What does a cross-correlation peak at lag -14 between daily ad spend and sign-ups tell you?

## What the cross-correlation function is Given two time series measured on the same clock, the **cross-correlation function** (CCF) is the correlation between them computed at every relative shift, or **lag**. Instead of one number, you get a curve: one correlation per lag, typically plotted with lag on the horizontal axis and the correlation on the vertical, with bands marking what is distinguishable from zero. A peak somewhere off zero says the two series match best when one is slid along the time axis relative to the other — a **lead-lag** relationship. ## Reading the sign This is the part that trips people up, and interviewers know it. There are two conventions in circulation, and they are exact mirror images: - `CCF(k) = corr(x_{t+k}, y_t)` — a negative peak means the *earlier* values of `x` line up with `y`, so `x` leads. - `CCF(k) = corr(x_t, y_{t+k})` — a *positive* peak means `x` leads. So "a peak at lag -14 between ad spend and sign-ups" is only interpretable once you say which series the lag indexes. Under the first convention, with spend as `x`, the peak at -14 says spend measured 14 days before a given day is what correlates with that day's sign-ups: **ad spend leads sign-ups by two weeks**, which is a plausible story for a consideration-phase product. Under the mirror convention the same number would say sign-ups lead spend, which would instead suggest budget is being allocated in response to demand. The safe habit is to confirm the direction with a sanity check you already believe — shift the series by hand and look at a plot — rather than trusting your memory of the convention. ## Why the peak can be an artefact A raw CCF between two business series is usually not trustworthy on its own, for two reasons. **Shared trend.** If both series grow over the year, they are positively correlated at essentially every lag. The CCF then sags in a broad hump rather than showing a sharp peak, and the exact location of the maximum is close to arbitrary — it can move by weeks between samples. **Autocorrelation within each series.** Ad spend today looks like ad spend yesterday; sign-ups today look like sign-ups yesterday. That internal memory smears any genuine relationship across many neighbouring lags and inflates the correlations, so the CCF shows a wide band of large values instead of a clean spike, and the sampling bands drawn around zero are too narrow to trust. **Shared seasonality.** If both series have a weekly rhythm — spend paused at weekends, sign-ups lower at weekends — the CCF will show peaks at multiples of 7 that reflect the calendar rather than any causal chain. ## Prewhitening The standard remedy is **prewhitening**. Fit a model to the input series (spend) that reduces it to something close to uncorrelated noise, then apply that *same* filter to both series, and compute the CCF on the filtered pair. Because the input's own memory has been stripped out, a peak in the resulting CCF reflects the transfer between the series rather than the internal structure of either one, and the significance bands become meaningful. Removing trend and seasonality first — by differencing, by subtracting a fitted seasonal component, or by including calendar terms — is the minimum version of the same idea. ## Other cautions - **Sampling interval sets the resolution.** Daily data cannot resolve a lead of a few hours, and rolling the data up to weekly buckets can hide a 3-day lead entirely or shift where the peak appears. - **A lead is evidence about ordering, not about mechanism.** That spend moves first is consistent with spend driving sign-ups, but also with both responding to something that moves spend first — a product launch calendar, for example, that raises budgets and then attracts users. - **Alignment is not a forecast.** A lead of 14 days is only exploitable if it is stable; check whether the same peak appears in a held-out stretch of the series before building anything on it. ## Interview framing A good answer names the direction confidently *and* immediately qualifies it: which convention, and what was done about trend, seasonality and autocorrelation before the plot was read.