Transcript A Multivariate Statistical Model of a Firm´s Advertising

A Multivariate Statistical Model of
a Firm’s Advertising Activities
and their Financial Implications
Oleg Vlasov, Vassilly Voinov, Ramesh Kini and Natalie
Pya
KIMEP, Almaty
Introduction
• This presentation describes a modification of the
well-known discrete multivariate probability
model for optimizing the efficiency of advertising
campaigns.
•
The model will permit to examine how the
profitability of an advertising campaign can be
maximized by statistically optimizing exposure
criteria subject to budget constraints and to
investigate the modalities of implementation of
the model so as to maximize the profitability of
the firm’s other activities.
• Computational problems associated with
conditional probabilities of the model will be also
discussed.
Introduction
• The advertising industry involves
"big money" and media planners have
an important job: to optimally allocate the media
budget and to make media plans as effective as
possible.
Introduction
• Several issues have to be addressed
in the process, however.
One important question is in which medium
(newspapers, television, radio, magazines, business
papers, direct mail, "outdoor media") an ad should be
placed to get the optimal effect.
Introduction
• The media planner would have to decide
how much of the budget should be
allocated to each medium, within each medium
and to particular media vehicles.
• Ads must be placed in different media to maximize
some exposure criterion without violating the overall
budgetary constraints.
Introduction
• Modeling a random vector X, describing,
say, the total number of exposures, two
correlations appear and cause problems.
The first is a within-vehicle correlation
and the second one is a between-vehicle
correlation.
• Another problem is the fact that knowing
the number of people exposed to
different media does not mean that we
know the number of people actually
reached by commercials. Nevertheless, a
strong positive correlation between that
number and profitability is only
to be expected.
Statistical Model
Statistical Model (continued)
• Consider a random vector
X  (X1 , X 2 ,..., X m 1 , X m ) T
with random components
X1 , X 2 ,..., X m1 , X m
that take arbitrary integer values.
The random variables
X1 , X 2 ,..., X m1
for
denote the firm’s expenditures
Statistical Model (continued)
and X m is the total exposure or
the impact of such exposures on the
firm’s profitability for the same period.
• It seems more reasonable to
consider X ,..., X
1
m
as continuous random variables, but
selecting a proper multivariate
model becomes problematic in this
case. On the contrary, quantizing
expenses by a reasonable amount,
say, $1000, will lead to the known
Statistical Model (continued)
Let observations be characterized by
vectors
a  (a 1 ,..., a m ) T , a j {~
a j1 ,..., ~
a jk j }, j  1,2,..., m,
with ~
a ij denoting integer midpoint
value of an interval of the range of
possible values of an observed
quantity.
Statistical Model (continued)
Denote by
a j  (a j1 ,..., a jm ) T , j  1,2,..., K
where K  k 1  k 2    k m,all values of a
defined by possible values of their
componentsa j .
Further, let p a , j  1,2,..., K be the
j
probability for obtaining vector
measurements a j , and  K
j1 p a j  1
Statistical Model (continued)
Further, let a random vector
X  (X 1 , X 2 ,..., X m )
T
take the value r  ( r1 ,..., rm ) T
if sums of observations of j th
components of vectors for, say,
n sequential dates are
K
 a ij l ai  r j
i 1
n min {~
a ji }  r j  n max {~
a ji }, j  1,2,..., m,
1ik j
1ik j
Statistical Model (continued)
where l a , i  1,..., K
i
denotes the number of observed
a i vectors
l ai
in a sample and the values of
are nonnegative
integers
such
that
K
i1 l ai  n
Then the probability that a random
vector X( rwill
take
a) Tdefinite value
,
r
,...,
r
1 2
m
r=
can be written down as
Statistical Model (continued)
P(X  r,p)  
n!
K
 l ai !
K
l ai
 p ai
i 1
i 1
where p  (p a ,..., p a ) T
i
K
is the vector of parameters and the
summation is performed over all
sets of nonnegative solutions
l ai , i  1,..., K
of the system of linear diophantine
equations  K
  a ij l ai  r j , j  1,..., m,
i 1
K
  l  n.
i 1 ai
Statistical Model (continued)
We shall consider P(X = r) as the joint
probability distribution function of a
multinomial type.
The model implicitly includes both
intra- and between-media
correlations as well as correlation of
exposure with expenditures by
media. This information is evidently
contained in parameters of the
model.
Statistical Model (continued)
Using observations for the chosen
period of time, we can extract that
information estimating parameters of
the model P(X = r), and, respectively,
the conditional probability to get a
specified total exposure or profit
given expenses by media:
P( X 1  r1 ,... X m  rm )
P( X m  rm | X 1  r1 ,..., X m1  rm1 ) 
 P( X1  r1,...X m  rm )
rm
Using, say, maximum likelihood
P( X m  rm |of
X 1  r1 ,..., X m1  rm1 )
estimates
, it is
possible to solve different
optimization problems aiming to
maximize the total exposure or
Computational
Problems
Computational Problems
Numerical calculations of conditional
probabilities may become
unachievable for a reasonable time on
a computer for nlarge
).
 15 samples (
In this case the limit distribution of the
model may be used. The system of the
K
first equations
 aijlai  rj , j  1,..., m,
i 1
Al  r
can beA
written
K in
mmatrix form as
~
T
where
isl  (l ,..., l matrix
of coefficients
a ij
)
ai
aK
and
Computational Problems
Under these notations forn  
the random vector X will have
asymptotically the multivariate
normal distribution
N m (nAp , nA  A T )
with the vector of meansnAp
the covariance matrixnA A T
where   D  pp T and D is the diagonal
paj
matrix with probabilities
on the
main diagonal.
Computational Problems
Using estimates of N m (nAp , nA  A T )
conditional probabilities
P( X m  rm | X 1  r1 ,..., X m1  rm1 )
may be evaluated with the help of, say,
technique proposed by Vijverberg.
Model Extensions
Model Extensions
We intend to extend the basic model
by:
• Including the competitive industry
(market-share, market penetration
and market expansion) dynamics in
the model;
• Making a distinction between flow
variables (media expenditures,
revenue inflows, cost outflows, etc.)
and state variables or stocks
(advertising goodwill and market
shares, etc.);
Model Extensions
• Introducing a two-tiered structure to
the competitive dynamic model so
that the rival firms’ media budgeting
and media allocation processes –
along with the other control
variables, e.g., prices, etc. – affect
their respective market shares, etc.,
and these in turn impact on the firms’
revenues, costs and bottom lines.
Expected Results
Expected Results
• A discrete multivariate probability model
measuring the efficiency of a certain
advertising campaign of a firm.
• A methodology and a software for
estimating parameters and conditional
probabilities to get a definite exposure or
profit given expenses by particular media
vehicles.
• A methodology and recommendations to
optimize exposure criteria subject to budget
constraints aiming maximization of the
profitability of an advertising campaign.
• Recommendations concerning applications
of the model for maximization of the
profitability of different firm’s activities
except advertising.
Questions?
Comments?