7
2 Model
In this section we propose a general form of a latent attr ition model that incorporates transaction
attributes. To keep terminology consistent with the empirical example in Section 3, we say that the
customer of the firm places orders for jobs, and the firm fills those orders by completing the jobs.
Thus, orders and jobs always occur in a pair, and are indexed by
k
. We assume that these jobs are
completed the instant the order is placed, so we index calendar time for orders and jobs by
t
. Without
loss of generality, we define a unit of calendar time as one week. The service was introduced to the
marketplace at time
t =
0 and
T
is the week of the end of the observation period. Let
t
1
be the week
of the customer’s first order, let
x
be the number of orders between times
t
1
and
T
, including that
first order at
t
1
, and let
t
k
be the time of order
k
. Therefore,
t
x
is order time of the final, observed
job. For clarity, we are suppressing the customer-specific indices on
t
and
x
in the model exposition.
Our baseline model is a variant of the BG/NBD model for non-contractual customer base analysis
(Fader, Hardie, and Lee 2005a). Immediately before the customer places an initial order at time
t
1
,
he is in an active state. While active, the customer places orders according to a Poisson process with
rate
. After each job (including the first one), a customer may churn, resulting in that order being
his last. With probability
p
k
, the customer churns after the
k
th
job and transitions from the active
state to the inactive state. Upon doing so, we assume that the customer is lost for good and will
not place any more orders, ever. If the customer does not churn, then the time until the next order,
t
k+1
t
k
, is a realization of an exponential random variable with rate
. We never observe directly
when, or if, a customer churns, although if a customer places
x
orders, he must have survived
x
1
possible churn opportunities.
For a customer who places
x
orders between times
t
1
and
T
, the joint density of the
x
1
inter-order times is the product of
x
1 exponential densities. For this customer, there could not
have been any orders between times
t
x
and
T
. This “hiatus” could occur in one of two ways. One
possibility is that the customer may have churned after job
x
, with probability
p
x
. Alternatively,
the customer may have “survived” with probability 1
p
x
, but the time of the next order would be
sometime after
T
. Thus, conditional on surviving
x
jobs, the probability of not observing any more