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The assumption $E(u)=0$ says that the distribution of the unobserved factors in the population is zero. It just doesn't make any sense to me. I really can't figure it out. In my textbook the assumption is illustrated by the following example:

$yield = B_0 + B_1fertilizer + u$

In $u$ factors like rainfall, land quality, and so on are contained. Why would the unobserved factors affecting the yield have an average of zero in the population of all cultivated plot?

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    – Silverfish
    Aug 18, 2016 at 20:30
  • $\begingroup$ If the expected value is different from zero then this non-zero value is 'caught' in the intercept B0 $\endgroup$
    – user83346
    Aug 18, 2016 at 21:38

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First of all, please notice that is false that "the distribution of the unobserved factors in the population is zero". The expected value (e.g. the mean) of the effects of the unobserved values is zero, which is not the same.

Assuming $E(u)=0$ is a matter of convenience and convention more than a strong underlying theoretical reason. In fact, you could choose to equal $E(u)$ to any (constant) given number and all the regression analysis wouldn't be very different than usual - the main difference is that you would get the constant $B_0$ with your constant subtracted.

Anyway, please notice that in the usual formulation all effects have zero expectation, even the effect of the observed factor ($fertilizer$). You can see that the regression line passes through the point $(mean(ferilizer),mean(yield))$ and for other points regression line just describes the effect on yield above or below mean fertilizer.

Furthermore, we shouldn't read $E(u)=0$ as meaning that the unobserved factors (e.g. rainfall) don't have a nett effect compared to zero. The regression line describes a prediction for the mean unobserved factors (e.g. for the mean rainfall), not a the prediction for zero value of unobserved factors (e.g. assuming no rainfall). Therefore, the unobserved factors (e.g. rainfall) don't have a nett effect compared to the mean.

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Since all those factors are not measured, you have no idea about:

1) The magnitude of their effect on the yield;

2) The direction of that effect.

For instance, for a given plot you can say that the yield is expected to be higher than average because the level of fertilizer is higher than average, but you have no idea whether the amount of rainfall has been higher or lower than the average precipitation. It can go either way. Therefore, your best guess is to say that, on average (across all the plots with varying level of rainfall), rainfall factor has zero effect. In technical terms, that statement corresponds to $E[u] = 0$.

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