What is Gauss–Markov Theorem: Why OLS Is BLUE (statistics tutoring)

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Gauss–Markov Theorem: Why OLS Is BLUE (statistics tutoring)
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The Gauss–Markov Theorem is a foundational result in regression analysis and statistics tutoring. It states that under a set of linear model assumptions, the Ordinary Least Squares (OLS) estimator is the Best Linear Unbiased Estimator (BLUE).

Students often struggle with the meaning of “best,” “linear,” and “unbiased,” and how the assumptions guarantee efficiency. For help with OLS, inference, or econometrics, visit the tutoring services page.

Gauss–Markov Theorem: Under the classical assumptions, OLS has the lowest variance among all linear unbiased estimators.

“Best” means minimum variance, “linear” means linear in the dependent variable, and “unbiased” means the estimator’s expectation equals the true parameter.

Why the Gauss–Markov Theorem Matters

The theorem is essential because it:

  • explains why OLS is the default estimator in statistics and econometrics
  • provides efficiency guarantees under mild assumptions
  • forms the basis for inference, confidence intervals, and hypothesis tests
  • is heavily tested in graduate statistics and econometrics courses

How the Gauss–Markov Theorem Works

  1. Start with the linear model:
    y = Xβ + ε.
  2. Assume the Gauss–Markov conditions:
    • Linearity in parameters
    • Random sampling
    • No perfect multicollinearity
    • Zero conditional mean: E[ε|X] = 0
    • Homoskedasticity: Var(ε|X) = σ²I
  3. Define the class of linear unbiased estimators:
    β̂ = Ay for some matrix A.
  4. Show OLS is unbiased:
    E[β̂_OLS] = β.
  5. Compute the variance of any linear unbiased estimator:
    Var(Ay) = Aσ²A’.
  6. Compare variances:
    Var(β̂_OLS) ≤ Var(β̂) for any other linear unbiased estimator.

Numerical Example

Consider a simple regression: y = β₀ + β₁x + ε.

OLS estimator for β₁: β̂₁ = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)².

Under homoskedasticity, its variance is: Var(β̂₁) = σ² / Σ(xᵢ − x̄)².

Any other linear unbiased estimator must have variance ≥ this value. This is the essence of the Gauss–Markov result.

Common Mistakes

  • Thinking Gauss–Markov requires normality (it does not)
  • Confusing homoskedasticity with independence
  • Believing OLS is BLUE under heteroskedasticity (it isn’t)
  • Mixing up “linear estimator” with “linear model”

Why This Matters in Regression Analysis

The Gauss–Markov Theorem underpins:

  • OLS efficiency
  • standard errors and inference
  • model diagnostics
  • econometric theory and applied statistics

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