Let us say we have a linear model $$ Y=\beta_0+\beta_1X+\varepsilon, \quad \varepsilon|X\sim i.i.N(0,\sigma^2). $$ Suppose estimation yields $(\hat\beta_0,\hat\beta_1,\hat\sigma^2)=(5,2,9)$.
Question 1: Shall one write $$ Y=5+2X+\hat\varepsilon $$ or $$ Y=5+2X+\varepsilon? $$ I believe the former makes more sense, since the coefficient estimates $(5,2)$ need not perfectly coincide with the true coefficients $(\beta_0,\beta_1)$ and so $\hat\varepsilon$ will not perfectly coincide with $\varepsilon$. On the other hand, I do not think I commonly see $\hat\varepsilon$ in such situations...
Question 2: What can be said about the estimated counterpart of $\varepsilon|X\sim i.i.N(0,\sigma^2)$? I think that $\hat\varepsilon$s are dependent, so one $i.$ from $i.i.N$ drops. Not sure whether conditioning on $X$ makes sense and if it does, what role it plays. The distribution has an
estimated imposed mean of $0$ and variance of $9$, but other than that, can we say anything more? What is the most we can say that would be correct?