Carryover (Adstock) Functional Forms

Summary

The adstock transformation captures advertising’s carryover (lag) effect by replacing current spend with a finite, normalized weighted average of current and past spend over $L$ periods. Jin et al. give two weight functions: geometric decay (effect peaks at the exposure period and decays by a retention rate ${\alpha }_m$) and delayed adstock (effect peaks ${\theta }_m$ periods later, a Gaussian-shaped radial kernel). Geometric adstock is the continuous analog of the Koyck distributed lag.

Overview

Some media build effect immediately (peak at exposure); others (e.g. brand TV) take time to peak. The adstock function transforms the raw spend time series so the regression sees the cumulative media effect rather than instantaneous spend. The maximum carryover duration $L$ truncates the window; for the chosen simulation parameters $L=13$ weeks approximates infinity (weights $<10^{-7}$ beyond 13 weeks), and $L=13$ is also used in the real-data application.

Main Content

Adstock transformation (finite-window, normalized)

$$\text{adstock}(x_{t-L+1,m},\dots ,x_{t,m};w_m,L)=\frac{{\sum }_{l=0}^{L-1}{w}_m(l)x_{t-l,m}}{{\sum }_{l=0}^{L-1}{w}_m(l)},$$

where $x_{t,m}$ is spend of channel $m$ at week $t$, $w_m(\cdot )\ge 0$ is a weight function, $L$ is the maximum carryover duration (common across media for simplicity). The denominator normalizes the weights so adstock is a proper weighted average (preserving the spend scale). A large $L$ approximates an infinite window. (Eq. 1)

Geometric decay weights

$$w_m^g(l;{\alpha }_m)={\alpha }_m^l,l=0,\dots ,L-1,0<{\alpha }_m<1.$$

${\alpha }_m$ is the retention rate of the ad effect from one period to the next. The effect peaks at the same period as the exposure ($l=0$) and decays geometrically. This is the discrete geometric distributed lag — the marketing analog of the Koyck lag (see Carryover Effects and Distributed Lags). (Eq. 2)

Delayed (peak) adstock weights

$$w_m^d(l;{\alpha }_m,{\theta }_m)={\alpha }_m^{(l-{\theta }_m{)}^2},l=0,\dots ,L-1,0<{\alpha }_m<1,0\le {\theta }_m\le L-1.$$

${\theta }_m$ is the delay of the peak effect: the weight is maximized at lag $l={\theta }_m$ rather than at $l=0$. The form is proportional to a normal density with mean ${\theta }_m$ and variance $-1/(2\log {\alpha }_m)$ — equivalently the radial (Gaussian) kernel used in local regression (Friedman, Hastie & Tibshirani 2009, p. 212). Setting ${\theta }_m=0$ recovers geometric decay. Other delayed forms (e.g. negative-binomial density, Hanssens et al. 2003) work too. (Eq. 3)

Intuition (Figure 1). With the same ${\alpha }_m=0.8$: geometric adstock decays monotonically from lag 0; delayed adstock with ${\theta }_m=5$ rises to a peak around lag 5 then falls off — a “pulse” of delayed response.

Examples

Simulation generating parameters (Sec. 5, Table 1)

The simulated dataset uses delayed adstock. Per-media: retention $\alpha =(0.6,0.8,0.8)$, delay $\theta =(5,3,4)$ for Media 1/2/3, with $L=13$. Estimation recovery (Sec. 6): the delay $\theta$ is estimated with low bias and only moderate uncertainty even in small samples, while $\alpha$ shows somewhat more bias/uncertainty in small samples but is precise for large samples. Adstock parameters are recovered better for stronger-signal media (Media 1, 2) than the weak Media 3.

Real-data preference for geometric adstock (Sec. 8)

In the shampoo application the delay $\theta$ is not estimated well (its posterior nearly equals the uniform prior), so the authors prefer the simpler geometric adstock; the more flexible delayed form does not improve fit. See MMM Model Selection and Application.

Connections

• Direct generalization of the Koyck / geometric distributed lag model: Carryover Effects and Distributed Lags. The retention rate ${\alpha }_m$ plays the role of the Koyck carryover coefficient $\lambda$.
• Feeds the combined model in Bayesian Media Mix Modeling - Overview (adstock applied before Hill shape transform).
• Prior choices for ${\alpha }_m$ (beta/uniform on $[0,1)$) and ${\theta }_m$ (uniform on $[0,L-1]$): Bayesian Estimation and Priors for MMM.

See Also

• Shape (Saturation) Effects
• Carryover Effects and Distributed Lags
• Bayesian Media Mix Modeling - Overview
• Index: Bayesian Media Mix Modeling