Advertising Adstock with Maximum Period Decay

Current advertising adstock transformations in all their forms assume an infinite decay function. This means 1 week of advertising can have an impact 100 weeks for initial airing. This is unrealistic.  In this article I’ll discuss a variation on advertising adstock called maximum period decay effect.

A typical decay factor for advertising adstock looks like the figure on the right. The graph will lead you to believe that after week 10 the adstock values are small, close to zero and can be treated as zeros. WRONG! These values are, indeed, close to zero but they aren’t zero. A human can ignore them but a computer won’t. In a regression analysis framework this will also causes a multicollinearity problems with multiple sequential variables as the adstock will continue after advertising is over and will function as two identical decreasing trend variables.

In my previous article on Advertising Adstock – Concept & Formula, advertising adstock was defined as
A_t = X_t + r·A_t-1 ; where r is the advertising adstock rate.

Mathematically speaking this formula can be rewritten as
A_t = X_t + r·X_t-1 + r²·X_t-2 + r³·X_t-3 + … + rⁿ·X_t-n ; where n is the maximum number of weeks available, i.e. n = t.

The mathematical short-hand notation is
A_t = Σ rⁱ·X_t-i ; 0 ≤ i ≤ maximum number of weeks available.

I redefine the maximum number of weeks available to be the maximum periods that week t will learn from the previous n weeks. So if we define n as 5, then the current advertising adstock will learn from the previous 5 weeks. This has many benefits as we limit the total effect of advertising. Coupons for example, aren’t expected to have unlimited decay. We also remove that ever-decaying trend factor and solve the multicollinearity problem for sequential variables.

The files below show this improvement of advertising adstock transformation with maximum period decay:
• Excel
• SAS
• R
• Python

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Advertising Adstock – Concept & Formula

Advertising adstock is a term used to measure the memory effect of advertising carried over from start of advertising.  For example, if a company advertises at a certain level in week 1, week 2 will have a portion of week 1 level. Week 3, in turn, will have a portion of week 2 level.  In other words, adstock is a percentage term that measures the decaying effect of advertising throughout the weeks.

The term that comes up often in response models where we try to measure the effect of advertising on sales or on purchase intent.  The models are usually regression based but are often published under names like Marketing Mix Models (MMM), Marketing Mix Optimization (MMO), Network-Effects and Hierarchical models.

The theory behind adstock is that marketing exposures build awareness in consumers’ minds.  That awareness doesn’t disappear right after the consumers see the ad but rather remains in their memory.  Memory decays over the weeks and hence the decay portion of adstock.

The formula for advertising adstock is A_t = X_t + adstock rate * A_t-1.

The files below show a simple implementation of advertising adstock transformation:
• Excel Adstock Transformation
• SAS Adstock Transformation
• R Adstock Transformation
• Python Adstock Transformation – coming soon