`EwBeta`#

Description#

EwBeta computes the exponentially weighted regression slope of x on y:

\[ \text{EwBeta}(x, y) = \frac{\operatorname{Cov}(x, y)}{\operatorname{Var}(y)} \]

This is a 2-input, 1-output function (FunctorBase<_, 2, 1>).

Argument-order convention#

The first argument is the dependent (target), the second is the regressor. Calling EwBeta(asset_returns, market_returns) returns the asset's beta with respect to the market, matching the CAPM definition and the existing RollingBeta convention.

This is the opposite of pandas (pandas.ols-style and most pandas docs use slope of y on x).

Parameters#

Specify exactly one of the following to set the smoothing factor alpha:

com: Center of mass. alpha = 1 / (1 + com)
span: Span. alpha = 2 / (span + 1)
halflife: Half-life. alpha = 1 - exp(-log(2) / halflife)
alpha: Directly sets the smoothing factor, 0 < alpha < 1

The first sample is NaN. Subsequent NaNs occur when the regressor has zero variance over the effective window.

Formula#

The bias-correction factor that EwVar and EwCov apply cancels between numerator and denominator, so EwBeta uses the simpler unbiased form. With \(\bar{x} = S_x / S_w\) and \(\bar{y} = S_y / S_w\):

\[ \text{EwBeta} = \frac{\dfrac{S_{xy}}{S_w} - \bar{x}\,\bar{y}} {\dfrac{S_{yy}}{S_w} - \bar{y}^2} \]

If the denominator is zero or non-positive, the output is NaN.

Identity check#

By definition EwBeta(x, y) == EwCov(x, y) / EwVar(y). Verified to ~1e-12 in the test suite.

NaN handling#

Policy: ignore. A NaN in any input at index t causes the function to skip that step: output at t is NaN and internal state is unchanged. Subsequent finite samples are processed as if step t had not occurred.

Examples#

Usage example#

import numpy as np
import pandas as pd
from screamer import EwBeta

rng = np.random.default_rng(0)
market = rng.standard_normal(500)
# An asset with true beta = 1.5 plus idiosyncratic noise
asset = 1.5 * market + 0.5 * rng.standard_normal(500)

# CAPM convention: dependent first, regressor second
beta = EwBeta(span=60)(asset, market)

# Validate against pandas cov/var
ref = (
    pd.Series(asset).ewm(span=60).cov(pd.Series(market)).to_numpy()
    / pd.Series(market).ewm(span=60).var().to_numpy()
)
np.testing.assert_allclose(beta, ref, equal_nan=True, atol=1e-10)

Reference#

Equivalent to pandas.Series.ewm(adjust=True).cov(market) / market.ewm(adjust=True).var(). Pandas does not expose an ewm.beta() method directly.

EwBeta#