The Regression That Lies: Granger and Newbold on Spurious Results

Here is an experiment you can run yourself in about four lines of code. Generate two completely independent random walks. Not correlated. Not related in any way. Just two separate sequences of cumulative coin flips, each knowing nothing of the other.

Now regress one on the other.

You will very often get a large, beautifully significant coefficient, a high R-squared, and a t-statistic that would let you publish in a good journal. The two series have literally nothing to do with each other, and the regression will tell you, with enormous statistical confidence, that they do.

In 1974, Clive Granger and Paul Newbold ran exactly this experiment, wrote down what happened, and detonated a large part of the empirical economics literature.

The problem: everyone was regressing trends on trends

By the 1970s, applied economics ran on regressions of one economic series on another. Consumption on income. Money on prices. Wages on productivity. And these regressions kept producing spectacular results: R-squared values above 0.9, coefficients significant at every conceivable level, relationships that looked like laws of nature.

Two things about these results should have set off alarms, and for a while they did not.

First, the results were suspiciously good. Economics is a noisy subject. Relationships that fit the data with 95% accuracy are not what you expect from human behaviour.

Second, the Durbin-Watson statistic was almost always terrible. That statistic measures whether a regression's errors are correlated over time, and a healthy value is around 2. These regressions routinely produced values near 0, meaning the errors were massively autocorrelated. This was widely noticed and widely shrugged at. It was treated as a nuisance to be patched, not a symptom of catastrophe.

Granger and Newbold argued it was a symptom of catastrophe.

The key idea via analogy: two balloons drifting in the same sky

Imagine two hot air balloons released far apart, each carried by its own independent air currents. Neither pilot can see the other. There is no connection between them whatsoever.

Now watch them over a long afternoon. Balloon A drifts steadily upward. Balloon B also drifts steadily upward, because that is what balloons do. If you now plot A's height against B's height, you will get an almost perfect straight line. The correlation will be enormous. You will conclude, with statistical certainty, that Balloon A is lifting Balloon B.

The relationship is entirely an artifact of the fact that both are drifting. Any two things that wander persistently in the same general direction, or even just wander persistently at all, will appear related. It is not that a relationship was measured badly. It is that a relationship was manufactured out of nothing.

The reason the standard statistics fail so badly is subtle but crucial. A regression's t-statistic assumes that as you gather more data, your estimate settles down toward a stable truth. But when both series are random walks, the errors of the regression are themselves a random walk. They never settle. They wander further and further from zero. The formula for the standard error, which is supposed to tell you how uncertain the coefficient is, assumes the errors are well-behaved and bounded. They are not. So the standard error is computed far too small, the t-statistic is computed far too large, and the test reports blazing confidence in a relationship that does not exist.

Worse: the problem gets worse with more data, not better. In an ordinary regression, more data means a more reliable answer. In a spurious regression, more data just means a more confidently wrong answer. This is the single most counterintuitive and most important part of the result.

Why it mattered

• It invalidated a large body of published work in one stroke. Any regression run on trending, non-stationary economic levels without checking the residuals was now suspect. Granger and Newbold effectively told a generation of economists that many of their results might be statistical mirages.
• It made the Durbin-Watson statistic a red flag rather than a footnote. The paper's practical advice was blunt: if your R-squared is high and your Durbin-Watson is low, you are probably looking at nonsense. That simple heuristic is still one of the fastest ways to catch a spurious regression.
• It forced the profession to think about stationarity. You cannot understand cointegration, unit root testing, or modern time-series econometrics without this paper. It is the problem that all of them exist to solve. Granger's own later work on cointegration, which won him a Nobel Prize, is essentially the constructive answer to the destructive result he published here: yes, regressing wandering series is usually nonsense, but there is a precise and important exception where it is not.
• It is the deepest trap in quantitative trading research. This is not a historical curiosity. Every time someone regresses a stock price on another stock price, or a strategy's cumulative PnL on some cumulative macro index, and finds a stunning relationship, they are at risk of rediscovering Granger and Newbold's result the expensive way. The fix is almost embarrassingly simple: regress changes, not levels. Returns, not prices. Differences, not cumulative sums.

The honest limitations

The result itself is not really contestable, but its interpretation has boundaries worth knowing.

• Differencing is not a free lunch. The obvious fix, regressing changes on changes, is safe but can throw away genuine long-run information. Two series that really are tied together in the long run (cointegrated) contain a valuable relationship in their levels, and blindly differencing destroys it. This is exactly the gap cointegration analysis was invented to fill.
• A low Durbin-Watson does not always mean spuriousness. It could mean an omitted variable, a misspecified functional form, or genuine serial correlation in a perfectly valid relationship. It is a warning light, not a diagnosis.
• The original demonstration was largely simulation-based. The full asymptotic theory explaining exactly why the t-statistics diverge came later, most notably from Peter Phillips in 1986. Granger and Newbold showed the fire; the theory that explained the chemistry followed.
• It does not tell you what is real. The paper is superb at telling you what is fake. It offers less guidance about how to find the genuine relationships buried in non-stationary data. That constructive project took another decade.

The one-line takeaway

Granger and Newbold showed that regressing one wandering series on another produces confident, significant, entirely meaningless results, and that gathering more data makes the lie stronger rather than weaker, which is why any quant who regresses prices instead of returns is not doing analysis, but generating fiction.