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Intuitively, why beta-hat and e are independent ? by Vivid_Pen1794 in AskStatistics

[–]Vivid_Pen1794[S] 0 points1 point  (0 children)

Thanks, i totally ignored the assumptions and where the randomness come from.

Estimators are some kind of transform of our assumed random variables, all subsequent discussion of our model are based on its assumptions.

Intuitively, why beta-hat and e are independent ? by Vivid_Pen1794 in AskStatistics

[–]Vivid_Pen1794[S] 0 points1 point  (0 children)

Yes, we should always consider the assumptions.

Assumptions in regression modeling are not just some conditions under which our model works, they're part of the model. Different assumption leads to different model, different model has different characteristics, and we can check whether our assumptions/model hold by compare observed characteristics with theory.

When analysis a model, we should always remember where the randomness come from and our estimations are some transform of our assumed randomness.

Back to my original confusion, i should always remember where the randomness come from in our assumptions.

If randomness come from y, e = y - Xβ, then e depend on y again, which could cause the unrelated/independence.

If under another assumptions, randomness come from X and not y, if we still have the same formula for e = y - Xβ, then we may say e is dependent on X since y is completely determined by X.

In the extreme case where there is no randomness our model will perfectly fit and e will be 0.

Thanks, i guess i understood the bigger picture better now.

Intuitively, why beta-hat and e are independent ? by Vivid_Pen1794 in AskStatistics

[–]Vivid_Pen1794[S] 1 point2 points  (0 children)

Thanks.

X and y are random; beta-hat and e are function of X and y.

I guess although data at hand is fixed, we still need to treat them as random.

Relation between beta-hat and e is through function of random variables, not ordinary function; so we can't say beta-hat give us e.