Why is it assumed that covariance equal to $0$ or independence between $y$'s in the simple linear regression model? I have started a book of econometrics and in the first pages are stated the assumption used in the linear regression model, which are :

The third one is the one I don't understand, why it has to be assumed? how can be verified if this assumption is not met? My professor explained it with an example : &quot;It is reasonable to say that the expenditure of a family with a certain income is independent of the one from another family with another income&quot;. What she says is okay to me, but it doesn't tell nothing about verifying the assumption when I have a set of datas, how can I have tuples of values in the case of the dependent variables $y_i$ and $y_j$ so that I can verify statistical independence or covariance equal $0$ ? Should I have the joint probability distribution of the variables? And why is it useful to assume one of these things, are not just the other assumptions sufficient? Any help is really appreciated.
The first assumption $E(y|x)=\beta_1+\beta_2 x$ identifies $\beta_2$. The roles of the rest are limited (if not nil) in terms of the identification of $\beta_2$ (identification = answer to the question 'what is $\beta_2$?').

Assumptions other than the first are all &quot;if&quot;s. They are useful for finding properties of OLS estimators, e.g., Gauss-Markov theorem, variance, efficiency, etc.

The third assumption should be written as $cov(y_i,y_j | x_1, \ldots, x_n)=0$ for $i\ne j$. Unconditionally, $cov(y_i,y_j)$ does not have to be zero. (If $x_i$ and $x_j$ are correlated, then $y_i$ and $y_j$ can be correlated.) But under the fourth assumption (nonrandomness), $cov(y_i,y_j)=0$ is equivalent to $cov(y_i,y_j | x_1, \ldots, x_n)=0$.

The fourth assumption is usually not well understood by beginners. It sometimes helps to state that $x_1$ is nonrandom, $x_2$ is nonrandom, and so on (instead of $x$ being nonrandom) in the context of repeated samples.

To the no autocorrelation assumption, it is usually regarded as sensible and acceptable to assume that individuals are independent each other. But it does not mean that it is true. Then what happens if it is violated? In fact, that's the question you want to ask, not why.

If you are doubt about the assumptions, the question goes this way: What happens to OLS and what should we do if the normality assumption is violated; what happens to OLS and what should we do if the heteroskedasticity assumption is violated; what happens to OLS and what should we do if the no autocorrelation assumption is violated; what happens to OLS and what should we do if $x_1, \ldots, x_n$ are random; what happens to OLS and what should we do if $E(u|x)\ne 0$?

So my answer to &quot;why no autocorrelation&quot; is: because it is a leading case and it is believed to be empirically relevant. And its violation does not necessarily mean the model is wrong. It only means that the unobserved determinants are correlated among individuals. The $\beta_2$ is  still the causal effect of $x$ on $y$. The identification of $\beta_2$ is the very starting point. Don't let the tail wag the dog. (That said, someone might want to identify $\beta_2$ by the no autocorrelation condition. This is rather unusual. $\beta_2$ is usually identified by $E(y|x)=\beta_1+\beta_2 x$.)

Related is, I don't think it is possible in general to completely account for autocorrelation by adding more $x$ variables. (For example, family members sharing the same experience but can this factor be sufficiently well observed?) More importantly, adding more variables to the right-hand side changes your model, by which you change the definition of your $\beta_2$. Sometimes you can do it (e.g., randomized experiments) but in most cases your definition of $\beta_2$ changes by changing your model.

My answer pertains to econometrics that tries to find causal effects. There can be other areas of statistics that have different approaches.