Multiple Regression & Inferenceintermediate

The Gauss-Markov Theorem

Adding MLR.5 homoskedasticity, $\text{Var}(u\mid x_1,\dots,x_k)=\sigma^2$ , to MLR.1 through MLR.4 completes the Gauss-Markov assumptions. Under them the Gauss-Markov theorem says OLS is BLUE, the best linear unbiased estimator, meaning it has the smallest variance among all linear unbiased estimators of the $\beta_j$ . BLUE is an efficiency claim, so it does not by itself justify the usual t and F tests.

Why it matters

Once errors are unbiased and equally noisy across all values of the regressors, no other linear unbiased recipe beats OLS on precision. That is the payoff of the homoskedasticity assumption. If the error variance changes with $x$ , OLS stays unbiased but loses the best-in-class title.

Formulas

MLR.5 homoskedasticity

\text{Var}(u\mid x_1,\dots,x_k)=\sigma^2

Error variance is constant across all regressor values.

OLS slope variance

\text{Var}(\hat{\beta}_j)=\frac{\sigma^2}{SST_j\,(1-R_j^2)}

R_j^2

is from regressing

x_j

on the other regressors; high

R_j^2

inflates the variance.

Worked examples

Scenario

You estimate `regress lwage educ exper tenure` and trust the default standard errors.

Solution

Those default standard errors are valid only under MLR.5. If a `estat hettest` rejects homoskedasticity, OLS is still unbiased but no longer BLUE, and you switch to `regress lwage educ exper tenure, robust` for valid inference.

NoteGauss-Markov concerns efficiency; robust standard errors fix inference, not the point estimates.

Common mistakes

✗Thinking Gauss-Markov needs normal errors. BLUE follows from MLR.1 to MLR.5 with no distributional assumption on $u$ .
✗Believing heteroskedasticity biases the OLS coefficients. It does not; it only voids the BLUE property and the default standard errors.
✗Reading BLUE as best among all estimators. It is best only within the class of linear unbiased estimators; a biased or nonlinear rule could have lower mean squared error.
✗Assuming Gauss-Markov licenses t and F tests. Those need either normality or a large-sample argument, which Gauss-Markov alone does not supply.

Revision bullets

•MLR.5 adds homoskedasticity, $\text{Var}(u\mid x)=\sigma^2$ .
•Under MLR.1 to MLR.5, OLS is BLUE (best linear unbiased estimator).
•BLUE is about efficiency, not unbiasedness or valid inference.
•Heteroskedasticity keeps OLS unbiased but breaks BLUE and the default standard errors.
•Valid t and F tests need normality or large samples on top of Gauss-Markov.

Quick check

Under the full Gauss-Markov assumptions, the OLS estimator is described as:

Which assumption is added to MLR.1 through MLR.4 to obtain the Gauss-Markov result?

If errors are heteroskedastic, what happens to OLS?

Connected topics

OLS Variance MLR assumptions CLM normality Multicollin.Heteroskedastic Robust SE

Sources

Wooldridge, Introductory Econometrics, Ch. 3
Wooldridge (2019), Introductory Econometrics: A Modern Approach, 7th ed., Sec. 3.4 (Gauss-Markov)
Greene, Econometric Analysis
Greene (2018), Econometric Analysis, 8th ed., Ch. 4

How to cite this page

Dr. Phil's Quant Lab. (2026). The Gauss-Markov Theorem. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-gauss-markov

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