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Gauss–Markov theorem: what a regression hedge ratio guarantees

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Start with the idea

If the inputs contain enough separate information, errors have mean zero given those inputs, and errors have equal variance with no correlation across observations, OLS has the smallest covariance among linear unbiased estimators. BLUE means best linear unbiased estimator.

Symbols, units & horizon

x_i,y_i: synchronous one-day instrument returns in basis points
i: daily observation index
x̄,ȳ: sample means
β̂: dimensionless estimated slope or return hedge ratio
α̂: estimated intercept in basis points per day
denominator must be positive

When and why to use this

Start here · the same idea in everyday words

Technical term	Everyday meaning	Concrete example
Residual	The observed value minus the fitted value.	If a model predicts 3 bp and the observed return is 5 bp, the residual is 2 bp.
Exogeneity	For any given input values, the model errors average to zero under the assumed probability model.	The inputs must not systematically conceal part of the error; running a regression does not prove this.
Homoskedasticity	The model errors have the same conditional variance across observations.	The assumed error spread does not grow merely because the input gets larger.

Unpack a technical sentence

Technical wording: Under a full-rank design and spherical error covariance, OLS is BLUE.

Read it as: If the inputs contain enough separate information, errors have mean zero given those inputs, and errors have equal variance with no correlation across observations, OLS has the smallest covariance among linear unbiased estimators. BLUE means best linear unbiased estimator.

A slower worked example · one step at a time

Use daily returns x=[−1,0,1] bp and y=[−1,1,3] bp. The mean of x is 0 and the mean of y is 1.
Subtract the means: centred x is [−1,0,1] and centred y is [−2,0,2].
Multiply matching centred entries and add: 2+0+2=4 bp². Add centred x squares: 1+0+1=2 bp². Divide 4 by 2 to get slope 2.
The intercept is mean y minus slope times mean x: 1−2 times 0=1 bp. At x=1 bp the fitted y is 1+2 times 1=3 bp.
If a new observation has x=1 bp and y=5 bp, its residual is 5−3=2 bp. One residual does not establish the model assumptions; inspect errors across the sample and evaluate later observations separately.

Self-assessed: Does “best” mean OLS is the best possible forecast model?

No. The theorem compares a defined class of estimators under defined assumptions. It is not a ranking of all possible forecasting methods.

Study loop: predict the result, work through the arithmetic, run the Python, then explain which assumption would make the result unreliable.

For one regressor with an intercept, centre both series. The OLS slope is centred cross-product divided by centred regressor sum of squares.
Assume a full-rank design, zero conditional mean errors and conditional error covariance equal to a common variance times the identity. OLS is then best linear unbiased: it has the smallest covariance among estimators linear in the outcomes and unbiased under the model. Gaussian errors are not required for this theorem.
Here “linear” refers to the estimator as a function of the outcomes. “Best” does not mean lowest forecast loss among every machine-learning model. Heteroskedasticity or serial correlation breaks the spherical-error premise; robust standard errors address inference under additional conditions, not omitted-variable bias.

Before reading the working, evaluate 4/2. Give the result in dimensionless slope.

The fitted return equation has intercept 1 bp per day and slope 2. This exact synthetic fit is a coding benchmark, not evidence of a profitable hedge.

\hat{β} = \frac{\sum_{i} (x_{i} - \overline{x}) (y_{i} - \overline{y})}{\sum_{i} (x_{i} - \overline{x})^{2}}, \hat{α} = \overline{y} - \hat{β} \overline{x}

Linear-model theorem conditional on design assumptions

Gauss–Markov theorem: what a regression hedge ratio guarantees

Use x=[−1,0,1] and y=[−1,1,3] bp. Their means are 0 and 1 bp.
Centred cross-products sum to (−1)(−2)+0×0+1×2=4 bp². Centred x squares sum to 1+0+1=2 bp².
Slope is 4/2=2 and intercept is 1−2×0=1 bp per day.

Work it by hand

The fitted return equation has intercept 1 bp per day and slope 2. This exact synthetic fit is a coding benchmark, not evidence of a profitable hedge.

An analogy to remember

A fitted ruler summarises how two measurements move together. The theorem compares the precision of particular ways to fit that ruler.

How this becomes a building block

A beta estimation component should return aligned observations and residual diagnostics alongside the fitted coefficients.

Python · run and reuse

Architecture pseudocode

# Python 3.10+; standard library only unless NumPy is imported.
# Synthetic teaching inputs; no historical performance claim.
import numpy as np

def ols_one_feature(x,y):
    # NumPy dependency; matched synchronous daily returns in bp.
    x,y = np.asarray(x,dtype=float),np.asarray(y,dtype=float)
    if x.ndim != 1 or x.shape != y.shape or x.size < 2 or not np.isfinite(x).all() or not np.isfinite(y).all():
        raise ValueError("Matching finite one-dimensional samples required")
    xc,yc = x-x.mean(),y-y.mean()
    denominator = xc@xc
    if denominator <= 0: raise ValueError("Regressor must vary")
    beta = (xc@yc)/denominator
    return float(y.mean()-beta*x.mean()),float(beta)

print(ols_one_feature([-1,0,1],[-1,1,3]))

# Replay the slower, synthetic example above. Units are stated in its steps.
def plain_language_example():
    from math import exp, e  # Standard library; no extra dependency.
    return {
        "slope": 4/2,
        "intercept": 1-2*0,
        "fitted": 1+2*1,
        "residual": 5-(1+2*1),
    }

print(plain_language_example())

Self-assessed: explain the calculation, its output units, and one condition under which it fails.

Use OLS to estimate return betas or a candidate hedge ratio, then check residual dependence, stability and instrument units. A return beta must be converted into tradable quantities using prices and contract multipliers. Conditional mean-zero errors are substantive. Endogeneity, asynchronous data or regime changes can invalidate the interpretation even if a regression runs successfully.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

# Python 3.10+; standard library only unless NumPy is imported.
# Synthetic teaching inputs; no historical performance claim.
import numpy as np

def ols_one_feature(x,y):
    # NumPy dependency; matched synchronous daily returns in bp.
    x,y = np.asarray(x,dtype=float),np.asarray(y,dtype=float)
    if x.ndim != 1 or x.shape != y.shape or x.size < 2 or not np.isfinite(x).all() or not np.isfinite(y).all():
        raise ValueError("Matching finite one-dimensional samples required")
    xc,yc = x-x.mean(),y-y.mean()
    denominator = xc@xc
    if denominator <= 0: raise ValueError("Regressor must vary")
    beta = (xc@yc)/denominator
    return float(y.mean()-beta*x.mean()),float(beta)

print(ols_one_feature([-1,0,1],[-1,1,3]))

# Replay the slower, synthetic example above. Units are stated in its steps.
def plain_language_example():
    from math import exp, e  # Standard library; no extra dependency.
    return {
        "slope": 4/2,
        "intercept": 1-2*0,
        "fitted": 1+2*1,
        "residual": 5-(1+2*1),
    }

print(plain_language_example())

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