Regression's nomenclature problem I'm reviewing the estimation of the parameters of a regression and a question arises regarding the nomenclature used by the book (Hansen's book).
The author considers it valid to write expressions such as the following:
$$E(e_i)=0$$
$$S(b)=E(y_i-x_ib)^2$$
$$b=(E(x_ix_i')^1E(x_iy_i))$$
where $x_i$ and $y_i$ are the observations for the i$th$ individual. $e_i$ is the error for the i$th$ individual. $b$ is the population parameter of the regression and $s(b)$ is the minimizer of the expected squared error.
Does it make sense to use the expected value for observations in these cases?- shouldn't it be used only in the model with random variables $X$ and $Y$?
For example, the expected value of the error of an observation should be the error of the observation and not 0.
 A: I don't understand your claim that the expected value of the error of an observation should be the error of the observation and not 0 - the claim is erroneous.
The expected value of $e_i$ is a number.  The error term $e_i$ is a random variable, whose possible values can be described by a distribution centered about the expected value $E(e_i)$ (in this case, 0).
I agree with you that the notation is not the best. If you denote the outcome variable by Y and the predictor variable by X, then you can use the following notation to refer to the observations you would expect to get from your $n$ subjects on these variables before you actually conduct the study and collect the data:  $(X_i, Y_i), i = 1, ..., n$.  You can formulate your model for these observations as: $Y_i = \beta_0 + \beta_1*X_i + \epsilon_i$, where $\epsilon_i$ is a random error term with expected value 0 and unknown variance $\sigma^2$.
Once the study is conducted and the data on $X$ and $Y$ are collected for the $n$ subjects, you can refer to the observed values of $(X_i, Y_i), i = 1, ..., n$ as $(x_i, y_i), i = 1, ..., n$. You are right that there is nothing random about these observed data values - they are known realizations of the random variables $(X_i, Y_i), i = 1, ..., n$. If something is fully known, it can't be random!
