imparts an infinite lag structure to this model, with the effect of the first Adstock term, approaching 0, as t tends to ∞.
This is a simple decay model, because it captures only the dynamic effect of advertising, not the diminishing returns effect.
This model is also approximately equivalent to an infinite distributed lag
model as shown below:
F r o m ( 1 ) w e h a v e , A t = T t + λ A t - 1
Recursively substituting and expanding we have,
A t = T t + λ T t - 1 + λ 2 T t - 2 + … λ n T t - n
Since λ is normally less than 1, λn will tend to 0 in limit as n tends to ∞. Therefore, this infinite polynomial distributed lag can be approximated by imposing a finite lag structure within an Almon distributed lag model (Almon
Log Decay Model
The Log Decay model applies a straightforward logarithmic distribution to the advertising variable
A t = L o g T t + λ A t - 1
This is a relatively inflexible non-linear specification of the Adstock model, as it doesn’t allow for varying saturation levels.
2.3.Negative Exponential Decay Model
The below formulation applies a negative exponential distribution to the basic Adstock formula, using two parameters, the ‘decay’ or lag weight parameter λ and the learning rate or saturation parameter ν.
At = 1-e (-ν Tt)
This model is comparatively more flexible as different values of the parameter ν can be empirically tested in a response model to correctly measure the level of current advertising saturation.
2.4.Logistic (S-Curve) Decay Model
Using a logistic distribution instead of negative exponential will impart an S- shape to the Adstock variable, implying an inflexion point or ‘threshold’ level