Endogeneity and exogeneity are topics that are mainly discussed in macroeconomics. We show that sales response functions (SRF) are exposed to the same problem if we assume that the control variables in a SRF refl
ect behavioral reactions of the supply side. The supply side actions are covering a 
flexible marketing component which could interact with the sales responses if sales managers decide to react fast according to new market situations. A recent article of Kao et al. (2005) suggested to use a class of production functions under constraints to estimate the sales responses that are subject to marketing strategies. In this paper we demonstrate this approach with a simple SRF(1) model that contains one endogenous variable. Such models can be extended by further exogenous variables leading to SRF-X models. The new modeling approach leads to a multivariate equation system and will be demonstrated using data from a pharma-marketing survey in German regions.Sales response functions, stochastic derivative constraints, simultaneous estimation, MCMC, pharma-marketing, model choice

Wolfgang Polasek

English

Research Papers in Economics

 
 
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Wolfgang Polasek 
Institute for Advanced Studies (IHS), Austria 
The Rimini Centre for Economic Analysis (RCEA), Italy 
 
 
 
ENDOGENEITY AND EXOGENEITY IN SALES 
RESPONSE FUNCTIONS Endogeneity and Exogeneity in Sales Response
Functions
Wolfgang Polasek1
IHS, Stumpergasse 56, A-1060 Vienna, polasek@ihs.ac.at
Summary. Endogeneity and exogeneity are topics that are mainly discussed in
macroeconomics. We show that sales response function (SRF) are exposed to the
same problem if we assume that the control variables in a SRF re
ect behavioral
reactions of the supply side. The supply side actions are covering a 
exible marketing
component which could interact with the sales responses if sales managers decide
to react fast according to new market situations. A recent article of [3] suggested to
use a class of production functions under constraints to estimate the sales responses
that are subject to marketing strategies. In this paper we demonstrate this approach
with a simple SRF(1) model that contains one endogenous variable. Such models
can be extended by further exogenous variables leading to SRF-X models. The new
modeling approach leads to a multivariate equation system and will be demonstrated
using data from a pharma-marketing survey in German regions.
Key words: Sales response functions, stochastic derivative constraints, simultane-
ous estimation, MCMC, pharma-marketing, model choice.
1 Introduction
Kao et al. (2005) [3] have proposed a simultaneous estimation of sales in de-
pendence of marketing inputs. The new idea behind this approach is that the
(optimal) expenditures, which are inputs in the SRF, might depend on the
current sales and should be estimated endogenously.
Polasek (2010) [5] has introduced a family of SRF(k) for a cross-sectional
sample where the parameter k denotes the number of endogenous input vari-
ables (like marketing expenditures or sales related covariates). In this paper
we show how to estimate a SRF(1) model that explains sales output by a
Cobb-Douglas type of function of marketing variables.
The main point of the current approach is the extension of the traditional es-
timation of a SRF to a system of observations, because some input variables
are jointly determined by the output. This problem was solved by system esti-
mation in macroeconomics over the last decades. New is the assumption that2 Wolfgang Polasek
the endogeneity of the inputs stems from an implied (stochastic) optimality
consideration, which allows the rst partial derivative of the SRF to follow a
normal distribution. We call this crucial behavioral assumption the stochastic
partial derivative (SPD) assumption. More extensions of these modeling con-
siderations lead to a model choice problem. The details of this model choice
problem need to be more worked out in future, but we will concentrate in
the next section on the model estimation part and we will use the marginal
likelihood criterion for the Bayesian model choice.
If the rst partial derivative of a response model is jointly determined with
the dependent variable, then these 2 equations imply a simultaneous equation
system, since the stochastic restrictions imply an endogeneity of the input
variables. There are 3 main implications that constitute a SRF-SPD model:
1. Sales (demand) model: a (production) function of the input variables X
explain ) output sales variables y plus noise.
2. Behavioral (supply) model + SPD assumptions: results of marketing eorts
are proxied by rst derivatives and ) follow a normal distribution.
3. Conditional on SPD and known SRF coecients: The SPD assumption
leads to ) stochastic regressors.
In the next section we describe the SRF(1) model together with a Bayesian
(MCMC) estimation procedure. Section 1 introduces the SRF(1)X(q) model
(a notation similar to notations in multiple time series) to indicate that a
SRF model does not need to have only endogenous variables, but can also
include q exogenous covariates. Section 2 explains the estimation procedure
and Section 3 discusses a regional sales response model example that involves
a German pharma-marketing data set for the years 2008 and 2007. In a nal
section we conclude.
2 The SRF(1) model with endogeneity
We consider the simplest possible model to demonstrate the endogeneity ef-
fects in the SRF(1) sales response function with one input variable x
y = 0x1e; (1)
where  is a N[0;2
y] distributed error term. By taking logs for the n cross-
sectional observations we get the following linear regression model
ln y  N[y = X;2
yIn] (2)
with the regression coecients  = (ln 0;1) and the regressor matrix X =
(1n : ln x) where 1n is a vector of ones and x is the cross-sectional decision
variable that will in
uence the sales y (a n1 vector) in the n regions. Thus
the model is of the type of a log linear production function as it is used
in macro-economics. The new assumption is the stochastic partial derivativeEndogeneity and Exogeneity in Sales Response Functions 3
(SPD) assumption, which implies an additional equation for the supply side
behavior:
@y=@x = 01x1 1e
Next, we make the assumption that the log derivatives across all units
ln yx = log(01) + (1   1)ln x (3)
are normally distributed with parameters that can be estimated:
ln yx  N[ln ;2
] or ln yx  N[ln 1n;2
In]: (4)
This means that the sales responses y and the decision variable x might
follow a restriction that allocates resources according to the rst derivative
of the SRF but the empirical observations across the n regions reveal some
noise. These stochastic 
uctuations across regions are captured by the mean
response , and the variance 2
 of the constraint measures the looseness or
strength of this optimality behavior in the model and can be interpreted as a
tightness parameter for the SPD restriction.
Adding the stochastic partial derivative (SPD) restrictions for the x regres-
sor into the SRF model imposes the behavioral assumption that the marketing
eorts are allocated in such a way that the marketing results (via the SRF
derivative) should be equal across the regional units: Additionally, the above
SPD assumption leads a normal distribution of the regressor x:
ln x  N[x;2
xIn] (5)
where the parameters x = (x;2
x) are determined by the SRF and SPD
parameters. This approach actually implies the endogeneity of x in the SRF
model and shows how the SPD assumption (i.e. the feedback from the sales
to the marketing control variables) in
uences the inference process to yield a
simultaneous estimation model.
To see how the SPD assumption translates to an assumption about the x
variable, we re-write the exponent of the SPD density (4) and use the log
derivative (3)
N[ln ;2
] / (ln yx   ln)2=2

= (log(01) + (1   1)ln x   ln)2=2
 =
=

log(01=)
1   1
  ln x
2
(1   1)2=2

/ N[ln x j x;2
x = 2
=(1   1)2]
with the mean x =
log(01=)
1 1 and variance 2
x =

2

(1 1)2 of log(x). (Note
that the Jacobian for ln x from  is just 1=j1   1j.)
Finally, we dene the SRF(1)-SPD model by the following sequence of
(conditional) normal densities:4 Wolfgang Polasek
Denition 1. The SRF(1)-SPD model y = 0x1eu is dened as the following
set of 3 log-normal densities:
ln y j SRF;  N[ln 0 + 1ln x;2
y]
) ln x j SPD;  N[(ln 0 + ln 1   ln )=(1   1);2
x]
ln yx j SPD;x  N[ln ;2
]: (6)
where 'SRF' stands for the functional form of the model and 'SPD' stands
for the yx is the rst derivative of the SRF(1) model and with the parameters
of the model given by  = (x;) = (;2
y;;2
). ")" denotes the derived
distribution for ln x.
An alternative way of writing the SRF(1) model is:
Demand : N[ln y j y;2
y]; y = ln 0 + 1ln x;
) Supply : N[ln x j x;;2
x]; x =
log(01=)
1   1
;
Behavior : N[ln yx j ln ;2
]:
It is important to note that while we assume independence between the 3 den-
sities in Denition 1, the parameters of the x distribution are not independent
and are functions of the parameters of the demand and behavioral equation.
The supply side of the model reacts to the market by 'targeting' the rst
derivative of the SRF model to get approximately equal results across all
regions. Making the assumption that the rst derivative of the SRF model
with respect to the control variable x is the marketing target of the supply
side has 2 implication for the modeling process: (1) the control variable x is
for the SRF estimation and therefore a simultaneous system for x and y has
to estimated. (2) The parameters of the x-density depend on the functional
form and the parameters of the behavioral equation, . Because the results
of marketing eorts on y can not be observed directly, this variable is a latent
variable and has to be proxied by the rst derivative that depends on the
functional form. Thus, the behavioral equation has to be included into the
model and is part of the simultaneous estimation process.
For statistical inference we can estimate the parameter vector  of the
system by maximum likelihood or by MCMC, assuming a prior density given
by p(). In the next section we outline the MCMC procedure.
2.1 MCMC estimation in the SRF(1) model
The MCMC estimation of the SRF(1) model requires the likelihood function
for D = (ln y;ln x) given by
l(D j ) = N[ln y j y;2
In]N[ln x j x;2
xIn]  J (7)
and J being the Jacobian from the transformation of  to x.Endogeneity and Exogeneity in Sales Response Functions 5
y = ln 0 + 1ln x and x = (ln 0 + ln 1   )=(1   1)
The prior density is assumed to be
p() = N[ j ;H]N[ j ;2
]
Y
j2fy;g
 [
 2
j j 2
jnj=2;nj=2]: (8)
Then the posterior distribution for  is given by p( j D) / l(D j )p():
Theorem 1 (MCMC in the SRF(1)-SPD model).
The MCMC iteration in the SRF(1)-SPD model with the likelihood function
(7) and the prior density (8) takes the following draws of the full conditional
distributions (fcd):
1. Starting values: set  = OLS and  = 0
2. Draw  from p( j ;2
)
3. Draw  by a Metropolis step (see [2]) from p[ j b;H]l( j y)
4. Draw  2
y from  [ 2
y j s2
y;ny]
5. Draw 
 2
 from  [
 2
 j s2
;n]
6. Repeat until convergence.
The proof can be found in Polasek (2010) [5]. The marginal likelihood of model
M is computed by the Newton-Raftery (1994) [4] formula
^ m(y j M) 1 =
1
nrep
nrep X
j=1
 
n X
i=1
ln l(Di j M;(j)
! 1
or ^ m =  m(l(D j M;(j)) 1)
where Di is the i-th data observation and with the likelihood given in (7) and
^ m is the -trimmed mean to avoid outliers.
3 Application: Sales response in pharma marketing
We apply the SRF(1) model with and without exogenous variables to regional
pharma sales in Germany for the year 2008. The model explains the regional
whole sales of the product M11 by the total sales U2008 of the company
across all regions. The posterior means (and SD) of the SRF(1) model for
M11pc = ln(U2008M11=Pop) by U2008pc (pc: per capita) is:
M11pc = 1:121 + 0:321 U2008pc R2 = 0:1502
(0:0827) (0:0222) y=1:2581;
with  = 27:8(1:678) and  = 1:26(0:021); acceptance rate = 50:8%. The
OLS t is quite similar to the posterior mean and the F-statistic (df =
1;1898) = 335:4101 is highly signicant (p-value = 0).
OLS Estimate Std.Err t-value Pr(>|t|); R^2=0.1502
Intercept 1.1237 0.0673 16.6984 0
X 0.3222 0.0176 18.3142 06 Wolfgang Polasek
Fig. 1. The beta coecients of visits and sales U2008 per capita (pc)
3.1 Exogenous variables: A SRF(1) model for regional pharma
sales
We extend the SRF(1) model for the sales of the M11 product M11pc =
ln(U2008M11=Pop) by including 2 more regressors: the number of visits for
the related product V 13pc = ln(visitsM13=Pop) and the purchasing power
potential ln(PPP). The posterior mean (and SD) of the Bayesian regression
estimation are:
M11pc = 1:152 + 0:22 U2008pc + 0:19 V 13pc + 0:19ln(PPP)
(0:2507) (0:0251) (0:0239) (0:0359)
with  = 22:478(2:268) and  = 1:225(0:020) (acceptance rate = 51%).
The OLS t is (with R2 = 0:1967;F(df = 3;1896) = 154:76; p-value=0)
and y = 1:2238:
M11pc = 1:1510 + 0:2197 U2008pc + 0:1893 V 13pc + 0:0962ln(PPP)
(0:1987) (0:0197) (0:0191) (0:0283)
The posterior distribution of the parameters are displayed in the next 2 gures:
We see that the exogenous variables have improved the t but the R2 is
still low, despite the signicant coecients. This is because the underlying
dispersion across regions is quite high, but positive slopes are present. Not
all exogenous variables t nicely with an SRF(1)-X model. The reasons for
the high volatility of the SRF in dependence of covariates have to be furtherEndogeneity and Exogeneity in Sales Response Functions 7
Fig. 2. SRF(1)X(2): Betas(intercept, U2008pc) on sales U2008M11pc
Fig. 3. SRF(1)X(2): Betas(visitsM13pc,PPP) on sales U2008M11pc8 Wolfgang Polasek
explored. If covariates can be found that reduces this volatility more likely ac-
curate forecasts on the eectiveness of varying the marketing control variable
x can be made. More on how regional heterogeneity can be modeled can be
found in Polasek (2010) [5].
4 Summary
In this paper we have proposed the SRF(1) model family with 1 endogenous
variable stochastic derivative constraints, as it was suggested by Kao et al.
(2005) [3]. The MCMC estimation of the model is quite straightforward, de-
spite the fact that the constraints lead to a simultaneous sales model. The class
of SRF models with stochastic partial derivative (SPD) constraints is quite

exible and allows the inclusion of exogenous variables denoted by SRF(1)-X
models. Future research will concentrate on the model selection procedure and
on the problem, if model choice can help to decide statistically what variables
in a SRF model are endogenous or exogenous. We have demonstrated this
approach by a regional pharmaceutical sale models for Germany. In Polasek
(2010) [5] it was shown that a whole SRF(p)-X(q) model family can be esti-
mated by MCMC and the SRF approach can be extended to a spatial sales
response model that takes the neighborhood structure of the observations or
other regional variations into account (see Baier and Polasek, 2009) [1].
References
1. Baier, D. and Polasek W. (2009), Marketing and Regional Sales: Evaluation of
Expenditure Strategies by Spatial Sales Response Functions, to appear in GFKL
2009 proceedings.
2. Chib, S. and Greenberg, E. (1995). Understanding the Metropolis-Hastings Al-
gorithm, The American Statistician, 49, 327-335.
3. Kao L.-J., Chiu C.-C., Gilbride T.J., Otter T., and Allenby G.M. Evaluating the
Eectiveness of Marketing Expenditures. Working Paper, Ohio State University,
Fisher College of Business, 2005.
4. Newton, M.A. and Raftery A.E. (1994). Approximate Bayesian Inference with
the Weighted Likelihood Bootstrap (with discussion), J. Royal Statistical Soc.,
Series B, 56, 3-48.
5. Polasek, W. (2010). Sales Response Functions (SRF) with Stochastic Derivative
Constraints. Working Paper, Institute of Advanced Studies, Wien, 2010.
6. Rossi P.E., Allenby G.M., and McCulloch R. Bayesian Statistics and Marketing.
John Wiley and Sons, New York, 2005.

Approximate Bayesian Inference with the Weighted Likelihood Bootstrap (with discussion),

Bayesian Statistics and Marketing.

Evaluating the Eectiveness of Marketing Expenditures. Working Paper,

Marketing and Regional Sales: Evaluation of Expenditure Strategies by Spatial Sales Response Functions, to appear in GFKL

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Endogeneity and Exogeneity in Sales Response Functions

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Endogeneity and Exogeneity in Sales Response Functions

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