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Proof of Unbiasedness for Ordinary Least Squares (OLS) Regression Coefficients

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The Formal Theorem

In the classical linear regression model defined by y=Xβ+ϵ \mathbf{y} = \mathbf{X}\beta + \mathbf{\epsilon} , where X \mathbf{X} is an n×k n \times k non-stochastic matrix of rank k k and E[ϵ]=0 E[\mathbf{\epsilon}] = \mathbf{0} , the OLS estimator defined by:
β^=(XTX)−1XTy \hat{\beta} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{y}
is an unbiased estimator of β \beta , such that:
E[β^]=β E[\hat{\beta}] = \beta

Analytical Intuition.

Imagine you are an explorer trying to find the exact center of a hidden temple, but your GPS is flickering due to atmospheric noise. In the language of statistics, the true center is β \beta , and the noise is ϵ \mathbf{\epsilon} . Ordinary Least Squares (OLS) is your navigation algorithm. Unbiasedness is a cinematic guarantee of the algorithm's integrity: it promises that while any single measurement might be 'off' due to random chance, the algorithm itself has no systematic 'lean.' If you were to repeat your journey across a thousand parallel dimensions and average your results, the noise would perfectly oscillate into nothingness, leaving you exactly at the temple's heart. Mathematically, this is achieved because the OLS estimator is a linear combination of the data; when we take the expectation, the linear operator passes through the constant matrices to confront the error term. Since the expected value of the error is zero, the 'noise' component vanishes, leaving the pure parameter β \beta standing alone in the light of mathematical truth.
CAUTION

Institutional Warning.

Students often confuse unbiasedness with efficiency. Unbiasedness only means the 'average' result is correct, not that the result is 'close' to the truth in a single trial. An estimator can be unbiased but still have such massive variance that individual estimates are practically useless.

Academic Inquiries.

01

Does the proof of unbiasedness require the assumption of Homoscedasticity?

No. Unbiasedness only requires E[ϵ]=0 E[\mathbf{\epsilon}] = \mathbf{0} and that X \mathbf{X} is independent of the errors. Homoscedasticity is required for the Gauss-Markov theorem regarding minimum variance (efficiency), but not for unbiasedness itself.

02

What happens if the regressor matrix X \mathbf{X} is stochastic?

If X \mathbf{X} is random, we require the stronger assumption that E[ϵ∣X]=0 E[\mathbf{\epsilon} | \mathbf{X}] = \mathbf{0} . If X \mathbf{X} and ϵ \mathbf{\epsilon} are correlated, the OLS estimator becomes biased and potentially inconsistent.

03

Why is the 'Full Rank' assumption critical for this proof?

The proof relies on the existence of the inverse matrix (XTX)−1 (\mathbf{X}^T\mathbf{X})^{-1} . If X \mathbf{X} does not have full column rank, the matrix is singular, and the OLS estimator cannot be uniquely computed.

Standardized References.

  • Definitive Institutional SourceGreene, W. H., Econometric Analysis.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Proof of Unbiasedness for Ordinary Least Squares (OLS) Regression Coefficients: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/applied-statistics/proof-of-unbiasedness-for-ordinary-least-squares--ols--regression-coefficients

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