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 : "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". 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.