Stationarity Tests and Why They Matter for Signals

A signal built on a non-stationary feature can show a strong in-sample fit that has nothing to do with a repeatable relationship.

A stationarity test checks whether a time series' statistical properties, its mean, its variance, and the way it correlates with its own past, stay constant over time rather than drifting. Signals built on non-stationary inputs routinely show strong relationships in a backtest that vanish out of sample, because the regression picked up a trend or a shifting level rather than a genuine, repeatable relationship.

A weakly stationary series has a constant mean, a constant variance, and an autocovariance between any two points in the series that depends only on the distance between them, not on where in time they fall. A price series does not have this property. Its variance grows with the length of the window examined and its mean depends heavily on the sample period chosen, which is the definition of a random walk rather than a stationary process.

Simple returns are usually much closer to stationary in their mean than price levels are, since the average daily return over most windows sits near a small, roughly constant number. Their variance is a separate story: it clusters into calm and turbulent periods rather than staying constant, so returns are closer to stationary in one property and not in another, and treating a series as fully stationary because one property checks out is a common shortcut that causes trouble later. This same distinction, a stationary mean sitting alongside a non-stationary variance, is central to mean-reverting strategy design, covered from the trading side in mean reversion and stochastic processes.

The Augmented Dickey-Fuller test, or ADF, sets its null hypothesis as the presence of a unit root, meaning the series is non-stationary. The test statistic is compared against a critical value, and the null is rejected, in favor of stationarity, only when the statistic is more negative than that critical value. A researcher who runs an ADF test and fails to reject the null has not proven the series is non-stationary. Failing to reject a null hypothesis is an absence of sufficient evidence against it, not proof that it is true.

A researcher applying this test picks a lag length for the underlying regression carefully, since too few lags leave autocorrelation in the test's own residuals and too many lags weaken its power to detect a genuine unit root. Automated lag-selection criteria such as the Akaike or Schwarz information criterion are the standard way to make that choice rather than guessing at a fixed number.