I'd have a starter-level question: When we say that we assume that E(e|X)=0 in an OLS setting, or claim that $E(\hat{\beta}_{OLS})=0$, which objects' pdfs are we using for the expectation operators? I mean what is in f(.), even if f(.) itself is undefined? I presume for E(e|X), it is f(e|X), but for the latter?
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Could you please explain what "f" means to you what "in f" is supposed to be?? – whuber May 06 '21 at 20:39
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A (joint) probability density function. My concern is that I am not sure of what. – lippi May 06 '21 at 23:37
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A search of this site for "regression distribution estimate" turns up an explanation at https://stats.stackexchange.com/questions/5135/interpretation-of-rs-lm-output/5138#5138. Are there any parts of your question it does not answer? – whuber May 07 '21 at 13:13
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I am mainly asking about whether $E(\hat{\beta}_{OLS})=\int_{-\infty}^{\infty} \hat{\beta}_{OLS}*f(\hat{\beta}_{OLS})d\hat{\beta}_{OLS}$ or if the integral expression should be written in some other way, but I believe this is not discussed in the post that you referred to. – lippi May 07 '21 at 21:44