Adstock: the assumption that last week's spend is still selling this week

Adstock is the transformation applied to a media series before a mix model is fitted: this week's model input is this week's spend plus a share of last week's input. That share is the retention rate, usually quoted as a half-life. It encodes a belief you impose rather than a fact the data hands you.

The transformation happens before any modelling decision anyone normally argues about. Choose a 1-week half-life and a channel's effect has nearly gone a fortnight later; choose 4 weeks and half its transformed weight sits after the campaign ended. Same spend, same sales, different answer.

What the transformation does to a spend series

Geometric adstock is 1 line: this week's adstocked value is this week's spend plus the retention rate times last week's. Retention rate and half-life are the same parameter in 2 units — 1 week is 0.50, 2 weeks roughly 0.71, 4 weeks roughly 0.84. Below, a 3-week flight of 100 a week, transformed 3 ways.

Two properties fall out of that arithmetic. Total transformed volume is raw spend divided by 1 minus the retention rate, so the 4-week version carries roughly 6 times the raw total and the 1-week version twice it. And in week 6 the long-memory series carries 151 units against 22 — nearly 7 times more media pressure available to explain any late sales.

Why the data usually cannot tell you the half-life

Fit a channel that ran at a steady level all year and the adstocked series is a near-constant multiple of the raw one. The model sees the same shape at every candidate half-life, and only the product of the coefficient and the long-run multiplier is pinned down: halve the coefficient, lengthen the memory to compensate, and the fitted contribution is identical.

What identifies a decay rate is variation the decay can explain: abrupt starts, hard stops, dark weeks, bursts of very different weight. A flighted television plan carries that information; a search budget held within a few percent of the same number every week carries almost none. It is the same identification problem behind a channel handed a negative coefficient.

What the choice does to the number a client sees

The damage is rarely in the total. It is in the split between channels and the timing of credit. Give brand media a 4-week memory and search a 1-week memory, and every sale in the fortnight after a burst has far more transformed brand pressure available to explain it. Reverse the priors and those weeks read as search converting demand that was already there.

That is a budget decision made by a parameter nobody chose deliberately. It is also why mix modelling and touch attribution disagree so often — different questions, different memories, as separated in mix modelling, touch attribution and experiments. Adstock is memory inside the model; a lookback window is memory inside the counting rule that produced the outcome column, which is the lookback window.

A half-life is a budget decision wearing the clothes of a preprocessing step, and it is usually made by whoever wrote the default in the script.

Choosing a half-life you can defend in the meeting

1. Set a candidate range per channel from the mechanism, not from a table. A channel intercepting existing intent decays in days; a channel building memory for a considered purchase plausibly decays over weeks.
2. Check whether the spend history can identify anything. Plot weekly spend per channel; if it never stops and never spikes, declare the half-life assumed rather than estimated, in writing.
3. Sweep the range and report the spread. Fit at each candidate, record the contribution each produces, and carry the minimum and maximum into the readout instead of the best fit alone.
4. Pin it with an experiment where the money justifies it. A geo holdout measures the post-flight tail directly and its interval bounds the plausible half-lives, provided it was powered — the failure examined in the test that came back saying nothing.
5. Freeze the assumption for the reporting year and log every change. A half-life edited between quarters silently restates history, and the restatement reads as a performance change.

Two mechanics deserve a sentence each. Delayed adstock, where the peak lands a week or 2 after the spend, fits media people notice and act on later; it adds a parameter you also cannot identify from a flat spend series. Saturation — the curve making the tenth unit worth less than the first — is a separate transformation applied in a documented order, not an alternative to adstock.

Teams fitting these models repeatedly end up with a small internal tool that stores the assumption set beside every result, which is why we treat it as a product build rather than a spreadsheet. An agency doing it across clients also has to keep those weekly tables in one warehouse without any client's data meeting another's, worked through in one warehouse, many clients who never overlap. Both sit inside the attribution, incrementality and mix work we do for marketing and advertising teams.

• mix modelling
• adstock
• measurement
• media