Stock Variables — Adstock — Dynamic Effects

One approach to capturing carry over effects is the use of stock variables, adstock for instance. Adstock implicitly distributes the amount of an advertising exposure over several periods. Advertising that is effective at a given time is equal to residual adstock (what is “left over” from previous advertising), plus learning (adstock gained from current advertising). Specifically, if A1, A2, ... At represent the advertising effort (GRP) in periods 1 to t, then the adstock is computed as follows:

$$ Adstock_t=\frac{1-r}{f(1-r)+r}(fA_t+rA_{t-1}+r^2A_{t-2}\, ...\, +r^{t-2}A_2+r^{t-1}A_1),$$ $$A_t=GRP_t.$$

Where, r is the retention (or decay) rate, and f (fade) is the impact in the first period. These two parameters may either be pre-fixed by the modeller or determined by the data.

Half-life, the time duration by which the advertising effort has had half its total effect, is a commonly used benchmark for setting f and r.


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