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 with random components that take arbitrary integer values. The random variables denote the firm’s expenditures for different media for a given period,

Statistical Model (continued) and 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 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.

Statistical Model (continued) Let observations be characterized by vectors with denoting integer midpoint value of an interval of the range of possible values of an observed quantity.

Statistical Model (continued) Denote by where,all values of a defined by possible values of their components. Further, let be the probability for obtaining vector measurements, and

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

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

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

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.

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

Computational Problems Under these notations for the random vector X will have asymptotically the multivariate normal distribution with the vector of means the covariance matrix where and D is the diagonal matrix with probabilities on the main diagonal.

Computational Problems Using estimates of conditional probabilities may be evaluated with the help of, say, technique proposed by Vijverberg.

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

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?