean and the posterior standard deviation in parentheses.

25

30
0

10

20

Frequency

40

50

60

Figure 1 Preference Change

0.0

0.2

0.4

0.6

0.8

1.0

Figure 2 Parameter Estimates in Switching Equation

25
20
0

5

10

15

Frequency

15
10
5
0

Frequency

20

25

30

2

30

1

−3

−2

−1

0

1

−2

26

−1

0

1

2

Figure 3 Preference Dynamics(a) ID#97 (frequent changing)
97
−0.5

0.0

0.5

1.0

B
D

23

2

1.0

−1.0

H

5
6

0

AC

0.0

1

F

0.5

G

E

−1.0

−2

−0.5

−1

4
1

−2

1
2
3
4
5
6

β
β

A
0
0
1
3
1
0

B
1
0
0
0
0
2

C
1
1
0
3
1
1

D
1
1
1
0
1
2

E
1
1
1
0
1
1

−1

F
1
1
1
0
1
0

0

G
0
1
1
0
0
0

H
0
1
1
0
1
0

1.237
0.226

27

1

f
-1.443-1.462-1.374-1.693-1.331-1.309

2

g
0.887
0.915
0.929
0.960
0.997
1.000

g
-0.4630.404
0.370-0.281
0.081
0.008

k
1.000
0.981
0.984
0.988
0.994
0.961

(b) ID#35 (moderate changing)
35
−1

0

1

2

2

2

−2

1

F
H

1

G
B
D
3

0

0

2
AC

−1

−1

E

−2

−2

1

−2

1
2
3
4
5
6

β
β

A
0
0
0
0
1
0

B
1
0
1
2
0
0

C
1
0
0
1
2
0

D
1
2
0
2
2
2

E
1
3
2
0
0
0

−1

F
0
0
0
0
0
2

0

G
0
1
2
0
0
0

H
0
0
1
1
1
2

-0.462
0.258

28

1

f
-0.485-0.421-0.068
0.413
1.066
1.583

2

g
-0.162
0.977
0.876
0.933
0.992
0.917

g
-0.987
0.212
0.483
0.359
0.130
0.400

k
1.000
0.561
0.506
0.462
0.433
0.297

(c) ID#15 (not changing)
15
−1

0

1

2

2

2

−2

1

G
1

F
H

B
D
1
0

0

AC

−2

−2

−1

−1

E

−2

1
2
3
4
5
6

β
β

A
0
0
0
0
0
0

B
1
1
0
0
2
0

C
1
0
0
0
0
0

D
1
1
3
3
4
4

E
1
1
0
0
0
0

−1

F
0
0
0
0
0
0

0

1

H
0
0
0
0
0
0

f
0.865
0.743
0.548
0.810
0.783
0.666

G
2
3
3
3
0
2

-1.147
0.278

29

2

g
0.967
0.942
0.840
0.768
0.815
0.723

g
0.254
0.335
0.542
0.640
0.579
0.690

k
1.000
0.156
0.198
0.204
0.173
0.152