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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    $\begingroup$ Let $u\sim\mathcal{N}(0,x^2)$ for instance: that has constant mean but since the variance of $u$ depends on $x,$ $(x,u)$ is not independent. You can find many more examples by searching for posts about weak stationarity of time series. This one is easy to simulate in R: u <- rnorm(1e3, 0, (x <- rnorm(1e3))^2); plot(x, u) $\endgroup$ – whuber Oct 27 '20 at 19:50
  • $\begingroup$ @whuber hahaha you went literal and introduce x into another moment. Evident my dear Watson... Does this even exist in practice? Will search what you suggest. $\endgroup$ – luchonacho Oct 27 '20 at 19:52
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    $\begingroup$ a finance example is any arch process. $\endgroup$ – mlofton Oct 27 '20 at 19:52
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    $\begingroup$ @mlofton Something like: the value of stocks do not depend on the weather but the variability of stocks do? $\endgroup$ – luchonacho Oct 27 '20 at 19:53
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    $\begingroup$ that's exogenous dependence rather than model dependence.arch says that variability is related to past variability. there are many introductory explanations of arch on the net. $\endgroup$ – mlofton Oct 27 '20 at 19:56

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