# For intuition, what are some real life examples of uncorrelated but dependent random variables?

In explaining why uncorrelated does not imply independent, there are several examples that involve a bunch of random variables, but they all seem so abstract: 1 2 3 4.

This answer seems to make sense. My interpretation: A random variable and its square may be uncorrelated (since apparently lack of correlation is something like linear independence) but they are clearly dependent.

I guess an example would be that (standardised?) height and height$^2$ might be uncorrelated but dependent, but I don’t see why anyone would want to compare height and height$^2$.

For the purpose of giving intuition to a beginner in elementary probability theory or similar purposes, what are some real-life examples of uncorrelated but dependent random variables?

In finance, GARCH (generalized autoregressive conditional heteroskedasticity) effects are widely cited here: stock returns $$rt:=(Pt−Pt−1)/Pt−1r_t:=(P_t-P_{t-1})/P_{t-1}$$, with $$PtP_t$$ the price at time $$tt$$, themselves are uncorrelated with their own past $$rt−1r_{t-1}$$ if stock markets are efficient (else, you could easily and profitably predict where prices are going), but their squares $$r2tr_t^2$$ and $$r2t−1r_{t-1}^2$$ are not: there is time dependence in the variances, which cluster in time, with periods of high variance in volatile times.

Here is an artificial example (yet again, I know, but “real” stock return series may well look similar):

You see the high volatility cluster around in particular $$t≈400t\approx400$$.

Generated using R code:

library(TSA)
garch01.sim <- garch.sim(alpha=c(.01,.55),beta=0.4,n=500)
plot(garch01.sim, type='l', ylab=expression(r[t]),xlab='t')