Consider the following population regression model:

$$y_{i} = \beta _{1} + \beta_{2}x_{i} + \epsilon _{i},$$

where $i=1,...,n$. Assume $\epsilon \sim iid$, with the pdf in equation: $f(\epsilon ) = \alpha \epsilon$ for $0\leq \epsilon \leq 1$.

I have the following data: $X = (3,1,2,2)^{'}$, and $Y = (3,0,1,4)^{'}$.

By calculating the estimated intercept and the slope, I have $b_{1} = -1$, and $b_{2}=1.5$.

Two classical assumptions are violated in this model: one is no mean-zero errors $E(\epsilon {_i})\neq 0$, and the other one is the errors not being normal.

From this, can I infer anything about the true intercept of the model? I think we could either be overestimating or underestimating the true intercept.

Also, as long as I have an intercept in the model, then is the OLS still BLUE? I think that our estimate of the intercept is biased, but the estimate for $\beta_{2}$ is still BLUE because we have an intercept in the model.

  • 1
    $\begingroup$ One of those "classical assumptions", about normality of the error distribution, is somewhat famously NOT used in proving OLS is BLUE. The Gauss-Markov theorem does not assume i.i.d. errors nor normally distributed errors, but uncorrelated (not necessarily independent) homoscedastic (not necessarily identically distributed) errors each with zero mean. (But I'm not yet sure what the answer to your final question is.) ${}\qquad{}$ $\endgroup$ Nov 6 '14 at 2:01

Note the Following.

Let $\bar{e}$ denote the mean of e. Then I can re-write your model as

$y_i = (\beta_1 + \bar{e}) + \beta_2x_i + (e_i - \bar{e})$.

This model is identical to yours and now has a mean-zero error term (though, as you point out, it is still not normally distributed), but the intercept will be "biased" by the mean of the original error.


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