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I have hard time understanding the LINEAR part. Found something like this:

Linear property of OLS estimator means that OLS belongs to that class of estimators, which are linear in Y, the dependent variable. Note that OLS estimators are linear only with respect to the dependent variable and not necessarily with respect to the independent variables. The linear property of OLS estimators doesn’t depend only on assumption A1 but on all assumptions A1 to A5.

But it doesn't tell me much, I still don't understand. Can you try to explain in some simple intuitive way?

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  • $\begingroup$ to clarify: are you asking why one is interested in BLUEs (best linear unbiased estimators) or why the standard OLS estimator is a linear estimator? $\endgroup$
    – Stefan
    Commented Feb 3, 2022 at 11:32
  • $\begingroup$ There is a widespread, very well understood mathematical definition of a linear function that applies here. Thus, it would be nice to have some clarification on your part concerning what you might be looking for that goes beyond that. $\endgroup$
    – whuber ♦
    Commented Feb 3, 2022 at 18:27
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    $\begingroup$ The last sentence of the quoted text does not make sense. It is a fact that OLS gives you estimates that are linear functions of the Y data, and this fact does not depend on any regression assumptions. $\endgroup$
    – BigBendRegion
    Commented Feb 3, 2022 at 20:02
  • $\begingroup$ @BigBendRegion Good observation. I found this quotation at albert.io/blog/ultimate-properties-of-ols-estimators-guide. Because its assumptions A2 (observations are randomly sampled) and A4 ("no multicollinearity") are also not required (and in fact cause a number of common misonceptions expressed in questions here on CV), I read no further and would suggest that nobody else bother to read that Web site either. $\endgroup$
    – whuber ♦
    Commented Feb 4, 2022 at 14:13

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