Example of mean independent variables but dependent still

Cross Validated Asked by luchonacho on December 11, 2020

Econometric textbooks often make the distinction between three types of independence:

  1. Stochastic independence: $mathrm{D}(u|x)=mathrm{D}(u)$
  2. Mean independence: $mathrm{E}(u|x)=mathrm{E}(u)$
  3. Linear independence: $mathrm{Cov}(u,x)=0$

with the one preceding being stronger and implying the subsequent. For instance, Wooldridge (2002, p.22) states:

We also need to know how the notion of statistical independence relates to conditional expectations. If $u$ is a random variable independent of the random vector $x$,
then $mathrm{E}(u|x)=mathrm{E}(u)$, so that if $mathrm{E}(u)= 0$ and $u$ and $x$ are independent, then $mathrm{E}(u|x) = 0$. The converse of this is not true: $mathrm{E}(u|x)=mathrm{E}(u)$ does not imply statistical independence between $u$ and $x$ (just as zero correlation between $u$ and $x$ does not imply independence).

It is easy to come with examples of uncorrelated random variables that are not (mean) independent. $Y=X^2$ is a classic one. Many questions on this site cover others (e.g. here).

What I’m struggling with is to think of an example of two variables which are mean independent but dependent more generally. I wonder whether this is merely a technical point or we are against a true possibility in science. I don’t even know how to start to simulate an example of such joint distribution. Independent on mean but dependent on variance? Maybe something from finance? Any ideas on this?

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