Stationarity, Differencing and Unit Roots

A time series is stationary if its statistical properties, such as its mean, variance and autocorrelation, do not change over time. Stationarity matters because most statistical methods assume it: if the average and volatility of a series keep shifting, relationships estimated in the past may not hold in the future. Prices are almost never stationary; returns are much closer. Many trading strategies, especially mean reversion and pairs trading, depend on finding series that really are stationary.

Types of stationarity#

In practice, "stationary" usually means weakly stationary.

Stationary vs non stationary#

Unit roots#

A random walk has a "unit root": shocks have a permanent effect, and the series does not return to a fixed mean. Many price series behave as if they have a unit root. Differencing them (taking changes) removes the unit root and produces a stationary series. See White Noise and Random Walks.

random walk: P_t = P_(t-1) + ε_t   (non stationary)
first difference: ΔP_t = P_t - P_(t-1) = ε_t   (stationary)

Testing for stationarity#

Using ADF and KPSS together gives a clearer picture, since they have opposite null hypotheses.

Half life of mean reversion#

For a stationary, mean reverting series modelled as AR(1), the half life estimates how long a deviation takes to shrink by half:

half life = -ln(2) / ln(φ)

where φ is the AR(1) coefficient. For φ = 0.94 per day, the half life is about 11 days. See Mean Reversion and ARIMA.

Making data stationary#

Why stationarity matters for traders#

• Spurious regressions: regressing non stationary series on each other produces false relationships. See Regression Analysis.
• Mean reversion strategies only work on series that actually revert.
• Machine learning features built from non stationary data can fail out of sample. See Feature Engineering.
• Regime changes break stationarity even in returns. See Structural Breaks and Regime Changes.

Frequently asked questions#

What does stationary mean in time series?#

That the series' statistical properties, such as its mean, variance and autocorrelation, stay constant over time.

Are stock prices stationary?#

No. Prices trend and wander, behaving like a random walk; returns are much closer to stationary.

How do you test for stationarity?#

With tests such as the Augmented Dickey Fuller (ADF) test, the Phillips Perron test and the KPSS test.

Next, learn how two non stationary series can stay together in Cointegration.