Rolling Hedge Ratio and Spread Z-Score

Approach

β_t     = Cov_w(y, x)_t / Var_w(x)_t          (pandas sample cov and var, ddof = 1)
spread_t = y_t − β_t x_t
z_t      = (spread_t − mean_w(spread)_t) / std_w(spread)_t

β_t is defined once `window` observations exist.
z_t needs a further `window` finite spreads, so it starts about 2·window observations in.
std = 0 → z is missing, not infinite.

How it sits next to cointegration

• A hedge ratio is only interesting as a mean-reverting spread if the series are cointegrated, or at least if the spread rejects a unit root. See the ADF page.
• The half-life of the spread is the usual way to choose the window that standardises z, rather than picking 60 because it is round.
• The same rolling slope, compiled and re-centred, is rolling_beta in rolling-window-numba.

Pitfalls

• No intercept. The spread is y − βx, not y − a − βx. If the two prices differ by a level, that level sits inside the spread and the z-score spends a long time absorbing it. Engle-Granger includes an intercept.
• Nested windows. z_t standardises a spread that already used a window to estimate β. Early z values are noisier than a z-score of a fixed-β residual.
• Look-ahead is not in the formulas, but it is easy to add. β_t uses prices through t. A position decided from z_t must not trade at the close of t. Lag the position by at least one bar, as on the Polars backtest page.
• Not a signal to trade. A z-score of 2 is two recent standard deviations. It is not an expected return and it is not a suggestion to buy or sell the spread.

Python reference implementation

Requires pandas. Tested before publication; the generator re-executes it on every rebuild. MIT licensed. Download pairs-zscore-signal.py · All snippets

import pandas as pd

def pair_zscore(y: pd.Series, x: pd.Series, window: int = 60) -> pd.DataFrame:
    """Rolling slope of y on x, the residual spread, and its rolling z-score.

    Sample covariance and variance (ddof=1), so the ratio matches OLS with an
    intercept-free slope. Illustrative only; not a trading signal.
    """
    if window < 2:
        raise ValueError("window must be at least 2")
    y, x = y.align(x, join="inner")
    beta = y.rolling(window).cov(x) / x.rolling(window).var()
    spread = y - beta * x
    sd = spread.rolling(window).std()
    z = (spread - spread.rolling(window).mean()) / sd.where(sd > 0)
    return pd.DataFrame({"beta": beta, "spread": spread, "zscore": z})

FAQ

Why does the z-score start so late?

The slope needs `window` rows, and the mean and standard deviation of the spread need another `window` finite spreads. With a window of 60 the z-score becomes available about 120 rows in, not 60.

Can I trade the sign of the z-score?

This function only standardises a spread. Whether the spread mean-reverts, whether both legs can be traded, and what it costs are separate questions. The output is not an order.