issn 0732-2399 eissn 1526-548X 04 2301 0050 doi 10.1287/mksc.1030.0046 ©2004 INFORMS
A Dynamic Changepoint Model for New Product
Sales Forecasting
Peter S. Fader
Wharton School, University of Pennsylvania, 3730 Walnut Street, Philadelphia, Pennsylvania 19104-6340, faderp@wharton.upenn.edu
Bruce G. S. Hardie
London Business School, Regent’s Park, London NW1 4SA, United Kingdon, bhardie@london.edu
Chun-Yao Huang
Department of Business Administration, 135 Yuan-Tung Road, Yuan Ze University, Jung-Li 32003, Taiwan, cyhuang@saturn.yzu.edu.tw
A
t the heart of a new product sales-forecasting model for consumer packaged goods is a multiple-event timing process. Even after controlling for the effects of time-varying marketing mix covariates, this timing process is not a stationary one, which means the standard interpurchase time models developed within the marketing literature are not suitable for new products.In this paper, we develop a dynamic changepoint model that captures the underlying evolution of the buying behavior associated with the new product. This extends the basic changepoint framework, as used by a number of statisticians, by allowing the changepoint process itself to evolve over time. Additionally, this model nests a number of the standard multiple-event timing models considered in the marketing literature.
In our empirical analysis, we show that the dynamic changepoint model accurately tracks (and forecasts) the total sales curve as well as its trial and repeat components and other managerial diagnostics (e.g., percent of triers repeating).
Key words: sales forecasting; trial/repeat; new product research; duration models; nonstationarity; changepoint models
History: This paper was received October 18, 2002, and was with the authors 4 months for 2 revisions; processed by Gary L. Lilien.
1. Introduction
For many new products, it is difficult to get an accu-rate read on their long-term potential based on only a few initial weeks of postlaunch (or test-market) sales data. Common problems include the following types of issues: (1) Significant promotional activity can artificially skew the initial sales levels, (2) early buyers may not exhibit typical purchasing rates, and (3) repeat-purchasing patterns may be hard to sort out from the voluminous amounts of first purchase (or trial) data (Ayal 1975, Lipstein 1961, Morgan 1979). It is therefore essential for practitioners to rely on for-mal models of new product sales to tease apart and understand each of these underlying components to create a valid sales forecast.
At the heart of a new product sales-forecasting model is a multiple-event timing process. For many behavioral processes besides new product sales fore-casting, researchers need to capture a series of interpurchase cycles while accommodating customer heterogeneity. In addition, they need to filter out the influences that exogenous factors, such as
promo-tional activities, may exert within and across these multiple purchase cycles.
One of the best-known contributions in this gen-eral area is a highly regarded paper by Gupta (1991), which carefully laid out a general framework to cap-ture the effects of time-varying explanatory variables in a multiple-event timing model. We will review some of the key technical aspects of Gupta’s paper later, but for the moment we want to call attention to his main empirical result. In applying a broad set of models to scanner panel data for regular ground coffee, Gupta observed that the most appropriate specification was a model that featured an Erlang-2 interpurchase process with gamma-distributed pur-chase rates (to account for customer heterogeneity), while also allowing for time-varying covariates such as price and promotion.
Let us consider the case of a new juice product, with the masked name of “Kiwi Bubbles,” which underwent a year-long test in two of IRI’s Behav-iorScan test markets prior to its national launch. Data on the purchasing of this new product were col-lected from 2,799 panelists. We fit Gupta’s
Figure 1 Sales Tracking Performance of Erlang-2/Gamma, Covariates Model 0 4 8 12 16 20 24 28 32 36 40 44 48 52 Week 0 10 20 30 40 Cum. Sales per 100 HH ... ... ... ... ... ... ... ... ... ... ... ... Actual ... ... ... ... ... ... ... ... ... . ... ... ... ... ... ... ... ... ... ... ... .. ... ... ... Predicted
2/gamma, covariates model to the first 6 months of purchasing data and then generate a forecast of purchasing behavior for the remaining 6 months of the test market. As shown in Figure 1, this model does a fine job of tracking sales within the 26-week calibration period, but it quickly (and significantly) veers away from the actual sales pattern in a hold-out forecasting period. This substantial overprediction (the Week-52 forecast is 23% higher than the actual cumulative sales level) suggests that the interpurchase times tend to be lengthening (slowing down) as buy-ers gain more experience with the new product. This bias is quite typical of what we have observed for numerous new products and will motivate much of the model development that follows in the next sec-tion.
Beyond this poor forecast, the Erlang-2/gamma, covariates model is even more troubling from a diag-nostic perspective. One of the commonly used man-agerial benchmarks for a new product is “percent triers repeating” (Clarke 1984, Rangan and Bell 1994), which provides a useful indication of how well the product has been accepted by its base of initial triers. In Figure 2, we see that the actual and predicted curves diverge quite drastically even in the middle of the calibration period. This means that the model
Figure 2 Tracking of Percent Triers Repeating by Erlang-2/Gamma, Covariates Model 0 4 8 12 16 20 24 28 32 36 40 44 48 52 Week 0 10 20 30 40 50 60 % Triers Repeating ... ... ... ... ... ... ... ... ... ... ... Actual ... ... ... ... ... ... ... ... ... ... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Predicted
is clearly not capturing the underlying purchase-to-purchase dynamics, even though the aggregate sales-tracking plot appears to behave acceptably well over the calibration period.
The main problem here is that the multiple-event timing process is not a stationary one. As customers move from one stage of purchasing to the next, there appears to be a shift occurring—in particular, early buyers are slowing down or dropping out completely. To be fair to Gupta (1991), he made no claims that his stationary models were suitable for new products and he conducted no empirical testing in this regard. How-ever, this points out the need for such a model—one that builds upon the basic template laid out in Gupta’s paper, while allowing for the kinds of dynamics that typically occur in the new product setting.
Over the years, many researchers have dealt with some of these dynamics by developing separate mod-els for trial and repeat behavior. The basic belief is that these two stages are behaviorally distinct, and that there is little or no “carryover” from the trial process that might improve (or otherwise affect) the performance of the repeat model. Therefore, while the trial-repeat decompositional approach has a long his-tory in marketing (e.g., Blattberg and Golanty 1978, Eskin 1973, Fourt and Woodlock 1960, Parfitt and Collins 1968, Pringle et al. 1982), it may suffer from three potential shortcomings:
• Separation of stages. Does the evolution of buy-ing patterns begin and end with the transition from the trial purchase to first repeat? Some buyers may continue their initial buying pattern through several “retrial” cycles before settling into a steady-state pat-tern (e.g., Aaker 1971). Other buyers may go through several evolutionary states (i.e., buying rates). In gen-eral, the repeat-buying component may be highly nonstationary, so a simple trial-repeat decomposition may be inadequate.
• “Dependence” between stages. As noted earlier, most traditional trial-repeat models do not allow any individual-level information about the first purchase process to come into play in the repeat model. As we transition from trial to repeat, the only piece of infor-mation we effectively retain about each person is that they made a trial purchase; we do not know when they made that purchase. However, it is known that early triers tend to be heavy category buyers (Morgan 1979, Taylor 1977); given “acceptance” of the new prod-uct, we could therefore expect them to have a higher repeat-buying rate than the later triers (who tend to be light category buyers). This failure to formulate a multiple-event timing model by first conditioning on the individual-level latent traits is known to result in biased inferences (Gupta and Morrison 1991).
• Parsimony. An integrated model requires fewer parameters than separate staged models, and may
therefore offer clearer, more interpretable diagnostic properties. Our most general model requires only four parameters (not including covariate effects), yet will be able to track and forecast the overall sales data extremely well. Furthermore, we will show that our model is also very effective at capturing each of the critical components (i.e., trial, first repeat, additional repeat), even though we do not develop an explicit, separate model for each of them.
A general class of models that can allow for some of these purchase dynamics is known as changepoint models. While they have not been very prevalent in the field of marketing, a number of statisticians have developed models that begin to allow for the type of ongoing evolution that may characterize new prod-uct sales patterns as shown and discussed above. These models are centered around an enduring, but stochastic, renewal process that lets the observation units (i.e., customers) change their parameters (buy-ing rates) at random points over time.
We adopt this basic framework, and also extend it in a novel way to accommodate an important distin-guishing aspect of the new product purchase process. We allow for the possibility that the changepoint pro-cess itself may evolve over time—hence, the name dynamic changepoint model. Specifically, our intuition and empirical observations suggest that changepoint occurrences (i.e., changes in buying rates) will be quite common in the early phase of a particular cus-tomer’s relationship with the new product, but these changes will occur much less frequently after the buyer has gained considerable experience with the product. In fact, the changepoint process may at some stage terminate completely, which would leave each buyer with a permanent, steady-state (but still hetero-geneous) purchasing rate. This steady-state scenario would be entirely consistent with the work of Gupta (1991), as well as that of many other researchers (Ehrenberg 1988; Jain and Vilcassim 1991, 1994; Morrison and Schmittlein 1988) who have shown the robustness and widespread nature of stationary (but heterogeneous) models for the purchasing of estab-lished products.
In the next section we introduce the general struc-ture of a changepoint process as it applies to the case of new product purchasing. We then discuss the notion of a dynamic changepoint process and develop the general form of our model, which includes sev-eral relevant nested versions such as the models ana-lyzed by Gupta (1991), as well as traditional (static) changepoint models. This is followed by an empir-ical analysis in which we examine the fit and fore-casting performance of the proposed model for two datasets with different repeat-purchasing characteris-tics. We conclude with a discussion of several issues that arise from this work and identify several areas worthy of follow-up research.
2. Model Development
Our goal is to develop a model of new product buy-ing behavior that incorporates the effects of marketbuy-ing mix variables and nonstationarity in buying rates at the individual consumer level. We obtain estimates of the model parameters using data collected in a test market environment, then generate an overall sales forecast as well as related managerial diagnos-tics for the new product. These data come from a con-sumer panel in which we track the purchasing of each panelist for the duration of the test market.
We expect to observe some nonstationarity in the underlying buying rates (after controlling for the tem-porary effects of marketing mix variables). The buy-ing rate reflects the consumer’s preference for the new product, and we would expect there to be some insta-bility in preferences as the consumer gains experience with it. For example, initial enthusiasm may dimin-ish and the consumer will return to substitute prod-ucts purchased before the launch of the new product; this would be reflected by a change to a lower value of the buying-rate parameter. This change might be extreme; i.e., the consumer drops the product from future consideration after the first purchase. Alterna-tively, the consumer may go through several “retrial” stages before deciding whether and how the product fits in with his regular purchasing habits.
As consumers gain more experience with the prod-uct, we would expect their preferences (and therefore their underlying buying rates) to stabilize to some extent. Consequently, a desired feature of the model is that it can capture the evolution towards a station-ary repeat-buying process as the product moves from being “new” to “established.”
It should be noted that preference changes need not be the only driver of changes in the underlying buy-ing rates; external “shocks” to the market, perhaps due to word-of-mouth influences or other unmea-sured covariate effects, can also lead to the occurrence of changepoints in the purchase sequence. However, it is not our goal to identify the specific drivers of each of the changepoints, but simply to capture these changes to make more accurate sales forecasts.
Before we formally develop our model of new product buying behavior, let us further explore the notion of nonstationarity. Suppose we have a sequence of observations y1 y2 yN ordered in time, with each observation being independent with density f yn n . If the sequence of
observa-tions is viewed as being stationary, we assume that n= ∀n. If n= ∀n, we say that the sequence is nonstationary.
According to the general changepoint model (e.g., Barry and Hartigan 1993, Henderson and Matthews 1993, Pievatolo and Rotondi 2000, Stephens 1994), this
nonstationarity is modeled by postulating that the sequence can be partitioned into q +1 contiguous sets,
y1 yw1 yw1+1 yw2 ywq+1 yN
Within each block, we assume n= wi for wi−1<
n ≤ wi, with changes in the parameter values
occur-ring at w1 w2 wq, which are called changepoints. Within the marketing literature, the logic of change-point models has been combined with traditional finite mixture methods to allow changes in segment membership over time (e.g., Böckenholt and Dillon 1997, Poulsen 1990). This set of models typically falls under the heading of hidden Markov models (MacDonald and Zucchini 1997), which can viewed as a specific subclass of changepoint process models. A hidden Markov model (HMM) is a discrete-time finite-state Markov model in which st, the state of the system at time t = 1 2 is not directly observ-able. Rather, we observe yt, which is linked to st by the stationary density f yt st . In many cases, includ-ing the model developed in this paper, we have a continuous-time process and/or there are an infinite number of possible latent states, in which case we look beyond the standard HMM framework to more general changepoint process models.
The standard changepoint problem is one of infer-ence: Given a single data sequence, the objective is to identify the number q and location w1 wq of the changepoints, along with the values of the
w1 wq. This differs from our modeling
prob-lem in two ways. First, given our interest in forecast-ing, we are not interested in identifying the precise times/instances of the changepoints in the observed data sequence. Rather, we need to determine whether there is any systematic structure in the pattern of the changepoints and then use this knowledge to create a forecast of the data sequence beyond yN.
Second, we are not modeling a single aggregate data sequence; rather, we have a separate sequence of observations for each panelist, which means we must explicitly capture the cross-sectional hetero-geneity in the parameters, as well as the temporal variation.
We will deal with this second point by assum-ing that the wi are distributed across the population
according to some heterogeneity distribution. The first point can be accommodated within the framework of a product partition model (Barry and Hartigan 1992, Howard 1965). In particular, we assume that the partition of the sequence y1 y2 yN is randomly
selected according to a product partition distribu-tion. Instead of assuming a constant probability of a changepoint occurring at each yn, which leads
to a geometric distribution for the length of the blocks within a given partition (e.g., Barry and
Hartigan 1993, Howard 1965, Yao 1984), we will allow these probabilities to follow a flexible structure that, for example, allows the probability of change in the underlying stochastic process to decrease as the consumer gains more experience with the new product. Forecasts can then be generated conditional on a given partition of future observations, and these forecasts can then be weighted and combined according to the probability of observing such a partition.
During the period 0 tc, where 0 corresponds to
the launch date of the new product and tc is the
cen-soring point that corresponds to the end of the model calibration period, we observe a consumer making K = J + 1 purchases of the new product at times
t0 t1 tJ. (By convention, j = 0 corresponds to the
trial purchase. We refer to j as the depth-of-repeat level.) We need to specify a model for the set of K =
J + 1 interpurchase times t0 t1− t0 tJ − tJ −1 and
the censored observation tc− tJ.
Let the underlying interpurchase times follow an exponential baseline distribution with marketing mix covariate effects incorporated using the standard pro-portional hazards framework:
htj j = jextj
≡ jAtj
where xt is the vector of marketing covariates at time t and the effects of these covariates.
By definition, the survivor function is given by
Stj tj−1 j = exp − tj tj−1 hu j du = exp −j tj 0 Au du − tj−1 0 Au du Assuming the time-varying covariates remain con-stant within each unit of time (e.g., a week in our empirical application), t 0 Au du = t≥1 Intt i=1 Ai + t − Intt AIntt + 1 ≡ Bt
where t≥1= 1 if t ≥ 1, 0 otherwise. The survivor func-tion of the with-covariates interpurchase time distri-bution is therefore
Stj tj−1 j = exp −jBtj tj−1 (1)
where Btj tj−1 = Btj − Btj−1 . Because Btj tj−1 captures the effects of the covariates between tj−1 and tj, the survivor function is not just a function of the covariate values at tj; it is a function of the covari-ate values for every time period (e.g., week) since the
last transaction. It follows that the pdf of the with-covariates interpurchase time distribution is
f tj tj−1 j = htj j Stj tj−1 j
= jA!j exp −jBtj tj−1 (2)
where !j is the time period (e.g., week) in which the
jth repeat purchase occurred, defined as
!j=
tj if tj is integer
Inttj + 1 otherwise
Following the standard discrete changepoint for-mulation, in which we assume that the j can change
immediately after the occurrence of the event of inter-est (i.e., the jth repeat purchase),1there are 2Kpossible
partitions of this sequence of observations. Let there be q ≤ K changepoints, and let w = wi , i = 1 q, be the set of changepoints, where wi corresponds to the depth-of-repeat level (i.e., repeat-purchase number) associated with the purchase occasion that immediately precedes a change in the buying-rate parameter. (If a change occurs immediately after the trial purchase, we have w1= 0.) Let "j be the
proba-bility that there is a change in a consumer’s under-lying buying rate following his jth repeat purchase. The probability of the partition w is therefore
Pw = j∈w "j j∈I−w 1 − "j (3) where I = 0 1 J .
The standard assumption of product partition mod-els is that "j = " ∀j. However, in the case of a new product, we expect that the probability of a consumer revising his preferences following a purchase (with the corresponding change in the underlying buying rate) will decrease as he gains more experience with the new product (i.e., makes additional purchases). To accommodate this, we assume that
"j= 1 − $1 − e−%j+1 j = 0 1 2 (4)
where & $ ∈ 0 1 and % ≥ 0. This parametric expres-sion is motivated by equivalent expresexpres-sions used in other papers (Eskin 1973, Fader and Hardie 2001, Kalwani and Silk 1980) to capture the evolution of repeat-buying patterns over time.
The relationship between "j and j is illustrated in
Figure 3 for three sets of values of $ and %. We note that as j increases, "j tends to 1 − $. Therefore, if $ = 1, the probability of a change tends to zero as a consumer moves to higher depth-of-repeat levels; in 1One behavioral “story” consistent with this is to assume that
con-sumption immediately follows purchase and that preference revi-sions would immediately follow consumption.
other words, the model evolves to a stationary process that would be consistent with the stabilization
✲
✛ wk1 ✲✛ wk 2×✲✛ wk 3✲✛×wk 4✲✛ wk 5✲✛ wk 6× ✲
0 t0 t1 t2 6
of consumer preferences. On the other hand, if $ < 1,
"j> 0 ∀j, which means that individual consumer
pref-erences will not stabilize; in other words, there is long-term nonstationarity in the marketplace. This general specification for "j results in what we call the
dynamic changepoint model. If % → , then "j is
inde-pendent of j and equals 1 − $ ∀j, and we call this the static changepoint model.
We incorporate heterogeneity in buying rates by allowing them to be distributed across the population according to a gamma distribution with shape param-eter r and scale paramparam-eter ):
g =)r+r r−1e−)
When a change in the underlying buying rate occurs, the consumer draws a new value of his buy-ing rate, independent of his previous one, from the same gamma distribution of purchase rates described above. This principle of independent sampling from a given mixing distribution was first raised in Howard’s “dynamic inference” model (Howard 1965) and was used by Sabavala and Morrison (1981) in their model of media exposure. More recently, it has been used by Barry and Hartigan (1992) and Yao (1984), among others, in Bayesian analyses of the changepoint problem.
To illustrate and convey the intuition of the pro-posed model, let us consider the following scenario of a consumer who makes three purchases of the new product in the first 6 weeks of it being on the market: trial at t0, first repeat at t1, and second repeat at t2. Given t0, t1, t2, we do not know whether the consumer ever revised his preferences and, if he did, how many times and at which points in time. For this consumer,
Figure 3 Probability of Change in Buying Rate by Depth of Repeat
0 1 2 3 4 5 6 7 8 9 10 11 12 13 Depth-of-Repeat Level (j) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 P(Change in Buying Rate) ...... ...... ...... ...... ...... ...... ...... ...... ...... ψ = 1.0, θ = 0.4 ... ... ... ... ...... ...... ... ...... ... ψ = 0.9, θ = 0.7 ψ = 0.8, θ → ∞
Table 1 Possible Partitions of Three Purchases Changepoint Occurs After
No. of Partition Trial 1st Repeat 2nd Repeat Changepoints
(i) 0 (ii) 1 (iii) 1 (iv) 1 (v) 2 (vi) 2 (vii) 2 (viii) 3
the number of changepoints could have ranged from zero to three. The set of eight possible partitions of this purchase sequence is given in Table 1.
Let us first assume that the consumer never revised his preferences in 0 6 (i.e., the first partition). Given the constant underlying buying rate (i.e., before we consider cross-sectional heterogeneity), the con-ditional likelihood function for this consumer is the product of the density and survivor functions; that is,
L = f t0 0 f t1 t0 f t2 t1 S6 t2
= 3A2 A4 A6 exp −B6 0
The corresponding unconditional likelihood function is obtained by accounting for heterogeneity in the underlying buying rate :
Lr ) = 0 L )rr−1e−) +r d = +r + 3 A2 A4 A6 +r · 1 ) + B6 0 3 ) ) + B6 0 r (5) This likelihood function is identical to the exp/gamma, covariates model from Gupta (1991).
Alternatively, suppose that the consumer revised his preferences following his second (first repeat) pur-chase (i.e., partition (iii) in Table 1, above). This parti-tion is defined by w = 1 . Let the purchasing rate a reflect the consumer’s initial preference for the new product, and b reflect the consumer’s revised prefer-ence following his first repeat purchase. Conditional on a and b, the likelihood function for this
con-sumer is therefore
La b
= f t0 0 a f t1 t0 a f t2 t1 b S6 t2 b
= 2
aA2 A4 exp −aBt1 0 bA6 exp −bB6 t1
We note that the change in buying rate results in a new value being drawn from the same underlying
gamma distribution. The unconditional likelihood function is obtained by accounting for heterogeneity in aand b: Lr ) = 0 0 La b )rr−1 a e−)a +r )rr−1 b e−)b +r dadb =+r + 2 A2 A4 +r 1 ) + Bt1 0 2 ) ) + Bt1 0 r ·+r + 1 A6 +r 1 ) + B6 t1 ) ) + B6 t1 r (6) Given the partition probability distribution defined by (3) and (4), the probability of observing the first partition with corresponding likelihood function (5) is
1−"0 1−"1 1−"2 , while the probability of
observ-ing the third partition with likelihood function (6) is
1 − "0 "11 − "2 . The overall likelihood function for
this consumer is computed by taking the weighted average of the likelihood function associated with each of the possible partitions, where the weights are the partition probabilities.
More generally, let Th= t0 t1 tJ be the set of
times at which consumer h, h = 1 H , makes his
K purchases of the new product in the period 0 tc.
For convenience, we suppress the household subscript for t0 tJ.
The exact nature of the likelihood function for con-sumer h depends on whether K = 0 or K > 0.
(i) If no purchase of the new product is observed (i.e., K = 0), this is because the consumer has not yet had the opportunity or need to make a trial pur-chase. Therefore, the likelihood function for a con-sumer making no purchases is
LTh =
)
) + Btc 0
r
(7)
which is simply the with-covariates survivor func-tion (i.e., the probability that no purchase occurred in
0 tc) mixed with the gamma distribution.
(ii) When K > 0, the possibility of there being changepoints in the sequence of interpurchase times emerges. As we cannot tell exactly when (or if) changes in the buying rates take place, we first for-mulate the likelihood function conditional on a given partition w with n ≤ K changepoints.
(a) For the case of no changepoints n = 0 , we have LTh w = J j=0 A!j +r + J + 1 +r · 1 ) + Btc 0 J +1 ) ) + Btc 0 r (8) (This is Gupta’s 1991 exp/gamma, covariates model.)
(b) For n > 0 changepoints, we have LTh w = J j=0 A!j n i=1 +r + w i− wi−1 +r · 1 ) + Btwi twi−1 wi−wi−1 ) ) + Btwi twi−1 r · +r + J − w n +r 1 ) + Btc twn J −wn · ) ) + Btc twn r (9)
Recalling the partition probability distribution defined by (3) and (4), the likelihood function for consumer h with K > 0 is simply the weighted average of the partition-specific likelihoods:
LTh =
s
LTh ws Pws (10)
where the summation is over the possible partitions indexed by s = 1 2 2K. (For K = 0, the likelihood
function is given by (7).) The overall sample log-likelihood function is given by
LL =H
h=1
ln LTh (11)
Equations (3), (4), and (7)–(11) define the model as fitted to a given dataset. Maximum likelihood estimates of the model parameters r ) $ % are obtained by maximizing the log-likelihood function given in (11). Standard numerical optimization methods are employed, using the MATLAB program-ming language, to obtain the parameter estimates. When we observe a large number of repeat purchases for the new product, the evaluation of the model log-likelihood function can become computationally complex given the need to evaluate each of the 2K
possible partitions of the purchase sequence. A dis-cussion of this issue and the examination of a recom-mended approach for minimizing computation time can be found in the appendix.
2.1. Properties of the Model
In its most general form, the model requires the esti-mation of 4 + p parameters, where p is the number of marketing covariates. It is a very flexible model that can capture many patterns of buying behavior. Examples of such buying phenomena include:
• “Traditional” stationary buying behavior. If "j= 0 ∀j (i.e., % → and $ = 1), we have the exp/gamma, covariates model considered by Gupta (1991). With the additional constraint that = 0, we have
the two-parameter exponential-gamma model of stationary repeat-buying behavior that is the timing counterpart of the NBD counting model (Gupta and Morrison 1991). The estimates of r and ) would equal to those obtained by fitting the NBD model to the data.
• The transition from a “new” to “established” product. If $ = 1 and % is finite, then "j→ 0 as j increases; that
is, the probability of seeing a change in the underly-ing buyunderly-ing rate tends to zero as a consumer moves to higher depth-of-repeat levels. This means that the ini-tial nonstationary buying process evolves to a station-ary process as the product becomes more established (i.e., when most buyers have made a large number of repeat purchases). Therefore, the model is consis-tent with the notion of nonstationary buying behavior during the early stages of a new product’s life and stationary buying behavior—as characterized by the NBD model—once it has become established in the marketplace.
• Long-term nonstationarity in repeat buying. When $ < 1, the probability of there being a change in the underlying buying rate will always be nonzero, which means that the repeat buying process is always non-stationary. If % → , "jis a constant 1−$. For finite %,
"j → 1 − $ as j increases; that is, the probability of
a change in the underlying buying rate tends to the constant 1 − $ as a consumer moves to higher depth-of-repeat levels. Such a model can easily capture the “leakage” of repeat buyers phenomena observed by East and Hammond (1996). If the underlying gamma distribution has an effective mode at zero (r ≤ 1), an ongoing low level of changes will see some con-sumers drawing a very small value of (which yields an almost zero probability of making a repeat pur-chase in the foreseeable future) on a given change, thereby “dropping out” of the market for the product of interest. Other researchers (e.g., Schmittlein et al. 1987) have proposed NBD-based models that include a “death” process. However our model is far more flexible, allowing for other forms of nonstationarity (e.g., “speeding up” and “slowing down” of latent purchase rates) beyond a simple “death” process. 2.2. Generating Sales Forecasts
To evaluate the tracking performance of the pro-posed model, or to use the model for forecasting sales beyond the model calibration period, it is necessary to generate sales numbers (i.e., counts) from this timing model. We are interested in a number of sales-related measures for the new product: (i) the cumulative trial sales by time t, T t ; (ii) the cumulative repeat sales by time t, Rt ; (iii) the total sales by time t, St , which is by definition equal to T t +Rt ; and (iv) the depth-of-repeat components of repeat sales. Defining
at least j repeat purchases of the new product by time t, we have Rt =
j=1Rjt . Our goal is to
gener-ate these numbers over the time interval 0 tf, where
tf denotes the end of the forecast period.
While we have a simple closed-form expression for expected cumulative trial sales,
E T t = H · 1 − ) ) + Bt 0 r
it is not possible to write out a closed-form expression for Rt , and consequently St . We therefore propose a simulation-based approach to computing the sales numbers.
Let the nonzero elements of the vector Nh denote
the times at which consumer h made his trial, first repeat, etc. purchases (if any at all). For a given indi-vidual, we simulate the elements of Nh in the follow-ing manner. We start by drawfollow-ing a value of from the gamma distribution. Using this value of and the actual values of the covariates, we simulate an interpurchase time off the exponential-with-covariates interpurchase time distribution. This gives us the con-sumer’s simulated value of t0, the time of his trial pur-chase. If t0> tf, the consumer is deemed to have made zero purchases of the new product by time tf and the procedure moves on to the next consumer. If t0≤ tf, we record the time of this purchase Nh0 = t0 and then draw a uniform random number to determine whether or not this purchase occasion corresponds to a changepoint: With probability 1 − "0 the consumer retains his value of and with probability "0 he obtains a new value from the underlying gamma dis-tribution. Another exponential-with-covariates inter-purchase time is then simulated and added to t0 to give us the consumer’s simulated value of t1, the time of his first repeat purchase. If t1> tf, the consumer is
deemed to have made only a trial purchase by time tf
and the procedure moves on to the next consumer. If t1≤ tf, we record the time of this first repeat purchase
Nh1 = t1 , and the whole process continues for this
consumer until tj > tf, at which time the procedure moves to the next consumer.
Once we have simulated Nh for all individuals, we can compute total sales and its components in the following manner: T t =H h=1 I 0 < Nh0 ≤ t Rjt = H h=1 I 0 < Nhj ≤ t Rt = j=1 Rjt St = T t + Rt
where I · is an indicator function that equals one if the logical condition is true, and zero otherwise.
We repeat this simulation, say 100 times, and take the average of the run-specific St , T t , etc. This simulation-based approach is used in the following empirical analysis.
2.3. Extension to the Basic Model
As with numerous other stochastic models of buyer behavior, our model is based on the assumption that individual consumer interpurchase times can be char-acterized by the exponential distribution. Two poten-tially troubling characteristics of this distribution are that it is memoryless (i.e., there is no influence of time since the last purchase) and that the mode of the dis-tribution is at zero (which means that the next pur-chase is most likely to occur immediately after the last one). Consequently, a number of researchers have proposed that the Erlang-2 distribution be used to model interpurchase times, as it allows for a more reg-ular purchase process (Chatfield and Goodhardt 1973, Herniter 1971, Jeuland et al. 1980). We therefore con-sider the case of Erlang-2-distributed interpurchase times as an extension to the basic model.
Using Gupta’s (1991) approach to incorporating the effects of time-varying covariates into the Erlang-2 distribution, the survivor function and pdf of the with-covariates interpurchase time distribution are given by
Stj tj−1 j = exp −jBtj tj−1 1 + jBtj tj−1
f tj tj−1 j = 2jA!j Btj tj−1 exp −jBtj tj−1
This results in new expressions for the partition-specific likelihood functions:
• For a consumer making no purchases in the cal-ibration period 0 tc: LTh = ) ) + Btc 0 r 1 + rBtc 0 ) + Btc 0 (12)
• For K > 0 purchases with no changepoints n = 0 , we have LTh w = J j=0 A!j Btj tj−1 · +r + 2J + 1 +r ) ) + Btc 0 r · 1 ) + Btc 0 2J +1 · 1 +r + 2J + 1 Bt) + Bt c tJ c 0 (13)
(This is Gupta’s 1991 Erlang-2/gamma, covariates model.)
• For n > 0 changepoints, we have LThw = J j=0 A!j Btjtj−1 ·n i=1 +r +2wi−wi−1 +r ) )+Btwitwi−1 r · 1 )+Btwitwi−1 2wi−wi−1 · +r +2J −wn +r ) )+Btctwn r · 1 )+Btctwn 2J −wn · 1+r +2J −wn BtctJ )+Btctwn (14)
Equations (12)–(14) replace Equations (7)–(9), respectively. Consequently, Equations (3), (4), and (10)–(14) define the model as fitted to a given dataset when we assume the underlying interpurchase times follow the Erlang-2 distribution.
We explore the value of this extension in the fol-lowing empirical analysis.
3. Empirical Analysis
We now examine the performance of the dynamic changepoint model (and its nested variants), using test market data for two new products that were tested using IRI’s BehaviorScan service prior to national launch. The first new product is “Kiwi Bubbles,” a masked name for a shelf-stable juice drink, aimed primarily at children, which is sold as a multipack of several single-serve containers. The sec-ond new product is “Four Seasons Biscuits,” a masked name for a snack product.
Focusing first on the Kiwi Bubbles dataset, we begin by discussing the fit of the dynamic change-point model and its variants in the calibration period. We then examine the out-of-sample forecasting per-formance of these models. This is followed by an exploration of the impact of the length of the model calibration period on model forecasting. To fur-ther illustrate and emphasize the performance of the dynamic changepoint model, we report the key results obtained from performing a similar analysis using the Four Seasons Biscuits dataset.
3.1. Model Fit
Prior to its national launch, the Kiwi Bubbles product underwent a year-long test conducted in two of IRI’s BehaviorScan test markets. We use panel data drawn from 2,799 panelists in the two markets. (As is typi-cally the case with new product launch data, we have
detailed purchasing information and marketing mix covariates only for the test brand.)
Using data for the 267 panelists who tried the new product by the end of Week 26, we wish to forecast the purchasing behavior of the whole panel (i.e., 2,799 panelists) to the end of the year (Week 52). That is, we fit the model specification to the first 6 months of purchasing data and generate sales forecasts for the whole year. We have information on the market-ing activity over the 52 weeks the new product was in the test market; this comprises a standard market-level scanner data measure of promotional activity (i.e., %ACV with any merchandising) and a measure of coupon activity. (The coupon variable is an aggre-gate measure of coupon activity, generated by IRI for modeling purposes, that reflects the face value and circulation of the coupon along with standard decays in the redemption rate.)
As noted above, the dynamic changepoint model developed in this paper nests a number of simpler models. First, we can vary the nature of the change-point process. Setting % → results in the static changepoint model in which the probability of there being a change in the underlying buying rate is not allowed to evolve as consumers gain experience with the product. Additionally, setting $ to 1.0 “turns off” the changepoint process and assumes that the under-lying buying rates are stationary. This corresponds to the with-covariates models developed in Gupta (1991). Second, we can remove the covariate effects, which is equivalent to setting to zero. Finally, we can replace the assumption of the underlying expo-nential distribution with the Erlang-2 distribution. We therefore have 12 different models’ specifications to consider. We fit these 12 models to the first 6 months of purchasing data for the Kiwi Bubbles new product. The results are summarized in Table 2.
Looking at the model log-likelihoods, we immedi-ately observe that the fit of the exponential model specifications strictly dominates their Erlang-2 coun-terparts. While this is contrary to the findings of Gupta (1991), this result is consistent with recent work on the modeling of trial purchasing for new grocery products, which finds strong support for the exponen-tial distribution (Fader et al. 2003, Hardie et al. 1998). Within the set of six exponential models, we observe that the inclusion of covariates results in a significant improvement in calibration-period model fit; however, the improvement in fit is less when we allow for more nonstationarity in the underly-ing buyunderly-ing rates: 5LL = 794 for the no-changepoint specification (Model 1 versus Model 4), 5LL = 479 for the static changepoint specification (Model 2 versus Model 5), and 5LL = 454 for the dynamic changepoint specification (Model 3 versus Model 6). For both the with- and without-covariates cases, the
Table 2 Model Estimation Results for the Kiwi Bubbles Dataset
Model Baseline Covariates Changepoint LL # Parameters r Coupon Promotion
1 Exponential N None −381240 2 0079 71375 (1) 2 Exponential N Static −377919 3 0049 26797 0750 3 Exponential N Dynamic −377198 4 0047 24057 0851 1144 4 Exponential Y None −373300 4 0076 138239 (1) 5182 0014 5 Exponential Y Static −373128 5 0066 97661 0912 5059 0012 6 Exponential Y Dynamic −372656 6 0061 80228 0966 1367 5204 0012 7 Erlang-2 N None −397344 2 0095 33094 (1) 8 Erlang-2 N Static −379752 3 0044 7243 0637 9 Erlang-2 N Dynamic −378396 4 0044 6937 0812 0836 10 Erlang-2 Y None −382423 4 0091 74370 (1) 2401∗ 0019 11 Erlang-2 Y Static −376054 5 0051 17086 0707 3180∗∗ 0010 12 Erlang-2 Y Dynamic −374786 6 0050 15872 0867 0910 3428∗∗ 0010
Note. All parameter estimates are significantly different from 0 at p < 0001, except those marked with∗p < 005 and∗∗p < 001 . dynamic changepoint specification provides a
signifi-cant improvement in fit over its no-changepoint coun-terpart. Compared to the Gupta (1991) exp/gamma, covariates specification (Model 4), our dynamic changepoint specification (Model 6) represents a sig-nificant improvement in calibration-period model fit (p = 00016 using the likelihood ratio test).
We also note that the dynamic changepoint model provides a significant improvement in fit over the static changepoint model, both in the case of the with-covariates specification (p = 0002 using the like-lihood ratio test) and the without-covariates specifica-tion (p < 0001 using the likelihood ratio test).
Substituting the estimates of $ and % in (3) yields the implied probabilities of a change in the underly-ing buyunderly-ing rate as a function of depth-of-repeat level (i.e., experience with the product). These are plotted in Figure 4. For the dynamic changepoint specifica-tion (Model 6), we see that there is a 28% chance that a consumer will change his purchase rate follow-ing his trial purchase; this is entirely consistent with the widely held view that trial purchases are different from repeat purchases and that trial rates are a poor predictor of repeat buying. There is a 10% chance that
Figure 4 Estimated Probability of Change in Buying Rate by Depth of Repeat 0 1 2 3 4 5 6 7 8 9 10 Depth-of-Repeat Level (j) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 P(Change in Buying Rate) ... static/without cov ... ... ... ... ... ... ... ... ...... ...... dynamic/without cov ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... static/with cov ...... ...... ... ...... ... ...... ...... dynamic/with cov
he will revise his buying rate following his first repeat purchase. This implies that there is a 65% chance that a consumer will not have revised his buying rate after two purchases of the new product. By the time three repeat purchases have been made, the change proba-bility has effectively reached its asymptotic value of 1 − $ = 35%.
3.2. Forecasting Performance
While the dynamic changepoint specification (Model 6) offers a statistically significant improve-ment in fit over both the static changepoint speci-fication (Model 5) and the stationary specispeci-fication (Model 4), the improvement in model fit is relatively small given the amount of data. However, it is dangerous to assess a forecasting model in terms of calibration-period fit, as the absence of a positive link between a model’s calibration (in-sample) fit and out-of-sample forecasting performance is well known (Armstrong 2001).
We assess the forecasting performance of each model specification using several standard measures. First, we index the year-end (cumulative) sales fore-cast against actual Week 52 cumulative total sales (Wk52 Index); indices greater than 100 represent an overforecast. We also consider MAPE (mean absolute percentage error) computed over the 26-week fore-cast horizon. We compute MAPE for total sales (MAPE-Tot) as well as for the trial (MAPE-TR), first repeat (MAPE-FR), and additional repeat (MAPE-AR) components of total sales. (We focus on this decom-position because managers are interested in the breakdown of total sales into these components—see, for example, Clarke 1984.) These results are summa-rized in Table 3.
The dominance of the dynamic changepoint spec-ification (Model 6) is now very clear, driven largely by its excellent performance in forecasting both of the repeat-sales components. Its forecasting performance is illustrated in Figure 5; observe that the model-based
Table 3 Model Forecasting Performance Results for the Kiwi Bubbles Dataset
Model Wk52 Index MAPE-Tot MAPE-TR MAPE-FR MAPE-AR
1 1307 170 6.2 344 232 2 960 27 1.9 236 169 3 932 56 2.0 31 139 4 1127 54 2.1 222 71 5 1049 25 2.0 217 50 6 1047 26 2.0 41 43 7 1485 285 9.5 448 417 8 868 89 1.9 192 326 9 865 100 2.2 58 252 10 1233 136 2.5 323 186 11 916 47 1.9 220 213 12 918 65 1.9 51 138
predictions provide an accurate tracking of both the total sales curve and its trial and repeat components. At year’s end, this is only a 5% error in the overall sales forecast.
Even though the level of additional repeat sales is relatively low at the end of the calibration period, it is evident that additional repeat will quickly bypass the other sales components, and will comprise the lion’s share of total sales in the period following Week 52. The ability of our model to accurately track and fore-cast this key component is, perhaps, the strongest indicator of its validity and usefulness.
Two other widely monitored measures of new product performance are “percent triers repeating” and “repeats per repeater” (Clarke 1984, Rangan and Bell 1994). At any point in time t, percent triers repeating is computed as R1t /T t , while repeats per repeater is computed as Rt /R1t . In Figures 6 and 7 we compare the actual development of these two measures with the predictions derived from the dynamic changepoint specification (Model 6) and the basic exponential-gamma, covariates specification (Model 4).
Comparing the tracks for Models 4 and 6 further illustrates performance of our dynamic changepoint
Figure 5 Predicted Sales
0 4 8 12 16 20 24 28 32 36 40 44 48 52 Week 0 10 20 30 Cum. Sales per 100 HH ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Actual ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Predicted Trial ... First Repeat .... .... .... .... .... .... .... .... .... .... .... .... ... ... Additional Repeat .... .... .... .... .... .... .... .... .... .... ... ... Total Sales ...... ... ...
Figure 6 Tracking Percent Triers Repeating
0 4 8 12 16 20 24 28 32 36 40 44 48 52 Week 0 10 20 30 40 50 60 % Triers Repeating ... ... ... ... ... ... ... ... ... ... ......... ... Actual ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Model 6 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Model 4
specification. To reinforce this, it is useful to con-trast Model 6 with the performance of the Erlang-2/ gamma, covariates specification (Model 10), both in terms of overall sales (Figure 1 versus Figure 5) and percent triers repeating (Figure 2 versus Figure 6). It is clearly important that a model incorporates a mech-anism for accommodating changes in the underlying buying rates, reflecting, among other things, prefer-ence changes as the consumer gains more experiprefer-ence with the new product.
3.3. Exploring Sensitivity to Calibration-Period Length
In the analysis presented above, we have used a 6-month calibration period, as this is the point at which these forecasts are typically made in practice. However, given the costs of testing (and the oppor-tunity costs of waiting too long), a manager con-ducting a test market often wants to know: “How many weeks are needed for an accurate forecast?” To explore the impact of the length of the model cali-bration period on forecasting performance, we recali-brated the dynamic changepoint process model (i.e., Model 6) using two shorter calibration periods: (i) the first 12 weeks of purchasing data, and (ii) the first
Figure 7 Tracking Repeats per Repeater
0 4 8 12 16 20 24 28 32 36 40 44 48 52 Week 0 1 2 3 4 Average # Repeat Purchases ... ... ... ... ... ... ... ... ... ... ... Actual ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Model 6 ... ... ... ... ... ... ... ... ... Model 4
Table 4 Model Estimation Results by Length of Calibration Period
Length (wk) LL (calib) LL (26 wk) r Coupon Promotion 12 −238734 −373405 0067 86573 1000 2353 5779 0009 20 −334371 −372917 0071 97682 1000 1747 4965 0011 26 −372656 −372656 0061 80228 0966 1367 5204 0012
20 weeks of purchasing data. The model estimation results are reported in Table 4, along with the results associated with a 26-week calibration period (taken from Table 2). To facilitate the comparison across calibration period lengths, we applied the 12- and 20-week parameter estimates to the first 26 weeks of purchasing data and computed the associated value of the log-likelihood function, LL (26 wk).
We observe that even between 20 and 26 weeks, there is movement in the values of the parameter estimates and this is reflected in the almost 3-point difference in the associated value of the 26-week log-likelihood function. From a managerial perspective, the impact of calibration-period length (i.e., length of the test market) is of greater interest. The fore-casting performance measures associated with the three calibration periods are therefore reported in Table 5.
We observe that while shortening the calibration period does not adversely impact our ability to track and forecast trial purchasing, it has a noticeable neg-ative impact on the repeat-purchasing aspect of the model which leads to error in the total sales forecast. For the case of Kiwi Bubbles we can say that using a model calibration period shorter than the standard 26 weeks has a substantial impact on forecast quality, but we are not in a position to generalize.
Just as Fader et al. (2003) have explored the sen-sitivity of trial forecast accuracy to the length of model calibration period across a number of datasets from different product categories, there is a need to examine the impact of calibration period length on our forecasts of the repeat-purchase component of new product sales. Clearly, any conclusions about calibration-period length will be dependent, in part, on the average interpurchase time, which will vary from category to category. Researchers examining this issue in the future should therefore try to select datasets from categories that vary systematically on such a measure.
Table 5 Model Forecasting Performance Results by Length of Calibration Period
Length (wk) Wk52 Index MAPE-Tot MAPE-TR MAPE-FR MAPE-AR
12 1375 187 21 206 398
20 1273 148 33 155 275
26 1047 26 20 41 43
3.4. Additional Analysis of Model Performance To further illustrate and emphasize the properties of the dynamic changepoint model, we briefly present the empirical results obtained from a similar analy-sis performed using the Four Seasons Biscuits dataset. Prior to its national launch, this snack product under-went a year-long test conducted in two of IRI’s Behav-iorScan markets, comprising a total of 3,953 panelists. However, the available data on repeat purchasing are much sparser than for Kiwi Bubbles, and thus this dataset represents a much tougher test for the dynamic changepoint model. First, the new product reached a relatively small number of triers by the end of Week 26 (148, compared to 267 for Kiwi Bubbles); second, the level of repeat purchasing per trier is much lower (1.06 repeats/trier over the 52 weeks, ver-sus 1.79 repeats/trier for Kiwi Bubbles). The combina-tion of these two factors leads to a dataset with only 77 repeat purchases in the calibration period (com-pared to 295 for Kiwi Bubbles). The marketing covari-ates provided with the dataset include the coupon and merchandising variables used in the Kiwi Bub-bles analysis, along with a measure of TV advertising (GRPs transformed using a standard exponentially smoothed “stock” variable, e.g., Broadbent 1984 to account for carryover effects), and a “Week 1” dummy variable suggested by the firm commissioning the test to accommodate unusual marketing activity during the launch week.
Limiting ourselves to the exponential model spec-ifications, we fit each of the six models to the first 26 weeks of purchasing data; the results are summa-rized in Table 6.
Once again, we observe the dominance of the dynamic changepoint specification (Model 6). In terms of forecast accuracy, the Week 52 index is 94.5. In terms of forecast period MAPE (computed over Weeks 27–52), the values for trial, first repeat, and additional repeat are 7.2, 9.8, and 15.5, respectively. Some of the errors in these sales components actu-ally cancel out, yielding a total sales MAPE of only 2.9. In contrast to the case of Kiwi Bubbles, there is greater error in the repeat-sales component; this should come as no great surprise given the smaller number of repeat purchases per trier in the model calibration period (77/148 = 052 for Four Seasons Bis-cuits versus 295/267 = 110 for Kiwi Bubbles).
It is interesting to compare the nature of the dynamic changepoint process between these two
Table 6 Model Estimation Results for the Four Seasons Biscuits Dataset
Model Covariates Changepoint LL # Parameters r Any Merch Dummy TV Coupon
1 N None −186489 2 0049 160900 (1) 2 N Static −184647 3 0018 25770 0527 3 N Dynamic −183973 4 0017 23576 1000 0494 4 Y None −182741 6 0049 329611 (1) 0014 0913∗∗ 0004ns 4668 5 Y Static −182580 7 0027 104648 0669 0013 0433ns 0002ns 2910† 6 Y Dynamic −181998 8 0023 68693 1000 0623 0013 0231ns 0002ns 2352ns
Note. All parameter estimates are significantly different from 0 at p < 0001, except those marked withns(not significant),†p < 010 , and∗∗p < 001 . datasets. Substituting the two sets of estimates of
$ and % in (3) yields the implied probabilities of a change in the underlying buying rate as a function of depth-of-repeat level, which are plotted in Figure 8. We observe a greater level of nonstationarity in the purchasing of Four Seasons Biscuits. There is a 54% chance that a consumer will change his purchase rate following his trial purchase and a 29% chance that he will revise his buying rate following his first repeat purchase. This implies that there is only a 33% chance that a consumer will not have revised his buying rate after two purchases of the new product (compared to 65% for Kiwi Bubbles). However, the parameter esti-mates suggest that the underlying purchasing evolves toward a stationary process $ = 10 , with a minus-cule chance of a changepoint occurring anytime after the sixth repeat purchase.
4. Conclusions
While certain “hot” topics come and go in the field of marketing research, there has always been inter-est (shared by academics and practitioners alike) in the issue of forecasting new product sales. One char-acteristic of new product purchasing is the existence of nonstationarity in the buying process, even after we have controlled for the effects of time-varying marketing-mix covariates.
Figure 8 Estimated Probability of Change in Buying Rate by Depth of Repeat 0 1 2 3 4 5 6 7 8 9 10 Depth-of-Repeat Level (j) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 P(Change in Buying Rate) ...... ...... ...... ...... ... “Kiwi Bubbles” ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... “Four Seasons”
Recognizing that existing approaches for handling these dynamics suffer from several shortcomings, we have proposed a dynamic changepoint model that captures the underlying evolution of the buying behavior associated with the new product. This model extends the basic changepoint framework, as used by a number of statisticians, by allowing the changepoint process itself to evolve over time. Additionally, this model nests a number of the standard multiple-event timing models that have previously been considered in the marketing literature. In our empirical analy-sis, we showed that the dynamic changepoint model accurately tracks (and forecasts) the total sales curve as well as its trial and repeat components and other managerial diagnostics (e.g., percent triers repeating). Beyond the context discussed so far in the paper (i.e., two new grocery products), it is worth discussing other relevant applications/extensions for the general type of methodology presented here. First, it is impor-tant to emphasize that the behavioral “story” behind our model is by no means limited to the grocery products setting. A similar pattern will likely emerge for other types of products and services (although the specific parameters that characterize the various components of the model will likely vary from one context to another). Likewise, the model might apply nicely to new consumers who are first encountering an existing product/service. For instance, as people first gain broadband access to the Internet and learn about various Web sites, their behavior over time should conform to the basic set of assumptions out-lined here; this would be a very promising area for future investigation.
As we run the model over multiple products/ services, it will be instructive to look for “meta-patterns” in the resulting model parameters. Our empirical analysis revealed one particular type of nonstationary behavior, but it would be useful to cat-alogue different forms of nonstationarity (and covari-ate effects) and begin to associcovari-ate them with product characteristics or other external measures. Many firms (e.g., ACNielsen BASES) attempt to database hundreds or thousands of products using simple sales summaries to enable early forecasts for future launches. Such a process can be greatly enhanced by
using the parameters from a complete (and behav-iorally plausible) model rather than relying strictly on summary statistics (such as repeats per repeater and the other measures we discussed earlier). As our field continues to make rapid advances with hierarchical Bayes methods, this task should become a workable possibility, even for practitioners, in the near future.
Reflecting on the assumptions underlying this model, the least appealing one is that when a change in the underlying buying rate occurs, the consumer’s new buying rate is independent of his previous one. A more plausible mechanism for modeling the evolu-tion in the buying rate would be to assume that wi+1
equals wi modified by a random component (e.g.,
Chernoff and Zacks 1964, Moe and Fader 2003). How-ever, such a mechanism is considerably more complex from a computational perspective and we leave the examination of this as an area of future research. It is worth noting that if this random shock was linked to the specific store environment in which each pur-chase of the new product occurred, we could accom-modate Anderson and Simester’s (2003) observation that deeper promotions increase repeat rates among first-time customers.
One potentially important limitation of the model developed in this paper is that it does not consider distribution build. A unique characteristic of con-trolled test market services such as BehaviorScan is the fact that they ensure complete retail distribution for the product of interest. In using this model in other test market settings, it would be desirable to modify the underlying formulation to accommodate the fact that retail distribution should be treated as a bottleneck that can prevent the trial and repeat pur-chasing of the product. (It is therefore inappropriate to simply treat distribution in the same way as other marketing covariates such as feature and display activity.) Having made such a modification, it would be natural to consider using the model for tracking a new product’s rollout across geographic regions. In such an application, we should also account for the endogeneity of distribution (e.g., Bronnenberg and Mela 2003, Elberse and Eliashberg 2003).
Another potentially important limitation of our model is the absence of any competitive effects. It is interesting to think about how new product entry can affect—and be affected by—existing mar-ket structures. (See Bronnenberg et al. 2000 for a recent review of this literature.) Also, we may desire additional diagnostic measures that provide insight into the source of volume for the new product (e.g., Chatterjee and Ramaswamy 1996). However, beyond these past approaches—mostly post hoc economet-ric models that were not intended for forecasting purposes—that other researchers have employed, we are intrigued by an extension of our product-specific
stochastic model to one that can simultaneously deal with the sales patterns and marketing activities of competitors.
Beyond the simple case of expressing the mea-sures of marketing activity relative to competition, one approach would be to build on a competing hazards framework. Alternatively, a dynamic discrete choice model could be used to model product choice given incidence, with a nonstationary timing process being used to model category purchasing. So, while we view our integrated model as offering a reason-ably accurate and managerially useful picture of the trial-repeat process for a given new product, we see it as just one step towards the creation of a “Holy Grail” model that builds in competition and other category-level phenomena to be able to anticipate the complete set of market dynamics that surround a new product launch. However, we must realize that when the forecasting of new product sales is the modeling objective, the competitive dimension may increase the range/magnitude of potential errors, since we need to condition the sales forecasts on forecasts of compet-itive actions (e.g., Alsem et al. 1989, Danaher 1994). (The quality of these forecasts depends on the qual-ity of the manager’s judgements concerning future competitive activity.) The ultimate performance of the model speaks for itself: The ability of our “noncom-petitive” model to forecast sales (and related manage-rial diagnostics) so accurately suggests that adding competitive effects may do little to improve pure sales forecasting accuracy.
Acknowledgments
The second author acknowledges the support of the London Business School Centre for Marketing.
Appendix: Issues in Parameter Estimation
At first glance, the task of maximizing the likelihood func-tion given in (11) appears to be a straightforward numerical optimization problem. However, looking closely at (10), we note that each evaluation of the likelihood function involves the evaluation of 2K partition-specific likelihood functions
for each consumer, where K is the number of purchases made by the consumer. For relatively long model calibration periods (e.g., 52 weeks) and/or categories with high aver-age purchase frequencies, it is not uncommon to observe consumers with 15 or more repeat purchases. For a con-sumer making 15 purchases, the likelihood functions associ-ated with each of the 215= 32786 separate changepoint pat-terns have to be evaluated. For average-sized panels, each evaluation of the overall likelihood function (11) therefore involves millions of computations, and the overall compu-tation time associated with maximizing the likelihood func-tion may become impractical.
One approach to minimizing computation time is to limit the number of possible changepoints for each consumer to M. If the number of purchases made by a consumer K is less than M, all the possible changepoint sets for that num-ber of purchases are evaluated. However, for a consumer
