# regression basic doubt

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?

• In your notation, $x_i(\beta + \epsilon_i) \neq x_i \beta + \epsilon_i$
– Jon
May 16, 2019 at 16:27
• @Jon Yes, that's why I have conditional expectations with regard to $X'X \epsilon$ and not $X' \epsilon$ Is that wrong way? How should I proceed? Also, there are two parts to the question, the first part has a normal OLS bivariate model May 16, 2019 at 16:36

You can search for random slope (or random coeficient) models, but this is normaly done with panel data, with a $$X$$ variable that varies through time for each individual. Otherwise, in your simple model, if you don't put any constraint on your coefficients, you will just end up saying that each $$Y$$ is 100% determined by $$X$$.
This applies to your second model as well. It assumes that there is no error term beside the error on the coefficient. Therefore, it just assumes that $$X(\beta+\epsilon_i)$$ completely determines $$Y$$, so nobody would ever use this to try to estimate individual-specific coefficients.
But if this is still the model you want to assume, then yes OLS gives a consistent estimate of the average effect $$\beta$$. It just means you have heterosckedasticity, so you should take this into account for inference.