What it means
Ordinary correlation tells you whether two series tend to rise and fall together at the same moment. Cross-correlation goes a step further by comparing one series with a time-shifted version of the other.
The shift is called the lag, and the result shows at which lag the relationship is strongest. This matters because business cause and effect rarely happen on the same day.
An advertising campaign may lift sales two weeks later, or a change in interest rates may affect property prices months afterwards. By testing different lags, an analyst can estimate how long the delay is.
The result is a number between -1 and +1 for each lag. A value close to +1 means the two series move together, a value close to -1 means they move in opposite directions, and a value near zero means there is little linear relationship.
Plotting the values across several lags gives a picture of the timing. Finance and operations teams use the idea for forecasting, risk management and planning.
A retailer might test whether online searches lead store sales, and an investment manager might check whether one asset's returns lead another's. The big warning is that correlation does not prove cause.
Two series can move together because of a third factor, by coincidence or because both follow a trend, so results should be tested and backed by business logic. It also needs enough data to be reliable.
With only a handful of observations, a high figure can arise by chance, which is why analysts prefer long data series and check for statistical significance.
In practice
Real-world examples.
Example
A consumer goods company compares weekly advertising spend with weekly sales. Cross-correlation shows the strongest link at a lag of two weeks, so the marketing team plans campaigns two weeks ahead of the peak. The company saves money by avoiding wasted spending in weeks when its advertising would arrive too early or too late.
Example
A portfolio manager tests whether the return on an oil price index leads the return on airline shares. A clear negative relationship at a lag of one month suggests rising oil prices tend to precede weaker airline returns. The manager treats the result as a hint for further research, not a trading rule, and checks it with other data.
Example
A property group compares changes in mortgage rates with new home sales. The relationship is strongest after several months, which helps the planning team schedule project launches. The group tests several lags before settling on a figure, and records how the relationship held up in previous cycles.
Formula
Calculation
Correlation (at lag 0) = Sum of (x deviation x y deviation) / square root of (sum of x deviation squared x sum of y deviation squared)
Worked example: monthly advertising spend x = 1, 2, 3, 4, 5 (in $10,000) and sales y = 2, 1, 4, 3, 5 (in $100,000). The mean of x is 3 and the mean of y is 3.
Deviations of x: -2, -1, 0, 1, 2. Deviations of y: -1, -2, 1, 0, 2.
Products: 2, 2, 0, 0, 4, which add up to 8.
Sum of squared x deviations = 4 + 1 + 0 + 1 + 4 = 10. Sum of squared y deviations = 1 + 4 + 1 + 0 + 4 = 10.
Correlation = 8 / square root of (10 x 10) = 8 / 10 = 0.8.
To test a lag, the analyst would shift the sales series by one month and repeat the calculation to see whether the result is higher or lower.Case study
Seen in the real world.
Brightline Retail is a fictional chain, and this case is illustrative only. Its finance team suspected that website traffic predicted store sales but did not know how far ahead.
An analyst calculated cross-correlation between weekly visits and sales for lags of zero to six weeks. The correlation peaked at 0.7 at a lag of two weeks, and was much lower for other lags.
The team used that two-week lead to adjust staffing and stock levels, ordering more goods when web traffic rose. They also warned colleagues that the pattern was a guide rather than proof, and they reviewed it each quarter. Over the next year, the forecast errors on weekly stock orders fell noticeably, and the analyst presented the method to other departments.
Watch out
Common mistakes.
- Treating a high correlation as proof that one series causes the other, when a third factor may drive both. Always ask whether a plausible business reason explains the link before acting on it.
- Using too little data, which can produce high correlations purely by chance. Longer data series and a test of statistical significance give more reliable conclusions.
- Ignoring trends and seasonality, which can create a false relationship between two series that both rise over time. Remove trends and seasonal patterns first, so that the analysis looks at genuine co-movement.
Questions
People also ask.
What is the lag in cross-correlation?
It is the amount of time by which one series is shifted before comparison, showing how long it takes for changes to feed through. Testing several lags shows which delay gives the strongest relationship.
How does it differ from ordinary correlation?
Ordinary correlation compares the series at the same time, while cross-correlation tests several time shifts. Both are measured on the same scale, but only cross-correlation tells you about timing.
What range of values is possible?
From -1, a perfect opposite relationship, to +1, a perfect matching relationship, with 0 meaning no linear link.
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