Stationarity and autocorrelation

Statistics learned on the past only tell you about the future if the process generating the data has not changed. A series whose statistical properties stay constant over time is called stationary.

Weak stationarity

A time series $X_t$ is (weakly) stationary if

1. its mean $\mathbb{E}[X_t] = \mu$ does not depend on $t$,
2. its variance is finite and constant, and
3. its autocovariance $\operatorname{Cov}(X_t, X_{t-k})$ depends only on the lag $k$, not on $t$.

Prices fail this: they drift and their variance grows with time. Returns are much closer to stationary, which is why almost all modelling is done on returns.

The random walk and the unit root

The simplest non-stationary model is a random walk, $P_t = P_{t-1} + \varepsilon_t$. More generally, consider

$$X_t = \phi X_{t-1} + \varepsilon_t$$

This is an AR(1) process. If $|\phi| < 1$ the series is stationary and mean-reverting, with a long-run variance of $\sigma_\varepsilon^2/(1-\phi^2)$. If $\phi = 1$ it has a unit root: shocks never decay, and the series wanders like a price.

The Augmented Dickey–Fuller (ADF) test checks the null hypothesis $\phi = 1$. Rejecting it (a sufficiently negative test statistic) is evidence the series is mean-reverting. The test is the standard first step in pairs trading, applied to the spread between two assets.

Half-life of mean reversion

For a stationary AR(1), a deviation decays by a factor $\phi$ each period. The time for half of it to disappear is

$$h = -\frac{\ln 2}{\ln \phi}$$

A spread with $\phi = 0.95$ per day has a half-life of about 13.5 days. That number tells you how long trades will last, how often you’ll trade, and therefore whether costs will eat the edge.

Autocorrelation

The autocorrelation at lag $k$ is

$$\rho_k = \frac{\operatorname{Cov}(X_t, X_{t-k})}{\operatorname{Var}(X_t)}$$

For returns, a positive $\rho_1$ means moves tend to continue (momentum); negative means they tend to reverse (mean reversion). Under the null of no autocorrelation, sample estimates have a standard error of roughly $1/\sqrt{T}$, so with 1,000 observations anything inside $\pm 0.063$ is indistinguishable from noise at the usual 95% level.

Why it matters for your P&L

If your strategy’s daily returns are autocorrelated, the usual $\sqrt{N}$ annualisation is off. A useful correction for the variance of a $q$-period sum is

$$\operatorname{Var}\Big(\sum_{t=1}^{q} r_t\Big) = q\,\sigma^2 \Big(1 + 2\sum_{k=1}^{q-1}\big(1 - \tfrac{k}{q}\big)\rho_k\Big)$$

Positive autocorrelation inflates long-horizon risk; negative autocorrelation (common in mean-reversion strategies) reduces it, which flatters the annualised Sharpe ratio.