数理解析研究所講究録
第 1804 巻 2012 年 144-161

144

The Latent Class Model of Brand Choice Behaviors
Incorporating Variety-Seeking and State
Dependence.

Apri124, 2012

Graduate School of Systems and Information, University of Tsukuba.
Tomohito Kamai
Department of School of Systems and Management, University of Tsukuba.

Yuichiro Kanazawa.

1

Introduction

Understanding the dynamic consumers’ brand choice behaviors is an interest to both retailers and manufacturers who plan their marketing strategies.

One of the major topic in this field includes the behavioral patterns seemingly persisting across more than one purchase occasion. They are referred

to as variety-seeking and state dependence.
The term “variety-seeking” is defined as the behavior such that the pur-chase of a brand will decrease the probability that the same brand will
be purchased again on the succeeding occasions. The “state dependence”,

sometimes called “inertia”, refers to the opposite behavior; the brand purchase will increase the probability of purchasing the same brand again on

145
the succeeding occasion (Bawa, 1990; Chintagunta, 1998).1
The existence of variety-seeking and state dependence plays an impor-tant roles in marketer’s decision making as Chintagunta (1998) states: “Froma managerial stand point, it would be important to know whether a brand’s

consumers are inertial or variety prone (Chintagunta, 1998, 254).” For example, the existence of a strong state dependence effect suggests that inducing
trial would be an effective marketing tactic; the promotional schemes suchas a free sampling would be more effective. The emphasis on brand retention
would be also important to marketers of this type of products (Chintagunta,
1998). On the other hand, the existence of variety-seeking behavior moti-

vates marketers to extend their product lines so that households’ brandswitching behaviors would benefit their own products (Seetharaman, 2004).
In summary, by correct understanding of the consumers’ brand choice
behaviors, the promotions can be effectively implemented to the right con-sumers with the right schemes. In this study, we will propose a model tocapture the complex behaviors of consumers who vary their choices without

any apparent reasons.

2

Literature Review

According to the taxonomy work on literature of varied behaviors of McAlister and Pessemier (1982), the variety-seeking behavior may not necessarily
be associated with a motivation for variety-seeking itself; this behavior is of-ten observed, for instance, when a household has multiple users for the same
product. The “real” or “pure” variety-seeking behavior we consider in thisstudy is a varied behavior motivated by a desire for an unfamiliar product ora stimulus associated with switching behavior. In this context, Givon (1984)

lWhen the purchase probability is affected by the last purchase (i.e., Markov process),
it is called “first-order” behavior. Accordingly, “zero-order” behavior refers to the pattern
in which the purchase probability is not affected by any previous choices.

146
used a stochastic model to express a utility associated with the switching
behavior itself in addition to the utility derived from the consumption of thespecific brands.
In contrast to variety-seeking behavior, some researchers found the “in-ertial behavior”, which is often referred to as “state dependence.” One of
the most influential paper in this stream is Guadagni and Little (1983), who
specified variables representing brand and size loyalties in the way that au-

thors assumed weighted sequential influences of past choices to the current

utility in the form of$x_{ijt_{t}}$$=$$\alpha_{b}\cdot x_{ij,t_{t}-1}$$+(1-\alpha_{b})$where$\alpha_{b}${ consumeri bought brand

$j$

at

$(t_{i}$$-1)$ -th occasion},

is a parameter and {statement} denotes an indicator function tak-ing the value unity if the statement is true.

Bawa (1990) proposed a “hybrid” behavior, which is characterized by

the behavior where consumers seem to exhibit inertial tendency for certain
period of time and then exhibit variety-seeking tendency once certain periodof time passes. The marginal utility in his model is specified to be thefunction of “run”, denoted by$r_{ij}(t_{i})$, which is the number of times thebrand had been continuously purchased up to -th occasion. The index$t_{i}$

and

3

$j$

$i$

denote consumer and product respectively.

The Specification of the Utility and the ModelIn constructing the model, we choose to use the brand loyalty variable of

Guadagni and Little (1983), which we will refer to it as “ $GL$ variable” henceforth, to express the state dependence part of the utility. Also, to capture

the effect of variety-seeking tendency, we choose to include run which was

defined in Bawa (1990). The purpose of including run is to “put brake” on
the $GL$ variable, which keeps increasing as long as the same brand is kept

147purchased. By including run, the consumer’s utility for the brand would

start to decline as a result of the repeated consumption of the same brandif run negatively affects utility.
Now if consumer is state dependent, the effects of both $GL$ variable andrun will be non-negative. The variety-seeking behavior will be detected by
the opposite signs; the effects of both $GL$ variable and run will be nonpositive. On the other hand, the hybrid behavior will be detected by the
positive coefficient of $GL$ variable and negative coefficient of run. Since the

combination of positive coefficient of $GL$ variable and negative coefficient of

run could indicate both variety-seeking and hybrid behaviors, depending on
the values of coefficient, we must scrutinize the results carefully before we
judge.

Now we write the utility of consumer $i=1,$

at occasion $t_{i}=1,$$U_{ijt_{i}}$where$x_{ijt_{i}}$$=$$T_{i}$$\ldots,$$\ldots,$$N$for brand $j=1,$$\ldots,$$J$as$x_{ijt_{i}}\beta_{\mathcal{S}}+\epsilon_{ijt_{i}}$,(3. 1)

is $1\cross R$ vector of the explanatory variables including a set ofdummy variables for brands except for one base brand, dummy variables forcoupon usage, the average coupon values redeemed, dummy variables forfeature and display, $GL$ variable and run. The$\beta_{s}$is corresponding$R\cross 1$

vector of parameters for segment . The segment is a subset into which$s$consumers are placed, where consumers in the same segment are assumed
to have homogeneous characteristics regarding preferences for brands andresponsiveness to the marketing variables. In our framework, we also assume

that consumers in the same segment show the same purchasing patterns
expressed in $GL$ variable and run. The term$\epsilon_{ijt_{i}}$is a random error termthat captures the effects of unobserved variables which follows i.i. . Gumbel$d$distribution.
In our study, we employ the latent class model which is one of the general models to incorporate heterogeneity across consumers. In the latent148
class model, it is assumed that there exist a finite and fixed number of segments where each consumer belongs to only one of the segments, and it isfurther assumed that consumers belong to the same segment over the periodof observation. The idea behind this type of model is that there exists anunderlying multi-dimensional distribution of parameters for intrinsic pref-

erences for brands and relative responsiveness to the marketing variables
which characterize consumers’ behaviors. In the latent class model, the underlying distribution of parameter is assumed to be discrete. Because thefinite representation of consumers’ characteristics of the latent class model
coincides well with the concept of segments, such a model is widely applied

to the marketing field. The major work of this field is Kamakura and Russel(1989). The other works using the latent class model include Bucklin et al.
(1998) and Gupta and Chintagunta (1994).

Now we will use the multinomial logit model framework for brand choice,and thelikelihood of panel data in our model is given by$\log$$l( \beta)=\sum_{s=1}^{s}\sum_{i=1t_{i}}^{N}\sum_{=1}^{T_{i}}\sum_{j=1}^{J}\{h_{i}(s)\cdot y_{ijt_{i}}\cdot\ln(\frac{\exp(x_{ijt_{i}}\sqrt{}S)}{\sum_{l=1}^{J}\exp(x_{i\iota t_{i}}\beta_{s})})\}$

where
$j$

$y_{ijt_{i}}$is the indicator variable taking value 1 if consumer buy brand$i$

at occasionof consumer$i$$t_{i}$

and$h_{l}\prime(s)$is the expected value of membership probability

to segment $s=1,$$\ldots,$$S$. For estimation, we used $EM$ algo-rithm assuming the consumers’ membership probabilities for each segment

as missing values.

4

Estimation

Now let us assume there are $s=1,$$S$$\ldots,$

segments in the sample. Obviously

as the number of segments is unknown nor can be observed, it must be
estimated. First define the relative sizes of segment$0<\pi_{S}\leq 1$

$s$

as

$\pi_{s}$such that

149
for all$s$and$\sum_{s=1}^{s}\pi_{S}=1$

.

(4.1)

In the latent class model, each consumer has different membership proba-

bilities for these segments, because membership probabilities are estimatedfrom their choice histories which differ across consumers. Now let us define
the random variablevalue one if consumer

for consumer , let

for $i=1,$$Y_{ijt_{i}}$$i$$y_{ijt_{i}}$$i$$\ldots,$

chooses brandbe entries of

$j$

$N$

at

$T_{i}\cross J$

and $t_{i}=1,$$t_{i}$$T_{i}$$\ldots,$, which takes-th occasion. In other words,

matrix$Y_{i}$(4.2)$Y_{i}=(\begin{array}{lll}y_{i11} \cdots y_{iJ1}| |y_{i1T_{i}} \cdots y_{iJT_{i}}\end{array})$

and let us denote each row as$y_{it_{i}}$. Assuming the$\epsilon_{ijt_{i}}$follows i.i. . extreme$d$value with respect to , we can express the probability that consumer$j$segment

chooses brand

$s$

$j$

$i$

in

at the -th occasion in the standard logit form
$t_{i}$

as$Pr\{(y_{i1t_{i}}, \ldots, y_{iJt_{i}})=(0, \ldots, 0,1,0, \ldots, 0)|S_{i}=s;\beta_{s}\}=\frac{\exp(x_{ijt_{i}}\beta_{S})}{\sum_{l=1}^{J}\exp(x_{ilt_{i}}\beta_{s})},$$\tilde{j-1} \overline{J-j}$(4.3)

where the random variable$S_{i}$

indicates which segments consumer$i$belongs

to, assuming we could observe the segment membership of consumer . For$i$brerity, we abbreviate (4.3) as$Pr(Y_{it_{i}}=j|S_{i}=s;\beta_{S})=\frac{\exp(x_{ijt_{i}}\sqrt{}S)}{\sum_{l=1}^{J}\exp(x_{ilt_{i}}\beta_{s})}$

,

(4.4)

henceforth.
The unconditional choice probability for brandconsumer$i$$j$of a randomly selected

can be obtained by integrating out the equation (4.3) by the

150
density in the population$\pi_{s}$as2$Pr(Y_{it_{i}}=j)=\int Pr(Y_{it_{i}}=j|S_{i}=s;\beta_{s})\cdot\pi_{s}ds$

Since the relative size of the segment$\pi_{s}$

.

(4.5)

is discrete, (4.5) is written as$Pr(Y_{it_{i}}=j)=\sum_{s=1}^{S}\pi_{8}\cdot Pr(Y_{it_{i}}=j|S_{i}=s;\beta_{s})$

.

(4.6)

This is a weighted average of logit formula evaluated at each mass point(segment), as pointed out by Kamakura and Russell (1989).

Now suppose that consumer$(H_{i1}, \ldots, H_{iT_{t}})$, where$H_{it_{i}}$$i$has the choice history defined as $H_{i}=$indicates the brand purchased at

Then the conditional choice probability that consumer

tory$H_{i}$

given that consumer$i$belongs to segment

$s$

$i$

$t_{i^{-}}th$occasion.

has the choice his-is written as(4.7)$Pr(H_{i}|S_{i}=s;\beta_{S})=\prod_{t_{i}=1}^{T_{t}}\prod_{j=1}^{J}\{Pr(Y_{it_{i}}=j|S_{i}=s;\beta_{S})\}^{y_{tjt_{i}}}$In the same manner as (4.6), the unconditional probability of randomly

selected consumer having the choice history$i$$H_{i}$

can be written as(4.8)$Pr(H_{i};\beta)=\sum_{s=1}^{s}\pi_{s}\cdot Pr(H_{i}|S_{i}=s;\beta_{s})$where$\beta$is $R\cross S$ parameter matrix(4.9)$\beta=(\beta_{1}, \cdots, \beta_{S})=(\begin{array}{llllll}\beta_{11} \cdots \beta_{1s} ’ \cdots \beta_{lS} | \beta_{r1} \cdots \beta_{rs} \cdots \cdots \beta_{rS} | \beta_{R1} \cdots \beta_{Rs} \cdots \cdots \beta_{RS}\end{array})$

Now if the segment memberships of consumers are completely known, thevector of parameters$\beta_{s}$can be estimated by the well known methods such

2The model of the form (4.5) is sometimes called mixed logit model and$\pi_{s}$is called

mixing distribution. The latent class model can be regarded as the special case of mixed
logit model where mixing distribution is discrete $(^{r}bain, 2003)$ .

151
as Newton-Raphson method. Here let us define for each the multinomial
indicator random variablewhich takes one if consumer belongs to$i$$z_{i}(s)$segment$\mathcal{S}$andof consumer

$i$

$0$

$i$

otherwise, assuming we know the membership probability

belonging to segment

given his $/her$ purchase history$s$$H_{i},$

that is $Pr(S_{i}=s|H_{i};\beta_{s})$ . Then this membership indicator random variables$z_{i}(s)$’s are entries of $N\cross S$ matrix $Z$ as$Z=(\begin{array}{l}z_{1}(\cdot)|z_{N}(\cdot)\end{array})=(\begin{array}{lll}z_{1}(1) \cdots z_{1}(S) z_{N}(1) \cdots z_{N}(S)\end{array})$

The row sums of the matrix $Z$ above are all 1.

Assuming we were able to observe $Z$ , the likelihood given the choicehistories of the all consumers under consideration is written$as^{3}$$L( \pi, \beta|H, Z) = \prod_{i=1}^{N}\prod_{s=1}^{s}\{\pi_{s}\cdot Pr(H_{i}|S_{i}=s;\beta_{s})\}^{z_{i}(s)},$where $H=(H_{1}, \ldots, H_{i}, \ldots, H_{N})$ is the choice history of all consumers inthe sample, $\pi=(\pi_{1}, \ldots, \pi_{S})$ is

Accordingly, the$\log$$1\cross S$

vector of relative sizes of segments.

likelihood could be written as$l( \pi, \beta|H, Z) = \sum_{i=1}^{N}\sum_{s=1}^{s}z_{i}(s)\cdot\ln(\pi_{s}\cdotPr(H_{i}|S_{i}=s;\beta_{s}))$$= \sum_{i=1}^{N}\sum_{s=1}^{S}z_{i}(s)\cdot\ln Pr(H_{i}|S_{i}=s;\beta_{s})+\sum_{i=1}^{N}\sum_{s=1}^{s}z_{i}(s)\cdot\ln\pi_{s}.$(4.10)
Now if we were able to observe $Z$ , we can estimate parameters by the traditional method. However, in reality, we cannot obtain the information$z_{i}(s)$.

In such a situation, the method called $EM$ algorithm may be implemented toobtain the estimate of$z_{i}(s)$along with the estimates of

$\pi$

and

$\beta$as explainedin the following subsection.

3The term
segment$s$$\pi_{s}\cdot Pr(H_{i}|S_{i}=s;\beta_{s})$and has choice history$H_{i}$

is unknown and has to be estimated.

is the joint probabihty that consumer$i$belongs to. Note, however, that the relative size of segment$\pi_{s}$

152
4.1

$EM$ algorithm

If the segment membership of consumers$Z$

were completely known, the$\beta_{s}$can be estimated by the algorithm described above. $EM$ algorithm takes
advantage of this fact and in the algorithm, the consumer’s membership to
the segment$z_{i}(s)$is assumed at first to be missing values and this value

is imputed by its “expectation” (to be explained below). Then the conditional likelihood is maximized based on the expected value of membership to

the segment. The consumers’ expected membership is then updated using
the updated likelihood. This cycle of “expectation” of membership to thesegment and “maximization” of likelihood is repeated until the likelihood

converges.
Now taking the expectation with respect to$z_{i}(s)$for the$\log$

likelihood(4.10), we have$E[l(\pi,\beta|H, Z)]$$=$$\sum_{i=1}^{N}\sum_{s=1}^{s}h_{i}(s)\cdot\ln Pr(H_{i}|S_{i}=s;\beta_{s})+\sum_{i=1}^{N}\sum_{s=1}^{S}h_{i}(s)\cdot\ln\pi_{s},$(4.11)

where$h_{i}(s)=E[z_{i}(s)]= \sum_{l=1}^{s}z_{i}(l)\cdot Pr(S_{i}=l|H_{i};\beta\iota)=Pr(S_{i}=s|H_{i};\beta_{s})$(4.12)

is the expected values of the indicator random variable
Since parameter$\beta$$z_{i}(s)$for $s=1,$

in (4.9) only appears in the first term and$\pi$$\ldots,$$S.$

only appears

in the second term on the right hand side of (4.11), they can be estimatedby maximizing $E[l(\pi, \beta|H, Z)]$ altemately.

Let us first look at the second term on the right hand side of (4.11).

Since we have the condition$\sum_{s=1}^{s}\pi_{s}=1$

from (4.1), the second term canbe maximized by the method of Lagrange multipliers given$L= \sum_{i=1}^{N}\sum_{s=1}^{s}h_{i}(s)\cdot\ln\pi_{s}-\lambda\{\sum_{s=1}^{S}\pi_{s}-1\}.$$\beta_{s}$

. Set

153
Then we have $(S+1)$ set of equations by partially differentiating
respect to$\pi_{S}$’s and$S$withand setting the resulting formulas as zero as$\lambda$(4.13)$\{\begin{array}{l}\frac{\partial L}{\partial\pi_{1}}=\frac{\Sigma_{i=1}^{N}h_{i}(1)}{\pi_{1}}-\lambda=0,:\frac{\partial L}{\partial\pi s}=\frac{\Sigma_{i=1}^{N}h_{i}(S)}{\pi s}-\lambda=0,\frac{\partial L}{\partial\lambda}=-\sum_{s=1}^{s}\pi_{s}+1=0.\end{array}$

From the first$L$equations in (4.13), we have,

(4.14)$\pi_{s}=\frac{1}{\lambda}\sum_{i=1}^{N}h_{i}(s)$(4.15)$\lambda=\frac{\sum_{i=1}^{N}h_{i}(s)}{\pi_{s}}$orfor $s=1,$$\ldots,$$S$. Substitute these equations into the last equation in (4.13)

to obtain$\frac{1}{\lambda}\sum_{i=1}^{N}h_{i}(1)+\cdots+\frac{1}{\lambda}\sum_{i=1}^{N}h_{i}(S) = 1$or$\sum_{i=1}^{N}(h_{i}(1)+\cdots+h_{i}(S))$$=$$\lambda$

or
$N$

$=$

$\lambda,$

since $h_{i}(1)+\cdots+h_{i}(S)=1$ . Therefore we have from (4.15)(4.16)$\pi_{s}=\frac{\sum_{i=1}^{N}h_{i}(s)}{N}$

for $s=1,$$s$$\ldots,$$S$. The solution (4.16) means that the relative size of segment

is the average of segment membership for

sample.

$\mathcal{S}$across all consumers in the

154
Now from (4.12), $h_{i}(s)=Pr(S_{i}=s|H_{i};\beta_{s})$ can be calculated using thedefinition of conditional probability$as^{4}$$h_{i}(s)= \frac{Pr(S_{i}=s,H_{i};\beta_{S})}{Pr(H_{i};\beta)}=\frac{\pi_{s}\cdot Pr(H_{i}|S_{i}--s;\beta_{s})}{\sum_{s=1}^{S}\pi_{s}\cdot Pr(H_{i}|S_{i}=s;\beta_{s})}$By substituting (4.17) for (4.16), we obtain.

(4.17)$\pi_{s}.$As for the first term of the right hand side of (4.11) for segment$s,$

the parameters can be estimated independently for each segment since thevectors of parameters$\beta_{s}$are independent across segments. Then the firstterm on the right hand side of (4.11) for segment$s$is written with thenotation similar to (4.7) as$l_{s}(\beta_{s}|H)$$=$$\sum_{i=1}^{N}h_{i}(s)\cdot\ln Pr(H_{i}|S_{i}=s;\beta_{s})$$= \sum_{i=1t_{i}}^{N}\sum_{=1}^{T_{i}}\sum_{j=1}^{J}\{h_{i}(s)\cdot y_{ijt_{i}}\cdot\ln Pr(Y_{it_{i}}=j|S_{i}=s;\beta_{s})\}.$(4.18)

To implement $EM$ algorithm, repeat the following steps.
$EM$ algorithm

Step 0.1 Set $t=0$ . Set the initial values$\pi_{s}^{(0)}=1/S$

for $s=1,$$\ldots,$$\hat{\beta}_{s}^{(t)}$and$\pi_{s}^{(t)}$for $s=1,$$S$$\ldots,$and set$S.$

Step 0.2 Set $s=1$ . For $i=1,$$Pr(Y_{it_{i}}=j|S_{i}=s;\beta_{S})$$\hat{\beta}_{s}^{(0)}$$\ldots,$$N$, calculate$h_{i}^{(t)}(s)$by calculating

using (4.3) first then (4.7) and (4.8) successively withand substitute these interim results for (4.17). Set $s=s+1$

and repeat Step 0.2 until $s=S.$

4Note that$i$$h_{i}(s)$in (4.17) can be interpreted as the posterior distribution of consumer’s membership probability for segment$s$with prior distribution

given segment membership $S_{i}=s$ as we mentioned earlier.

$\pi_{s}$and hkelihood$H_{i}$

155
Step 0.3 Calculate $E[l^{(t)}(\pi^{(t)},\hat{\beta}^{(t)}|H, Z)]$ using (4.11).

Step 1 Set $s=1$ . Renew
Step 2 Estimate$\hat{\beta}_{s}^{(t+1)}$$\pi_{s}^{(t+1)}$

from (4.16) using$h_{i}^{(t)}(s)$.

by maximizing (4.18) with (4.3) and$h_{i}^{(t)}(s)$obtainedpreviously. The actual maximization is done by the scoring or NewtonRaphson method.

Step 3 Renew $Pr(Y_{it_{t}}=j|S_{i}=s;\beta_{S})^{(t+1)}$ by substituting$\sqrt(t+1)\wedge S$

obtainedin Step 2.
Step 4 Calculate

for $i=1,$$\ldots,$$N$$h_{i}^{(t+1)}(s)$from (4.17) with the renewed$\hat{\beta}_{s}^{(t+1)}$

and$\pi_{S}^{(t+1)}$. Set $s=s+1$ and goto Step 1. If $s=S$ , goto Step 5.

Step 5 Calculate $E[l^{(t+1)}$ $(\pi^{(t+1)},\hat{\beta}^{(t+1)}|H, Z)]$ using (4.11). If$E[l^{(t+1)}$$(\pi^{(t+1)},\hat{\beta}_{s}^{(t+1)}|H, Z)]$and$E[l^{(t)}(\pi^{(t)}, \sqrt(t)\wedge \mathcal{S}|H, Z)]$

are close enough,for example, less than small prescribed constant , stop the iteration as the$\epsilon$

expected$\log$

likelihood is maximized. Else set $s=1$ and $t=t+1$ , andreturn to Step 1.

5

Empirical Results

We use ERIM database, the panel data of U. . households in Sioux Falls,
$S$

$SD$

and Springfield, $MO$ which was collected from lst week of 1986 to 34th

week of 1988. ERIM database is the data collected by the now-defunct

ERIM division of A.C. Nielsen on panels of households in Sioux Falls andSpringfield for academic research.5

We choose ketchup category for our empirical analysis for the following

5We acknowledge the James M. Kilts Center, University of Chicago Booth School ofBusiness for letting us use the data.

156
reasons: First, since we are interested in consumers’ brand choice behaviors with the possible presence of state dependence and $/or$ variety-seeking
behaviors, product categories in which a consumer exhibits strong genuine

preference to specific brand are not suitable because a consumer would buy

the specific brand anyway. Secondly, the products that are purchased withrelatively high frequency are preferable, since we calibrate the effects of pastbrand choices on current purchasing occasion. In other words, the productsthat are purchased on irregular intervals would not suit as consumers may

forget the brands they purchased on the previous occasion. After screening
data, we have 137 households with 1,504 purchase records.

The summary statistics of the five SKUs analyzed in this study is listed
in Table 5.1. “Coupon Usage” indicates the number of times coupon wasused, and “Display” and “Feature” indicate the number of times they were

promoted conditional on SKU being purchased. “Mean Value of Coupons”
indicates the average value of coupons when they were used.

Table 5.1: Summary statistics of SKUs under study.
SKU

Share

Mean Price per oz.

Mean Value of Coupons

Coupon Usage$D\iota$splay

FeatureHeinz 32 oz.

31.70%

3.37

1.24

37.94%

11.52%

43.82%

Heinz PLS 28 oz.

15.80%

4.38

2.41

33.09%

16.73%

34.55%

Hunt’s PLS & GLS 32 oz.

14.30%

3.22

1.30

32.57%

11.93%

36.70%Del Monte 32 oz.

6.40%

2.87

1 (K)

7.20%

11.20%

36.(K)%

Contro132 oz.

5.00%

2.65

1.64

3.77%

5.66%

24.53%

To calibrate the effectiveness our model, we tested the two other mod-els; the model which only uses marketing variables as explanatory variable(Model 1); the model which incorporates $GL$ variable along with the mar-keting variables (Mode12). The third model is our proposal model which
incorporates $GL$ variable and run in addition to marketing variables (Model

3 . We have determined the number of segments based on AIC. The number$)$of segments is chosen to be four because no significant increase in AIC isobserved for Model 3 when the number of segments is increased from four

157
to five as shown in Table 5.2. Comparing three different models with six
segments, our proposal model has the lowest AIC value.
Table 5.2: AIC of the three models with different numbers of segments.
Model 1

Mode12Mode13

2 segments

1310.60

1058.19

1041.80

3 segments

948.52861.27835.644 segments

842.64805.61796.745 segments

819.09796.54794.21

6 segments

809.53803.84813.02The estimated parameters of Mode13 are presented in Table 5.3. The

coefficients of the brands indicate intrinsic preferences for the brands relative
to Contro132 ounce which is used as the base brand. All the coefficientsare consistent with our intuitions, i.e., all coefficients of price are negative,
those of coupons are positive, and those of display and feature are positivein all segments. As for $GL$ variable and run, they show interesting patterns
which would have not been discovered if run was not included in the model.

Only Segment 2 has an insignificant (absolute -value less than 2) coefficient$t$

for run. The negative coefficients of run across three segments imply thatthe marginal utility of the same brand decreases as a result of repeatedpurchases of the same brand over time.

Now to see the behavioral patterns regulated by the combination of $GL$variable and run, we calculated the logit probabilities for each SKU andsegment, assuming the situation where consumers repeatedly purchase thesame brands five times in row. In the calculation, we used the average prices
of SKUs and assuming no promotions took in place during the period. The
results are shown in Table 5.4. For example, the number at $t=3$ is thepurchasing probability- of the SKU being-purchased given two-consecutive
purchases of that SKU. In Table 5.4, we have the information of state depen-

dence or variety-seeking tendency of each segment for each SKU. Segment

158

Table 5.3: The parameters for the Mode13.
Heinz 32 oz.

Heinz PLS 28 oz.
Hunt’s PLS & GLS 32 oz.
Del Monte 32 oz.

Price
Coupon
Display

Feature$GL$

Run

segment 1

segment 2

segment 3

segment 4-0.076

4.279

1.885

1.437

(0.0148)(0.0183)(0.0122)(0.0155)

1.311

2.661

1.727

2.399

(0.0072)(0.0070)(0.0065)(0.0073)

0.027

0.296

2.713

-2.530(0.0068)(0.0050)(0.0097)(0.0055)

1.176-2.1411.071-1.154(0.0071)(0.0014)(0.0043)(0.0057)-0.848-0.735-0.666-2.509(0.0701)(0.0769)(0.0682)(0.0713)

2.792

5.015

3.287

5.471

(0.0211)(0.0207)(0.0183)(0.0202)

3.419

3.903

3.855

4.750

(0.0070)(0.0067)(0.0067)(0.0072)

5.622

2.504

2.938

5.894

(0.0126)(0.0121)(0.0119)(0.0127)

4.303

0.743

1.737

5.607

(0.0136)(0.0115)(0.0102)(0.0126)$*_{-}o.127$-0.222-0.1230.

$505$$\frac{(0.0891)(0.0663)(0.0519)(0.0529)}{SizeofSegments0.2610.2340.2890.217}$$\underline{\underline{Total{\rm Log} Likelihood-365.81}}$$*90\%$ level significance with t-value-1.917.

All the other coefficients were significant at the 0.05 level.
The numbers in parentheses are standard errors.

159
1 and 4 exhibit state dependence tendencies while segment 2 and 3 exhibit
variety-seeking tendencies.
Table 5.4: Logit Probability of Purchase: Mode13
Heinz 32 oz.

Segment 1

Segment 2

Segment 3

Segment 4

$t=0$

88%

87.6%

206%

335%

$t=1$

56.1%

91.7%

42.4%

93.6%

$t=2$

$74$

1%

404%

95.3%

$t=3$

86.7%

90.5%

37.7%

96.2%

$t=4$

93.1%

897%

34.7%

96.8%

$t=5$

964%

88.8%

313%

97.1%

9%

$91$

$\frac{He\grave{1}nzPLS28oz.SegmentlSegment2Segment3Segment4}{t=015.0\% 83\% 90\% 7.1\%}$

$t=1$

70.1%

12.4%

219%

687%

$t=2$

84.5%

11.7%

20.5%

75.3%

$t=3$

92.3%

109%

188%

79.4%

$t=4$

96.1%

100%

16.8%

81.9%

$t=5$

980%

92%

14.8%

83.5%

$\frac{Hunt^{)}sPLS\ GLS32oz.SegmentlSegment2Segment3Segment4}{t=011.2\% 18\% 52.5\% 0.9\%}$

$t=1$

62.4%

28%

75.8%

21.6%

$t=2$

79.5%

27%

742%

276%

$t=3$

89.4%

25%

720%

32.6%

$t=4$

946%

23%

693%

363%

$t=5$

97.2%

20%

660%

38.8%

$\frac{DelMonte32oz.SegmentlSegment2Segment3Segment4}{t=047.4\% 02\% 12.8\% 90\%}$

$t=1$

92.2%

$0$

3%

29.4%

740%

$t=2$

96.5%

$0$

3%

27.7%

79.8%

$t=3$

98.4%

$0$

3%

25.5%

83.3%

$t=4$

99.2%

03%

23.1%

85.5%

$t=5$

99.6%

02%

20.5%

86.7%

$\frac{Contro132oz.SegmentlSegment2Segment3Segment4}{t=017.6\% 21\% 5.1\% 49.5\%}$

$t=1$

739%

32%

13.2%

$t=2$

869%

30%

123%

97.5%

$t=3$

935%

28%

111%

98.0%

$t=4$

96.8%

25%

99%

983%

$t=5$

98.3%23%8.6%98.5%

96.6%

The information in Table 5.4 can be used as a starting point for brandmanagers to plan their marketing strategies and promotional activities. For
example, since consumers in segment 1 exhibit strong state dependence, Del

160
Monte may need to ensure it has enough amounts of promotions to retain

consumers in this segment since Del Monte is preferred by segment 1 most.
As consumers have low coefficients for coupon and display but have highcoefficient for feature, Del Monte would want to increase feature to retain

consumers from Segment 1.

6Discussion

Overall, our model achieves the best AIC compared to the previously proposed models with a fair number of significant variables, indicating that thehouseholds are heterogeneous in their behavioral patterns over time. It gives
important connotations for marketers, because the model without state dependence and variety seeking effects would be misleading in constructing

the strategy and planning promotions as pointed out by previous researchsuch as Keane (1997).

Unfortunately, the hybrid behavior was not detected in our analysis.
This may be because most of consecutive purchases of the same SKU are

three at most; about 90% of purchases in the data are shorter than threeruns. Also, the products like ketchup, where the bottle is consumed througha relatively long period of time, a satiation effect may start during theconsumption period and that may lead households to switch brand, i.e.,the hybrid behavior is hidden as a result of the large package sizes of the
ketchup. By using the products which are consumed in a relatively shortperiod of time the hybrid behavior may has been detected.

For future researches, the model presented in this study can be tested
using different data sets for the validity of the model. The new variable toexplain state dependence and variety-seeking behaviors can be constructed

ae well. From a microeconomic perspective, the budget constraint can beincorporated in the model because households may switch brands dependingon their budget at each shopping trip.

161
References
Bawa, Kapil (1990) ‘Modeling inertia and variety seeking tendencies in brandchoice behavior.’ Marketing Science 9(3), 263-278

Chintagunta, Pradeep K. (1998) ‘Inertia and variety seeking in a model ofbrand-purchase timing.’ Marketing Science 17(3), 253-270
Givon, Moshe (1984) ‘Variety seeking through brand switching.’ Marketing

Science 3(1), 1-22

Guadagni, Peter M., and John D. C. Little (1983) ‘A logit model of brandchoice calibrated on scanner data.’ Marketing Science 2(3), 203-238
Kamakura, Wagner A., and Gary J. Russell (1989) ‘A probabilistic choicemodel for market segmentation and elasticity structure.’ Journal of Marketing Research 26(4), 379-390
Keane, Michael P. (1997) ‘Modeling heterogeneity and state dependence in

consumer choice behavior.’ Journal of Business and Economic Statistics
15(3), 310-327
McAlister, Leigh, and Edgar Pessemier (1982) ‘Variety seeking behavior: Aninterdisciplinary review.’ Joumal of Consumer Research 9(3), 311-322
Seetharaman, P.B. (2004) ‘Modeling multiple sources of state dependence
in random utility models: A distributed lag approach.’ Marketing Science23(2), 263-271

Train, K. (2003) Discrete choice methods with simulation (Cambridge Univ
Pr)