Estimating Promotion Response When Competitive Promotions Are Unobservable Sangkil Moon North Carolina State University Wagner A. Kamakura Duke University Johannes Ledolter University of Iowa April 8, 2006 * Sangkil Moon is Assistant Professor of Marketing at College of Management, North Carolina State University. Wagner Kamakura is Ford Motor Co. Professor of Global Marketing at Fuqua School of Business, Duke University. Johannes Ledolter is Maxwell C. Stanley Professor of Management Sciences at the Henry B. Tippie College of Business, University of Iowa. Moon’s contact information is: email: [email protected], phone: (919)515-1802, fax: (919)515-6943. Kamakura’s contact information is: email: [email protected], phone: (919)660-7855, fax: (919)684-2818. Ledolter’s contact information is: email: [email protected], phone: (319)335-3814, fax: (319)335-0297.
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Estimating Promotion Response When Competitive Promotions Are Unobservable
Sangkil Moon North Carolina State University
Wagner A. Kamakura
Duke University
Johannes Ledolter University of Iowa
April 8, 2006
* Sangkil Moon is Assistant Professor of Marketing at College of Management, North Carolina State University. Wagner Kamakura is Ford Motor Co. Professor of Global Marketing at Fuqua School of Business, Duke University. Johannes Ledolter is Maxwell C. Stanley Professor of Management Sciences at the Henry B. Tippie College of Business, University of Iowa. Moon’s contact information is: email: [email protected], phone: (919)515-1802, fax: (919)515-6943. Kamakura’s contact information is: email: [email protected], phone: (919)660-7855, fax: (919)684-2818. Ledolter’s contact information is: email: [email protected], phone: (319)335-3814, fax: (319)335-0297.
Estimating Promotion Response When Competitive Promotions Are Unobservable
ABSTRACT
This study addresses a problem commonly encountered by marketers who attempt to assess the impact of their sales promotions: the lack of data on competitive marketing activity. In most industries, competing firms may have competitive sales data from syndicated services or trade organizations, but seldom have access to data on competitive promotions except, if any, at a high level of aggregation; competitive promotion data are rarely available at the customer level. Promotion response models in the literature either have ignored competitive promotions focusing on the focal firm’s promotions and sales response or have considered the ideal situation where the analyst has access to full information about each firm’s sales and promotion activity. We propose a random-coefficients hidden Markov promotion response model, which takes the unobserved promotion level (i.e., promotion or no promotion) by competitors in the same product category as a latent variable driven by a Markov process to be estimated simultaneously with the promotion response model. This allows us to estimate cross-promotion effects by imputing the level of competitive promotions, even though these promotions are not directly observable. First, we test the proposed model on synthetic data through a Monte Carlo experiment. Then, we apply and test the model to actual prescription and sampling data from two main competing pharmaceutical firms in the same therapeutic category. The two tests clearly show that, in comparison with several benchmark models, our random-coefficients hidden Markov model (HMM) successfully imputes competitive promotions when they are completely missing and, accordingly, reduces biases in the own and cross promotion parameters. Furthermore, our model provides better predictive validity than the benchmark models.
1
Researchers in marketing have devoted considerable attention to the estimation
of promotion response models to assess the effects of promotional efforts by a brand on
its own and competitors’ sales (see Hanssens, Parsons, and Schultz 2001 for a
comprehensive review). A vast literature already exists in such areas as the effects of
advertising (Dekimpe and Hanssens 1995; Lodish et al. 1995), personal selling (Parsons
and Abeele 1981), sales promotion (Blattberg & Wisniewski 1989), and price discounts
(Farris and Quelch 1987) on brand sales at the national and market levels. Most studies
on the development and application of promotion response models in marketing focus on
packaged-goods for which data on sales and marketing efforts by the leading brands
competing in each product category are generally available from syndicated sources. This
wealth of data allows researchers to develop elegant and sophisticated models of brand
competition that provide estimates of own and cross-elasticities. Under this situation, the
brand manager can draw inferences about the impact of her marketing efforts on her own
brand and the competitor brands, and, furthermore, about her vulnerability to competitive
efforts (Cooper 1988). Disaggregate analyses using single-source data provide an even
clearer picture of cross-competitive effects, allowing the manager to tailor her efforts to
each individual household based on individual-level estimates of own and cross-
elasticities.
The recent developments in promotion response modeling briefly reviewed above
are of great value to the packaged-goods industry and to a few other industries where full
information is available on the sales and promotional efforts by the major competing
brands at the market or national level. On the other hand, these response models are of
limited value to industries where each firm has access to sales data of its competitors
2
from syndicated sources, but cannot obtain information on the competitors’ promotional
efforts. For example, a manufacturer of consumer electronics may have monthly sales
data for all competing brands at each regional market from syndicated sources such as
GfK, but will not know about slot allowances or other promotions by its competitors.
This is a very common situation for most manufacturers selling their products through
distributors or retailers. As another example, a manufacturer of pharmaceuticals may
have prescription data for all competing brands at the physician level from syndicated
sources such as NDC or IMS, but only knows which physicians were visited (or detailed
as these visits are named by the industry) and/or received free samples from its own sales
force. (Some syndicated sources provide competitive data on detailing and sampling, but
only for a small sample and typically on a rotating panel.) This is also a typical situation
to any firm trying to assess the effectiveness of its salespeople, when it might have data
on the clients’ purchases in the product category but not about their exposure to the
competitors’ salespeople. Furthermore, there is a growing emphasis on customer
relationship management (CRM), where marketing effort is customized at the customer
level. In most CRM implementations, the firm has a wealth of information regarding its
own contacts with individual customers and customers’ responses to these contacts, but a
dearth of data on the contacts these customers might have had with the firm’s
competitors. In other words, many firms do not have access to the data about their
competitor’s marketing efforts that would make it possible for them to apply a wide
variety of the promotion response models found in the marketing literature. One solution
(the most commonly used) would be to ignore competitive efforts and estimate a model
using only the firm’s own marketing efforts. It is, however, well known that ignoring a
3
predictor will lead to biased estimates of the response parameters, when the missing
predictor is correlated to the observed one (Goldberger 1991, pp.189-190). Therefore,
this simple and common solution will not only prevent the brand manager to have
estimates of promotion cross-elasticities, but also lead to biased estimates of her own
promotion elasticities.
The main purpose of our study is to propose and test an approach aimed at
resolving the problem of unobserved data on their competitors’ promotional efforts. We
propose a hidden Markov model (HMM) that can effectively estimate the impact of both
own and cross-promotional efforts even when competitive promotion data are not
available. The model simultaneously imputes the missing competitive promotional efforts
while estimating the own and cross-promotional effects on sales. The Markov process of
the unobserved competitive promotions allows us to estimate cross-promotional effects
through our imputation of the competitive promotions. Our goal in this study is to
introduce this idea of estimating a full cross-effects promotion response model while
imputing the competitive promotions. It achieves the ostensibly simple but inherently
difficult goal of estimating promotion cross-effects for a firm that has only sales data
about its competitors in a market. In the following section, we introduce the HMM. We
then extend the model into a Random-Coefficients HMM that estimates both unobserved
heterogeneity and cross-promotional effects at the individual client level. These models
are compared with alternative formulations on synthetic data followed by a real-world
data application estimating effects of promotion on physician prescription behavior.
Finally, we provide discussion and directions for future research.
4
A HIDDEN MARKOV MODEL OF RESPONSE
TO UNOBSERVABLE COMPETITIVE PROMOTIONS
Our goal is to estimate how individual clients (e.g., physicians) respond to
marketing communications when a brand manager knows her own marketing behavior,
that is, whether each client is exposed to the brand’s own promotions. However, the
manager usually has no information on when the clients is exposed to competitive
promotions. This is a common situation encountered in various industries including the
pharmaceutical industry. In the pharmaceutical industry, a firm can obtain sales data for
each physician (i.e., client) from syndicated services (e.g., NDC or IMS), but only knows
when their salespeople visited each physician. For the purpose of clear illustration, we
consider the case of a firm with only one major competitor in a duopoly market. This
model can be extended to a market where there is any number of competitors. In such a
general case, we can infer the general competitive promotion level rather than the only
competitor’s promotion behavior in a duopoly market. We discuss this generalization
later.
For each client n, we want to estimate the following system of response functions
in a duopoly market:
(1a) natntnantnananat exzy +++= δβµ
(1b) nbtntnbntnbnbnbt exzy +++= δβµ ,
where
• yndt = sales by client n for brand d (where a denotes the focal brand and b denotes the competitor brand) in month t,
• znt = variable indicating whether client n is exposed to promotion by the focal firm in
month t (znt = 1) or not (znt = 0),
5
• xnt = variable indicating whether client n is exposed to promotion by the competitor in month t (xnt = 1) or not (xnt = 0),
• µ, β, δ = response parameters to be estimated, • enat ~ N(0, σna
2) and enbt ~ N(0, σnb2) are regression residuals, and
• σnab is the covariance of the two regression residuals, ena and enb.
If y, z, and x were all observed, the estimation of the system in Equation 1 would be
straightforward as a system of Seemingly Unrelated Regressions (SUR). The focal firm a,
however, does not observe the promotion efforts by the competitor at the client level, and
therefore xnt is unavailable. If the focal firm ignores this competitive promotion, it
obviously will not be able to assess how the competitor affects its own sales. Moreover,
ignoring competitive promotion leads to a biased assessment of its own promotion effects
(βna) if there is a correlation between its promotion (Zn) and the competitor’s unobserved
promotion (xn).
We treat the unobserved competitive promotions xnt as missing data to be imputed
at the same time when we estimate the SUR system of response functions in Equation 1.
For this purpose, we assume that the unobserved competitive promotions (xnt) follow a
two-state, first-order Markov-switching process with initial state probabilities Πn = (πn0,
where D indicates all the observed data points. In our case, D is a combination of the
focal brand promotion (z) and both brands’ prescriptions (ya and yb). It is assumed that,
conditional on X, the transition probabilities (q and p) are independent of both the
response model parameters and the data D. Conditional on D and X, regressions are a
system of Seemingly Unrelated Regressions (SUR) with known dummy variables, D and
X. These conditioning features of the model allow us to employ the simulation tool of
Gibbs-sampling for Bayesian inference. Furthermore, it is assumed that the collection of
regression parameters for client n, θn = {µna, βna, δna, σna, µnb, βnb, δnb, σnb, σnab}, follows
the following multivariate normal distribution (MVN) across all the clients.
(6) θn | D, X ~ iid MVN(γp, Σp),
where γp and Σp are defined as the population means and variances-covariances (Gelman
et al. 1995; Allenby and Rossi 1999).
Based on the KN algorithm, we apply three key changes to the basic model that
leads to our Random-Coefficient HMM. First, our estimation incorporates a random-
coefficients component into the KN procedure to pool information across all the clients in
the dataset. This extension is required to make full use of the observed data. Typically,
we can only obtain data of a limited time length for each client. Since we must estimate
the promotion state (promotion / no promotion) of each time point, model estimation
makes extremely high demands on the data. By extending the individual-level KN
procedure into a hierarchical model to pool data across clients, we can improve
estimation accuracy significantly. This point is clearly shown in our tests using synthetic
data and actual prescription data. Second, our model consists of a system of regressions,
11
whereas the KN procedure considers only a single regression. When the explanatory
variables of the two regressions are identical, as is the case in our model, the GLS and
OLS estimators coincide and each of the two regressions can be estimated separately in
our model1 (Goldberger 1991, p.327). Third, the original KN procedure generates state-
specific regression parameter estimates, whereas our model is simplified to generate
state-common estimates for the purpose of parsimony.
To implement the Gibbs-sampling estimator, we need to derive the distributions of the
blocks of each of the above T + K variates conditional on all the other blocks of variates
at the individual client level. Furthermore population regression parameters are estimated
in a separate step because it is a random-coefficients model. Using arbitrary starting
values for the model parameters, the following four steps can be repeated until
convergence occurs. This is a significant extension of the KN procedure, which does not
include Step 1 below. Specific details in each step are provided in Appendix.
Step 1: Generate population regression parameters φp = {µpa, βpa, δpa, σpa, µpb, βpb, δpb, σpb, σpab} from g(φp | D, X, θ), where θ is a collection of all the individual clients’ regression parameters. Step 2: Generate each xt from g(xt | Xt-, µa, βa, δa, σa, µb, βb, δb, σb, σab, q, p, D), t = 1,2, …, T, where Xt- = { x1, …, xt-1, xt+1, …, xT}’ refers to a vector of X variables that excludes xt. Step 3: Generate the transition probabilities, q and p, from g(q, p | X). Step 4: Generate individual clients’ regression parameters µa, βa, δa, σa, µb, βb, δb, σb, σab from g(µa, βa, δa, σa, µb, βb, δb, σb, σab | D, X).
1 This, however, does not mean that the two regression residuals are independent; the correlation of the two regression residuals has an impact on estimating the hidden state of the competitive promotion as shown in the Appendix.
12
TESTING THE PROPOSED PROMOTION MODEL ON SYNTHETIC DATA
Before applying our proposed promotion response model to actual data, we
present results from a Monte Carlo simulation study where we generate synthetic data
based on known parameter values and compare different formulations of the promotion
response model in terms of their ability to recover the “true” response parameters and to
impute the promotion activity of the competing brand. Furthermore, models are
compared in their predictive performance.
Data Generation
For each client n, we draw “true” transition probabilities qn and pn from the
uniform distribution of U[0.05, 0.95]. Then, we draw promotion response parameters µna,
βna and δna for the first regression in Equation 1 from iid normal distributions φ(µ=10,
σ2=52), φ(5, 2.52) and φ(-3, 22), respectively, and for the second regression (µnb, βnb and
δnb) from iid normal distributions φ(10, 52), φ(-3, 22) and φ(5, 2.52), respectively. All
these parameter values are chosen to be close to their counterparts in the prescription
dataset used in our empirical analysis later in this paper. Given these parameters for each
client n, we then generate the synthetic promotion and sales time series for 50 time
periods (i.e., t = month) using the following steps:
1. Using the drawn transition probabilities for client n, qn and pn, we draw a series of the binary promotion variable (promotion / no promotion) for the competitor brand, xnt. The initial states for each client are drawn from a Bernoulli distribution with promotion probability 0.7. This series is assumed to be unobserved in the model estimation, but used to generate synthetic data here. On the other hand, a series of the binary promotion variable for the focal brand, znt, is generated from Bernoulli draws with proportions varying for each client from the uniform distribution of [0.05, 0.95]. It is assumed that the series of the focal firm’s promotion variable are observed in the model estimation.
2. Draw the two regression residuals eat and ebt from φ(0, 2.52) and φ(0, 2.32),
respectively.
13
3. Using the promotion predictors and residuals for the month as well as the regression
coefficients for the client, generate the sales of the two brands for the client and the month.
4. Correlations between the two promotion variables, xnt and znt, are computed.
According to the correlation, we draw 200 clients with a low correlation (0.0 ~ 0.3) and other 200 clients with a high correlation (0.3 ~ 0.6). The two client groups are used to consider situations when the bias caused by ignoring competing promotions would be low or high. It is known that the correlations of the two predictor variables can affect the magnitude of coefficient estimation biases in a misspecified model (Goldberger 1991, pp.189-190). Besides, the client-specific correlation between the two regression residuals ranged from -0.41 ~ 0.35 in the synthetic data.
Benchmark Models
In order to assess the performance of the two versions of our proposed hidden
Markov promotion response model (i.e., the individual HMM and the random-
coefficients HMM), we consider three benchmark models as follows.
• Seemingly Unrelated Regressions (SUR): The system of regressions in Equation 1 under the assumption that we know the promotions for both the focal and competing brands for each client. This is the ultimate benchmark as it uses full information about the competitor’s promotion.
• Regressions Without Competitor: The most naïve benchmark, ignoring the
unobservable promotions by the competing brand. • Latent Variable Regressions: The competitor’s promotion is taken as unobservable
and imputed as a time independent latent variable. This model is estimated as a random-coefficients model, and the only difference between this benchmark model and our proposed random-coefficients HMM model is that the former model does not link the competitor brand’s unobserved promotion to past promotions, an essential part of our HMM model. Our random-coefficients HMM model is expected to be more effective in recovering the unobserved promotion values because it utilizes additional information from past unobserved promotions through the hidden Markov process.
Performance Measures
Our proposed models have three main purposes: a) to estimate own and cross-
promotion effects for both the focal and competing brand (parameter recovery), b) to
14
impute the unobservable promotions by the competitor (imputation of unobservable
promotion), and c) to predict sales level based on the intended promotion activity by the
focal brand and the projected promotion for the competing brand (predictive fit).
Therefore, we compare our two proposed HMM models with the three benchmark
models on multiple performance criteria addressing these three objectives as follows.
a) Parameter recovery
Mean Absolute Deviation ∑=
−=N
nnn NMAD
1/|ˆ| θθ
Mean Absolute Percentage N
MAP
N
n n
nn∑=
−
= 1 |||ˆ|
θθθ
,
where nθ and nθ̂ are actual and estimated response coefficients (µ, β, and δ) for client n.
b) Imputation of unobservable promotion
Hit Ratio HR = % of correct imputations of the missing promotions
Average Predicted Probability NT
xAPP
N
n
T
tntt
/)(
1
1
*
∑∑
=
=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=γ
,
where )( *ntt xγ is the posterior probability of the competitor’s promotion to client n at
period t.
c) Predictive fit
K-step-ahead Mean Squared Error ( )∑=
−−=N
nTKTnnT NyyKMSE
1
2,, /)ˆ()( ,
K-step-ahead Mean Absolute Percentage Error
Nyyy
KMAPEN
n nT
TKTnnT /|ˆ|
)(1
,,∑=
−
⎟⎟⎠
⎞⎜⎜⎝
⎛ −= ,
15
where TKTny ,,ˆ − is the K-step ahead forecast produced for client n at time point T-K for
time point T based on the observed promotion by the focal firm and projected promotion
by the competitor at time point T-K.
Empirical Results on Synthetic Data
Overall Performance of the two proposed HMM models (Individual HMM and
Random-Coefficients HMM) and three benchmarks (SUR, Regressions Without
Competitor, and Latent Variable Regressions) on synthetic data is compared in Table 1 in
terms of the three main goals (parameter recovery, imputation of unobservable promotion,
and predictive fit).
TABLE 1 ABOUT HERE
Since the SUR model uses all the available information, including data on the
promotions by the competitor, it shows the best parameter recovery in terms of the mean
absolute deviation (MAD) and the mean absolute percentage deviation (MAP) from the
“true” parameters. Results from the naïve model (Regression Without Competitor) show
that ignoring promotions by the competitor not only prevents us from estimating the
effect of the competitor’s promotion but also substantially increases the biases in the
parameter estimates for the focal brand. The random-coefficients Latent Variables
Regressions model allows us to impute the missing promotions by the competitor and to
estimate its effect on the dependent variable. However, despite the fact that the
individual-level parameters are obtained through shrinkage, the bias is as high as with the
naïve model. Parameter recovery for the Individual HMM is fairly similar to the Latent
Variables Regression model probably because estimating a model for each individual
makes too heavy demands on the limited available data. Much better results are obtained
16
with the Random-Coefficients HMM, showing the benefits of combining the first-order
Markov structure on the unobserved promotions and individual-level estimates of the
response coefficients through shrinkage.
Table 1B shows that the performance of the Random-Coefficients HMM is also
clearly superior to Latent Variable Regressions and Individual HMM in terms of its
ability to impute the unobservable promotions by the competitor. The Individual HMM
shows essentially the same performance in imputing the missing data as the random-
coefficients Latent Variable Regression model, again suggesting that the HMM model
makes too heavy demands on the data when it is estimated at the individual level. Table
1B also shows that in most tested cases, the Random-Coefficients HMM is second only to
the full-information model (SUR) in terms of predictive fit.
We study two groups of simulated clients in Tables 1A and 1B based on the
correlation between the unobserved and observed promotion variables, xnt and znt.
Specifically, individuals are divided into a low correlation group (0.0 ~ 0.3) and a high
correlation group (0.3 ~ 0.6). This division is for empirically confirming that the biases
on the focal brand’s promotion variable coefficients (βa and βb) are significantly
amplified for the high correlation case when the competitor brand’s promotion
information is ignored as in the Regression Without Competitor model due to omitted-
variable bias (Goldberger 1991, pp.189-90). A comparison of the MAP measures of βa
and βb between the two correlation groups confirms this bias. Specifically, for the
Regression Without Competitor the MAP of βa for the high correlation group (32%) is
twice as large as the same measure for the low correlation group (16%). The bias increase
is even larger for βb in the high correlation sample with the MAP of 118% compared to
17
44% for the low correlation sample. Such amplified biases in βa appear to have
influenced the predictive fit for the focal brand (Brand a). Again, for the Regressions
Without Competitor, the MAPE(1) measures (102%) for the high correlation physician
group is much larger than the same measure (84%) for the low correlation group. The
same pattern is observed for the MAPE(2) measures. Most importantly, in the large
correlation group, our Random-Coefficient HMM can significantly reduce the biases
relative to the naïve model for both βa (MAP= 32% to 21%) and βb (MAP = 118% to
68%). This bias correction results in better predictive performance for our Random
Coefficient HMM, as well.
In general, our performance tests are consistent and robust across all the three
Notes: APP = Average Predicted Probability; MSE = Mean Square Error; MAPE = Mean Absolute Percentage Error; Brand a is the focal brand and Brand b is the competitor brand.
34
Table 2 SUMMARY STATISTICS OF THE PRESCRIPTION DATA
Brand Variable Data Description
Prescription Mean 5.07 Standard Deviation 6.44
Sampling Binary Data (Sampling / No Sampling) Proportion 53.6%
Focal Brand (Brand a)
Detailing (The Number of Monthly Visits Per Physicians) Mean 1.20 Standard Deviation 1.13
Prescription Mean 2.86 Standard Deviation 3.70
Sampling Binary Data (Sampling / No Sampling) Proportion 70.8%
Competitor Brand (Brand b)
Detailing Data Not Available Notes: The actual brand names cannot be released due to confidentiality request of the data provider.
35
Table 3A PARAMETER RECOVERY ON ACTUAL PHYSICIAN PRESCRIPTION DATA
MAPE(2) 60% 126% 124% 107% 103% Notes: APP = Average Predicted Probability; MSE = Mean Square Error; MAPE = Mean Absolute Percentage Error; Brand a is the focal brand and Brand b is the competitor brand.
36
Table 4 SUMMARY OF THE PARAMETER ESTIMATES ACROSS PHYSICIANS
Notes: SUR = Seemingly Unrelated Regressions; RCHMM = Random-Coefficients HMM.
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
SURRCHMM
-12
-10
-8
-6
-4
-2
0
2
-10 -8 -6 -4 -2 0 2
SURRCHMM
38
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