Everywhere in the literature, I have seen that while deriving the variance of the least squares slope estimate $Var(\hat \beta_1) = \dfrac{\sigma ^2}{SS_{xx}}$, we always assume that $X_i$s are fixed i.e., they are deterministic in nature. Can we derive the variance formula without this assumption? (Assuming we already have the result $E(\hat \beta_1) = \beta_1$
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https://stats.stackexchange.com/questions/183986/derivation-of-ols-variance/184015#184015 – Christoph Hanck Mar 12 '21 at 07:49
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@ChristophHanck Thanks for the link. Do you mean that if $X_i$s are also random, then the formula for variance should be $\sigma^2 E(\dfrac{1}{SS_{xx}})$ instead of $\dfrac{\sigma ^2}{SS_{xx}}$? – Avneesh Khanna Mar 12 '21 at 08:30
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Yes. An (unconditional) variance is a population parameter, hence a constant, and if the $X_i$ are random variables, so is $SS_{xx}$, so that it cannot be the unconditional variance. – Christoph Hanck Mar 12 '21 at 09:08
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@ChristophHanck Okay thanks a lot! – Avneesh Khanna Mar 12 '21 at 09:09