Advertising adstock is the memory effect of advertising carried over from start of advertising; e.g., if a company advertises at a certain level in week 1, week 2 will have a portion of week 1 level, and so on. Adstock is a percentage term that measures the decaying effect of advertising throughout the weeks.

The idea behind this is that marketing exposures build awareness in consumer’s minds. That awareness doesn’t disappear right after the consumers see the ad but rather remains in their memory. Memory decays over the weeks and hence the decay portion of adstock.

There are two dimensions to advertising adstock:

• decay or lagged effect.
• saturation or diminishing returns effect.

Advertising Lag: decay effect

Exposure to ads build awareness in the consumer market, resulting in sales. Each new exposure increases awareness to a new level. The decay level of adstock eventually reduces awareness to its base level, until this decay is reduced by new exposure. This decay is expressed in terms of 'half-life' of the ad. A two-week half-life means that it takes two weeks for the awareness of an ad to decay to half its present level.

The formula for advertising adstock is

where $A_t$ is the adstock at time $t$, $T_t$ is the value of the advertising variable at time $t$, and $\lambda$ is the decay or lag weight parameter.

Advertising Saturation: Diminishing Returns effect

Increasing the amount of advertising increases the percent of the audience reached by the advertising, hence increases demand, but a linear increase in the advertising exposure doesn’t have a similar linear effect on the demand. Each incremental amount of ad causes a progressively lesser effect on demand increase. This is advertising saturation. Saturation only occurs above a threshold level that can be determined by Adstock Analysis.

Adstock can be transformed to an appropriate nonlinear form like the logistic or Negative Exponential Distribution, depending upon the type of diminishing returns or saturation effect the response function is believed to follow.

A little example in R

 1# Program Name: Adstock Transformation
 2# Written By  : Gabriel Mohanna
 3# Date Created: Feb 23, 2014
 4# Narrative   : A simple advertising adstock transformation.
 5
 6# Define Adstock Rate
 7adstock_rate = 0.50
 8
 9# Create Data
10advertising = c(117.913, 120.112, 125.828, 115.354, 177.090, 141.647, 137.892,   0.000,   0.000,   0.000,   0.000,  0.000,   0.000,   0.000,   0.000,   0.000,   0.000, 158.511, 109.385,  91.084,  79.253, 102.706,  78.494, 135.114, 114.549,  87.337, 107.829, 125.020,  82.956,  60.813,  83.149,   0.000,   0.000,  0.000,   0.000,   0.000,   0.000, 129.515, 105.486, 111.494, 107.099,   0.000,   0.000,   0.000,  0.000,   0.000,   0.000,   0.000,   0.000,   0.000,   0.000,   0.000)
11
12# Calculate Advertising Adstock
13# Credit: http://stackoverflow.com/questions/14372880/simple-examples-of-filter-function-recursive-option-specifically
14adstocked_advertising = filter(x=advertising, filter=adstock_rate, method="recursive")
15
16# Alternative Method Using Loops Proposed by Linh Tran
17adstocked_advertising = numeric(length(advertising))
18adstocked_advertising[1] = advertising[1]
19for(i in 2:length(advertising)){
20    adstocked_advertising[i] = advertising[i] + adstock_rate * adstocked_advertising[i-1]
21}
22
23# Graph Data
24plot(seq(1,length(advertising)), advertising, type="h", 
25     xlab="Time (Usually in Weeks)", ylab="Advertising", 
26     ylim=c(0, max(c(advertising, adstocked_advertising))), 
27     frame.plot=FALSE)
28lines(adstocked_advertising)