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} + r^{2}·X_{t-2} + r^{3}·X_{t-3} + … + r^{n}·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 r^{i}·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