If have a simple bivariate regression model:
$ Y_i= x_i \beta + \epsilon_i $
where $i$ are the number of observations.
How do I test for the hypothesis that the OLS coefficient $\beta$ does not change across observations??
Another Question is, if instead i have the model: $ Y_i= x_i (\beta + \epsilon_i) $
then, the OLS estimator will still be consistent right?
My logic:
$\hat\beta$ = $(X'X)^{-1}X'Y$
$\hat\beta$ = $(X'X)^{-1}X'(X (\beta + \epsilon)) $
$\hat\beta$ = $\beta + (X'X)^{-1}X'(X\epsilon)) $
Now, if $E(\epsilon \vert X)$ = 0;
taking probability limits, can we say that,
$E(X'X\epsilon)$ = $E_x(E(X'X\epsilon \vert X))$
$E(X'X\epsilon)$ = $E_x(X'XE(\epsilon \vert X))$
$E(X'X\epsilon)$ = $0$
Are we allowed to do that?