Radar Perene / Archive / science

Spurious regression: series that move together for nothing

◦ Index methodology v2.2 (working papers with DOI). See the methodology.

Science

In 1926, the British statistician George Udny Yule published a paper with a title that still stings: Why do we sometimes get nonsense-correlations between time-series? His example became a classic: between 1866 and 1911, the proportion of Church of England marriages and the country's mortality rate showed near-perfect correlation. Neither influenced the other; both simply declined across the decades, each for its own reasons. Two escalators going down side by side — and a century later, financial markets are still riding them hand in hand with conclusions that do not exist.

A spurious regression is a statistically strong relation between two series that have no real link at all — an artifact produced, in most cases, by trends or persistence the two carry at the same time. Time is the universal third factor: almost everything that grows, grows together; almost everything that withers, withers in company. The strength of the measured relation is not, by itself, evidence of any relation.

If the problem were only a Victorian curiosity, it would not deserve this page. It is, in fact, one of the most demolishing results of modern econometrics — and a trap set exactly where market research most likes to step.

Why time manufactures kinship

The coup de grâce came in 1974, when Clive Granger and Paul Newbold ran an experiment of brutal honesty: they generated pairs of series by lottery, built to have no relation whatsoever, and regressed one on the other. In numbers: in the paper's simulations, roughly three out of every four regressions between those independent series produced statistics that, by the conventional reading, would point to a "significant" relation. Three out of four — between series manufactured to be strangers to each other.

The mechanism is not mystical. Trending series, or strongly persistent ones — where today's value inherits almost everything from yesterday's — violate the premises that give classical tests their meaning. The test keeps spitting out numbers; it is the numbers that stop meaning what the textbook promises. And market series are the problem's perfect habitat: prices, indices, monetary aggregates and GDPs are almost all persistent, almost all trending over long windows. Whoever looks for correlations among them will find them. The scientific question was never "is there a correlation?" — it is "does the correlation survive once you remove the inertia the series share?".

This trap's kinship with another the house has described is evident: whoever tests many pairs until one "works" stacks spurious regression on top of p-hacking — two distinct mechanisms manufacturing, together, the same illusion of discovery.

The antidote econometrics built

The literature did not stop at the diagnosis. The general prescription is well known and can be described without formulas: test whether the series are rooted in their own trends before regressing them; work with changes instead of levels when appropriate; and, when the question is precisely about a long-run relation between levels, use the machinery of cointegration — the test of whether two series share the same anchor, a theme that earns its own page on this trail. Above all, the methodological defense repeats the mantra of this series' other pages: hypothesis declared before the test, plausible mechanism declared before the correlation, and suspicion proportional to the finding's convenience.

That is the spirit of the house's editorial filter: a correlation without control and without mechanism does not become a published sentence. The ruler is described in why we never write "you should" — and it blocks, in practice, exactly the kind of finding Yule dismantled a century ago.

A pair from the house's own archive

The house archive keeps an instructive example of a strong relation that carried no information — not by lottery, but by construction. In more than one episode, the ratio between commodities measured in reais and the Ibovespa jumped to the top of its own history, and the hasty reading would have been "enthusiasm for commodities." The published record says otherwise: the jump was the exchange rate inside the ruler. Commodities quoted in dollars, converted into a weakening currency, rose in reais by arithmetic — and the ratio rose with them, no enthusiasm anywhere in sight. The episode is documented in the commodities top that was currency; the archive recorded the equivalent in April 2016. "It looked like enthusiasm; it was accounting" — the original record's sentence is the definition of a spurious relation written without jargon.

The example teaches the gesture that matters: facing two glued lines, the house's first question is not "what does one say about the other?", but "what moves them at the same time?". When the answer is "the same variable inside both" or "the same trend beneath both", the gluing is mechanical — and the headline it suggested dies before being born.

Frequently asked questions

Does a high, long-lasting correlation prove nothing?

It proves the series share something — possibly just a trend or a common factor. It is raw material for a hypothesis, never a verdict. The gluing's duration increases the temptation, not the evidence.

How is a spurious regression detected?

The classic signs: a very strong relation between levels that vanishes in changes; residuals that behave like the series themselves; a fit too good for a mechanism nobody can name. The formal diagnosis uses unit-root and cointegration tests — described in broad strokes on this trail.

Does working with changes always solve it?

No. It removes the common trend, but it can throw away exactly the long-run relation that mattered — and it does not protect against common factors that also change. It is a tool, not an absolution.

Does the problem exist outside time series?

Kinship manufactured by a third factor exists in any data. The spurious-by-trend version is typical of series over time — and it is why the problem is endemic in finance and macroeconomics.

Escalators descending together are half the problem; the other half is the most famous pair in the Brazilian market — and the arrow the headline draws between them: the Ibovespa and the dollar, what correlation doesn't tell →

House readings: the record free of manufactured kinship is in today's note, in the Daily; the gluings time has dismantled, in the precedents, in the Atlas.

Running a specific pair of series through this diagnosis is an exercise the house conducts on request.