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 height2 might be uncorrelated but dependent, but I don’t see why anyone would want to compare height and height2.

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?

Answer

In finance, GARCH (generalized autoregressive conditional heteroskedasticity) effects are widely cited here: stock returns rt:=(PtPt1)/Pt1, with Pt the price at time t, themselves are uncorrelated with their own past rt1 if stock markets are efficient (else, you could easily and profitably predict where prices are going), but their squares r2t and r2t1 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):

enter image description here

You see the high volatility cluster around in particular t400.

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')

Attribution
Source : Link , Question Author : BCLC , Answer Author : luchonacho

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