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

(3)

Since λ is normally less than 1, λ^{n }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

1965)

.

2.2.

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

(4)

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.

A_{t }= 1-e ^{(-ν Tt) }

+

λA

_{t-1 }

(5)

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

5