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Gauss-Markov Theorem

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Gauss-Markov Theorem

3.1 The BLUE Property

Gauss-Markov Theorem: Under the classical assumptions (linearity, zero mean errors, homoscedasticity, uncorrelated errors), the OLS estimator β^\hat{\boldsymbol{\beta}} is BLUE — Best Linear Unbiased Estimator.

Assumptions

  1. Linearity: y=Xβ+εy = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}
  2. Zero mean: E[ε]=0E[\boldsymbol{\varepsilon}] = 0
  3. Homoscedasticity: Var(εi)=σ2\text{Var}(\varepsilon_i) = \sigma^2 (constant variance)
  4. Uncorrelated errors: Cov(εi,εj)=0\text{Cov}(\varepsilon_i, \varepsilon_j) = 0 for i≠ji \neq j
  5. Full rank: X\mathbf{X} has full column rank

Variance of OLS Estimator

Var(β^)=σ2(XTX)−1\text{Var}(\hat{\boldsymbol{\beta}}) = \sigma^2 (\mathbf{X}^T\mathbf{X})^{-1}
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