The note discusses the most commonly used types of marketing mix model (MMM) specifications: the additive and multiplicative models. We discuss the strength and weaknesses of each. We argue that multiplicative models offer a more realistic representation of reality than simpler additive models.

MMM specification can have a simple linear additive form as presented in the equation below. For simplicity we use only two independent variables A and P, denoting advertising and price, respectively, to model sales S

 St = α0 + α1Pt + α2At + εt (1)

where parameters α’s are estimated coefficients of independent variables on sales. Subscript t denotes time periods and εt stand for errors.

The main advantage of this specification is its simplicity. If synergies and interactions are not explicitly captured by variables, the model estimates the discrete impact of each variable on sales. The contribution of each factor to sales in period t can be easily calculated as the product of explanatory variable and respective coefficient, e.g. α2At stands for the contribution of advertising to sales in period t.

The coefficient captures an absolute effect of unit change in explanatory variable on modeled KPI:

α = ΔDependent_Variable/ΔExplanatory_Variable.

This means that unit increase in advertising (A) drives α units impact on sales (S).

The additive model implies a constant absolute effect of each additional unit of explanatory variable. Therefore, they are suitable only if brand sales take place in the stable environment and they are not affected by a range of interacting factors. But is this assumption realistic? Let’s take a price effect as an example. If a brand starts with a certain base sales level and later doubles its distribution, that leads to a significant increase in sales, the absolute effect of unit price change will not be the same. As sales increase on the back of stronger distribution, the absolute effect of price change will also increase proportionally to the rise in sales. The same holds for seasonal sales. The effect of the same change in explanatory variable – price, advertising, etc – will be larger at the peak of the season compared to the lower season. Also, as the price point increases, price sensitivity might not stay the same. A similar line of argument applies to other factors.

Modelers are aware of the described non-linearities and they have been looking at other model forms that help to overcome the limitations of additive models. The alternative model forms are semi-logarithmic and logarithmic models, referred to as multiplicative models.

### Multiplicative models

The multiplicative model derives its name from the fact that independent variables of marketing mix are multiplied together. This is done in two ways. First, the exponents of factor contributions are multiplied, or second, the independent variables are multiplied. We look at these two forms separately.

### Semi-logarithmic models

In the first type of multiplicative models – the exponents of explanatory factors are multiplied.

 St = exp(α0) · exp(α1Pt) · exp(α2At) · exp(εt) (2)

Products of exponents can be rewritten as

 St = exp(α0 + α1Pt + α2At + εt) (3)

Logarithmic transformation linearizes the model form, which in turn can be estimated as an additive model.

 ln(St) = α0 + α1Pt + α2At + εt (4)

The only difference between the additive and semi-logarithmic model is that the latter one has the modelled variable S expressed as a logarithmic transformation. However, the implications are significant.

The semi-logarithmic model provides benefits in several critical aspects.

First, each factor – the explanatory variable in the model – works on top of what has been already achieved by other factors. The response coefficients, interpreted as % change in sales to unit change in explanatory variables,

100 · α = %ΔDependent_Variable/ΔExplanatory_Variable,

are closer to real life situations. As brand sales get stronger on the back of improved distribution, or temporarily due to the season or price promotions, advertising uplift will be proportionally stronger compared to the situation, when sales are at lower levels.

Second, the point above implies that the modeled variable is driven by interactions of all factors in the model. In other words, explanatory factors have synergistic effect on the modeled variable. In many situations, the variables of marketing mix indeed interact and their simultaneous effect enhances sales more than the sum of the two effects occurring alone.

In some, the semi-logarithmic model is flexible enough to capture relationships with non-linear shapes.

### Logarithmic models

In the second type of multiplicative model, explanatory variables are multiplied together.

 St = exp(α0) · (Pt ^ α1) · (At ^ α2) · exp(εt) (5)

We can linearize the model through logarithmic transformation.

 ln(St) = α0 + α1 ln(Pt) + α2 ln(At) + εt (6)

In this case, all model variables are subject to logarithmic transformation. Benefits of this model are very similar to those of semi-logarithmic model. The main difference lies in the interpretation of response coefficients. While in the semi-logarithmic models α’s stand for % change in sales in response to unit change of explanatory variable, here they denote elasticity α = %ΔDependent_Variable/%ΔExplanatory_Variable, i.e. % change in sales in response to 1% change in explanatory variable. This implies constant elasticity of sales to explanatory factors. The constant elasticity feature is sometimes pointed out as a drawback of the logarithmic model. It is up to modeler’s considerations if the assumption of constant elasticity is optimal.

The following tables summarize three discussed functional forms. We assume a simple model with one dependent and one explanatory variables, y and x, respectively, with β as response coefficient.

Table 1: Summary of MMM functional forms

 Model form Dependent Variable Independent Variable Interpretation of β Marginal effect of Δx Additive y x β = Δy / Δx β Semi-logarithmic ln(y) x 100 · β = %Δy / Δx y · β Logarithmic ln(y) ln(x) β = %Δy / %Δx y · β / x

In semi-logarithmic models, elasticity cannot be directly estimated but can be calculated from the coefficient as β · X for every time period. It increases in absolute value with the explanatory variable.

### Which model to use

The right model form depends on what sort of dynamics we expect in the model.

First, what is a likely pattern of modeled variable response to unit change in explanatory factors? And second, what interactions between explanatory variables do we expect?

Additive model form is suitable only if unit increase in explanatory variables generates the same response in the modeled variable regardless the sales level dynamics. This means that for example if β = -100, unit price increase drives 100 unit decline in sales. This relation holds also when base sales double or fall to half of its original level.

This model is therefore suitable only if the values of explanatory factors move within the contained range and when the base level of model metric is relatively flat. Additive models are not appropriate for brands with large variation in modelled KPI, e.g. seasonal brands or brands with structural breaks in sales.

In addition, the additive model does not capture any possible synergies between explanatory variables. If such synergies exist, their parts correlated with individual explanatory variables are allocated to them. However, a proportion of variation may stay unexplained – as part of residuals. Such models suffer from omitted variable bias and autocorrelation of residuals. One way to fix this is to construct interaction variables and estimate synergies through the constructed variables. This requires good knowledge of model dynamics, variable pre-processing and/or lots of testing, which synergies are significant.

For the reasons above, semi-logarithmic models became popular in MMM. All it requires is to apply logarithmic transformation on the dependent variable.

On the plus side, a response to unit change in explanatory variable is more flexible. The absolute level of response increases proportionally with the modeled variable.

The synergy effects are integral part of semi-logarithmic models. The magnitude of synergies between variables rises with the overlapping positive impact of individual explanatory factors and with the number of explanatory variables in the model.

Logarithmic models offers similar benefits as semi-logarithmic in terms of synergies and more flexible response, compared to additive models.

The main aspect in which the logarithmic model differs from the semi-logarithmic one is that estimated coefficients are elasticities, allowing comparisons of response coefficients between models.

The choice between the two therefore depends on the objectives of modeling and the type of questions we want to answer.

If MMM is focused on measuring the effect of advertising, media variables are subject to several transformations to capture carryover effect and diminishing returns. With multiple transformations, their response coefficients contain little information and generally are not reported. Therefore, in this case, semi-logarithmic model is a sufficient and more practical alternative.

On the other hand, price and promotion discount elasticities are of the main interest in MMM focused on price sensitivity and effectiveness of store activity. In these models, the key variables are not subject to a range of transformations. Logarithmic models are more optimal to use.

### Factor Decomposition of Additive and Semi-Logarithmic Model

Factor decomposition means quantification of incremental contributions of individual factors to the modeled variable. This is the main output of MMM essential for assessing the effect of all marketing activities and calculating return on marketing investment (ROI).

The procedure is straightforward for additive models. We take Equation 1 introduced above as a starting point.

Given that some variables, such as price and distribution and some external factors are continuous over time and they enter the dataset with positive values their base effect at the beginning of the modelled period or alternatively their average effect should be included in the base. In Equation 1, a price is the type of variable where the negative effect is present from the first time period.

Essentially, starting value of α1Pt in period t=1 or the average value of this term is subtracted from each α1Pt to get rebased contribution of its change to sales. All subtracted parts are then added to estimated α0 to get correct base sales level. For most of ATL marketing activities, such as advertising, price promotions, displays, the entire α2At accounts for the effect of advertising in period t.

This gives us a model form with rebased factor contributions

 St = base + Contributions_of_Pricet + Contributions_of_Advt + εt (7)

Where Contributions_of_Price and Contributions_of_Adv stand for the contributions of price and advertising to sales, respectively.

As argued above, semi-logarithmic functional form is a preferred form for MMM. Given that the model is multiplicative, its factor decomposition is less straightforward.

One of the approaches used in the industry is the bottom-up approach where in the first step individual net parts of factor contributions are separated from the base. In the second step, unaccounted part of the model variation is distributed among factors in a weighted manner.

This approach provides a good approximation of the actual effect of individual factors, while keeping all benefits of multiplicative model.

### To Sum up

It is important to understand the implications of functional form on the results. The functional form determines the level of flexibility and the depth of insights the model provides. It affects how much of modeled variable variation is attributed to individual explanatory factors. In more complex environments, simple additive model cannot well reflect the market dynamics.

Given that it is possible to automate model decompositions; multi-step procedure should not deter marketing mix modelers from using appropriate functional forms. Models that more accurately reflect the market - leads not only to more precise ROIs, they also enable more reliable projections and scenario simulations of new marketing strategies.

Sydney, 26 March 2013,

Elena Yusupova, PhD