Spurious Regression

A classic signature of spurious regression is an artificially high coefficient of determination (R²)—often exceeding 0.90—combined with a Durbin-Watson statistic near zero. This indicates strong apparent explanatory power alongside severe positive autocorrelation in the residuals.

• Granger and Newbold (1974) demonstrated that random walks routinely produce R² > 0.95
• A Durbin-Watson statistic below 1.0 strongly suggests non-stationary residuals
• The combination signals that the model is capturing stochastic trends rather than genuine relationships

Spurious regression occurs exclusively when both the dependent and independent variables are non-stationary—typically integrated of order one, I(1). The regression exploits shared stochastic trends rather than causal linkages.

• Variables with unit roots drift over time without mean reversion
• Differencing the series to I(0) often eliminates the spurious relationship entirely
• Testing for stationarity via ADF or KPSS tests is a prerequisite before interpreting regression output

In a valid regression, residuals should be white noise—stationary with constant variance. In spurious regressions, residuals inherit the non-stationarity of the underlying series, causing them to wander persistently away from zero.

• Residual-based cointegration tests (Engle-Granger) check whether residuals are stationary
• If residuals are I(1), the regression is spurious regardless of t-statistic significance
• Plotting residuals over time often reveals long swings and trending behavior

Standard errors in spurious regressions are severely underestimated, causing t-statistics to explode and null hypotheses to be rejected far too often. Phillips (1986) proved that as sample size increases, the probability of falsely finding significance approaches 1.

• Conventional critical values become meaningless for I(1) variables
• The t-ratio diverges at rate √T rather than converging to a stable distribution
• Bootstrapping or specialized asymptotic theory is required for valid inference with non-stationary data

The only scenario where regressing non-stationary series yields meaningful results is when they are cointegrated—sharing a long-run equilibrium relationship. In this case, a linear combination of the I(1) variables is itself I(0).

• Engle-Granger two-step method tests for cointegration by examining residual stationarity
• Johansen's trace test identifies multiple cointegrating vectors in multivariate systems
• Cointegrated regressions are superconsistent—estimates converge to true values at rate T rather than √T

When cointegration is absent, the standard remedy is to difference non-stationary variables to stationarity before modeling. This transforms the analysis from spurious levels to meaningful changes.

• First-differencing I(1) variables produces I(0) series suitable for standard inference
• The trade-off is loss of long-run information—only short-run dynamics are captured
• For financial applications, log returns (differenced log prices) are naturally stationary and avoid spurious regression entirely

The foundational example from Udny Yule (1926) demonstrating that two independent random walks will frequently exhibit high R² values and significant t-statistics when regressed against each other. In modern finance, this manifests when regressing the cumulative price level of Apple (AAPL) against a completely unrelated series like cumulative rainfall in London. Both series contain stochastic trends (unit roots), causing the residuals to be non-stationary and violating the Gauss-Markov theorem. The Durbin-Watson statistic will typically be close to zero, signaling severe autocorrelation and invalid inference.

A common pitfall: regressing the S&P 500 level on a macroeconomic variable like US GDP or industrial production in levels rather than differences. Both series are I(1) processes that trend upward over decades. The regression will appear to show a strong predictive relationship, but the residuals will be highly persistent. The correct approach is to test for cointegration using the Engle-Granger or Johansen procedure. If no cointegration exists, the model must be specified in first differences (log returns) to achieve stationarity and valid inference.

The critical distinction: if two price series are cointegrated, the regression is not spurious but represents a genuine long-run equilibrium. For example, Coca-Cola (KO) and PepsiCo (PEP) share prices may both be I(1), but a linear combination is I(0) — stationary. This forms the basis of statistical arbitrage. A spurious regression is ruled out by testing residuals with the Augmented Dickey-Fuller (ADF) test. If residuals reject the unit root null, the relationship is valid and tradable via mean-reversion strategies.

At tick-level or 1-minute bars, spurious regression risk intensifies due to microstructure noise and non-synchronous trading. Regressing the mid-quote of two ETFs that track different sectors can produce spuriously high correlation during overlapping trading hours simply because both respond to the same market-wide order flow shock. The Epps effect — the decay of correlation estimates as sampling frequency increases — is a related phenomenon. Mitigation requires synchronized sampling, Hayashi-Yoshida estimators, or aggregating to lower frequencies where the signal-to-noise ratio improves.

A rigorous method to distinguish spurious from genuine relationships: simulate 10,000 random walks with the same length as your sample and regress them against each other. The distribution of t-statistics from these null regressions will be centered far from zero — typically with a 5% critical value around 3.0 instead of the standard 1.96. If your observed t-statistic falls within this inflated null distribution, the relationship is likely spurious. This Granger-Newbold (1974) simulation approach remains the gold standard for educating analysts on the severity of the problem.

A modern manifestation: regressing daily stock returns on Twitter sentiment scores or news volume counts that are themselves non-stationary. If the sentiment index is constructed as a cumulative sum of positive-minus-negative mentions, it becomes an I(1) process. Regressing it on another trending variable like Bitcoin price will produce spurious results. The fix: use daily changes in sentiment (I(0)) or apply a VECM if cointegration is established. Always check the order of integration of alternative data before modeling.