Understanding Marketing Mix Modelling: Part I

Introduction

According to Nielsen, Marketing mix modeling (MMM) is the use of statistical analysis to estimate the past impact and predict the future impact of various marketing tactics on sales. These marketing tactics typically consist of controlling certain variables, often summarized as the four Ps of marketing: product, price, promotion and place(McCarthy 1978). Some examples of variables related to the product are newness of the product, differentiation from competitors, etc. Price refers to the product’s list price and incentives such as discounts and deals. Promotion refers to advertising and displays. Place refers to the delivery of the product measured by variables such as distribution, availability and shelf-space (Tellis 2006).

This article attempts to provide a mathematically rigorous introduction to MMM, and is targeted towards readers with some quantitative background, but no previous experience in marketing science or econometrics. We mainly consider the effect of price and promotion in this article as these factors are readily quantifiable. We begin by formulating the regression problem. We then incorporate certain stylized facts about the impact of price and promotion on sales into the model. We then show how this analysis can e performed in R using a well-known dataset.

Problem Formulation

A common methodology employed to study the effect of a response variable (sales, in this case) on certain explanatory variables (the four Ps) is multivariate linear regression. Let \(y_t\) denote the sales at time \(t\). Typically, sales volume in units sold is used and not revenue from sales. This is because the revenue from sales is dependent on the price, which is an explanatory variable. Let \(x_{it}\), \(i = 1, \ldots, K\) denote the \(K\) explanatory variables. These could include the price, measures of ad exposure, shelf placement, store location etc. To estimate the effect of each of the explanatory variables on the sales volume, we fit a regression model given by \[ \tag{1} y_t = \beta_0 + \sum_{i = 1}^{K}\beta_ix_{it} + \varepsilon_t \] where \(\beta_i\), \(i=0,\ldots,K\) denote the regression coefficients. Assuming the Gauss-Markov assumptions hold, the ordinary least squares (OLS) estimators for \(\beta_i\), denoted by \(\hat{\beta}_i\) are the best linear unbiased estimators. In practice, however, this is rarely the case since the process \(y_t\) is seldom stationary or independent. In a later section we see how we can adjust for this using a practical example.

There are several key stylized facts about the effect of advertising on sales

1. Exposure to advertising today has effects on consumers’ purchasing decisions in the future (Carryover Effect)
2. The effect of ad exposure on purchasing decisions decays with time (Carryover Effect)
3. The effect of ad exposure on sales is subject to the law of diminishing returns (Shape Effect)
4. Changing the level of ad exposure brings about a relative change in sales volumes (Shape Effect)

There are several other effects that can be taken into consideration like those of competition, content and advertising media(Tellis 2006). However, to avoid complicating the model, we focus on these effects for now. Let us look at these one by one and express them mathematically

Carryover Effects

Consider the model in \((1)\). Let us simplify the model, wherein the sales at time \(t\) can by explained by a constant base value and the additional effect of ad exposure at time \(t\), measured by a variable \(x_{t}\). This is denoted by \[ y_t = \beta_0 + \beta_1 x_t + \varepsilon_t \] We can incorporate Effects 1 and 2 into the above model by incorporating the lagged effects of all previous ad exposures. This model is given by \[ y_t = \beta_0 + \beta_1 s_t + \varepsilon_t \\ s_t = x_t + \lambda s_{t-1} \] where \(s_0\) is typically set to \(0\).

Shape Effects

To clearly express Effect 3, we rewrite the above equation as follows \[ y_t = \beta_0 + \beta_1 x_t + \lambda \beta_1 s_{t-1} + \varepsilon_t \] The above expression implies that the rate of change of sales volumes with respect to ad exposure in the same time period is constant, i.e, \(\partial y_t/ \partial x_t = \beta_1\). However, Effect 3 implies that this rate tapers off as we increase ad exposure and therefore the following must hold true \[ \lim_{x_t \to \infty} \frac{\partial y_t}{\partial x_t} = 0 \] Several functions satisfy this property(Joy 2006), e.g, the log function and the logistic function. We pick the logistic function with parameter \(\alpha\) for its flexibility. The model now becomes \[ \tag{2} y_t = \beta_0 + \beta_1 s_t + \varepsilon_t \\ s_t = \frac{1}{1+ e^{-\alpha x_t}} + \lambda s_{t-1} \] Now, \((2)\) implies that the increase in sales from increase in ad exposure is independent of \(y_t\). In other words, it is absolute. This means that increasing ad exposure by one unit cause the sales to increase by a certain amount irrespective of the magnitude of \(y_t\). For example, assume \(\beta_0 = 0\), \(\beta_1 = 1\), \(\alpha = 1\), \(\lambda = 0\). Let us increase \(x_t\) to \(x'_t\). Assuming the increment is small enough, We can approximate the new sales value by \[ y'_t \approx y_t + \frac{\partial y_t}{\partial x_t}(x'_t - x_t) \] If \((x'_t - x_t) = 0.1\), and \(x_t=1\), we have \(y'_t - y_t \approx 0.2\). Therefore, sales increase by 0.2 units regardless of whether \(y_t\) is 100 or 1000. In practice, we usually see that sales increase in a relative manner, i.e by a percentage rather than in absolute terms. To take this into account we employ a logarithmic transformation on the dependent variable and obtain \[ \log y_t = \beta_0 + \beta_1 s_t + \varepsilon_t \\ s_t = \frac{1}{1+ e^{-\alpha x_t}} + \lambda s_{t-1} \] It is trivial to check that \({\partial y_t}/{\partial x_t}\) indeed depends on \(y_t\) after this transformation.

Impact of Price

In order to bring price into the model, we rely on the price elasticity of demand, a measure of responsiveness of the quantity of the good demanded to a change in its price. We can argue that sales volumes directly reflect the demand for the product, assuming that production can keep up with demand linearly. The price elasticity \(E\) is expressed as \[ E = \frac{\partial q_t/q_t}{\partial p_t /p_t} \] where \(p_t\) is the price of the product at time \(t\) and \(q_t\) is the quantity of the product demanded. Assuming sales volumes are directly proportional to quantity demanded, we can write \(y_t = k q_t\), where \(k\) is a constant of proportionality. Using this in the above expression we have \[ \frac{\partial \log y_t}{\partial p_t} = kE \log p_t \] We can readily merge \(kE\) into one term and include the above expression into the regression model. Finally, we have \[ \log y_t = \beta_0 + \beta_1 s_t + \beta_2 \log p_t +\varepsilon_t \\ s_t = \frac{1}{1+ e^{-\alpha x_t}} + \lambda s_{t-1} \] We see that the model has 5 parameters that need to be estimated, and it is nonlinear in \(\lambda\) and \(\alpha\). In what follows we fix \(\lambda\) and \(\alpha\) to regain linearity in the parameter space. To fix \(\lambda\) we use the empirical observation that the half-life of an ad copy is approximately 2.5 weeks. This yields \(\lambda=0.25\). We will simply assume \(\alpha=2\), though in reality, this is calibrated more carefully.

An Example

We use the Sliced Cheese Dataset from the bayesm package to illustrate how to fit the model.

library(bayesm)
data("cheese")
summary(cheese)

                             RETAILER        VOLUME            DISP             PRICE      
 BALTI/WASH - SAFEWAY            :  68   Min.   :   231   Min.   :0.00000   Min.   :1.320  
 BALTI/WASH - SUPER FRESH        :  68   1st Qu.:  1990   1st Qu.:0.00000   1st Qu.:2.457  
 BIRMINGHAM/MONTGOM - KROGER     :  68   Median :  3408   Median :0.04737   Median :2.703  
 BOSTON - STAR MARKET            :  68   Mean   :  4771   Mean   :0.11428   Mean   :2.869  
 BUFFALO/ROCHESTER - TOPS MARKETS:  68   3rd Qu.:  5520   3rd Qu.:0.16600   3rd Qu.:3.203  
 BUFFALO/ROCHESTER - WEGMANS     :  68   Max.   :148109   Max.   :1.00000   Max.   :4.642  
 (Other)                         :5147                                                     

The data consists of weekly observations of the volume of sales of sliced cheese over different retailers in the United States. Here, VOLUME corresponds to \(y_t\), DISP to \(x_t\) and PRICE to \(p_t\). We are not yet looking at hierarchical models, so we focus on one retailer, say Dominick’s in Chicago. We then apply the required transformations to obtain the regression dataset. these are shown below.

library(dplyr)

adstockTransform <- function(x){
  stats::filter( 1/(1+exp(-2*x)), 0.25, method = "recursive")
}

mmm.data <- cheese %>% filter(RETAILER == 'CHICAGO - DOMINICK') %>%
  select(-RETAILER) %>% 
  mutate(log.volume = log(VOLUME), log.price = log(PRICE), adstock= adstockTransform(DISP)) %>%
  select(-VOLUME, -DISP, -PRICE)

After checking the ACF and PACF of log.volume to ensure stationarity and independence, we fit a linear model

fit <- lm(log.volume ~ adstock + log.price, data=mmm.data)
summary(fit)

Call:
lm(formula = log.volume ~ adstock + log.price, data = mmm.data)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.53274 -0.13505 -0.02216  0.08934  0.58237 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  11.6611     0.3029   38.50   <2e-16 ***
adstock       0.6719     0.2615    2.57   0.0128 *  
log.price    -2.9680     0.1442  -20.58   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.2468 on 58 degrees of freedom
Multiple R-squared:  0.9016,    Adjusted R-squared:  0.8982 
F-statistic: 265.6 on 2 and 58 DF,  p-value: < 2.2e-16

From the above, we see that the regression coefficients are statistically significant with \(R^2 = 0.8982\). as expected, we can infer that ad exposure has a net positive effect on sales volumes. Furthermore, we find that the price elasticity is estimated to be -2.96, indicating that increasing the price has a negative impact on sales.

In Part II, we will see how we can estimate the same model without fixing \(\lambda\) and \(\alpha\) using a Bayesian approach. In Part III, we will construct a hierarchical model to guage the impact of another P, Place on the sales volume.

References

Joy, J. V. 2006. “Understanding Advertising Adstock Transformations.” https://ssrn.com/abstract=924128.

McCarthy, J. E. 1978. Basic Marketing: A Managerial Approach. Homewood, IL: RD Irwin.

Tellis, G. J. 2006. “Modeling Marketing Mix.” In The Handbook of Marketing Research, edited by M Grover R. & Vriens, 506–22. Thousand Oaks, CA: SAGE Publications Ltd.