Stationarity

Stationarity is a property of a time series whose statistical characteristics stay the same throughout its history. A stationary series fluctuates around a fixed average with a stable degree of variability, and the relationship between its values at different times depends only on how far apart they are, not on when they occur. A series lacking these features is called non-stationary.

The concept matters because most statistical methods assume it. Averages, correlations and regression coefficients estimated from a stationary series describe something lasting, while the same quantities estimated from a non-stationary series may be meaningless. In finance, asset prices are generally non-stationary and returns are approximately stationary, which is why quantitative work is done on returns and spreads and not on price levels.

A series is strictly stationary if the joint probability distribution of any set of its values is unchanged when the whole set is shifted in time. This is hard to verify, so applied work uses the weaker form, called weak or covariance stationarity, which requires three conditions:

• The mean of the series is the same at every point in time.
• The variance is finite and the same at every point in time.
• The covariance between two values depends only on the number of periods separating them.

Common causes of non-stationarity are a trend, a level that wanders without limit, seasonal patterns and abrupt structural breaks.

The simplest model that can be either stationary or not is the first-order autoregression:

• $y_t$ is the value of the series at time $t$.
• $c$ is a constant.
• $\phi$ is the coefficient on the previous value.
• $\varepsilon_t$ is a random shock with mean zero and constant variance.

The series is stationary when $|\phi| \lt 1$. Shocks then fade over time and the series is drawn back toward a long-run mean of $c / (1 - \phi)$. When $\phi = 1$ the model becomes a random walk: shocks never fade, there is no long-run mean and the variance grows without limit. This case is described as having a unit root.

With $c = 2$ and $\phi = 0.8$, the long-run mean is $2 / (1 - 0.8) = 10$. If the series stands at 15, the expected next value is $2 + 0.8 \times 15 = 14$, one fifth of the way back. The half-life of a deviation, the time for it to shrink by half, is $\ln(0.5) / \ln(0.8)$, about 3.1 periods.

The standard tests look for a unit root. The Dickey-Fuller test and its extension, the augmented Dickey-Fuller test, take non-stationarity as the null hypothesis and reject it if the evidence of reversion is strong enough. The KPSS test reverses the roles and takes stationarity as the null, so the two are often run together.

A non-stationary series can often be made stationary. Taking differences, the change from one period to the next, removes a unit root; a series that becomes stationary after one round of differencing is called integrated of order one. This is the integration step in an ARIMA model. Taking logarithms stabilizes variance that grows with the level, and a log difference of prices is a return.

Regressing one non-stationary series on another can produce a high coefficient of determination and apparently significant coefficients even when the two are unrelated. Clive Granger and Paul Newbold described this problem of spurious regression in 1974. Two random walks will often look strongly correlated by chance.

The exception is cointegration. Two non-stationary series are cointegrated if some combination of them is stationary, which means they share a common long-run path. In that case a regression between them is meaningful.

The search for a stationary series is the core of mean reversion strategies. In pairs trading, neither share price is stationary, but the trader looks for a pair whose spread is. A stationary spread has a level to return to, so a large deviation, often measured as a z-score, is treated as a signal, and the half-life indicates how long a position may need to be held.

Stationarity is equally relevant to forecasting models. A model trained on past data assumes that the relationships it learned will persist, and inputs are usually transformed into stationary form, such as returns or ratios, before they are used.

Stationarity is an assumption about the future as well as a description of the past. Tests have low power in short samples and can fail to distinguish a slowly reverting series from a random walk. A relationship that was stationary for years can break when a company is acquired, a regulation changes or a market enters a new regime. Financial returns also show periods of calm and turbulence, a feature modelled by GARCH, so their short-run variability is not constant even where the long-run variance is.