How do two unrelated series come to look related?
The stationarity lesson stated the mechanism; this one runs it, first on random numbers and then on real prices, because a demonstration you have performed is the only kind that survives the next persuasive chart.
Two random walks
Take two streams of independent normal draws and cumulate each into a walk. Measured on 2026-09-04 with a fixed seed and 2,500 steps: the correlation of the two levels was −0.42, the correlation of the steps −0.014. Change the seed and the level correlation lands anywhere between −0.9 and +0.9; the step correlation stays within a few hundredths of zero. Nothing connects the two series, and a chart of the two levels will show them tracking each other, or mirroring each other, for hundreds of steps at a time.
This is not a rare accident. Two random walks have a level correlation that is typically large, in one direction or the other, because both wander and any two wanderers share long stretches of drift. Yule described it in 1926 as "nonsense correlations"; the mechanism is that a random walk's level has no fixed mean to be correlated around.
Two real series
SPY.US and USDTRY.FOREX over the ten years to 3 September 2026, measured on 2026-09-04: each correlated 0.95 with a day counter, and through it strongly with each other. Their daily changes: near zero. The share of an American index fund and the price of a dollar in lira share a decade of trending, and nothing else.
Real examples with more plausible stories are everywhere, and their plausibility is the trap. Global money supply and a stock index; a country's debt and its bond yields; two shares in one sector. Each pair of levels correlates highly; each pair of returns may or may not; and the story attached to the level correlation is, nine times in ten, a story about two series that both went up.
The three checks
Difference first. Correlate returns or changes, never levels, unless the ratio of the levels has a mean it returns to. Look at the scatter, not the line chart. Two trending lines on one chart with two axes is the format in which most spurious correlations are published; a scatter of changes shows a cloud or it shows a slope. Ask what the correlation was in the other half of the sample. A relationship that exists in 2016–2021 and not in 2021–2026 is a description of the first window, and the stocks-and-bonds lesson showed even a real relationship failing that test.
Where it costs money
A pairs trade on two prices with a 0.9 level correlation and no return correlation is a position in two random walks that happened to drift together; it will drift apart on the same logic, with no force pulling it back. A hedge sized on a level regression is the same error with a beta attached. And a risk model whose correlation matrix was built on price levels rather than returns has been fed the nonsense correlations of every pair it contains.
In the data
The real-series check is two /eod/ pulls joined on date. The random-walk check needs no endpoint: 2,500 draws from any normal generator, cumulated, correlated twice — once before cumulating and once after. Five lines of code, and the most useful demonstration in this course.
Try it now
- Generate two random walks of 2,500 steps and compute the two correlations. Run it five times with five seeds and write down the five level correlations and the five step correlations. Then draw the conclusion, in a line you would defend.
- The real pair, whose levels agreed with a clock:
- Find any chart you have seen this month that overlays two series on two axes. Name which of the three checks it fails.