General interpretation of Marketing Mix models

[andycrellin]

Hi,

This question is about interpreting Marketing Mix Models, to present to an average "senior management"-type group.

I'm trying to understand the process of going from a model to presentation for Marketing Mix modelling, and I'm hoping someone can give me some pointers. Specifically, I have a model that has log and non-log terms and also a lag of the dependent, so I'm trying to understand how to interpret the coefficients correctly to produce views such as :
- An optimal media mix (some but not all of my media get into the model)
- What proportion of sales each component is responsible for
- Return on media investment
- Diminishing returns curves, and optimal point of spend

My model looks like the following (although it actually has many more terms, for the sake of this question, I want to know how to deal with these _types_ of terms), with S being sales, A being some base var like average temperature, and B, C, D being various media volumes. B' has also been AdStocked (i.e. contains a proporton of t-1's value to model the slower decay of the effect of something like TV or Radio)

log(S_t)=β_0+λlog(S_[t−1])+β_1A_t+β_2log(B′_t)−β_3log(C_t)+β_4log(C_t∗D_t)+ϵ_t

I Have some specific questions that might help give the discussion direction (and I understand that I have the challenges of interpreting a multiplicative equation rather than a linear one!):

1. Do I need to change the way I interpret coefficients β, considering I have the autoregressive term, λlog(S_[t−1]), in there?
2. How do I get from these coefficients to a "due to" view - e.g. the proportion of S due to A or B, etc? I know that we end up a multiplicative relationship (because of the log-log nature of the model) but I've seen tantalising references to doing this, but haven't found a solid method. Various presentations on Marketing Mix Modelling have shown the decomposition as a stacked chart summing to total sales, which is what I'm aiming for.
3. Can the effect of DD on SS be estimated, even though it is only in as part of an interaction?
4. How do I calculate the optimal marketing mix? Is it simply the ratios of the coefficients, or is special treatment required?
5. Do I need to treat β2β2 in a special way to understand the effect of media BB, considering it has been "AdStocked"?
6. If I didn't have the autoregressive term in there, would any of the answers to 2 to 5 differ in a significant way?

I'm a generalist rather than a statistician, so am looking for real-world methods and processes if possible, rater than statistically rigorous treatments.

The question is also set on Cross Validated here, with much better formatting!: http://stats.stackexchange.com/question ... -mix-model

Many thanks for any help!

Andy.