In this post, one of the answers provides the following information about the assumptions of linear regression in the case of random design (as opposed to fixed design):
The usual regression model is $Y=X\beta+\varepsilon$ and the assumpitons are:
- Homoscedasticity, $E[\varepsilon^2|X]=\sigma^2$
- No serial correlation, $E[\varepsilon_i,\varepsilon_j|X]=0$
Note that there is a subscript on the error terms in the last assumption. Does this mean this assumption is not relevant to the population variables, but only to the sample data?
To make it clearer what I'm asking, consider the linear regression model with respect to population variables vs sample variables. Denote by $X^p$, $Y^p$, and $\varepsilon^p$ the population variables for which the regression model holds: $$ Y^p = f(X^p) + \varepsilon^p. $$
Consider drawing $n$ samples from the population $(X_1^s,Y_1^s),(X_2^s,Y_2^s),\dots,(X_n^s,Y_n^s)$. The linear regression model applied to the samples is $$ Y_i^s = f(X_i^s) + \varepsilon_i^s, \quad \quad i=1,2,\dots,n. $$ where the errors are $\varepsilon_i$ all have the same CDF as the population error $\varepsilon^p$. In the samples we have multiple error observations $\varepsilon_i$ so the 'no serial correlation' assumption makes sense. But for the population model, we just have a single random variable for the error $\varepsilon^p$, so it seems we can't speak of 'no serial correlation' with respect to the population?
I have assumed that we can frame the linear regression model with respect to both the sample data and the population we have drawn the sample from. However, if I have made some fundamental mistake in drawing a distinction between the regression model applied to the population vs the sample please let me know, I am still getting to grips with the details of statistical models.