Spurious Regression

Here is a result that has burned generations of analysts. Take two completely unrelated series that each wander like a random walk, the number of storks in Denmark and the S&P 500, say, and regress one on the other. Standard regression output will often hand you a large R-squared and a t-statistic far past the usual significance cutoff. Everything looks like a strong relationship. There is none. This is spurious regression, and the culprit is that both series are nonstationary. When your variables trend or wander without settling down, the arithmetic that makes ordinary least squares and its t-tests trustworthy quietly falls apart.

Why it happens

Ordinary regression inference rests on the residuals being well-behaved, roughly independent noise around the line. When you regress one random walk on another, the two just happen to drift, and any two drifting things will often drift together for long stretches by pure chance. OLS eagerly fits a line through that shared wandering and reports a great fit. But the residuals are themselves a wandering, highly autocorrelated series, not clean noise, so the standard errors are badly understated and the t-statistic is inflated. The formula that says "a t-stat above 2 is unlikely under the null" simply does not hold when the inputs have unit roots.

Worked example: the Granger–Newbold experiment

Granger and Newbold ran the definitive demonstration. Generate two independent random walks, $X_t = X_{t-1} + u_t$ and $Y_t = Y_{t-1} + v_t$, where $u_t$ and $v_t$ are separate, unrelated noise, there is no link between the series by construction. Then regress $Y$ on $X$ and test whether the slope is zero.

If OLS inference worked, you would wrongly reject the (true) null of no relationship about $5\%$ of the time. In their simulations you reject it roughly 75% of the time at the $5\%$ level. A single such regression might report R-squared around $0.7$ and a t-statistic near $8$, numbers that would normally scream "rock-solid relationship", for two series that share nothing at all. The lesson is stark: on nonstationary data, a beautiful regression is weaker evidence than an ugly one, because the beauty is exactly what nonstationarity fakes.

The warning signs and the fix

• The classic tell. A very high R-squared paired with a low Durbin–Watson statistic (well below 2) is the fingerprint. Low DW means the residuals are strongly autocorrelated, exactly what you get when leftover wandering has not been captured, and a Durbin–Watson far below the R-squared is Granger and Newbold's rule of thumb for "this regression is spurious."
• Difference first. Regress changes on changes rather than levels on levels. Differencing a random walk yields stationary noise, and if the two series really are unrelated, the differenced regression correctly finds nothing.
• The cointegration exception. Sometimes two nonstationary series genuinely move together in the long run, their difference is stationary even though each wanders. That is Cointegration, and it is the one case where a levels regression is legitimate (and is the statistical backbone of pairs trading). Test for it explicitly rather than assuming it.

Spurious regression is the time-series face of correlation without causation, a direct consequence of ignoring Stationarity, and the reason unit-root testing and Cointegration are mandatory steps before regressing financial series on each other.