Adstock Methodology

1. Methodology Overview

Adstock uses a frequentist Marketing Mix Modeling (MMM) framework built on Ridge regression with geometric adstock transformations. The methodology follows the same foundational econometric approach used by Google's Meridian and Meta's Robyn, adapted for speed-to-insight and accessibility.

The model ingests daily time-series data of marketing spend and revenue, applies carryover (adstock) transformations to capture lagged advertising effects, and fits a regularized linear model with non-negativity constraints to decompose revenue into channel-level contributions.

2. Data Requirements

• Input format: Daily time-series data with a date column, revenue (target variable), and one or more spend_<channel> columns representing daily marketing spend per channel.
• Channel auto-discovery: The engine automatically identifies channels via column prefixes. spend_ columns are treated as spend features; impr_ columns as impression features. Both are included as model predictors.
• Minimum recommended data: 120–180 days for a 5-channel model, providing approximately 10–20 observations per estimated parameter. Shorter datasets will produce results but with reduced statistical reliability.
• Holiday support: An optional binary is_holiday column (0 or 1) allows the model to control for holiday-driven revenue spikes that are not attributable to marketing spend.

Deployment flexibility: Adstock can be deployed within a client's own cloud environment (Azure, GCP). Marketing data never needs to leave the client's infrastructure, satisfying enterprise security and data governance requirements.

3. Adstock Transformation (Carryover Effect)

Note: In the Adstock dashboard, adstock is referred to as “carryover” — a plain-English term for the same concept. The industry-standard term “adstock” is used throughout this methodology page for consistency with academic literature and frameworks like Google Meridian and Meta Robyn.

Advertising impact is not instantaneous. An ad exposure on day t can continue to influence consumer behavior on days t+1, t+2, ... with diminishing effect. The adstock transformation models this carryover using geometric decay:

• α (decay rate) controls how much of the previous day's transformed value carries forward. Higher α = longer-lasting effect.
• Current implementation: Geometric decay applied uniformly across all channels with α = 0.5. This is equivalent to a half-life of exactly 1 day — 50% of the effect persists after 1 day, 25% after 2 days, 12.5% after 3 days.
• Purpose: Captures the lagged impact of advertising exposure on consumer behavior, allowing the model to attribute revenue to spend that occurred in prior periods.

Each adstocked column then enters the standardized feature matrix described in section 5, alongside the seasonality and trend controls described in section 4, before being passed to Ridge regression.

This is the industry-standard geometric adstock formulation used in both Google Meridian and Meta Robyn. See: Robyn Analyst's Guide, Google Meridian documentation.

4. Seasonality and Trend Controls

Marketing spend is rarely the only driver of revenue variation. Without explicit controls for time effects, the model can misattribute trend, weekly cycles, and annual seasonality to whichever marketing channel happens to correlate with them. v2 adds four control families to the design matrix that are estimated jointly with the marketing coefficients:

• Linear trend: A single integer time index (0, 1, 2, …, n−1) captures gradual long-run drift in the baseline — business growth, brand strengthening, or category-wide tailwinds and headwinds.
• Yearly seasonality: A Fourier expansion with order 4 (4 sine/cosine pairs at period 365.25 days). This captures smooth annual cycles — tax-season demand, summer slowdowns, Q4 build-up — without requiring a calendar of named events.
• Weekly seasonality: A Fourier expansion with order 2 (2 sine/cosine pairs at period 7 days). This absorbs day-of-week patterns common in DTC and retail data — weekend lifts, Monday troughs.
• Holiday flag: The optional is_holiday binary column retains its previous role as a step-function control for known one-day events.

Fourier seasonality is the same approach used by Facebook's Prophet library and is standard practice for smooth periodic effects. Estimating these controls jointly with marketing features — rather than applying a pre-detrending step — lets the Ridge penalty arbitrate between competing explanations for any given week's revenue.

5. Feature Standardization

The Ridge L2 penalty λ∑_jβ_j² treats every coefficient on the same scale. If one feature is measured in dollars (with values in the thousands) and another is a unit-bounded Fourier sine wave, the penalty effectively shrinks the dollar-scale coefficient much harder — not because that feature is less important, but because a coefficient producing the same revenue impact is much smaller in raw units. This is the classic scale-bias problem in regularized regression.

v2 corrects this by z-scoring every feature in the design matrix before fitting:

Ridge then sees features on a common scale, and the penalty is applied uniformly. After fitting, the standardized coefficients are back-transformed to original feature units so that contribution math (βj × xj[t]) and ROI calculations remain in spend dollars and revenue dollars:

The intercept absorbs the mean shift so predictions are unchanged. End users see coefficients and contributions in the same units they always did; the standardization is a numerical-stability layer that lives entirely between adstock and Ridge.

6. Ridge Regression Model

Model Specification

• Regularization: Ridge (L2) penalty with λ = 1.0 to address multicollinearity between channels that scale spend simultaneously. The L2 penalty minimizes ∑β_j², shrinking correlated coefficients toward each other rather than allowing one to dominate arbitrarily.
• Positive coefficient constraint: All channel coefficients are constrained to be ≥ 0, preventing the model from assigning negative revenue impact to marketing spend. This is an econometric prior: we assume marketing activity does not reduce revenue.

Why Ridge over OLS

Marketing channels are frequently collinear — spend moves together during peak seasons, promotions, and campaign flights. Ordinary Least Squares (OLS) produces unstable, high-variance coefficient estimates in the presence of multicollinearity, often yielding negative coefficients or wildly inflated values. Ridge regularization shrinks unstable coefficients toward zero, producing more reliable and interpretable estimates.

See Google Meridian documentation and Meta Robyn Analyst's Guide for comparative discussion of regularization approaches in MMM.

7. Two-Stage Residualized Fit

The positivity constraint described in section 6 (βj ≥ 0) is appropriate for marketing features — we don't want the model claiming that spending money on Facebook destroyed revenue. But it is the wrong constraint for the trend and seasonality controls: a downward trend or a negative weekend coefficient is meaningful information that the model should be free to discover.

scikit-learn's Ridge applies positive=True to all coefficients or none. To allow positive-only marketing coefficients alongside unconstrained controls, v2 uses a two-stage residualized fit:

• Stage 1 — Controls (unconstrained): Fit Ridge on the control features only (linear trend, Fourier yearly, Fourier weekly, holiday flag) to predict raw revenue. Coefficients are unconstrained, so trend and seasonal components can be positive or negative as the data dictates.
• Stage 2 — Marketing (positive-constrained): Subtract the Stage 1 fit from revenue to produce a residual series. Fit a second Ridge with positive=True on the adstocked marketing features against this residual, with no intercept (already absorbed in Stage 1).

The two coefficient sets are then combined and back-transformed jointly to original units. This pattern — sometimes called residualization or partialling-out — is also used in Meta Robyn for the same reason.

Tradeoff

Marketing coefficients are estimated only on the signal that survives after the controls have absorbed everything they can. If a channel's spend pattern is strongly correlated with the trend or with annual seasonality, some of its true effect will be absorbed by the control terms in Stage 1, and Stage 2 will produce a downward-biased marketing coefficient. This is a known property of two-stage approaches and the reason a fully joint Bayesian fit (planned for v3) is preferred when computational budget allows.

8. Attribution and Revenue Decomposition

For each observation (day), total revenue is decomposed into the following components:

• Baseline (β₀): The model intercept, representing revenue that would occur with zero marketing spend — organic demand, brand equity, and other non-marketing factors.
• Channel contributions: For each channel j, daily revenue contribution is βj × hj[t], where hj[t] = Hill(x'j[t]; Kj, Sj) is the adstock-decayed and Hill-saturated spend feature. This isolates the non-linear revenue attributable to each channel on each day.
• Structural Controls & Exogenous Seasonality: Rather than treating periodic demand fluctuations as unexplained residual noise or conflating them with marketing lift, seasonality is parameterized as an explicit structural feature in the design matrix. Fourier expansions (annual harmonics with order N=3 and weekly harmonics with order N=3) plus statutory holiday indicators act as exogenous demand controls, absorbing predictable calendar cycles and macro trend shifts independently of advertising spend.
• Residual (ε): The remaining difference between actual and predicted revenue after accounting for marketing channels, structural baseline, and explicit seasonality controls. This residual captures true stochastic variance and unmodeled external shocks (e.g., unanticipated competitive actions, severe macroeconomic disruptions, or supply chain bottlenecks) rather than predictable seasonal cycles.

By explicitly framing seasonality as an exogenous structural control rather than an unexplained residual, Adstock ensures that cyclical surges (such as Q4 holiday spikes or weekend shopping lifts) are not misattributed to contemporaneous marketing spend or discarded into residual variance. The residual gap thus represents purely unmodeled variance after accounting for structural seasonality and marketing effects.

9. ROI Calculations & Budget Reallocation

• Average ROI per channel:
  ROI_j = Total Attributed Revenue_j ÷ Total Spend_j
  
  Measures overall historical efficiency. ROI ≥ 1.0 indicates the channel generated more revenue than it cost across the full analysis period.
• Marginal ROI per channel (Active Hill Saturation Causal Formulation):
  mROI_j(x_j) = β_j · Hill'(a_j; K_j, S_j) · 1 / (1 − α_j)
  
  where β_j is the channel's regression coefficient, α_j is the geometric carryover decay rate, and aj = xj / (1 − αj) is the steady-state adstocked daily spend.
  
  Marginal return on the next dollar of spend is governed by the analytical first derivative of the Hill saturation function:
  Hill'(a; K, S) = S · K^S · a^S−1 / (K^S + a^S)²
  
  In Adstock's active production engine, the shape exponent is set to S = 1.0 and the half-saturation parameter K_j is pegged to each channel's median positive adstocked spend (Kj = median(adstocked_spendj)). For S = 1.0, the derivative simplifies directly to:
  Hill'(a; K, 1.0) = K / (K + a)²
  
  At historical median spend (aj = Kj), this derivative evaluates to 1 / (4 · Kj). As spend scales up (aj >> Kj), the derivative monotonically declines toward zero, modeling realistic diminishing marginal returns. Because a marginal dollar of new spend propagates through geometric carryover as 1 + α + α² + … = 1 / (1 − α), the long-run marginal ROI incorporates the carryover multiplier 1 / (1 − α).
• Elimination of Naive Linear Attribution Models: In legacy linear attribution models without saturation, the derivative is treated as a constant (d(revenue)/d(spend) = β / (1 − α)). Under constant marginal returns, unconstrained mathematical optimization is degenerate: 100% of marginal capital collapses into whichever single channel has the highest point estimate, regardless of market saturation, client capacity, or estimation variance. Adstock eliminates linear attribution assumptions in favor of the active Hill saturation formulation, ensuring that marginal returns continuously taper as spend expands.
• Convex SLSQP Optimization to Equal-Marginal-Return Equilibrium: With diminishing returns active, Adstock utilizes Sequential Least Squares Programming (SLSQP) to find the optimal budget allocation vector x* that maximizes total portfolio revenue:
  max_x ∑_j Revenue_j(x_j)   subject to   ∑_j x_j = Total Budget,   L_j ≤ x_j ≤ U_j
  
  At unconstrained mathematical optimality, the Kuhn-Tucker conditions guarantee that marginal ROI is equalized across all active channels:
  mROI₁(x₁*) = mROI₂(x₂*) = … = mROI_m(x_m*) = λ*
  
  Capital is automatically channeled into under-saturated channels with high marginal efficiency until their returns taper to equilibrium with the rest of the portfolio.
• Bootstrap Stability Shrinkage & Upper-Bound Constraints: Point estimates with high sampling variance are directly bounded during optimization. Adstock computes the coefficient of variation (CV = σ/μ) and minimum coefficient across 5 overlapping 80% bootstrap subsample time folds:
  S_j = 1 / (1 + (CV_j / 0.50)²)
  
  These stability statistics are enforced directly as optimization boundary constraints [Lj, Uj]:
  • High Confidence (CV < 0.40 and fold min > 0): Consistent parameter recovery. Optimized freely up to standard allowable upper bounds (e.g., 150%–200% of current spend).
  • Borderline Confidence (0.40 ≤ CV < 0.60 and fold min > 0): Moderate signal variation. Upper bound strictly capped at a +5% exploratory increase to safeguard capital.
  • Insufficient Confidence / Hard Gate (fold min ≤ 0.0 OR CV ≥ 0.60): Upper bound locked to current spend ($0 budget expansion). Spend is held flat until empirical incrementality tests verify true incremental lift.
  • Unverified Tier (missing stability stats): Channels lacking subsample folds default conservatively to a +5% exploratory cap.
  • Donor Channels (mROI < 1.00): Upper bound locked to current spend; lower bound allows trimming toward a 10% spend floor to release capital for profitable channels.
• Gating Rationale & The Non-Negativity Constraint: Because Adstock solves Ridge regression under a strict non-negative coefficient constraint (β ≥ 0), a fold minimum of 0.0 does not indicate a “small positive effect.” It indicates that the non-negativity constraint was actively binding — meaning an unconstrained fit on those subsamples would have estimated a negative coefficient. On those time slices, empirical evidence for any marketing lift vanished entirely. Concentrating real dollars into a channel whose signal disappears across time periods is a catastrophic MMM failure mode. Withholding budget increases and prompting an incrementality test protects client capital while driving empirical verification.
• Linear Mode Fallback Heuristic: When non-linear saturation is explicitly disabled by configuration (enable_saturation=False), Adstock falls back to an excess-return weighting heuristic (Excess Returnj = max(0, mROIj − 1.0)) scaled by continuous stability shrinkage S_j. This heuristic prevents linear point estimate collapse when running in legacy linear mode, but the non-linear Hill saturation model with SLSQP is Adstock's primary production engine.

10. Model Evaluation Metrics

• R² (Coefficient of Determination — r2_in_sample): Measures the proportion of revenue variance explained by all inputs in the model — marketing channels plus the v2 control terms (linear trend, Fourier yearly + weekly seasonality, holiday flag). In Adstock's two-stage architecture, controls absorb periodic and trend fluctuations first, allowing marketing features to compete cleanly on residual signal. The resulting r2_in_sample directly underpins the headline Analysis Confidence tier (High: ≥ 0.70, Medium: ≥ 0.40, Low: < 0.40).
• MAPE (Mean Absolute Percentage Error): Measures average daily prediction accuracy as a percentage of actual revenue. Displayed to end users as “Prediction Accuracy” (calculated as 100 − MAPE) to improve interpretability for business audiences without the common confusion surrounding R².

Note: While r2_in_sample governs model evaluation thresholds, the main dashboard abstracts raw R² values into intuitive confidence badges and MAPE-based accuracy, ensuring executive stakeholders interpret reliability without statistical misinterpretation.

11. Lookback Window Analysis

• Model training: The model is always trained on the full dataset to maximize statistical reliability. Coefficients are estimated once and do not change when a lookback window is selected.
• Window slicing: Lookback windows (7, 30, 90 days) filter the model's output — specifically, the daily contributions and daily spend — to show how channel ROI varies across recent time periods. This enables detection of performance trends without sacrificing training-set size.
• Reliability gates: Metrics are suppressed when data within a window is insufficient for meaningful calculation. Per-channel gates require minimum non-zero spend days and minimum total spend thresholds:
  • 7-day window: ≥ 5 non-zero spend days and ≥ $500 total spend
  • 30-day window: ≥ 15 non-zero spend days and ≥ $500 total spend
  • 90-day window: ≥ 30 non-zero spend days and ≥ $500 total spend
  • All Data: no gates applied
• Carryover note: Short-window ROI calculations include carryover effects from spend that occurred before the window start date. A channel may show attributed revenue in a window even if spend within that window was zero, due to adstock decay from prior periods.

12. Credibility and Industry Context

The foundational methodology — adstock transformation + regularized regression + positive constraints — is shared with Google's Meridian and Meta's Robyn, the two most widely adopted open-source MMM frameworks.

Key Difference from Google Meridian

Meridian uses a Bayesian hierarchical framework with geo-level partial pooling, prior specification on parameters, and posterior sampling via MCMC. This produces credible intervals on all estimates but requires significantly more computational resources and analyst expertise. Adstock uses a frequentist approach optimized for speed-to-insight and accessibility.

Key Difference from Meta Robyn

Robyn uses Nevergrad for multi-objective hyperparameter optimization, evaluating thousands of model candidates across a search space of adstock parameters, saturation curves, and regularization strengths. This produces a Pareto-optimal set of models but requires significant computational time and analyst judgment to select a final model. Adstock uses a single-model specification with analyst-validated parameters.

Structural Independence

As an independent measurement platform with no media inventory to sell, Adstock has no structural incentive to bias attribution toward any channel. This is a meaningful differentiator: Google and Meta's open-source MMM tools, while methodologically rigorous, are developed by organizations with inherent conflicts of interest as both media sellers and measurement providers.

Comparison to Enterprise MMM Vendors

For enterprise MMM vendors (Nielsen Marketing Cloud, Analytic Partners, Measured), engagements typically require 3–6 months and $200K–$500K+ annually. Adstock delivers comparable foundational methodology at a fraction of the cost and timeline.

13. Model Evaluation & Confidence Badges

Adstock translates econometrically rigorous parameter estimates into actionable, risk-managed reallocation decisions through canonical evaluation schemas and a standardized CMO confidence framework.

Canonical Evaluation Metrics

• r2_in_sample (In-Sample Goodness-of-Fit): Quantifies the proportion of revenue variance explained by the joint two-stage model (controls + marketing). Instead of presenting raw decimal statistics that can confuse executive stakeholders, Adstock directly maps r2_in_sample into three operational trust tiers:
  • High Confidence (≥ 0.70): Robust variance capture; recommendations can be executed with high trust.

  • Medium Confidence (≥ 0.40): Acceptable directional fit; recommended for guided budget allocation and experimental validation.

  • Low Confidence (< 0.40): Weak fit indicating dominant unmodeled external drivers; recommendations are directional and require incrementality testing.
• marginal_roi (Marginal Return on Investment): Measures the expected incremental revenue returned per additional dollar spent at current daily volume, calculated analytically via the Hill saturation first derivative:
  marginal_roi = β_j · Hill'(a_j; K_j, S_j) · 1 / (1 − α_j)
  
  Unlike average ROI (which reflects historical aggregate efficiency), marginal_roi governs forward-looking budget optimization: channels with marginal_roi > 1.00 return positive net profit on the margin, while channels with marginal_roi < 1.00 lose money on the next dollar.

Three-Tier CMO Confidence Framework & Badges

To protect client capital from noisy point estimates, Adstock evaluates parameter stability across 5 overlapping 80% bootstrap subsample time folds. The resulting stability statistics map directly to five standardized badges across the dashboard:

Active in Production

• Hill saturation curves: Non-linear diminishing returns via Hill transformations pegged at channel median positive adstock (S = 1.0, K = median). Marginal ROI adjusts dynamically with spend volume, enabling convex SLSQP optimization to equal-marginal-return equilibrium.
• Per-channel carryover (adstock) decay: Independent retention rates configurable per channel (e.g., longer decay for brand CTV, shorter for direct search).
• Three-tier confidence boundary constraints: Subsample bootstrap stability directly bounds SLSQP optimization ceilings, protecting marketing capital from noisy channel estimates.

Targeted for Future Releases

• Bayesian inference (PyMC): Joint posterior distributions and credible intervals on coefficients and ROI point estimates.
• Unconstrained joint Hill parameter learning: End-to-end estimation of S and K parameters from spend experimentation, extending the pegged Hill baseline.
• Incrementality calibration: Joint optimization anchored to geo-lift tests and holdout causal priors.
• Time-varying coefficients: Capturing evolving macro effectiveness and seasonal shifts.
• Geo-level hierarchical modeling: Multi-market regression with partial pooling.