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Is this true?

OLS is the BLUE in every linear model

No — it is false.

The counterexample

$Y_{i} = β x_{i} + ε_{i}$ with $V a r (ε_{i}) = σ^{2} x_{i}^{2}$

Gauss–Markov assumes constant variance; here weighted least squares (the mean of $Y_{i} / x_{i})$ has smaller variance.

The kind of mistake this is

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

Drill statements like this

statistics regression

Others that fail the same way

From Linear Models and Multivariate › Gauss–Markov, regression, ANOVA basics

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