Does homocedasticity and conditional mean of error equal 0 imply the error is independent from the explanatory variable? Let $Y = X \beta + e $ be a regression model, where $E[e|X] = 0$ and $E[e^2|X] = \sigma^{2}$. Does this mean that $e$ is independent of $X$?
If so, how to prove it? If not, are there some counterexamples?

Here are some terse hints.

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*A lurking assumption here is a the choice of probability model used for the random variables. Ask yourself whether these random variables are Gaussian, or something else. What is their joint distribution?


*$X$ and $e$ will be independent if-and-only-if $\mathbb{E}[g(e)f(X)] = \mathbb{E}[g(e)] \mathbb{E}[f(X)]$ for all measurable functions $g$ and $f$.


*If you find that there exists a pair of measurable functions that does not satisfy this criterion, then you will have found that $X$ and $e$ are not independent.


*Does $\mathbb{E}[e|X]=0 \land \mathbb{E}[e^2|X]=\sigma^2$ preclude the existence of such pairs of functions?
