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Suppose I've specified a linear regression model: $$ Y = \beta_0 + \beta_1 X + \epsilon $$ where $\beta_0$, $\beta_1$ are the population parameters. My question is: why are these parameters populational? My intuition is that they are calculated based on all the possible realizations of the random variables $X$ and $Y$. The coefficient $\beta_1$ is $E(XX')^1E(XY)$ where the expected values ​​are calculated based on all the possible realizations of both variables and their probabilities.

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The linear regression model that you specified can be view as a linear approximation of expected value of $Y$ on $X$. In many presentation about regression this fact is not clearly affirmed but it seem me very relevant. So, as usual in mathematical statistics, you have some parameters $\theta$s that characterize the joint distribution $(Y,X)$. These are population parameters or, equivalently, the exact/true parameters. You can see $\beta$s as transformations of $\theta$s, so you can see the $\beta$s as population parameters also.

Changing point of view we can think about statistical inference. So, very briefly, we can collect data from $Y$ and $X$ and estimate a regression in order to analyze some associations of interest (note: estimated approximation of expected value of $Y$ given $X$). Here you achieve the estimated parameters $\beta_{est}$ that seem me what you have in mind (est stand for estimator). Moreover you can interested in many features of $\beta_{est}$, for instance efficiency. In fact $\beta_{est}$ have its variance while $\beta$ is a constant. This happen precisely because $\beta$ collect population parameters and $\beta_{est}$ their estimators.

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