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Proof of Unbiasedness of the OLS Estimator: E(β̂) = β

Master the rigorous proof of OLS estimator unbiasedness, \( E(\hat{\boldsymbol{\beta}}) = \boldsymbol{\beta} \). Understand critical assumptions, geometric intuition, and common pitfalls for robust linear modeling.

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The Formal Theorem

Given the multiple linear regression model y=Xβ+ϵ \mathbf{y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\epsilon} , where y \mathbf{y} is an n×1 n \times 1 vector of observations, X \mathbf{X} is an n×k n \times k design matrix of full column rank, β \boldsymbol{\beta} is a k×1 k \times 1 vector of unknown parameters, and ϵ \boldsymbol{\epsilon} is an n×1 n \times 1 vector of random errors satisfying the Gauss-Markov assumption E(ϵ∣X)=0 E(\boldsymbol{\epsilon} | \mathbf{X}) = \mathbf{0} , the Ordinary Least Squares (OLS) estimator β^ \hat{\boldsymbol{\beta}} , defined as β^=(XTX)−1XTy \hat{\boldsymbol{\beta}} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y} , is an unbiased estimator of β \boldsymbol{\beta} , formally stated as:
E(β^)=β E(\hat{\boldsymbol{\beta}}) = \boldsymbol{\beta}

Analytical Intuition.

Imagine being a master marksman aiming for a luminous bullseye that represents the true, unknown population parameter β \boldsymbol{\beta} . Each time you fire, you collect a sample and calculate an Ordinary Least Squares estimate, β^ \hat{\boldsymbol{\beta}} . While your hand might shake slightly (representing the random error term ϵ \boldsymbol{\epsilon} ), causing each individual bullet to land slightly off the bullseye, the unbiasedness property guarantees something profound: if you were to fire an infinite number of shots, the *average* landing point of all those bullets would be precisely the center of the bullseye. The inherent randomness ϵ \boldsymbol{\epsilon} doesn't systematically push your aim high or low, left or right; it simply introduces symmetrical, zero-mean deviations around the true target. Therefore, our OLS estimator β^ \hat{\boldsymbol{\beta}} is a tool that, on average, provides an accurate reading of the underlying reality, ensuring E(β^)=β E(\hat{\boldsymbol{\beta}}) = \boldsymbol{\beta} .
CAUTION

Institutional Warning.

Students often confuse E(ϵ)=0 E(\boldsymbol{\epsilon}) = \mathbf{0} with the critical E(ϵ∣X)=0 E(\boldsymbol{\epsilon} | \mathbf{X}) = \mathbf{0} . The latter, strict exogeneity, ensures errors are uncorrelated with *all* regressors, guaranteeing finite-sample unbiasedness. The former, merely zero unconditional mean error, is insufficient if X \mathbf{X} is correlated with ϵ \boldsymbol{\epsilon} .

Academic Inquiries.

01

Does the unbiasedness property hold if the regressors X \mathbf{X} are stochastic (random variables) rather than fixed?

Yes, provided the strict exogeneity assumption E(ϵ∣X)=0 E(\boldsymbol{\epsilon} | \mathbf{X}) = \mathbf{0} holds. This conditional expectation accounts for X \mathbf{X} being stochastic by ensuring that, for any realization of X \mathbf{X} , the errors average to zero, thus allowing the law of iterated expectations to yield E(β^)=β E(\hat{\boldsymbol{\beta}}) = \boldsymbol{\beta} .

02

Is an unbiased estimator always preferred over a biased one?

Not necessarily. While unbiasedness is desirable, it doesn't consider estimator variance. A slightly biased estimator with much lower variance might be preferred, especially in terms of Mean Squared Error (MSE). This is known as the bias-variance trade-off, crucial in advanced estimation theory.

03

What happens to unbiasedness if there is perfect multicollinearity among the regressors?

Perfect multicollinearity means the design matrix X \mathbf{X} does not have full column rank, rendering XTX \mathbf{X}^T \mathbf{X} singular. Consequently, its inverse (XTX)−1 (\mathbf{X}^T \mathbf{X})^{-1} does not exist, and the OLS estimator β^ \hat{\boldsymbol{\beta}} cannot be uniquely computed, thus making the concept of its unbiasedness moot.

04

How does omitted variable bias specifically break the unbiasedness of OLS?

Omitted variable bias occurs when a relevant variable, correlated with both an included regressor and the dependent variable, is left out of the model. Its effect is absorbed into the error term ϵ \boldsymbol{\epsilon} , making ϵ \boldsymbol{\epsilon} correlated with the included X \mathbf{X} . This violates E(ϵ∣X)=0 E(\boldsymbol{\epsilon} | \mathbf{X}) = \mathbf{0} , causing E((XTX)−1XTϵ) E((\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \boldsymbol{\epsilon}) to be non-zero, leading to a biased β^ \hat{\boldsymbol{\beta}} .

Standardized References.

  • Definitive Institutional SourceWooldridge, J. M. (2019). Introductory Econometrics: A Modern Approach (7th ed.). Cengage Learning.

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Derivation of the Variance-Covariance Matrix of the OLS Estimator: Var(β̂) = σ²(X'X)⁻¹

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Proof of Unbiasedness of the OLS Estimator: E(β̂) = β: Visual Proof & Intuition. Retrieved from https://www.nicefa.org/library/general-linear-models-/proof-of-unbiasedness-of-the-ols-estimator--e--------

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