Half-Life of Mean Reversion

Formula

Δy_t = a + g · y_{t−1} + ε_t
φ = 1 + g

half-life = ln(2) / −ln(1 + g)     if −1 < g < 0
          = ∞                       if g ≥ 0
          undefined (NaN)           if g ≤ −1  (oscillatory or explosive)

Continuous approximation, small g:  ln(2) / −g
Example: φ = 0.9 → g = −0.1 → half-life = ln(2) / −ln(0.9) ≈ 6.58 periods.
The ADF page uses this discrete definition, not ln(2) / −g.

Where the number is used

• As a holding-period scale for a spread that has already passed a stationarity check. See ADF and cointegration. A half-life of six days and a half-life of six months are different research problems.
• As a way to set the window on a rolling z-score. A window much shorter than the half-life mostly fits noise; a window of many half-lives averages different regimes.
• Hurst is a related but different summary of the same wish: H below one half suggests anti-persistence. It is not a half-life.

Pitfalls

• Use the discrete form. Dividing ln 2 by −g treats g as a continuous-time speed. The regression you just ran is discrete. This function matches the half-life already published on the ADF page, ln 2 / −ln(1+g).
• The regression is not a proof of mean reversion. A short sample of a random walk often produces a slightly negative g. Pair the half-life with an ADF or cointegration test, and expect the estimate to move when you add data.
• An intercept is included. The mean that the series reverts to is −a/g, not necessarily zero. Forcing the regression through the origin changes g.
• Not a trading rule. A finite half-life is a sample statistic. It does not say the spread will keep reverting, or that a position sized to it is desirable.

Python reference implementation

Requires numpy. Tested before publication; the generator re-executes it on every rebuild. MIT licensed. Download half-life-mean-reversion.py · All snippets

import numpy as np

def half_life(spread) -> float:
    """Discrete mean-reversion half-life, ln(2) / -ln(1+g), from Δy = a + g y_lag.

    Infinite when g >= 0 (no mean reversion). NaN when g <= -1.
    Same definition as half_life on the ADF page.
    """
    y = np.asarray(spread, dtype=float)
    if y.ndim != 1 or y.size < 3 or not np.all(np.isfinite(y)):
        raise ValueError("need at least three finite observations")
    lag = y[:-1]
    if float(np.var(lag)) == 0.0:
        return float("inf")
    X = np.column_stack([np.ones(lag.size), lag])
    g = float(np.linalg.lstsq(X, np.diff(y), rcond=None)[0][1])
    if g >= -1e-12:
        return float("inf")
    if g <= -1.0:
        return float("nan")
    return float(np.log(2.0) / -np.log1p(g))

FAQ

Why not use −ln(2) / g, which is the formula many notes quote?

That is the continuous-time limit. The regression Δy_t = a + g y_{t−1} implies a discrete autoregressive coefficient φ = 1 + g, and a deviation multiplies by φ each period. Halving means φ^h = 1/2, so h = ln(2) / −ln(φ). At φ = 0.9 the continuous shortcut is about 6.93 periods and the discrete half-life is about 6.58.

What half-life should a pairs trade use?

This page does not pick one. It estimates a half-life from the series you pass. Whether the residual is stationary, whether both legs can be traded, and what it costs are separate questions; a finite estimate is not a signal to trade the spread.