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From all the content I have read, ordinary least squares (OLS) assumes that the errors are normally distributed i.i.d. I still cannot find any content that gives a bit more insight on what can be inferred about the distribution of (Xi, yi).

From what I read here it seems like having i.i.d assumptions on (Xi, yi) is a much stronger assumption and here the author says that independent samples of (Xi, yi) is sufficient.

I would like to check whether the only assumption on (Xi, yi) is that they are independently sampled? Does this mean one could sample (Xi, yi) from a i.n.i.d (independent but not identically distributed) distribution and as long as the errors are i.i.d OLS would work?

spie227
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  • Your first question regarding what the distributional assumption on the error implies about the marginal distribution of $y_i$ is answered here. Nontrivial assumptions about the joint distribution of $(X_i, y_i)$ are not usually made in linear regression; instead $X_i$ is usually treated as fixed (i.e., non-random, without a probability distribution of its own). – Durden Feb 25 '24 at 17:24

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