of GRPs before diminishing returns set in. Below this threshold, the logistic
function imparts exponential returns.
At = 1/ (1+e
(-ν Tt)) + λA
As in the negative exponential model, the parameter ν can be used to model different saturation levels.
3. Half-Life Estimation
Advertising half-life, η is calculated in the same manner as estimating decay half-life for radioactive substances:
A s s u m e t h a t i n t i m e - p e r i o d t + n , A t w o u l d h a v e d e c a y e d t o A t / 2 . T h e r e f o r using equation (1) and assuming that no new advertising is present (so the e
first term in the equation equals zero),
At+1 = λAt
In time period t+n, where n is the value of the half-life η, we have,
= A t / 2
Therefore from (3) and (4), and recursively substituting,
A t / 2 = λ n A t
Finally taking n as the value of the Half-Life η, we have,
η = Log (0.5)/Log (λ)
Different values for λ can be empirically tested in an econometric model to estimate the half-life for an advertising program.
For example, a λ of 0.25 implies a half-life of 2.4 weeks.
Adstock transformation is an efficient and effective technique to incorporate nonlinear and dynamic advertising effects in sales response models. The alternative option of building a dynamic nonlinear model is both computationally expensive and complex to estimate.