A Multivariate Statistical Model of A Multivariate Statistical Model of a Firm’s Advertising Activities a Firm’s Advertising Activities

and their Financial Implicationsand their Financial Implications

Oleg Vlasov, Vassilly Voinov, Ramesh Kini and Natalie Pya

KIMEP, Almaty

IntroductionIntroduction

• 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 addressedin the process, however.

One important question is in which medium(newspapers, television, radio, magazines, businesspapers, direct mail, "outdoor media") an ad should beplaced to get the optimal effect.

Introduction• The media planner would have to decide

how much of the budget should beallocated to each medium, within each mediumand to particular media vehicles.

• Ads must be placed in different media to maximizesome exposure criterion without violating the overallbudgetary 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 onlyto be expected.

Statistical ModelStatistical Model

Statistical Model (continued)

• Consider a random vector

with random components

that take arbitrary integer values. The random variables

denote the firm’s expenditures for different media for a given period,

X Tm1m21 )X,X,...,X,X(

m1m21 X,X,...,X,X

1m21 X,...,X,X

Statistical Model (continued)

and is the total exposure or the impact of such exposures on thefirm’s profitability for the same period.

• It seems more reasonable to consider

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 and well-understood discrete model.

mX

m1 X,...,X

Statistical Model (continued)

Let observations be characterized by

vectors

with denoting integer midpointvalue of an interval of the range ofpossible values of an observed

quantity.

m,1,2,...,j },a~,...,a~{a ,)a,...,a(jjk1jj

Tm1 a

ija~

Statistical Model (continued)

Denote by

where ,all values of a

defined by possible values of theircomponents .

Further, let be theprobability for obtaining vector measurements , and

K1,2,...,j ,)a,...,a( Tjm1jj a

m21 kkkK

ja

K1,2,...,j ,pj

a

ja 1pK1j ja

Statistical Model (continued)

Further, let a random vector

take the value

if sums of observations of j th components of vectors for, say,n sequential dates are

Tm21 )X,...,X,X(X

Tm1 )r,...,r(r

K

1ijij rla

ia

m,1,2,...,j },a~{maxnr}a~{minn jiki1

jjiki1 jj

Statistical Model (continued)

where denotes the number of observed

vectors in a sample and the values of are nonnegative integers such that

Then the probability that a random vector X will take a definite value

r = can be written down as

K1,...,i ,li

a

iai

la

nlK1i ia

Tm21 )r,...,r,r(

Statistical Model (continued)

where is the vector of parameters and the

summation is performed over all sets of nonnegative solutions

of the system of linear diophantine equations

K

1i

l

K

1i

i

i

p!l

!nia

a

a

p)r,P(X

T)p,...,p(Ki aap

K1,...,i ,li

a

K

1i

K

1ijij

.nl

m,1,...,j ,rla

ia

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:

Using, say, maximum likelihood estimates of , it is possible to solve different optimization problems aiming to maximize the total exposure or

profit.

mrmm

mmmmmm rXrXP

rXrXPrXrXrXP

),...(

),...(),...,|(

11

111111

),...,|( 1111 mmmm rXrXrXP

Computational Computational ProblemsProblems

Computational Problems

Numerical calculations of conditional probabilities may become unachievable for a reasonable time on a computer for large samples ( ). In this case the limit distribution of the model may be used. The system of the first equations

can be written in matrix form as where is matrix of

coefficients and

rAl A mK

ija~ T)l,...,l(Ki aal

15n

K

ijij rla

1

m,1,...,j ,ia

Computational Problems

Under these notations for

the random vector X will have asymptotically the multivariate normal distribution

with the vector of means the covariance matrixwhere and D is the

diagonal matrix with probabilities on the main diagonal.

n

)n,n(Nm TAAAp

Apn

TAAn

TppD

jpa

Computational Problems

Using estimates of

conditional probabilities

may be evaluated with the help of, say, technique proposed by Vijverberg.

)n,n(Nm TAAAp

),...,|( 1111 mmmm rXrXrXP

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?