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While studying Linear Least Square, there was a sentence saying that the Linear Least Square becomes Best Linear Unbiased Estimator. I am confused about the way it says.

Does it mean that the Linear Least Square is biased if we do not apply the Gauss-Markov assumptions or we have to test those Gauss-Markov assumptions? Do we usually have the Gauss-Markov assumptions set before we perform Linear Least Square Methods?

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I am not sure I understand your question completely. Under spherical errors (uncorrelated, homoscedastic) OLS is BLUE. What happens when the assumption fails? Still assuming errors have zero mean, OLS will be unbiased but not generally efficient. Is the assumption testable? Yes. Do we have it 'set', I am not sure what you mean. Wiki on OLS is quite informative.

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