Short answer
Adstock models the carry-over effect of advertising - the fact that a TV ad seen this week still drives sales next week. Saturation models diminishing returns - the fact that the tenth million pounds in a channel does less than the first. Both are essential ingredients in any credible MMM.
Advertising rarely converts in the week it is seen. Adstock is the mathematical transformation that spreads a media impulse across future weeks with a decay rate. Geometric adstock uses a single decay parameter (retention rate); Weibull adstock allows a delayed peak, which fits TV and brand media better. Half-life ranges from one to eight weeks depending on channel and message.
The first pound in a channel finds a highly-motivated audience cheaply. The hundred-millionth pound has already reached them and is chasing marginal viewers. Saturation captures this curve, usually via a Hill function or Michaelis-Menten form. Without saturation, MMM produces straight-line ROI and encourages infinite spend - which is nonsense.
Adstock and saturation together determine the marginal ROI curve for every channel - and marginal ROI is what drives budget optimisation. A model with wrong curves produces confidently wrong reallocations. Modern MMM (Meridian, Robyn, PyMC-Marketing) fits these shapes with Bayesian priors and validates them against experiments.
Digital performance channels have short adstock (half-life under 2 weeks) and saturate fast. TV and brand have longer adstock (4-8 weeks) and saturate more slowly. Sponsorship and OOH have the longest carry-over. Priors from prior campaigns and industry benchmarks are essential where data is thin.
Bayesian MMM, calibrated with experiments, refreshed monthly, delivered as a decision system.
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