# of GRPs before diminishing returns set in. Below this threshold, the logistic

function imparts exponential returns.

A_{t }= 1/ (1+e

^{(-ν Tt)}) + λA

t-1

(6)

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),

A_{t+1 }= λA_{t }

(7)

In time period t+n, where n is the value of the half-life η, we have,

A_{t+n }

= A t / 2

(8)

Therefore from (3) and (4), and recursively substituting,

A t / 2 = λ n A t

(9)

# And,

λ^{n}=1/2

(10)

Finally taking n as the value of the Half-Life η, we have,

η = Log (0.5)/Log (λ)

(11)

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.

# 4. Conclusion

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.

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