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In linear regression, the assumptions are linearity, independence, (sometimes) normality, (sometimes) homoscedasticity.

But when people talk about independence, sometimes they say we need independent observations (e.g. https://math.stackexchange.com/questions/1530571/linear-regression-model-assumptions) and sometimes they say we need independent error terms (e.g. What is a complete list of the usual assumptions for linear regression? )

My question is, are these two somehow equivalent? I guess that is the case if we do not treat $x$ as random variables, but is this the correct connection?

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Independent observation and independent error terms are not equivalent

Independent observation means to say that in linear regression say y = b0 + b1x1 + b2x2...bnxn --> x1 in not related to x2, x2 is not related to x3 and so on. If x1 has high correlation with x2 then there would be double counting in estimation of y

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