Unveiling the Association between the Transaction Timing,
Spending and Dropout Behavior of Customers
Abstract
The customer lifetime value combines into one construct the transaction timing,
spending and dropout processes that characterize customer purchase behavior. Recently, the
potential association between these processes, either at the individual customer level (i.e.
intra-customer association) or between customers (i.e. inter-customer association) has received
increasing attention. In this paper, we propose to jointly unveil all possible types of association
between these processes using copulas, a flexible statistical tool to capture the association
between probability distributions of different nature. Conceptually, we investigate the behavioral
rationales that underlie the various types of association, and empirically, we investigate the
existence and direction of these associations across four distinct product categories, namely
online music albums sales, securities transactions, utilitarian and hedonic fast-moving consumer
good retail sales.
For all product categories, we find a substantial amount of association both at the intraand inter-customer level. For the product categories where the association is the strongest, we
find that the association enhances the predictive performance of the model. Overall, we find that
frequent buyers tend to spend on average more per transaction than others, and, for half of the
categories, they also have a shorter lifetime. In turn, large buyers have, on average, a longer
lifetime than others. At the intra-customer level, we find that the extent to which customers show
so-called compensating purchase behavior (i.e. adapt their purchase quantities to the time since
the last purchase) varies across product categories and across customers. Using these empirical
findings, we show how firms can improve their targeted marketing decisions.
KEYWORDS: Association, Copula, Customer Metrics, Customer Lifetime Value,
Dropout, Pareto/NBD Model.
1
1. Introduction
As firms are complying with increasingly stricter marketing budget constraints, customer
lifetime value (hereafter, CLV) has become a popular customer valuation metric in marketing
research and practice (Berger et al. 2002, Gupta et al. 2004, Kumar & Venkatesan 2006, Rust et
al. 2004, Venkatesan & Kumar 2004). This metric integrates three key decision processes a
customer makes: (i) the transaction timing process, or when to buy, (ii) the spending process, or
how much to spend,1 and (iii) the dropout process, or when to become permanently inactive.
Together, these decisions determine the cash flows that firms can expect from each customer
over her lifetime and guide the allocation of marketing efforts across the customer base
(Venkatesan et al. 2007).
While these three purchase decisions have long been assumed to be independent of each
other (Fader et al. 2005), recent research is exploring the extent to which they actually interact.
An overview is provided in Table 1. In a non-contractual setting, the Pareto/NBD models the
flow of transactions over time, accounting for dropout (Reinartz & Kumar 2000; 2003,
Schmittlein et al. 1987, Schmittlein & Peterson 1994), while a separate Gamma/Gamma
sub-model accounts for the monetary value of the transactions (Fader et al. 2005). In this context,
an association between the processes can occur either at the intra-customer level, because the
timing at which a customer makes a purchase interrelates with the value of this transaction (e.g. a
long interpurchase time may lead to a large purchase, as in Jen et al. 2009), 2 and/or at the
inter-customer level, because the expected number of transactions, transaction value and/or
customer lifetime correlate across customers (e.g. frequent buyers may make, on average,
1
The spending process captures how much a customer spends per transaction. Usually, the CLV framework does
not make a distinction between the number of units bought per transaction (quantity) and the price per unit, but
considers the aggregate total (dollars) value of a transaction. For simplicity, we will refer to the notion of quantity
per transaction or value per transaction interchangeably.
2
Note that the intra-customer association with the dropout does not exist as dropout only occurs once.
2
lower-value transactions than occasional buyers). In other words, the former association is a
temporal/longitudinal association at the customer level, while the latter is a cross-sectional
association at the customer base (or firm) level.
Table 1: Literature overview of the association between the CLV processes
Intra-customer
Inter-customer association
Setting
Industry
association
References (by
Timing -
Timing -
Timing -
Spending -
publication date)
spending
spending
drop out
drop out
x
x
x
Boatwright et al. (2003)
x
Borle et al. (2008)
Abe (2009)
x
Non-contractual
Online grocery
Contractual
Direct marketing
Non-contractual
Online music retailer, department
Non-contractual
B2B (office supply) and B2C (direct
store, music chain
Jen et al. (2009)
x
marketing in health & beauty)
Singh et al. (2009)
Abe (2013)
x
x
x
(Non-)contractual
x
x
x
Non-contractual
Department store, music chain
Non-contractual
Online music retailer, hypermarket
Non- contractual
Online music retailer, financial
Romero et al. (2013)
x
x
Our study
x
x
x
x
Direct marketing
securities, hedonic FMCG,
utilitarian FMCG
In Table 1, we indicate, for each study, the type of association considered, whether it is a
contractual or non-contractual setting and the industry under study. At the intra-customer level,
Boatwright et al. (2003) make purchase timing dependent on lagged quantity using a generalized
Poisson specification, while Jen et al. (2009) model the contemporaneous association between
the timing and quantity decisions using a bivariate log-normal distribution. At the inter-customer
level, Abe (2009) allows for a cross-sectional association between the timing and the dropout
processes using a bivariate normal distribution, while Borle et al. (2008), Singh et al. (2009) and
Abe (2013) use a trivariate normal distribution to capture the association between all three
processes, either in a contractual or non-contractual setting. Finally, Romero et al. (2013) use a
partially hidden Markov model to forecast CLV while allowing the purchase timing and
3
spending processes to be correlated between activity states at the intra and inter-customer level
(keeping the independence assumption within a given state).
Overall, these studies jointly provide evidence for the existence of the four types of
association between the timing, spending and dropout decisions. They show that ignoring them
substantially biases customer purchase behavior and CLV predictions (Jen et al. 2009). Based on
the current literature, we identify a number of remaining open questions, which our paper aims to
address. First, the existence of different types of association simultaneously requires a method to
model all four associations jointly. So far, previous methods have only considered a subset of
these associations, which can bias predictions and managerial recommendations. Second, there is
still a large amount of confusion regarding the distinct interpretations of the various types of
association. These associations underlie different purchase behaviors with distinct behavioral
rationales. For instance, the intra-customer association between the timing and spending
processes reflects the existence of compensating purchase patterns at the individual customer
level (i.e. whether a customer who defers or hastens his purchase compensates by buying more
or less). In contrast, the inter-customer association between these processes mirrors the customer
heterogeneity in shopping behavior (e.g. routine vs. quick shoppers, Bell and Lattin, 1998). All
studies so far have shown the predictive gains attached to modeling association but have not paid
attention yet to the behavioral rationales that underlie the various types of association, nor to the
implications of these different behavior types for companies. This paper addresses these
important questions.
On the methodological side, we replace the independent distributions for each customer’s
interpurchase time and spend per transaction by a joint distribution (intra-customer level) and
specify a joint distribution for the transaction rates, spending rates and dropout rates across
4
customers (inter-customer level). To link these distributions, we use copulas (Danaher & Smith
2011; Danaher & Hardie 2005) which are able to “couple” distributions of different family and
unmask the true strength of dependence between two processes, which a classical correlation
coefficient (e.g. Pearson) would not identify. We show that incorporating all associations
improves predictions of customer purchase decisions.
On the conceptual side, we develop a typology of the different levels (intra- or
inter-customer) and directions (positive or negative) of association that possibly relate the
transaction timing, spending and dropout processes to each other. In contrast to prior work, we
explain how these associations translate in terms of purchase behavior and why they are likely to
occur. We rely on the comprehensive utility maximizing framework (Chintagunta, 1993) that
governs the tradeoffs that customers make when making purchase decisions. Customers decide
whether to buy, when to buy and how much to buy based on a number of common underlying
drivers, such as their unobserved reservation price, their stockpiling cost, transaction cost, budget
constraints, specific consumption needs or preferences, sensitivity to (competitive)
marketing-mix activities, to cite only a few (see e.g. Gupta 1988). In addition, their decisions
directly interdepend on each other (e.g. for a same consumption level, a customer can buy small
quantities often or large quantities occasionally). Based on this conceptual framework, we
explain how our typology can guide managerial decisions.
In order to investigate how our framework generalizes, we apply our model to four
customer transaction databases, representative of different product categories and/or industries.
The first one contains music albums sales at an online retailer (CDNOW).3 The second includes
securities transactions at a major financial institution. The last two data sets concern the retail
3We
thank Bruce Hardie who kindly provided us the data set.
5
industry; one contains transactions of a utilitarian FMCG, the other of a hedonic FMCG.
The remainder of the paper is organized as follows. In Section 2, we define the intra- and
inter-customer association between the transaction timing, spending and dropout processes and
investigate how they characterize different purchase behaviors and underlying behavioral
rationales. In Sections 3 and 4, we explain the copula methodology and show how to incorporate
it into the stochastic buyer behavior framework by Fader et al. (2005). In Sections 5 and 6, we
describe the four data sets and apply our methodology. Section 7 concludes with a number of
important managerial implications, several limitations and suggestions for future research.
2. Inter- and Intra-Customer Association
We observe transactions made by customers over a given time span. Each customer is
characterized by the time preceding each of his/her transactions (i.e. interpurchase time) and the
value of these transactions (i.e. units per transaction x price per unit). Over the complete time
range, each customer is also characterized by an individual transaction rate (i.e. expected number
of transactions per time unit or transaction frequency), spending rate (i.e. expected dollar value
per transaction) and dropout rate (i.e. hazard for this customer to become permanently inactive,
given that the customer is still active). In a non-contractual setting, we do not observe dropout
but estimate it from the transaction stream. At the intra-customer level, we are interested in the
temporal association between the interpurchase time and the transaction value, which varies
across customers. At the inter-customer level, we explore three cross-sectional associations
(between each pair of the three processes), which is defined at the customer base level (i.e. one
parameter for all customers). Below, we provide a formal definition of each association as well
as some behavioral interpretations and underlying rationales for each in turn.
6
2.1. Definitions
Intra-customer association between the transaction timing and spending process: This
association captures the degree to which the time preceding a given transaction (interpurchase
time) relates to the value of this transaction. A positive (respectively negative) association
indicates that a customer who delays his/her purchase (i.e. shows a longer interpurchase time
than usual) will buy in larger (respectively smaller) quantities than usual on that purchase. The
intra-customer association is customer-specific.
Inter-customer association between the transaction timing and spending processes: This
association captures the relation between how much customers spend on average per transaction
and the number of purchases they make. A positive (respectively negative) association indicates
that customers who spend on average less (respectively more) per transaction than other
customers also purchase less frequently than others.
Inter-customer association between the transaction timing and dropout processes: This
association captures the degree to which the timing at which customers purchase on average is
related to their lifetime. A positive (respectively negative) association indicates that customers
who purchase on average less often (respectively more often) than other customers also drop out
later than others.
Inter-customer association between the spending and dropout processes: This association
captures the relation between how much customers spend on average per transaction and their
lifetime. A positive (respectively negative) association indicates that customers who spend on
average less (respectively more) per transaction than other customers drop out later than others.
7
2.2. Behavioral Interpretations and Underlying Rationales
Each association (direction) corresponds to a specific purchase behavior. In Table 2, we
summarize the possible scenarios and we discuss each of them in turn. For each, we also review
the potential drivers.
Table 2: Summary of possible association directions and behavioral interpretations
Direction/
Intra-customer association
Significance/
Drivers
Timing-spending
Compensating behavior (e.g. a
delayed purchase leads to larger
Positive
quantities)
Purchase acceleration or
deceleration (e.g. a delayed
purchase leads to smaller
Negative
quantities)
Promotion proneness (+),
inventory and/or budget
Potential drivers
constraints (+), consumption
(and direction of
expandability (-),wear-out in
the effect)
variety seeking (-), satiation (-),
state dependence (-), learning
effects (-), addiction (-)
2.2.1.
Inter-customer association
Timing-spending
Timing-dropout
High- vs. low-
Complementary
potential segments
segments
(frequent/large
(frequent/short-life
buyers vs.
buyers vs.
infrequent/small
infrequent/long-life
buyers)
buyers)
Complementary
High- vs. low-
segments
potential segments
(frequent/small
(frequent/long-life
buyers vs.
buyers vs.
infrequent/large
infrequent/short-life
buyers)
buyers)
Spending-dropout
Complementary
segments
(large/short-life buyers
vs. small/long-life
buyers)
High- vs. lowpotential segments
(large/long-life buyers
vs. small/short-life
buyers)
Heterogeneity in
shopping patterns
driven by the tradeoff
between acquisition
and transaction
Heterogeneity in
Heterogeneity in
loyalty, satisfaction,
loyalty, satisfaction,
trust, commitment
trust, commitment
(dis-)utility
Intra-customer association between the transaction timing and spending processes
Customers with a positive association are customers who show temporal deviations from
their average buying patterns that reflect compensating behaviors (Jen et al. 2009). When they
defer (resp. hasten) their purchase, they tend to compensate by buying in larger (resp. smaller)
quantities. In contrast, customers with a negative association are customers who show either an
8
increasing or a decreasing demand for a given product (category). In the former case, they start
purchasing earlier and in larger quantities, pointing out to purchase acceleration. In the latter
case, they start buying later and in smaller quantities, indicating purchase deceleration. Finally,
customers with a non-significant association do not show any systematic relation between how
much and when they buy. Below, we review a number of reasons or situations when these
different purchase behaviors occur.
Behavioral rationale for compensating behaviors (i.e. positive association)
A number of factors are likely to drive compensating purchase behaviors (i.e. positive
intra-customer association). First, purchase decisions are influenced by price promotions and
other marketing-mix variables. These activities often lead to pre- and post-promotion sales dips
(Van Heerde et al. 2004), because customers adjust their purchase timing and quantity decisions
based on the promotion calendar. They postpone their purchase until the next promotion
(pre-promotion dip), buy more during a promotion, and stockpile (post-promotion dip) for future
consumption (Hendel and Nevo 2003). This consumer inventory model is likely to translate into
compensating purchase patterns. Therefore, proneness to price promotions and marketing
activities should have a positive effect on the intra-customer association. We expect the product
categories and the customers the most susceptible to pre- and post-promotion dips to show a
positive association. Macé and Nelsin (2004) find that they mainly concern the storable,
frequently promoted and high-budget share product categories and customers large households
who own a car and work (i.e. have a limited time).
Given the first point, inventory constraints are likely to create some interdependence
between the timing and quantity decisions (Blattberg et al. 1981; Meyer and Assuncao, 1990). In
case of limited storage capacity, a long delay between two purchases will require the consumer
9
to buy in larger quantities. Meyer and Assuncao (1990) explain that purchases reflect a rational
policy among consumers for minimizing the long-term costs of consuming and storing goods.
Likewise, the degree of storability or perishability of a good should also increase the association
(Wansink and Deshpande 1994). In the same vein, individual budget constraints also affect the
joint decision process. If the budget is limited, delaying a purchase will give the customer a
larger financial freedom to buy in larger quantities. The smaller the available budget, the higher
the interdependence between purchase quantity and timing decisions would become.
Third, the degree to which customers adjust their consumption over time (i.e.
consumption expandability or flexibility) also affects the interaction between both processes.
Prior research provides support for the inclusion of an inventory-based consumption function in
a model for purchase incidence and quantity (Ailawadi and Neslin, 1998) and raises the notion of
endogenous consumption (Sun, 2005). In some cases, customers adjust their consumption to the
level of their inventory (Ailawadi et al. 2007, Bell et al. 1999). We expect that, if a customer
does not adjust her consumption based on her inventory, she will show a more positive
association than in case of flexible consumption. Inventory-based consumption has been found to
be most pronounced for high-convenience products (Chandon and Wansink, 2002) and for
versatile and substitutable product categories (Sun 2005, Wansink and Deshpande 1994).
Moreover, Krishna (1992, 1994) shows that, when customers stockpile a preferred brand, they
tend to consume more of it. As a consequence, consumption expandability will moderate the
intensity of compensating behaviors. This notion corroborates the finding of Jen et al. (2009) that
customers with a positive association exhibit a constant spending stream while customers with a
non-significant association show variable spending streams.
10
Behavioral rationale for purchase deceleration or acceleration (i.e. negative association)
Purchase deceleration or acceleration has been found to be related to prior consumption
(Seetharaman et al. 1999). On the one hand, customers can experience wear-out in variety
seeking (McAlister 1982) as their marginal utility to consume a given good decreases with
consumption (Hartmann 2006). Customers may satiate themselves with previous chosen goods
or brands and switch brands in a quest for variety (McAlister 1982) or stop buying from a given
product category. Enduring effects of prior consumption can be long lasting, depending on the
consumer and the product. For instance, when people visit a tourist attraction, the experience
provides utility for a lifetime to some customers (Hartmann 2006). Such cases are sometimes
referred to as negative state dependence. Negative state dependence leads to purchase
deceleration.
On the other hand, prior purchase of a given brand can increase the probability of the
brand to be chosen again (Howard and Sheth 1969). We then talk about positive state
dependence, or inertia. Such positive state dependence is usually caused by preference change,
which creates a form of loyalty (Dubé et al. 2010), or by customer learning. The learning
literature indicates that, when they start buying a new product, customers, which are first
unfamiliar with the brand, gradually gain knowledge and experience (learning) with the good,
leading to purchase acceleration. Acceleration is likely to occur in particular for new products or
new product categories, experience goods that require trials to evaluate quality, products for
which customers gain efficiency in usage over time. Addiction is an extreme form of positive
state-dependence. Finally, Seetharaman et al. (1999) find that the longer the interpurchase time,
the lower the inertia becomes. The latter can lead to purchase deceleration.
11
2.2.2.
Inter-customer association between the transaction timing and spending processes
A positive association points to the existence of a dual customer base, in which a share of
the customers buys more frequently and in larger quantities than another share (who buys at a
lower frequency and in smaller quantities). The former segment represents a potentially higher
source of revenues for the firm than the latter. In contrast, a negative association translates into
complementary segments, in that a share of the customers buys in larger quantities but at a lower
frequency than another share of customers who buys in smaller quantities but at a higher
frequency. The first segment insures a sporadic but substantial stream of income while the other
offers frequent and small revenues to the company. Finally, a non-significant association
indicates that all four segments co-exist. In other words, the amount a customer purchase on
average is not informative as to how often she purchases and vice versa.
Behavioral rationale for the existence of different customer segments
The customers’ different profile in terms of shopping patterns is a key driver for the
association between timing and spending decisions at the inter-customer level. The role of
shopping behavior variables, how often one visits the store and his/her average expenditure per
trip has been found to influence his/her purchase behavior at the store (e.g., Ainslie and Rossi,
1998; Manchanda et al., 1999). Rhee and Bell (2002) make a distinction between occasional and
heavy, frequent and light, frequent and heavy, or occasional and light shoppers. In the retailing
sector, most research confirms the existence of, on the one hand, a segment of so-called regular,
routine shoppers (e.g. once a week trip), and on the other hand, a segment of so-called frequent,
quick shoppers (Bell and Lattin 1998, Kahn and Schmittlein 1989; Kim and Park, 1997). The
former segment tends to buy more at each purchase occasion than the latter one who buys more
frequently instead. Such segmentation leads a negative inter-customer association.
12
Transaction costs are known to drive shopping frequency decisions and are therefore
likely to influence the direction and intensity of inter-customer association one firm might
observe. Baltas et al. (2010) develop a household production framework that explains how
households organize their shopping trips. Shopping pattern decisions are often seen as being
rooted in a “household production” framework (see, e.g., Baltas et al. 2010; Urbany et al. 1996;
Vroegrijk et al. 2013). Within this framework, consumers’ shopping decisions depend on the
(dis)utility derived from the shopping activity as such (the transaction utility; Chintagunta et al.
2012; Gupta and Kim 2010), which competes with other household activities for the same
resources. Because shopping involves an allocation not only of money but also of other scarce
resources (e.g., time, effort), monetary as well as nonmonetary benefits and costs must be taken
into account (see, e.g., Chintagunta et al. 2012). In line with the utility maximization theory,
households are expected to select the shopping pattern that provides the most overall (acquisition
and transaction) utility. For instance, Kim and Park (1997) find that routine shoppers have higher
opportunity costs (e.g. they are time pressed) and lower storage costs and the lowest budget
constraints, which enforce more regular patterns, while quick shoppers are more price-sensitive
and exploit price promotions which create irregular shopping intervals. In other contexts
however, it is possible that the customers with the lowest opportunity costs (e.g. time constraints)
also have the lowest budget constraints. It could, for instance, be the case when the customers of
a company show different preference structures (Vilcassim and Jain 1991), e.g. different levels
of involvement. When buying their favorite brand, customers might buy more often and lot more
than customers who do not have a strong relation with the brand. When the relationship with the
brand or the store matters for the consumer, we expect to see a positive association.
13
2.2.3.
Inter-customer association between the transaction timing and dropout processes
A positive association points to the existence of complementary segments, in that a share
of the customers buys at a lower frequency but for a longer period of time than another share of
customers who buys at a higher frequency for a shorter period of time. The first segment insures
the company a long-term stream of income while the other offers a short-term revenue boost. In
contrast, a negative association translates into a dual customer base, in which a share of the
customers buys frequently and for a long period while another share buys at a lower frequency
and a shorter time. In this case, the former segment is clearly preferred over the latter one.
Behavioral rationale for the existence of different customer segments
According to Reinartz and Kumar (2003), purchase timing and customer lifetime are
likely to depend on each other. They suggest that a high purchase frequency translates a strong
relationship between the firm and the customer, which leads to longer lifetimes. This mechanism
would result into a negative association. However, Morgan and Hunt (1994) suggest that this
mechanism requires the interactions between the firm and the customers to remain satisfactory.
Moreover, Kim and Rossi (1994) observe that consumers with a high purchase frequency are
more demanding and price sensitive. They are more likely to search for cheaper alternatives and
re-evaluate their relation with the firm. In this context, frequent buyers could actually drop out
earlier than infrequent buyers if they face a negative experience. If this is the case, the
association will be less negative or even become positive. The BG/NBD model proposed by
Fader et al. (2005b) makes a similar assumption that a customer’s probability to drop out is tied
with her purchase occurrence. After any transaction, the customer “flips a coin” whether to
become inactive with a certain probability. The more frequently she buys, the more often the
coin will be flipped.
14
2.2.4.
Inter-customer association between the transaction spending and dropout processes
A positive association points to the existence of complementary segments, in that a share
of the customers makes small-quantity or low-value purchases over a long lifetime while another
share of customers spends a lot on each purchase over a short lifetime. The first segment insures
a long-term stream of income while the other offers a short-term revenue boost to the company.
In contrast, a negative association translates into a dual customer base, in which a share of the
customers spends a lot and for a long period while another share spends little and a shorter time.
Behavioral rationale for the existence of different customer segments
Customer satisfaction, commitment and/or loyalty are likely to underlie both the spending
and dropout decisions of customers, feeding an association between both processes. On the one
hand, Bolton (1998) and Bolton and Lemon (1999) find that more satisfied customers have
longer relationships with their service providers as well as higher usage levels of services. It
would translate into a segment of longer-life customers making larger purchases and a segment
of shorter-life customers making small purchases. However, at the same time, customers who
spend a large budget in a given product category are also likely to make involved purchase
decisions (Reinartz and Kumar, 2003) and are more price-sensitive (Kim and Rossi 1994). They
check competing offerings, are sensitive to their experience with the company and re-evaluate
their dropout decisions in a more elaborate way than small buyers. This mechanism can lead the
large buyers to have shorter lifespan at the firm than the small buyers, which would reduce or
make negative the association between spending and dropout.
15
2.3. A Visual Comparison: Inter vs. Intra
In order to provide the reader with a visual representation of the difference between the
inter- and intra-customer associations, Figure 1 exhibits the transaction value (in Euros) in
function of the interpurchase time (in weeks) of transactions made by five imaginary customers.
Transactions made by different customers are represented with different symbols, while the
average interpurchase time and transaction value for each customer is depicted in bold.
[INSERT Figure 1 ABOUT HERE]
When we focus on the inter-customer association level, we observe that the bold symbols
show a positive slope, indicating a positive association between the average interpurchase times
and the average transaction values. As the average interpurchase time is inversely proportional to
the transaction rate, it translates into a negative association between transaction rate and
transaction value. At the intra-customer level in contrast, we observe three customers showing a
positive association (the crosses, lozenges, and stars), a customer with a zero association (the
squares), and a customer with a negative association (the circles) between the interpurchase
times and the transaction values.
3. Copulas
A copula capture the association between two random variables 𝑋 and 𝑀 with given marginal
distributions 𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥) and 𝐺(𝑚) = 𝑃(𝑀 ≤ 𝑚) . For example, 𝐹 may be an
exponential distribution and 𝐺 a log-normal. The joint distribution function is given by
𝐻(𝑥, 𝑚) = 𝑃(𝑋 ≤ 𝑥, 𝑀 ≤ 𝑚). Obtaining an explicit expression for the joint distribution 𝐻 is
generally cumbersome, motivating the use of copulas. The Sklar's theorem (Sklar 1959) yields
that, for any 𝐹 and 𝐺, there always exists a copula function 𝐶 such that
16
𝐻(𝑥, 𝑚) = 𝐶(𝐹(𝑥), 𝐺(𝑚)).
(1)
The copula function 𝐶 is assumed to be known up to an unknown parameter 𝜃. If 𝑓, 𝑔 and ℎ
are the probability density functions corresponding to 𝐹, 𝐺 and 𝐻, the copula density function
𝑐 verifies
ℎ(𝑥, 𝑚) = 𝑐(𝐹(𝑥), 𝐺(𝑚))𝑓(𝑥)𝑔(𝑚).
(2)
Various families of copulas exist. The simplest one is the independent copula, which
assumes the independence between 𝑋 and 𝑀, given by 𝐶(𝐹(𝑥), 𝐺(𝑚)) = 𝐹(𝑥). 𝐺(𝑚). The
corresponding copula density is then equal to one. One can find a plethora of other copulas in the
literature (see Nelsen 2006, for more detail). Among the most common, there is the Gaussian
copula, which corresponds to a multivariate Normal distribution. In turn, the Gumbel copula only
allows for a positive association between 𝑋 and 𝑀, while the Frank copula does not allow for
extreme association values. More details on these various copulas are given in Appendix A.
In order to illustrate the peculiarity of each copula, Figure 2 reports the contour plots
corresponding to the joint distribution of two standard-normal random variables that have a
Spearman rank correlation equals to .50, when specifying (i) an independent copula (upper-left
plot), (ii) a Gaussian copula (upper-right plot), (iii) a Gumbel copula (lower-left plot), and (iv) a
Frank copula (lower-right plot). While the joint distribution corresponding to the Gaussian
copula is bivariate normal, the joint distribution corresponding to the Gumbel copula is
asymmetric. In turn, the joint distribution corresponding to the Frank copula yields a weaker
association in the tails compared to the Gaussian copula. We select the most appropriate copula
based on goodness-of-fit, using the Bayesian Information Criterion (BIC).
[INSERT Figure 2 ABOUT HERE]
17
One of the main advantages of copulas is that they can fit complex association structures
without affecting the marginal distributions. This is particularly interesting when the
distributions of the single random variables are well-known. The usefulness has been recently
demonstrated in a few marketing applications. For instance, Meade and Islam (2003) and Sriram
et al. (2010) introduce copulas to model the association between the time of adoption of related
technologies. Another marketing issue where multivariate distributions prove useful is the
modeling of household's purchase timing or incidence across related - complementary or
co-incidental - product categories, such as pasta and pasta sauce (Chintagunta and Haldar 1998),
laundry detergent and fabric softener (Manchanda et al. 1999), or bacon and eggs (Danaher and
Hardie 2005). The latter use the Sarmanov family of distribution (Sarmanov 1966), which is a
special form of copulas. The Sarmanov family of distribution has also been used by Park and
Fader (2004) and Danaher (2007) to model the association between multiple websites' browsing
patterns, and more recently by Schweidel et al. (2008) to account for the correlation between
acquisition and retention times across customers. Danaher and Smith (2011) demonstrates that
the Sarmanov family is more limited in its ability to model even moderate-sized correlation
levels than the copulas described in this section. Finally, copulas have also been used on a
regular basis in other research fields, in particular in finance (Cherubini et al. 2004, Glasserman
and Li 2005). We refer to Danaher and Smith (2011) for an extensive overview of copulas.
Equations (1) and (2) extend immediately to the trivariate case, where the association
between a set of three variables needs to be modeled. For the trivariate case, we focus on
elliptical copula, i.e the normal copula and the Student copula with 5 degrees of freedom,
allowing for fatter tails. The Gumbel and Frank copula can be generalized to the trivariate case,
but result in an expression more complex to manipulate. Therefore, we do not consider them at
18
the inter-customer level. It is important to note that using a normal copula does not imply that the
trivariate distribution is multivariate normal; the marginal distributions remain to be specified
separately. The parameter of an elliptical copula is not a univariate scalar 𝜃, but a matrix of the
form
1
𝑅(𝜃) = (𝜎𝜆,𝜇
𝜎𝜆,𝜈
𝜎𝜆,𝜇
1
𝜎𝜇,𝜈
𝜎𝜆,𝜈
𝜎𝜇,𝜈 )
1
(3)
where the three parameters in 𝜃 = (𝜎𝜆,𝜇 , 𝜎𝜆,𝜈 , 𝜎𝜇,𝜈 ) are between -1 and 1, but such that 𝑅(𝜃) is
a positive definite matrix. In the implementation of the method, we reparametrized 𝑅 as a
function of three other parameters, all varying without restriction in the interval [0,1] and still
ensuring positive definiteness of the correlation matrix (Pinheiro and Bates 1996).
4. Modeling CLV using Copulas
For customer 𝑖, with 1 ≤ 𝑖 ≤ 𝑛, we observe transactions at time points 𝑡𝑖0 < 𝑡𝑖1 < ⋯ < 𝑡𝑖,𝑥𝑖
with 𝑥𝑖 the number of repeated transactions. The first transaction is made at time point 𝑡𝑖0 , the
last transaction is made at 𝑡𝑥𝑖 ≤ 𝑇𝑖 , where 𝑇𝑖 indicates the time point at which the CLV needs
to be computed. The monetary values of the repeated transactions are given by 𝑚𝑖1 , … , 𝑚𝑖,𝑥𝑖 .
The monetary value of the first transaction at 𝑡0 is not used to model the CLV since it might be
atypical of future purchases. For customer 𝑖, we compute the interpurchase time between
transaction 𝑗 − 1 and 𝑗
𝐼𝑃𝑇𝑖𝑗 = 𝑡𝑖,𝑗 − 𝑡𝑖,𝑗−1
𝑓𝑜𝑟 𝑗 = 1, … , 𝑥𝑖 ,
(4)
and the recency of transaction 𝑥𝑖
𝑅𝑖 = 𝑇𝑖 − 𝑡𝑥𝑖
(5)
for 1 ≤ 𝑖 ≤ 𝑛. The unobserved time of dropout of customer 𝑖, or the time to “death,” is denoted
19
by 𝜏𝑖 . After 𝑡𝑖,0 + 𝜏𝑖 , the customer defects and makes no more transactions.
The assumptions on the marginal distributions of the spending process 𝑚𝑖,𝑗 , the
interpurchase time process 𝐼𝑃𝑇𝑖,𝑗 , and the time to death 𝜏𝑖 are identical as in Fader et al.
(2005). The interpurchase time of a customer follows an exponential distribution with parameter
𝜆𝑖 , such that the total number of purchases in a unit time interval follows a Poisson distribution
with expected value 𝜆𝑖 . On the other hand, the dollar value of a customer's transaction follows a
gamma(𝑝, 𝜈𝑖 ) distribution, with mean 𝑝/𝜈𝑖 . Finally, the time to death follows an exponential
distribution with parameter 𝜇𝑖 , such that the expected lifetime equals 1/𝜇𝑖 . We call 𝜆𝑖 the
transaction rate, 𝑝/𝜈𝑖 the spending rate, and 𝜇𝑖 the dropout rate of customer 𝑖.
These transaction, spending and dropout rates are considered as random variables, as is
common in a two-level hierarchical model. As in Fader and Hardie (2007), we allow for both
observed and unobserved customer heterogeneity. We denote 𝑉𝑖𝜆 , 𝑉𝑖𝜈 , and 𝑉𝑖
𝜇
the values of
time-invariant covariates for customer 𝑖 that we use to explain the transaction, spending and
dropout rates. Covariates include customer socio-demographics as well as company-related
customer characteristics. We specify
𝜆𝑖 = 𝜆0,𝑖 exp(𝛽𝜆′ 𝑉𝑖𝜆 )
(6)
𝑝/𝜈𝑖 = (𝑝/𝜈0,𝑖 )exp(𝛽𝜈′ 𝑉𝑖𝜈 )
(7)
𝜇
𝜇𝑖 = 𝜇0,𝑖 exp(𝛽𝜇′ 𝑉𝑖 ),
(8)
where the random effect 𝜆0,𝑖 follows a gamma(𝑟, 𝛼) distribution, the random effect 𝜈0,𝑖 a
gamma(𝑞, 𝛾) distribution, and the random effect 𝜇0,𝑖 a gamma(𝑠, 𝛽) distribution. According to
((6) and (7)), a positive 𝛽𝜆 indicates that the transaction rate increases when the value of a
covariate increases. Similarly, a positive 𝛽𝜈 implies that the expected transaction value
20
increases when the value of a covariate increases. Finally, a positive 𝛽𝜇 indicates an increase of
the dropout rate when a covariate increases. Note that we do not include time-varying covariates
into the mode. Including time-varying covariates would allow the model to capture exogeneous
effects having a potential impact on the customer bahavior. We leave it as a possible extension of
our model.
We extend the original CLV model in two ways. First, we introduce two levels of copula
to model both the inter- and intra-customer associations. Second, we specify a customer-specific
intra-customer association by accounting for both observed and unobserved customer
heterogeneity. A complete listing of all model assumptions is given in Appendix B.
Note, for completeness, that the fact that the original model by Fader et al. (2005)
assumes independent priors for the various decision processes does not imply that the posteriors
are independent (see Fader et al. 2010 for a discussion on this).
4.1. Modeling the Association Structure
The inter-customer association is the cross-sectional association between the transaction
rate 𝜆𝑖 , the spending rate 𝑝/𝜈𝑖 , and the dropout rate 𝜇𝑖 . Since the marginal distribution of the
three rates were specified before, it remains to model the association using a trivariate copula
𝑐𝑖𝑛𝑡𝑒𝑟 with parameter 𝜃𝑖𝑛𝑡𝑒𝑟 = (𝜎𝜆,𝜈 , 𝜎𝜆,𝜇 , 𝜎𝜈,𝜇 ) (see equation (3)). At the lower level of the
hierarchical model, we define the intra-customer association as the association between 𝐼𝑃𝑇𝑖,𝑗
and 𝑚𝑖,𝑗 , for each customer 𝑖. This association is captured through a copula 𝑐𝑖𝑛𝑡𝑟𝑎 with
parameter 𝜃𝑖𝑛𝑡𝑟𝑎,𝑖 . We parameterize the copula such that a positive 𝜃𝑖𝑛𝑡𝑟𝑎,i
indicates that a
transaction preceded by a longer interpurchase time (compared to the customer mean) is likely to
be of higher value (compared to the customer mean).
21
4.2. Modeling Customer Heterogeneity
Similar to the transaction, spending and dropout rates, we allow for both observed and
unobserved customer heterogeneity in the intra-customer association 𝜃𝑖𝑛𝑡𝑟𝑎,𝑖 , by specifying
𝜃𝑖𝑛𝑡𝑟𝑎,𝑖 as a function of customer covariates 𝑉𝑖
𝑓(𝜃𝑖𝑛𝑡𝑟𝑎,𝑖 ) = 𝛽𝑖𝑛𝑡𝑟𝑎 ′𝑉𝑖 + 𝜀𝑖 ,
(9)
2
for a suitable link function 𝑓 and with 𝜀𝑖 following a 𝑁(0, 𝜎𝑖𝑛𝑡𝑟𝑎
). The link function ensures
that 𝜃𝑖 stays within the bounds of the copula family. 4 The parameter 𝛽𝑖𝑛𝑡𝑟𝑎 captures the effect
of a covariate on the strength of the intra-customer association. It can be used to identify
customer segments with different intra-customer association directions and intensities.
4.3. Estimation of the Model Parameters
The parameters to be estimated (listed in Appendix B) are collected in a vector 𝛉. The
log-likelihood of 𝛉 given the observed interpurchase times 𝐼𝑃𝑇𝑖,𝑗 , the recency 𝑅𝑖 and the
transaction values 𝑚𝑖,𝑗 , for 1 ≤ 𝑖 ≤ 𝑛, and 1 ≤ 𝑗 ≤ 𝑥𝑖 , is given by
𝑛
(10)
log𝐿 = ∑ log ∫ 𝐿𝑖 (𝜆, 𝜇, 𝜈, 𝜃𝑖𝑛𝑡𝑟𝑎 )𝑔𝑖 (𝜆, 𝜇, 𝜈, 𝜃𝑖𝑛𝑡𝑟𝑎 )𝑑𝜆 𝑑𝜇 𝑑𝜈 𝑑𝜃𝑖𝑛𝑡𝑟𝑎 ,
𝑖=1
where 𝑔𝑖 is the joint distribution of the customers' individual parameters. Each individual
log-likelihood is given by
𝑥𝑖
𝐿𝑖 (𝜆, 𝜇, 𝜈, 𝜃𝑖𝑛𝑡𝑟𝑎 ) = ∏ 𝑃(𝜏𝑖 > 𝐼𝑃𝑇𝑖𝑗 )𝑓(𝐼𝑃𝑇𝑖𝑗 |𝜆)𝑔(𝑚𝑖𝑗 |𝜈)
𝑗=1
𝑐𝜃𝑖𝑛𝑡𝑟𝑎 (𝐹(𝐼𝑃𝑇𝑖𝑗 |𝜆, 𝜇), 𝐺(𝑚𝑖𝑗 |𝜈)) ⋅ 𝑓(𝑅𝑖 |𝜆, 𝜇).
Here, 𝑓(⋅ |𝜆) and 𝐹(⋅ |𝜆) are the density and cdf of the distribution of the interpurchase times,
4In
particular,𝑓(𝜂′𝑉𝑖 ) = 2/𝜋arctan(𝜂′𝑉𝑖 ) for the Gaussian copula, 𝑓(𝜂′𝑉𝑖 ) = exp(𝜂′𝑉𝑖 ) + 1 for the Gumbel
copula and 𝑓(𝜂′𝑉𝑖 ) = 𝜂′𝑉𝑖 for the Frank copula.
22
and 𝑔 and 𝐺 are the density and cdf of the distribution of the monetary values.
The individual log-likelihood can be factorized as a product of three terms. Given the
properties of the exponential distribution, we obtain
𝑥𝑖
𝐴𝑖 (𝜆, 𝜇): = ∏ 𝑃(𝜏𝑖 > 𝐼𝑃𝑇𝑖𝑗 )𝑓(𝐼𝑃𝑇𝑖𝑗 |𝜆, 𝜇) ⋅ 𝑓(𝑅𝑖 |𝜆, 𝜇)
𝑗=1
=
𝜆𝑥𝑖 +1
𝜇+𝜆
(11)
𝜇𝜆𝑥𝑖
exp(−(𝜆 + 𝜇)𝑇𝑖 ) + 𝜇+𝜆 exp(−(𝜆 + 𝜇)(𝑇𝑖 − 𝑅𝑖 )),
as in Schmittlein et al. (1987). Furthermore, using (A3) from Appendix B, and denoting 𝑚
̅ 𝑖 the
average of the monetary values of the repeated transactions, and log(𝑚𝑖 ) the average of their
log-transformations, we have
𝑥𝑖
𝐵𝑖 (𝜈): = ∏ 𝑔(𝑚𝑖𝑗 |𝜈)
𝑗=1
log𝐵𝑖 (𝜈) = (𝑝 − 1)log(𝑚𝑖 ) − 𝑥𝑖 𝑚
̅ 𝑖 𝜈 + 𝑝𝑥𝑖 log(𝜈) − 𝑥𝑖 logΓ(𝑝).
(12)
In order to compute 𝐵𝑖 , it is not sufficient to know the sample average 𝑚
̅ 𝑖 . We conclude that
𝐿𝑖 (𝜆, 𝜇, 𝜈, 𝜃𝑖𝑛𝑡𝑟𝑎 ) = 𝐴𝑖 (𝜇, 𝜆)𝐵𝑖 (𝜈)𝐶𝑖 (𝜆, 𝜇, 𝜈, 𝜃𝑖𝑛𝑡𝑟𝑎 )
with 𝐴𝑖 defined in (11), Bi defined in (12), and
𝑥𝑖
𝐶𝑖 (𝜆, 𝜇, 𝜈, 𝜃𝑖𝑛𝑡𝑟𝑎 ) = ∏ 𝑐𝜃𝑖𝑛𝑡𝑟𝑎 (𝐹(𝐼𝑃𝑇𝑖𝑗 |𝜆, 𝜇), 𝐺(𝑚𝑖𝑗 |𝜈)).
(13)
𝑗=1
Following the semi-parametric maximum likelihood approach for copula estimation (see
for example Genest et al. 1995), we approximate 𝐶𝑖 by
𝑥𝑖
𝐶̂𝑖 = ∏ 𝑐𝜃𝑖𝑛𝑡𝑟𝑎 (𝐹̂𝑖𝑗 , 𝐺̂𝑖𝑗 ),
𝑗=1
where 𝐹̂𝑖𝑗 is the rank of 𝐼𝑃𝑇𝑖𝑗 among all other interpurchase times of the same customer, and
23
similarly for 𝐺̂𝑖𝑗 . Since the above quantity only depends on 𝜃𝑖𝑛𝑡𝑟𝑎 , and 𝐴𝑖 and 𝐵𝑖 only depend
on the other individual parameters, we can decompose the log-likelihood in (10) as
log𝐿 ≈ ∑𝑛𝑖=1 log ∫ 𝐴𝑖 (𝜆, 𝜇)𝐵𝑖 (𝜈)𝑔𝑖 (𝜆, 𝜇, 𝜈)𝑑𝜆 𝑑𝜇 𝑑𝜈 + log ∫ 𝐶̂𝑖 (𝜃𝑖𝑛𝑡𝑟𝑎 )𝑔𝑖 (𝜃𝑖𝑛𝑡𝑟𝑎 )𝑑𝜃𝑖𝑛𝑡𝑟𝑎 .
Since the expression above involves an integration, which cannot be solved analytically, we use
Simulated Maximum Likelihood (SML) (see e.g. Green 2003, pp. 590-594). We draw
𝑠
(𝜆𝑖𝑠 , 𝜇𝑖𝑠 , 𝜈𝑖𝑠 , 𝜃𝑖𝑛𝑡𝑟𝑎,𝑖
) , for 𝑠 = 1, … , 𝑆 = 1000 from the joint distribution of the individual
parameters, and approximate
𝑛
𝑆
𝑛
𝑆
𝑖=1
𝑠=1
𝑖=1
𝑠=1
1
1
log𝐿 ≈ ∑ log( ∑ 𝐴𝑖𝑠 𝐵𝑖𝑠 ) + ∑ log( ∑ 𝐶̂𝑖𝑠 ),
𝑆
𝑆
(14)
where 𝐴𝑖𝑠 , 𝐵𝑖𝑠 , 𝐶𝑖𝑠 are defined in (11), (12), and (13), taking 𝜆 = 𝜆𝑖𝑠 , 𝜇 = 𝜇𝑖𝑠 , 𝜈 = 𝜈𝑖𝑠 and
𝑠
𝜃𝑖𝑛𝑡𝑟𝑎 = 𝜃𝑖𝑛𝑡𝑟𝑎,𝑖
. The parameters involved in the first term of (14) are different from those in the
second term, allowing for two separate maximization problems. The fact that the likelihood can
be split in two parts allows us to study both levels of association separately. In cases where
individual transaction data are not available, only the inter-customer association can be
estimated. Also note that for customers making no repeat purchase, the model boils down to the
traditional Pareto/NBD model.
We then estimate 𝛉 by maximizing the simulated log-likelihood in (14). Standard errors
are retrieved from the Hessian of the log-likelihood. The random generation of the
𝑠
(𝜆𝑖𝑠 , 𝜇𝑖𝑠 , 𝜈𝑖𝑠 , 𝜃𝑖𝑛𝑡𝑟𝑎,𝑖
) is based on the same random uniform number for every 𝑠, ensuring enough
smoothness of the log-likelihood function.
Once the (hyper)parameters in 𝛉 are estimated, following the empirical Bayes principle,
it is possible to simulate from the posterior distribution of the individual parameters using the
independent Metropolis-Hastings algorithm (see Appendix C for more detail). In the empirical
24
application, we are particularly interested in the intra-customer association parameter 𝜃𝑖𝑛𝑡𝑟𝑎,𝑖 .
We obtain a point estimate for 𝜃̂𝑖𝑛𝑡𝑟𝑎,𝑖 as the average of the outcomes of the simulated Markov
Chain.
4.4. CLV Prediction
The customer lifetime value at horizon 𝐻 for customer 𝑖 , 𝐶𝐿𝑉𝑖,𝐻 , is given by the
discounted sum of all profits that will be generated during the next 𝐻 time units by this
customer. Let 𝑇 denote the time at which the prediction is made. Future transactions take place
at times 𝑇 + 𝑡1 , 𝑇 + 𝑡2 , …. with corresponding transaction values 𝑚i,𝑡1 , 𝑚i,𝑡2 , …. Adapting Gupta
et al. (2004), we define
𝐶𝐿𝑉𝑖,𝐻 = ∑
𝑡𝑗 ≤𝐻
𝑚i,𝑡𝑗
(1 + 𝑑)
𝑡𝑗 ,
(15)
where 𝑑 is the discount rate. Note that the 𝐶𝐿𝑉𝑖,𝐻 defined in (15) is a random variable.
Once the model is estimated, it is possible to simulate future transactions for every
customer, resulting in the simulated distribution of the CLV over the next 𝐻 periods. Our
approach is similar to Singh et al. (2009). Details are provided in Appendix C. The average over
the simulated distribution yields a prediction of the expected CLV for a given customer. Since
we simulate the whole distribution of 𝐶𝐿𝑉𝑖,𝐻 , it is also possible to construct prediction intervals.
5. Data
Our analysis is based on four data sets. Below, we describe each of them in turn and provide
some summary statistics of the various data characteristics in Table 3.
5.1.1. Online Music Album Sales
The CDNOW data contains the individual transactions of 2,357 customers on the online
music site CDNOW. The data cover 78 weeks for a sample of customers who made their
25
first-ever purchase at the website during the first quarter of 1997. We use the first 39 weeks of
1997 for the estimation of the model parameters and keep the remaining 39 weeks as hold-out
sample to assess the predictive performance of the model.
5.1.2. Financial Securities Transactions
The securities transaction data are provided by a major international financial service
institution.5 The data contain securities transactions made by 3,472 randomly-selected customers
who made their first transaction between January 2001 and December 2003. Transactions
include the purchase and selling of stocks at this bank between January 2001 and December
2005.6 Based on a discussion with the firm management, we consider a profit margin of 0.1%
of the transaction and an annual discount rate of 12%. We keep the last two years of transactions
(from January 2004 to December 2005, 𝐻 = 24 months) as hold-out sample to assess the
predictive performance of the models. The data also contain socio-demographics and
company-related customer characteristics that we use to model observed customer heterogeneity.
Customer characteristics include a continuous customer age variable (which will be
mean-centered in the model estimation), as well as a dummy variable living area accounting for
the type of area where a customer is living. This variable takes the value one when the customer
lives in the suburb of a city, and zero otherwise. As company-related customer covariates, we
include a customer lifetime variable (also mean-centered in the model estimation), where the
lifetime is defined as the number of time units since the first transaction of the customer. In
addition, we also include a dummy variable taking value one when the bank is the customer's
primary bank.
5
Due to data confidentiality agreements, we are unable to divulge more details about the companies providing us
the data on the financial securities, functional/utilitarian FMCG and hedonic FMCG.
6Note that we remove the stock transactions part of an automated pension plan from the data set.
26
5.1.3. Utilitarian Fast-Moving Consumer Good
The third data set is provided by a major packaged goods manufacturer and contains
transactions of a FMCG that can be classified as utilitarian product (e.g. detergent, toilet paper,
pet food) according to Dhar and Wertenbroch (2000) or Khan et al. (2005). The data contain
purchases made between March 2007 and December 2009 by the 2,968 customers who reported
their first transaction to the data provider between March 2007 and December 2008. All these
customers live in the same European country. We keep the last year of purchase (from January
2009 to December 2009, H = 12 months) as hold-out sample. The data also contain the size of
the household (which will be mean-centered in the model estimation) that we use to model
observed customer heterogeneity. There is a substantial amount of customer heterogeneity in the
RFM variables, as the large standard deviations confirm.
5.1.4. Hedonic Fast-Moving Consumer Good
The fourth data set is provided by same packaged goods manufacturer than the other
FMCG data set and contains transactions of a FMCG that can be classified as a hedonic product
(e.g. ice cream, chocolate bar, wine, beer) according to Dhar and Wertenbroch (2000) or Khan et
al. (2005). The data contain purchases made between January 2007 and August 2011 by the
5,682 customers who made their first transaction between January 2007 and December 2008.
These customers live in the same European country as for the other FMCG. We keep the last 20
months of purchase (from January 2010 to August 2011, 𝐻 = 20 months) as hold-out sample.
The data also contain the size of the household (which is mean-centered in the model estimation)
that we use to model observed customer heterogeneity.
27
Table 3: Summary statistics of the various data characteristics
Mean
Standard
deviation
Minimum
Maximum
Online music albums
Number of repeated transactions
Recency (in weeks)
Average transaction value (in euros)
Lifetime (in weeks)
1.04
6.85
14.08
32.72
2.19
10.73
25.76
3.33
0.00
0.00
0.00
27.00
29.00
38.43
299.64
38.86
4.73
9.57
2.38
48.89
0.43
34.91
0.90
7.24
11.14
2.91
16.18
0.49
0.78
0.30
0.00
0.00
0.00
6.00
0.00
33.53
0.00
48.00
35.07
19.73
99.00
1.00
36.43
1.00
7.85
11.69
928.64
16.01
2.49
6.65
7.31
959.93
5.21
1.16
0.00
0.00
0.00
1.00
1.00
19.00
19.00
6469
20.00
8.00
2.30
11.40
436.75
22.75
2.34
2.47
10.10
337.60
8.06
1.08
0.00
0.00
0.00
13.00
1.00
12.00
34.00
2093.00
36.00
8.00
Financial securities
Number of repeated transactions
Recency (in weeks)
Average transaction value (in euros)
Age (in years)
Living area (Dummy, City suburb = 1)
Current lifetime (in weeks)
Primary bank (Dummy)
Utilitarian FMCG
Number of repeated transactions
Recency (in months)
Average transaction value (in eurocent)
Current lifetime (in months)
Household size
Hedonic FMCG
Number of repeated transactions
Recency (in months)
Average transaction value (in eurocent)
Current lifetime (in months)
Household size
6. Empirical Findings
For each data set, we estimate a model with and without association at inter- and/or
intra-customer level. First, we assess whether the various data contexts show evidence of
significant intra- and/or inter-customer association, by comparing the fit of the various model
specifications. Second, we report the parameter estimates, including the association parameters,
and the forecasts obtained by the best-fitting models.
6.1. Model fit comparisons
Model fit comparisons are reported in Error! Reference source not found. (in bold,
the best-fitting model). For each specification, we select the copula (Gauss, Student, Gumbel or
28
Frank) that provides the best fit.
Table 4: Model fit comparisons
No
association
Intra-customer
association only
Inter-customer
association only
Intra- and
inter-customer
association
-13,688
7
Null model
27,431
Frank
-13,660
9
56 (<0.001)
27,390
Gauss
-13,641
10
94 (<0.001)
27,360
Frank & Gauss
-13,609
12
158 (<0.001)
27,312
-67,828
19
Null model
135,840
Gauss
-67,495
25
666 (<0.001)
135,233
Student
-67,149
22
1,358 (<0.001)
134,512
Gauss & Student
-66,815
28
2,026 (<0.001)
133,902
-223,974
9
Null model
448,039
Student
-219,306
12
9336 (<0.001)
438,733
Gauss
-219,977
12
7994 (<0.001)
440,075
Student & Gauss
-219,002
15
9944(<0.001)
438,155
-136,484
9
Null model
273,053
Frank
-135,340
12
2,288(<0.001)
270,794
Gauss
-135,000
12
2,968(<0.001)
270,114
Frank & Gauss
-134,928
15
3,112 (<0.001)
269,998
Online music albums
Log-Likelihood
Number of parameters
Likelihood ratio t-stat (p-value)
BIC (with n=2457)
Financial securities
Log-Likelihood
Number of parameters
Likelihood ratio t-stat (p-value)
BIC (with n=16434)
Utilitarian FMCG
Log-Likelihood
Number of parameters
Likelihood ratio t-stat (p-value)
BIC (with n=23294)
Hedonic FMCG
Log-Likelihood
Number of parameters
Likelihood ratio t-stat (p-value)
BIC (with n=13054)
For all the datasets, we find that both levels of association improve the model penalized
fit (see the values of the Bayesian Information Criterion BIC) and are jointly significant
(according to a likelihood ratio test comparing a model with both association vs. the null model
without association). For the online music album sales, the Frank copula at the intra-customer
level and a Gauss copula at the inter-customer level offer the lowest BIC. For the financial
securities, a Gauss copula at the intra-customer level and a Student copula at the inter-customer
level offer the lowest BIC. For the utilitarian FMCG goods, the best penalized fit is given by the
Student copula at the intra-customer level and a Gauss copula at the inter-customer level. For the
hedonic FMCG product category, the Frank copula at the intra-customer level and a Gauss
copula at the inter-customer level give the lowest BIC. For completeness, we also estimated the
29
same models without covariates and find that, for all datasets, the models with covariates and
both levels of association yield the lowest BIC.
In Appendix D, we report the parameter estimates, with their respective standard errors,
obtained for the best-fitting models. Below, we discuss the specific association results.
6.2. Estimated intra-customer associations between timing and spending
In Figure 3 to Error! Reference source not found., we plot the distribution of the
intra-customer association for all the data sets.
[INSERT Figure 3 to Error! Reference source not found. ABOUT HERE]
For the online music sales at CDNOW (Figure 3), the intra-association is significantly
positive but small for most customers. There is little heterogeneity between customers as the
small 𝜎̂𝑖𝑛𝑡𝑟𝑎 (= 0.14, standard deviation = .26) in Appendix D testifies. Our findings corroborate
the correlation of .11 reported in Fader et al. (2005) and is in line with the findings of Boatwright
et al. (2003) for an online grocery retailer. The compensating purchase behavior of the customers
in this product category can be driven by the fact that customers allocate a relatively fixed budget
to the purchase of music CD. They might also be likely to adjust their purchase timing and
quantity decisions to the calendar of price promotions.
For the financial securities (Figure 4), we find both positive and negative associations
between the interpurchase time and the amount spent over time. There is a substantial amount of
unobserved heterogeneity in the intra-customer association as the large 𝜎̂𝑖𝑛𝑡𝑟𝑎 (= 1.85, standard
deviation = .07, see Appendix D) testifies. We find that 20% of the population exhibits
compensating purchase behavior (i.e. a significantly positive association), and 12% shows
purchase acceleration or deceleration (i.e. significantly negative association). The remaining
30
68% have a non-significant association. The age of the customer has a significant effect (= -.38,
standard deviation = .12) on this association. The older the customer becomes, the more negative
the association between the interpurchase time and the amount spent at the next transaction. In
other words, young customers are more likely to show compensating purchase patterns, possibly
because of budget or time constraints. In contrast, older customers tend to show purchase
acceleration, possibly because they are gradually learning how to make good investment
decisions (learning effects); or purchase deceleration, either because their budget or interest (e.g.
wear-out or satiation effect) in the stock market decreases over time.
For the utilitarian product (Figure 5), we find a significant positive association for 15% of
the customers and a significant negative association for only 5% of the customers. The individual
consumption of this category is by nature non-expandable (e.g. toilet paper consumption) and
any deviation, for instance due to marketing activities, is temporary and compensated by
stockpiling. The product we investigate has a very long expiration time (1 to 5 years in this
product category) and the demand in this product category is known to be sensitive to price
promotions and to show substantial post-promotion dips due to its storage propensity. This
explains why a substantial amount of customers show compensating behavior. Interestingly,
household size has a positive effect on the association. Larger households tend to have larger
storage capacity, and are therefore more likely to exhibit compensating behavior, which
corroborates our interpretation that stockpiling is at play for these customers. The small fraction
customers show acceleration or deceleration. The latter are likely to be customers who gradually
enter or quit the product category (from or to a substitute category 7 ). Interestingly, the
association we find is similar to the one reported by Jen et al. (2009) in another utilitarian
7
Due to data confidentiality, we cannot reveal the product category but it is useful to mention the existence of an
important substitute category.
31
category (office supply products).
For the hedonic good (Figure 6), only a very small fraction (about 1%) of customers
show a significant association, all others have a non-significant association. Related to the nature
of the product category, the large majority of the customers adjust their consumption to the
quantities they bought (consumption expandability). These products are harder to resist and their
purchase and consumption is more impulsive than e.g. utilitarian goods. When bought in larger
quantities, the customer has the tendency to consume more of it. As a consequence, the quantity
bought has little influence on the next purchase timing. Inventory-based consumption is known
to be frequent in such product categories (Ailawadi and Neslin, 1998) and our results suggest
that marketing activities have the potential to increase customer spending in this category.
Interestingly, the household size plays no influence in this category.
6.3. Estimated inter-customer associations between timing and spending
In Table 5, we report the parameters that capture the inter-customer association between
the timing, spending and dropout processes (see equation 3), as well as their standard errors.
Overall, the four product categories offer consistent results. Customers show a positive
inter-customer association between their timing and spending decisions, a positive or
non-significant association between the timing and dropout decisions, and a negative or
non-significant association between the spending and dropout decisions.
For all four product categories, higher purchase frequencies are associated with larger
purchase amounts. The associations are the strongest for both FMCGs and relatively smaller for
the CDNOW data and for the financial securities. Our results are in line with the positive
association found by Abe (2013) for a large chain of music CDs and contrasts with the negative
association he found for a Japanese department store in the same study. A negative association is
32
also found by Borle et al. (2008) or Singh et al. (2009) for a membership-based direct marketing
company. As pointed out by Abe (2013), the associations are clearly context-dependent and
should be investigated on a case-by-case basis.
Table 5: Estimated inter-customer associations
Timing-spending
Timing-drop out
Spending-drop out
Online music albums
Sigma (standard error)
0.28*** (0.03)
0.43*** (0.09)
-0.17** (0.08)
Financial securities
Sigma (standard error)
0.15*** (0.03)
0.67*** (0.05)
0.03 (0.04)
Utilitarian FMCG
Sigma (standard error)
0.50*** (0.03)
0.13 (0.34)
-0.38** (0.19).
Hedonic FMCG
Sigma (standard error)
0.76*** (0.03)
0.02 (0.30)
-0.64** (0.30)
***Significant at the 1% probability level, ** Significant at the 5% probability level,, * Significant at the 10% probability level
Our results indicate that we can find, in the four product categories, a segment of frequent
and large buyers, from which firms could derive a large share of their revenues. These customers
are possibly more involved and committed with the brand or product category, making their
purchase utility higher and/or their transaction costs lower, compared to the other customers
(Vilcassim and Jain 1991). Note that the estimated association parameters have the same sign
and significance levels if the models are estimated without covariates.
6.4. Estimated inter-customer associations with dropout
The association between the customers’ propensity to drop out and their other purchase
decisions, i.e. when to buy and how much to buy, is also relatively consistent across the four
product categories. Overall, we find a positive association between the customers’ propensity to
dropout and their purchase frequency. Frequent buyers have, on average, a higher propensity to
dropout (and a shorter lifetime) than the other customers, as also found by Singh et al. (2009) for
their membership-based direct marketing dataset. The association is significant for the music
33
albums sales and for the financial securities, but not significant for both FMCG categories. The
positive association is in line with the idea that each purchase occasion is an opportunity for the
customer to re-evaluate her choice for a certain brand or store, and in turn influences her decision
to drop out (Fader et al. 2005). The non-significant association for the FMCGs is also in line
with the findings of Abe (2009, 2013).
In turn, the association with spending is negative and significant for all categories, except
for the financial securities. This result corroborates with the findings of Bolton and Lemon
(1999) that large buyers tend to be involved in their purchase decisions and more loyal.
6.5. Forecasting Results
In this section, we evaluate the predictive performance of the model that accounts for all
types of association vs. the model that assumes independence between the timing, spending and
dropout decisions. For all four data sets, we make all predictions on a separate hold-out sample,
as described in the data section. In Table 6, we evaluate the accuracy of the CLV forecasts at the
individual customer level by comparing the root mean squared error (RMSE) and the correlation
between the actual and predicted values (correlation). We expect the benefits of accounting for
the association to be most visible for the product categories where the associations are the
strongest (i.e. the utilitarian FMCG showed a substantial intra- and inter-customer association
between timing and spending; and the hedonic FMCG showed the highest inter-customer
association between timing and spending) and, in particular, on the customers that show the
stronger intra-customer association. In order to investigate this point, we also compute the
RMSE and correlation for the top 10% of the customers with the strongest intra-customer
association (in absolute value).
34
Table 6: Hold-out forecasting comparisons for the full sample (left panel) and the top 10%
segment in terms of intra-customer association intensity (right panel).
Full sample
No association
Intra- and
inter-customer
association
Top 10% segment
No association
Intra- and
inter-customer
association
Online music albums
RMSE
Correlation
70.07
.55
71.07
.53
149.66
.42
146.57
.46
Financial Securities
RMSE
Correlation
38.93
.42
38.69
.40
35.33
.35
32.11
.52
Utilitarian FMCG
RMSE
Correlation
9,951.35
.66
9,744.09
.66
11,169.79
.55
10,637.34
.57
Hedonic FMCG
RMSE
Correlation
2,255.94
.68
2,116.38
.68
3,600.58
.58
3,349.66
.58
In line with our expectations, the models with association substantially outperform the
models that assume independence for both the utilitarian and hedonic FMCGs. As expected as
well, the difference is more pronounced for the customers who show the strongest association
(right panel). Finally, the results are mixed for the other two product categories. For the
CDNOW data, the low association previously reported by Fader et al. (2005) and confirmed by
our analysis has no influence on the accuracy of the CLV forecasts for the large majority of the
customers. For the financial securities data, the model without covariate and with association
offers the best out-of-sample forecasts.8
Besides the individual CLV forecasts, we also compare the aggregate predictions of the
key components of CLV in Figure 7 to Figure 12: (i) the number of transactions made by each
customer during the hold-out period, (ii) the probability that a customer makes at least one
8
For this product category, we use the models without covariates as their inclusion decreases the out-of-sample
predictive performance of both the models without and with association.
35
transaction during the hold-out period, and (iii) the average amount per transaction during the
hold-out period. As done in Fader et al. (2005) and Abe (2009), we divide the customer base into
customer groups based on their number of transactions during the estimation period (i.e. 8
groups: no transaction, 1 transaction, ... , 7 or more transactions). The dotted lines depicts the
actual values , the dashed lines the predictions without association, and the thick solid lines the
predictions accounting for association.
[Insert Figure 7 to Figure 12 about here]
We find that modeling the association mostly helps with the predictions of the spending
process (see bottom left panels). The improvement is most visible for the securities and the
hedonic product categories (see bottom left panel). In addition, the association also helps with
predicting the number of transactions (i.e. timing process), in particular in the hedonic category.
It is important to realize that, while the gains in predictive performance are not very large, the
main benefits of modeling the association in CLV also lies in the understanding of the behavioral
rationales underlying the associations as well as in the managerial implications, which are
developed in the next section.
7. Managerial Recommendations, Limitations, and Future Research
The direction and intensity of the intra- and inter-customer association between the
timing, spending and dropout decisions provide many insights to improve firms’ strategies.
Among them, we focus on an important trade-off that firms struggle with, i.e. how maximize
their short-run cash flows while stabilizing their long-run revenues. For instance, the customers
who bring largest revenues might not be the ones who stay the longest with the firm and those
who stay long might not bring sufficient revenues to the firm, as our empirical analysis before
36
has shown. In practice, firms often sacrifice short-run incomes in order to ensure a more
long-term steady influx of revenues. The advantages of stability are numerous. It facilitates
production planning, reduces inventory costs, avoids liquidity issues, and allows the firm to
make long-run investments. In this section, we illustrate how companies can leverage the
information on the various associations to address this point. We do so using the results we
obtained on the utilitarian product category, which shows both significant and large
inter-customer association between timing and spending and between spending and drop out, as
well as a significant and highly heterogeneous intra-customer association. A similar analysis can
be made for the other categories.
From the previous analysis, we learned, given the positive inter-customer association
parameters, that the utilitarian buyers are of two types. The first type is a frequent, large and
short-lived buyer. Such customers form a source of short-run high revenues stream for the
company. The second type is made of infrequent, small and long-lived buyers. In contrast to the
former group, they represent a longer-run but smaller source of revenues for the company.
Furthermore, at the intra-customer level, we learned that 15% of the customers tend to show
compensating purchase patterns, that 5% of them show sign of purchase acceleration or
deceleration, while the remaining 80% exhibit non-compensating purchase behaviors. From the
stabilization vs. maximization perspective, the first group of customers (positive association)
offers steady cash flows and be an interesting group to acquire and retain. Their compensating
behavior leads to a stable revenue stream on average (i.e. return to the mean). At the same time,
their compensating purchase behavior makes them no good candidate for temporary marketing
actions such as price promotion. In contrast, the last group (non-significant association) forms a
potential source of additional short-run revenues. If the company can induce them to spend more
37
or earlier than usual at a given purchase occasion (e.g. with a marketing action), they would not
compensate for it at the next purchase and the company will be able to generate additional
revenues.
Both the inter- and intra-customer association can be used in combination by firms in
order to decide how much to focus on the stabilization of their long-term cash flows or on the
maximization of their short-term revenues. In Table 7, we portrait the various types of customers
based on the association results along the dimensions of interest. Firms interested in stabilizing
their long-term cash flows should maintain a sufficient number of customers in the first quadrant.
Given that these customers generate comparatively little revenues, a critical mass of them is
needed to ensure the firm’s survival. Alternatively, firms might be better off composing a mix of
long-lived customers from the first quadrant combined with a more frequently renewed set of
shorter-lived but higher-revenues customers from the fourth quadrant. In order to stabilize their
long-term cash flows furthermore, firms can then choose, within each quadrant, to focus on the
customers with a positive intra-customer association. Their return-to-the-mean behavior offer
steady revenues. Additionally, they can also target the customers with a non-significant
intra-customer association with short-run marketing actions, e.g. price promotions, to generate
additional revenues. In any case, they should avoid the customers with a negative association
who create a lot of instability.
In contrast, firms interested in maximizing short-run revenues should focus in priority on
the customers in the fourth quadrant. Among them, it is crucial that they focus on the customers
with a positive or a non-significant association. The purchase acceleration or deceleration
behavior of the customers with a negative association is unlikely to be profitable given the short
expected lifetime of these customers.
38
A hybrid of both strategies can be attractive. Firms could for instance focus on the
customers with a non-significant association in the first quadrant, which would ensure a
long-term stream of revenues with temporary additional revenues generated by marketing
actions, and in addition, on the small group of short-lived customers with a positive association
in the fourth quadrant, which will in turn boost the overall revenues of the firm.
In sum, the direction and intensity of the inter-customer association can guide firms in
understanding and monitoring which customers contribute sustainably or temporarily to their
cash flow stream and which customers contribute to the size of their revenues, and as such, be
able to make better decisions in terms of which customers to maintain and grow and which
customers to target with additional marketing incentives.
Table 7: Repartition of intra-association as a function of purchase frequency and quantity
for the repeat customers of the Utilitarian FMCG dataset
Small buyers
(less than 7.10 euros per transaction)
(median dropout = 0.0502)
Large buyers
(more than 7.10 euros
transaction)
(median dropout = 0.0338)
per
Infrequent buyers
(less than one transaction every 1.90
months)
Median dropout = 0. 0542
Positive intra-association: 14%
Non-significant association: 82%
Negative intra-association: 4%
Median dropout = 0. 0343
Positive intra-association: 13%
Non-significant association: 81%
Negative intra-association: 6%
Frequent buyers
(more than one transaction every
1.90 months)
Median dropout = 0. 0368
Positive intra-association: 15%
Non-significant association: 75%
Negative intra-association: 10%
Median dropout = 0. 0337
Positive intra-association: 21%
Non-significant association: 72%
Negative intra-association: 7%
7.1. Conclusions, limitations and future research
While the Pareto/NBD, Gamma/Gamma framework developed by Fader et al. (2005) has
been successfully adopted as a powerful customer valuation method for non-contractual settings,
the potential association between the transaction, the spending and the dropout processes, both at
the intra- and inter-customer levels, calls for the development of a conceptual and
methodological framework to better understand customer purchase behavior. In this paper, we
39
propose an integrative approach, based on copulas, to account for both levels of associations
between the transaction, spending and dropout processes and investigate the behavioral
rationales that underline these phenomena.
Despite our efforts, a number of limitations should be mentioned and raises suggestions
for future research. First, while we do explore the association across a variety of product
categories, the data we use for each application are unfortunately limited. In particular, we do not
have marketing-mix information, either competitive or market-related variables. For instance, for
the financial securities data, we do not have information on the stock market fluctuations, which
may admittedly have a sizeable influence on customers' purchase and selling behavior. As such,
we might overestimate the effect of the other variables. Future research that would incorporate
marketing-mix variables could certainly provide additional managerial insights. Second, our
method captures the contemporary association between the various purchase decisions but it
overlooks the possibility of lead-lag relationships between the variables. Future research could
benefit from adapting our copula framework to the context where longer lags would be
considered. Note however that it would substantially complicate the estimation.
To conclude, the major advantage of the copula-extended model resides in its possibility
to reveal and quantify the association between transaction, spending and dropout processes from
an observed database of customers' transactions. This is of key importance when deciding which
customers to target with marketing efforts and should allow us to obtain better
return-on-investment than an approach that ignores the association in the data.
40
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Figure 1: Transaction values vs. interpurchase times of several purchases made by five
imaginary customers, each represented by a different symbol. Customer averages are
reported in bold.
46
Figure 2: Contours plots corresponding to the joint distribution of two standard-normal
random variables with Spearman rank correlation equals to .50, for (i) an independent
copula (upper-left plot), (ii) a Gaussian copula (upper-right plot), (iii) a Gumbel copula
(lower-left plot), and (iv) a Frank copula (lower-right plot).
47
2500
Frequency
2000
1500
1000
500
0
0.05
0.055
0.06
0.065
0.07
0.075
Intra Spearman Correlation
0.08
0.085
0.09
Figure 3: Histogram of the estimated intra-customer association for the online music
album sales. The x axis reports the spearman correlation.
48
1600
1400
1200
Frequency
1000
800
600
400
200
0
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
Intra Spearman Correlation
0.3
0.4
0.5
Figure 4: Histogram of the estimated intra-customer association for the securities
transactions. The x axis reports the spearman correlation.
49
1200
1000
Frequency
800
600
400
200
0
-0.6
-0.4
-0.2
0
0.2
0.4
Intra Spearman Correlation
0.6
0.8
Figure 5: Histogram of the estimated intra-customer association for the utilitarian
product. The x axis reports the spearman correlation.
50
3000
2500
Frequency
2000
1500
1000
500
0
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
Intra Spearman Correlation
0.15
0.2
0.25
Figure 6: Histogram of the estimated intra-customer association for the hedonic product
dataset. The x axis reports the spearman correlation.
51
Predicted probability of at least one transaction
Predicted number of transactions
6
# Transactions
Probability
0.6
0.4
0.2
0
0
1
2
3
4
5
6
# Transactions in Observation Weeks
4
2
0
7+
0
1
2
3
4
5
6
# Transactions in Observation Weeks
Predicted transaction value
Predicted CLV
2.5
25
20
1.5
CLV
Value
2
1
15
10
0.5
0
7+
5
0
1
2
3
4
5
6
# Transactions in Observation Weeks
0
7+
0
1
2
3
4
5
6
# Transactions in Observation Weeks
7+
Figure 7: Securities transactions dataset - Actual (dotted line) and predicted values for the
hold-out period as a function of the number of transactions in the estimation period for the
independent model (dashed line) and the copula-extended model for the full sample (thick
solid line).
Predicted probability of at least one transaction
Predicted number of transactions
6
# Transactions
Probability
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
# Transactions in Observation Weeks
4
2
0
7+
0
Predicted transaction value
1
2
3
4
5
6
# Transactions in Observation Weeks
7+
Predicted CLV
25
20
1.5
CLV
Value
2
1
0.5
0
15
10
5
0
1
2
3
4
5
6
# Transactions in Observation Weeks
7+
0
0
1
2
3
4
5
6
# Transactions in Observation Weeks
7+
Figure 8: Securities transactions dataset - Actual (dotted line) and predicted values for the
hold-out period as a function of the number of transactions in the estimation period for the
independent model (dashed line) and the copula-extended model for the top 10% segment
(thick solid line).
52
Predicted probability of at least one transaction
Predicted number of transactions
8
# Transactions
Probability
1
0.5
0
0
1
2
3
4
5
6
# Transactions in Observation Weeks
6
4
2
0
7+
0
Predicted transaction value
7+
Predicted CLV
800
6000
600
CLV
Value
1
2
3
4
5
6
# Transactions in Observation Weeks
400
2000
200
0
4000
0
1
2
3
4
5
6
# Transactions in Observation Weeks
0
7+
0
1
2
3
4
5
6
# Transactions in Observation Weeks
7+
Figure 9: Hedonic product dataset - Actual (dotted line) and predicted values for the
hold-out period as a function of the number of transactions in the estimation period for the
independent model (dashed line) and the copula-extended model for the full sample (thick
solid line).
Predicted probability of at least one transaction
Predicted number of transactions
8
# Transactions
Probability
1
0.5
0
0
1
2
3
4
5
6
# Transactions in Observation Weeks
6
4
2
0
7+
0
Predicted transaction value
1
2
3
4
5
6
# Transactions in Observation Weeks
7+
Predicted CLV
1000
6000
600
CLV
Value
800
400
2000
200
0
4000
0
1
2
3
4
5
6
# Transactions in Observation Weeks
7+
0
0
1
2
3
4
5
6
# Transactions in Observation Weeks
7+
Figure 10: Hedonic product dataset - Actual (dotted line) and predicted values for the
hold-out period as a function of the number of transactions in the estimation period for the
independent model (dashed line) and the copula-extended model for the top 10% segment
(thick solid line).
53
Predicted probability of at least one transaction
Predicted number of transactions
10
# Transactions
Probability
1
0.5
0
0
1
2
3
4
5
6
# Transactions in Observation Weeks
8
6
4
2
0
7+
0
1500
15000
1000
10000
5000
500
0
7+
Predicted CLV
CLV
Value
Predicted transaction value
1
2
3
4
5
6
# Transactions in Observation Weeks
0
1
2
3
4
5
6
# Transactions in Observation Weeks
0
7+
0
1
2
3
4
5
6
# Transactions in Observation Weeks
7+
Figure 11: Utilitarian products dataset - Actual (dotted line) and predicted values for the
hold-out period as a function of the number of transactions in the estimation period for the
independent model (dashed line) and the copula-extended model for the full sample (thick
solid line).
Predicted probability of at least one transaction
Predicted number of transactions
10
# Transactions
Probability
1
0.5
0
0
1
2
3
4
5
6
# Transactions in Observation Weeks
8
6
4
2
0
7+
0
Predicted transaction value
1
2
3
4
5
6
# Transactions in Observation Weeks
7+
Predicted CLV
10000
1000
CLV
Value
1500
5000
500
0
0
1
2
3
4
5
6
# Transactions in Observation Weeks
7+
0
0
1
2
3
4
5
6
# Transactions in Observation Weeks
7+
Figure 12: Utilitarian products dataset - Actual (dotted line) and predicted values for the
hold-out period as a function of the number of transactions in the estimation period for the
independent model (dashed line) and the copula-extended model for the top 10% segment
(thick solid line).
54
APPENDIX A: Copula Density Functions
Here we list the probability density functions of the copula distributions used in this paper.
Gaussian copula A first family of copula that can be found in the literature is the Gaussian or
normal copula, with density function
1
1
(A.1)
𝑐𝜃 (𝐹 , 𝐺 ) =
exp (− 𝜓′(𝑅(𝜃 )−1 − 𝐼2 )𝜓),
2
1/2
2
(1 − 𝜃 )
where −1 < 𝜃 < 1 , 𝜓 = (Φ−1 (𝐹 ), Φ−1 (𝐺 ))′ , Φ is the univariate standard normal
distribution function, 𝑅(𝜃) is the correlation matrix
1 𝜃
(A.2)
𝑅(𝜃) = (𝜃 1 ),
and 𝐼2 is the identity matrix of size 2. This copula permits both positive and negative
association between the variables. Values of 𝜃 equal to −1, 0 and 1 correspond to the
minimal value of negative association, independence, and the maximum of positive association.
When combined with two normal marginal distributions, the joint distribution is bivariate
normal.
Gumbel copula The density of the Gumbel (or logistic) copula is given by
𝑐𝜃 (𝐹 , 𝐺 )
𝐶(𝐹 , 𝐺 )[log(𝐹 )log(𝐺 )]𝜃 −1
=
[((−log(𝐹 ))𝜃
𝐹 𝐺 [(−log(𝐹 ))𝜃 + (−log(𝐺 ))𝜃 ]2−1/𝜃
(A.3)
1/𝜃
+ (−log(𝐺 ))𝜃 )
+ 𝜃 − 1]
where 1 ≤ 𝜃 < ∞ is the assocation parameter.
Error! Reference source not found. is
The
copula
distribution
in
1/𝜃
𝐶(𝐹 , 𝐺 ) = exp{−((−log(𝐹 ))𝜃 + (−log(𝐺 ))𝜃 )
}.
The limiting case 𝜃 = 1 gives independence while for 𝜃 tending to ∞, one obtains a
perfect dependency.
Frank copula The density of the Frank Copula is given by
(A.4)
𝜃 (1 − 𝑒 −𝜃 )𝑒 −𝜃 (𝐹 +𝐺 )
𝑐𝜃 (𝐹 , 𝐺 ) =
,
[(1 − 𝑒 −𝜃 ) − (1 − 𝑒 −𝜃 𝐹 )(1 − 𝑒 −𝜃 𝐺 )]2
where −∞ < 𝜃 < ∞. This copula permits both positive and negative association between the
variables. Values of −∞, 0 and ∞ correspond to the values of smallest negative association,
independence, and the largest positive association respectively.
Trivariate Student copula Let (𝑋0 , 𝑌0 , 𝑍0 ) be a trivariate multivariate t-distribution with 𝑑𝑓
degrees of freedom, and correlation matrix 𝑅. Denote 𝐹𝑋0 , 𝐹𝑌0 , 𝐹𝑍0 the respective marginal
distribution. We say that a trivariate random variable (𝑋, 𝑌, 𝑍) has a Student copula with 𝑑𝑓
degrees of freedom and parameter 𝑅 if the distribution of (𝐹𝑋 (𝑋), 𝐹𝑌 (𝑌), 𝐹𝑍 (𝑍)) is the same as
of (𝐹𝑋0 (𝑋0 ), 𝐹𝑌0 (𝑌0 ), 𝐹𝑍0 (𝑍0 )). For 𝑑𝑓 = ∞, we find back the trivariate Gaussian copula.
55
APPENDIX B: The Copula-Extended CLV Model
The model we consider is a two-level model. Conditions A1-A4 below describe the transactions
process of an individual customer, i.e. at the intra-customer level. Conditions A5-A9 describe the
model at the second level, i.e. at the inter-customer level.
A1: If a customer is alive, the interpurchase times 𝐼𝑃𝑇𝑖,𝑗 are exponentially distributed with
parameter 𝜆𝑖 .
A2: The unobserved time to death 𝜏𝑖 of a customer is exponentially distributed with parameter 𝜇𝑖 .
A3: The transaction values 𝑚𝑖𝑗 are gamma(𝑝, 𝜈𝑖 ) distributed, with constant shape parameter 𝑝,
and rate parameter 𝜈𝑖 . In particular 𝐸[𝑚𝑖𝑗 ] = 𝑝/𝜈𝑖 .
A4: Conditional on the individual parameters 𝜆𝑖 , 𝜇𝑖 and 𝜈𝑖 , we assume that 𝜏𝑖 is independent of
the interpurchase times and the transaction values. Furthermore the couples (𝐼𝑃𝑇𝑖𝑗 , 𝑚𝑖𝑗 ) are
independent for 1 ≤ 𝑗 ≤ 𝑡𝑥𝑖 , but we allow for an association between 𝑚𝑖𝑗 and 𝐼𝑃𝑇𝑖,𝑗 . This
association is modeled by a copula 𝑐𝑖𝑛𝑡𝑟𝑎 depending on a parameter 𝜃𝑖𝑛𝑡𝑟𝑎,𝑖 .
A5: Heterogeneity for the transaction rate is modeled as
𝜆𝑖 = 𝜆0,𝑖 exp(𝛽𝜆′ 𝑉𝑖𝜆 ).
Here the 𝑉𝑖𝜆 are the values for the covariates of customer 𝑖, and 𝜆0,𝑖 is a random effect
following a gamma(𝑟, 𝛼) distribution.
A6: Heterogeneity for the dropout rate is modeled as
𝜇
𝜇𝑖 = 𝜇0,𝑖 exp(𝛽𝜇′ 𝑉𝑖 ).
𝜇
Here the 𝑉𝑖 are the values for the covariates of customer 𝑖, and 𝜇0,𝑖 is a random effect
following a gamma(𝑠, 𝛽) distribution.
A7: Heterogeneity for the expected transaction value is modeled as
𝑝 𝑝exp(𝛽𝜈′ 𝑉𝑖𝜈 )
𝐸[𝑚𝑖𝑗 ] = =
.
𝜈𝑖
𝜈0,𝑖
Here the 𝑉𝑖𝜈 are the values for the covariates of customer 𝑖, and 𝜈0,𝑖 is a random effect
following a gamma(𝑞, 𝛾) distribution. The parameter 𝑝 is fixed.
A8: The random effects 𝜆0,𝑖 , 𝑝/𝜈0,𝑖 and 𝜇0,𝑖 follow a trivariate copula 𝑐𝑖𝑛𝑡𝑒𝑟 with parameter
𝜃𝑖𝑛𝑡𝑒𝑟 . Or, conditional on the covariates, the transaction rate, the spending rate and the
dropout rate follow a trivariate copula 𝑐𝑖𝑛𝑡𝑒𝑟 with parameter 𝜃𝑖𝑛𝑡𝑒𝑟 .
A9: Heterogeneity for the individual association parameter 𝜃𝑖𝑛𝑡𝑟𝑎,𝑖 is modeled as
′𝑉𝑖
𝑓(𝜃𝑖𝑛𝑡𝑟𝑎,𝑖 ) = 𝛽𝑖𝑛𝑡𝑟𝑎
+ 𝜀𝑖 , for a suitable link function 𝑓 and with 𝜀𝑖 following a
2
𝑁(0, 𝜎𝑖𝑛𝑡𝑟𝑎 ). The covariates are in the vector 𝑉𝑖 , for each customer 𝑖.
The (hyper)parameters of this model are in the vector
2
𝛉 = (𝑟, 𝛼, 𝑠, 𝛽, 𝑝, 𝑞, 𝛾, 𝛽𝜆 , 𝛽𝜇 , 𝛽𝜈 , 𝜃𝑖𝑛𝑡𝑒𝑟 , 𝛽𝑖𝑛𝑡𝑟𝑎 , 𝜎𝑖𝑛𝑡𝑟𝑎
).
Conditions A4 and A9 describe the intra-association between the spending and the transaction
process. Condition A8 deals with the inter-assocation. Conditions A1-A2 are standard for the
Pareto/NBD model and conditions A5-A7 introduce the covariates in the model in the same way
as in Fader & Hardie (2007). For identification purposes, there is no constant term allowed in
𝜇
𝑉𝑖𝜆 , 𝑉𝑖𝜏 , 𝑉𝑖 .
56
APPENDIX C: Prediction
Below, we outline (1) how we can simulate from the posterior distribution of the individual
model parameters, i.e. the transaction rate, spending rate, dropout rate, and intra-customer
association (2) how the CLV over the next 𝐻 time units can be predicted from the estimated
copula-extended model. We use the notations of Appendix B.
For every customer 𝑖, we (1) first generate values from the posterior distribution of the
individual parameters, then (2) simulate future transaction data, and compute a value of CLV. By
repeating this 𝑀 times, the distribution of 𝐶𝐿𝑉𝑖,𝐻 is simulated. A point estimate is obtained by
averaging over the simulated CLV values.
∗
(1) We generate a set of individual parameters (𝜆∗𝑖 , 𝜇𝑖∗ , 𝜈𝑖∗ , 𝜃𝑖𝑛𝑡𝑟𝑎,𝑖
) from the estimated posterior
density 𝑔𝑖 (𝜆, 𝜇, 𝜈, 𝜃𝑖𝑛𝑡𝑟𝑎 |𝑑𝑎𝑡𝑎). Using the Empirical Bayes principle, we replace unknown
(hyper)parameters by their estimates. We sample from the posterior distribution using the
independent Metropolis-Hastings algorithm with the estimated prior 𝑔𝑖 (𝜆, 𝜇, 𝜈, 𝜃𝑖𝑛𝑡𝑟𝑎 ) as
proposal density.
This results in the following iterative scheme:
1. Let (𝜆̃𝑖 , 𝜇̃𝑖 , 𝜈̃𝑖 , 𝜃̃𝑖𝑛𝑡𝑟𝑎,𝑖 ) be the generated values of the individual parameters in
step 𝑘 of the Markov Chain.
2. Generate (𝜆0 , 𝜇0 , 𝜈0 ) from the estimated prior distribution specified in
2
assumptions (A5)-(A8). Generate 𝜀𝑖 from a 𝑁(0, 𝜎̂𝑖𝑛𝑡𝑟𝑎
), see (A9).
3. Compute
𝜆𝑖 = 𝜆0 exp(𝛽̂𝜆′ 𝑉𝑖𝜆 )
𝜇
𝜇𝑖 = 𝜇0 exp(𝛽̂𝜇′ 𝑉𝑖 )
𝜈𝑖 = 𝜈0 exp(−𝛽̂𝜈′ 𝑉𝑖𝜈 )
𝜃𝑖𝑛𝑡𝑟𝑎,𝑖 = 𝑓 −1 (𝛽̂𝑖𝑛𝑡𝑟𝑎 ′𝑉𝑖 + 𝜀𝑖 )
4. Compute
𝐿𝑖 (𝜆𝑖 , 𝜇𝑖 , 𝜈𝑖 , 𝜃𝑖𝑛𝑡𝑟𝑎,𝑖 )
𝑃 = min (1,
).
𝐿𝑖 (𝜆̃𝑖 , 𝜇̃𝑖 , 𝜈̃𝑖 , 𝜃̃𝑖𝑛𝑡𝑟𝑎,𝑖 )
The likelihood 𝐿𝑖 is given in equation(14). The Markov Chain takes then in
∗
step 𝑘 + 1 the values (𝜆∗𝑖 , 𝜇𝑖∗ , 𝜈𝑖∗ , 𝜃𝑖𝑛𝑡𝑟𝑎,𝑖
) with probability 𝑃 , while with
probability 1 − 𝑃 the previous values are kept.
(2) For every generated set of individual parameters, we simulate a future transaction stream
and the corresponding CLV. First we compute the posterior probability to be alive, 𝑃(𝜏𝑖 >
𝑇𝑖 |𝑑𝑎𝑡𝑎), given the individual parameters, as
1
(A.5)
𝑝𝑖 (𝜆, 𝜇) =
,
𝜇
1+
(exp((𝜆 + 𝜇)𝑅𝑖 ) − 1)
𝜆+𝜇
see Schmittlein et al. (1987).
Then we proceed as follows
57
1. We draw a value from a uniform distribution on [0,1]. If this value is larger
than 𝑝𝑖 (𝜆∗𝑖 , 𝜇𝑖∗ ) , we set 𝐶𝐿𝑉 ∗ = 0 and consider the customer as ``death.''
Otherwise, we continue to simulate the transaction process.
2. We draw the time of death 𝜏 ∗ from an exponential distribution with parameter
𝜇𝑖∗ .
3. Set 𝑡 ∗ = 0 and 𝐶𝐿𝑉 ∗ = 0. While 𝑡 ∗ ≤ 𝐻 and 𝑡 ∗ ≤ 𝜏 ∗
a) Draw a value (𝑈1∗ , 𝑈2∗ ) from the copula distribution with parameter
∗
𝜃̂𝑖𝑛𝑡𝑟𝑎,𝑖
.
b) Compute 𝐼𝑃𝑇 ∗ as the inverse of an exponential cdf with parameter 𝜆∗𝑖 at
𝑈1∗ .
c) Compute 𝑚∗ as the inverse of a the cdf of a 𝑔𝑎𝑚𝑚𝑎(𝑝̂ , 𝜈𝑖∗ ) at 𝑈2∗ .
d) Update 𝑡 ∗ ← 𝑡 ∗ + 𝐼𝑃𝑇 ∗ .
e) Update 𝐶𝐿𝑉 ∗ ← 𝐶𝐿𝑉 ∗ +
𝑚∗ ×𝑚𝑎𝑟𝑔𝑖𝑛
∗
(1+𝑑)𝑡
.
Recall that 𝑑 stands for the discount rate, and the 𝑚𝑎𝑟𝑔𝑖𝑛 is fixed.
As such, we obtain 𝑀 = 4000 draws from the posterior distribution of the individual
parameters and of 𝐶𝐿𝑉𝑖,𝐻 . The first 2000 values of the Markov Chain are discarded as burn-in
period.
58
APPENDIX D: Parameter estimates (and standard deviations) for the best models
Online music albums
No
With
association association
Inter-Association
𝜎𝜆,𝜈
𝜎𝜆,𝜇
𝜎𝜇,𝜈
Intra-Association
Constant
Household size
Living Area
Lifetime
Primary Bank
Age
𝜎𝑖𝑛𝑡𝑟𝑎
Transaction
Process
𝑟
𝛼
Household size
Living Area
Lifetime
Primary Bank
Age
Dropout Process
𝑠
𝛽
Household size
Living Area
Lifetime
Primary Bank
Age
Spending Process
𝑝
𝑞
𝛾
Household size
Living Area
Lifetime
Primary Bank
Age
Financial securities
No
With
association
association
Utilitarian FMCG
No
With
association
association
Hedonic FMCG
No
With
association
association
-
0.28 (0.03)
0.43 (0.09)
-0.17 (0.08)
-
0.15(0.03)
0.67(0.05)
+0.03(0.04)
-
0.50 (0.03)
0.13 (0.34)
-0.38 (0.19)
-
0.76 (0.03)
0.02 (0.30)
-0.64 (0.30)
-
0.63 (0.14)
-
-
0.14 (0.26)
-
-0.07(0.12)
-0.02(0.08)
0.01(0.07)
0.05(0.12)
-0.38(0.12)
1.85(0.07)
-
0.09 (0.26)
0.18 (0.07)
1.94 (0.15)
-
0.48 (0.09)
0.40 (0.43)
1.35 (0.17)
0.55 (0.04)
10.58 (0.74)
-
0.46(0.04)
7.32 (0.59)
-
1.18 (0.04)
2.35 (0.16)
0.03 (0.04)
0.50 (0.04)
-0.30 (0.07)
0.03 (0.05)
1.04(0.05)
1.84(0.16)
0.06(0.04)
0.33(0.04)
-0.20(0.07)
0.03(0.07)
3.60 (0.18)
6.84 (0.33)
-0.32 (0.06)
-
3.44 (0.66)
6.51 (1.31)
-0.24 (0.19)
-
1.33 (0.04)
11.22 (0.37)
-0.27 (0.06)
-
1.16 (0.04)
9.58 (0.37)
-0.27 (0.10)
-
0.61 (0.04)
0.22 (0.04)
2.10 (0.29)
18.04 (3.93)
0.97(0.08)
6.89(1.20)
0.02 (0.00)
0.00 (0.00)
0.01 (0.07)
0.00 (0.01)
11.69 (1.19)
-
4.30 (1.69)
-
0.10 (0.05)
-0.40 (0.06)
-0.27 (0.10)
-0.47 (0.09)
0.16(0.05)
-0.56(0.06)
-0.24(0.10)
-0.56(0.09)
0.35 (0.12)
-
0.73 (0.83)
-
38.91 (3.22)
4988.73
(7.08)
-0.60 (0.22)
-
28.76 (3.62)
4984.32
(9.89)
-1.20 (0.57)
-
6.25 (1.19)
3.74 (0.27)
6.55 (1.10)
3.31 (0.16)
1.20 (0.06)
3.13 (0.13)
8.15 (0.92)
1.73(0.04)
2.60(0.09)
4.05(0.20)
15.44 (4.14)
-
11.54 (2.63)
-
-0.03 (0.03)
0.01 (0.03)
-0.38 (0.05)
0.82 (0.05)
-0.04(0.04)
0.04(0.04)
-0.33(0.04)
0.82(0.04)
0.77 (0.06)
2.85 (0.14)
2654.83
(322.55)
-0.46 (0.07)
-
1.48 (0.02)
2.02 (0.06)
754.64
(62.75)
-0.44 (0.26)
-
3.06 (0.18)
4.05 (0.18)
510.05
(55.22)
-0.03 (0.06)
-
1.71 (0.02)
5.05 (0.22)
1066.48
(61.93)
-0.05 (0.06)
-
59