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

A rigorous derivation of the Variance-Covariance matrix for the OLS estimator, exploring the geometric impact of data configuration on statistical precision.

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The Formal Theorem

Let the linear model be Y=Xβ+ϵ Y = X\beta + \epsilon , where Y∈Rn Y \in \mathbb{R}^n , X∈Rn×p X \in \mathbb{R}^{n \times p} is a matrix of full column rank, β∈Rp \beta \in \mathbb{R}^p , and ϵ∼(0,σ2In) \epsilon \sim (0, \sigma^2 I_n) . The OLS estimator β^=(X′X)−1X′Y \hat{\beta} = (X'X)^{-1}X'Y has a variance-covariance matrix given by:
Var(β^)=E[(β^−β)(β^−β)′]=σ2(X′X)−1 \begin{aligned} \text{Var}(\hat{\beta}) = E[(\hat{\beta} - \beta)(\hat{\beta} - \beta)'] = \sigma^2 (X'X)^{-1} \end{aligned}

Analytical Intuition.

Imagine the OLS estimator β^ \hat{\beta} as a camera lens attempting to focus on the true parameter β \beta through the 'fog' of random noise ϵ \epsilon . The matrix X′X X'X acts as our 'information manifold'; it encapsulates how much data we have and how well-distributed it is across the features. If the features in X X are highly correlated, the matrix X′X X'X becomes near-singular, effectively shrinking the determinant and blowing up the variance—like trying to sharpen a blurred image with a lens that has too little surface area. The term σ2 \sigma^2 represents the intrinsic 'fuzziness' or volatility of the underlying process. Thus, the variance of our estimate is a tug-of-war between the inherent noise of the universe σ2 \sigma^2 and the strength of our data configuration (X′X)−1 (X'X)^{-1} . A well-conditioned X X spreads the variance out, ensuring that β^ \hat{\beta} remains tightly clustered around β \beta , providing the stability required for rigorous statistical inference.
CAUTION

Institutional Warning.

Students frequently conflate the variance of the residuals σ^2 \hat{\sigma}^2 with the variance of the parameters Var(β^) \text{Var}(\hat{\beta}) . Remember: σ^2 \hat{\sigma}^2 measures noise in data, while Var(β^) \text{Var}(\hat{\beta}) measures uncertainty in our estimated coefficients.

Academic Inquiries.

01

What happens if X is not full rank?

If X X is not full rank, X′X X'X is singular and non-invertible. This signifies perfect multicollinearity, meaning the parameters are not uniquely identifiable.

02

Does this derivation assume normality of errors?

No. The variance-covariance derivation requires only the Gauss-Markov assumptions (constant variance and uncorrelated errors); normality is only required for exact finite-sample inference.

03

Why is (X′X)−1 (X'X)^{-1} called the information matrix?

In the context of Likelihood theory, X′X/σ2 X'X/\sigma^2 is the Fisher Information matrix. Its inverse is the Cramer-Rao lower bound, representing the minimum possible variance for an unbiased estimator.

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). Derivation of the Variance-Covariance Matrix of the OLS Estimator: Var(β̂) = σ²(X'X)⁻¹: Visual Proof & Intuition. Retrieved from https://www.nicefa.org/library/general-linear-models-/derivation-of-the-variance-covariance-matrix-of-the-ols-estimator--var----------x-x---

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