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.
