Econometric Methodology

A formal mathematical overview of the econometric models and algorithms powering SpotLift.

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1. High-Frequency Impulse Attribution vs. MMM

Linear television attribution operates on two distinct analytical horizons:

1. Macro Econometrics (Media Mix Modeling / MMM): Solves long-term adstock carryover, baseline sales, diminishing returns, and macro seasonality using weekly or monthly aggregated regressions (Bayesian MMM, Robyn, Meridian).
2. High-Frequency Temporal Attribution (SpotLift): Measures the immediate, unconfounded impulse response on digital telemetry in the seconds and minutes directly following a broadcast airing.

SpotLift focuses exclusively on high-frequency temporal attribution, delivering spot-by-spot precision for creative testing, station selection, daypart optimization, and immediate conversion response.

Broadcast Airing (t0) ──► Immediate Spike [t0, t0 + W) ──► Diurnal Decay
                                ▲
                          Baseline (Bt)

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2. Dynamic Diurnal Baseline Algorithm

Accurate TV attribution requires an unbiased estimate of the counterfactual: How many visitors would have visited the digital property if this spot had not aired?

A simple moving average or pre-spot minute average fails because web traffic exhibits strong diurnal patterns (e.g. traffic naturally rising by 4% per minute at 19:30 or falling sharply at 23:30).

Formulation

SpotLift implements a decay-weighted cyclic rolling diurnal baseline:

$$B(t)=\alpha (t)\cdot {\overline{Y}}_{\text{dow},\text{hour},\text{minute}}$$

Where:

• ${\overline{Y}}_{\text{dow},\text{hour},\text{minute}}$ is the historical median traffic for the given day-of-week, hour, and minute across non-airing control windows.
• $\alpha (t)$ is an adaptive flight-level scaling factor computed from the uncontaminated 30-minute pre-spot period:

$$\alpha (t)=\frac{\sum _{s\in [t-35,t-5)}Y(s)}{\sum _{s\in [t-35,t-5)}\overline{Y}(s)}$$

::: important Pre-Spot Baseline Masking Notice that the calibration window ends at $t-5$ minutes before airing, not $t-0$. This intentional 5-minute pre-spot buffer prevents broadcast clock drift or pre-roll sponsor teasers from polluting the baseline calibration. :::

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3. Half-Open Response Window Integration

To avoid double-counting at boundary minutes, SpotLift defines the response integration window as a half-open interval:

$$\mathcal{W}(t_0)=[t_0,t_0+W)$$

For a standard $W=6$ minute window airing at 20:15:00:

• Evaluated minutes: 20:15, 20:16, 20:17, 20:18, 20:19, 20:20
• Minute 20:21 belongs to the subsequent interval.

The total incremental lift $\mathrm{\Delta }$ is integrated as:

$$\mathrm{\Delta }=\sum _{t\in [t_0,t_0+W)}max(0,Y(t)-B(t))\text{or}\sum _{t\in [t_0,t_0+W)}(Y(t)-B(t))$$

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4. Signed Lift & Zero Rectification Bias

Most commercial attribution tools artificially clamp negative minute lift to zero before aggregating (lift = max(0, obs - base)).

The Rectification Bias Theorem

If web traffic has natural random noise ${ϵ}_t\sim \mathcal{N}(0,{\sigma }^2)$ around baseline $B(t)$, clamping negative observations creates an upward bias:

$$\mathbb{E}[max(0,Y_t-B_t)]=\mathbb{E}[max(0,{ϵ}_t)]=\frac{\sigma }{\sqrt{2\pi }}>0$$

Over a 500-spot flight, clipping negative noise rectifies random fluctuations into fake positive lift, overstating total campaign return by 15% to 35%.

By default (clip_negative=False), SpotLift preserves signed net lift across spots:

• Ineffective spots in low-response dayparts correctly report zero or slightly negative net lift.
• Aggregated campaign-level lift sums to an unbiased econometric total.

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5. Collision Detection & Attribution

During prime time (20:00–22:30), television schedules frequently exhibit broadcast collisions: two or more spots air simultaneously or within overlapping response windows across different channels.

Station A (RTL):       [20:15:00 ─── 20:21:00]
Station B (ProSieben):       [20:17:30 ─── 20:23:30]
Overlapping Region:          |█████████|

SpotLift classifies spots into:

• Isolated Single Spots: Air with $>W$ minutes separation from any other spot. Attributed with exact diurnal integration.
• Collision Clusters: Air within $\le W$ minutes of an adjacent spot.

Collision Attribution Methods in Open-Core

SpotLift provides two analytical methods in the open-core library:

1. weighted_share (Default): Evaluates continuous exponential decay response curves for each colliding spot, weighting credit by spend and elapsed time from each airing:

$$w_i(t)={\text{Spend}}_i\cdot \exp (-\lambda (t-t_i))\cdot \mathbb{I}(t\ge t_i)$$

2. naive_proportional: Splits the combined cluster lift in proportion to spot media spend:

$${\text{Lift}}_i=\text{Total Cluster Lift}\cdot \frac{{\text{Spend}}_i}{\sum _j{\text{Spend}}_j}$$

Need Exact Mathematical Separation?

For prime-time campaigns with heavy multi-channel overlaps, SpotLift Pro implements unsupervised Shifted-Gamma Expectation-Maximization (EM) Deconvolution, estimating distinct shape and scale parameters for each creative and station using GRP priors.

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6. Unit Economics: CPIV & CPKE

SpotLift computes normalized unit cost metrics for every individual spot:

Cost per Incremental Visitor (CPIV)

$${\text{CPIV}}_i=\{\begin{array}{ll}\frac{{\text{Net Spend}}_i}{{\text{Incremental Lift}}_i}& \text{if }{\text{Incremental Lift}}_i>0\\ \text{None}& \text{if }{\text{Incremental Lift}}_i\le 0\end{array}$$

Cost per Key Event (CPKE)

When conversion telemetry is provided (see the Conversion-Rate Bridge), SpotLift bridges session lift to conversion events:

$${\text{CPKE}}_i=\frac{{\text{Net Spend}}_i}{{\text{Key Event Response}}_i}$$