# Solved – 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?

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In finance, GARCH (generalized autoregressive conditional heteroskedasticity) effects are widely cited here: stock returns $$r_t:=(P_t-P_{t-1})/P_{t-1}$$, with $$P_t$$ the price at time $$t$$, themselves are uncorrelated with their own past $$r_{t-1}$$ if stock markets are efficient (else, you could easily and profitably predict where prices are going), but their squares $$r_t^2$$ and $$r_{t-1}^2$$ 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 $$tapprox400$$.

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') ``

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# Solved – 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 $$r_t:=(P_t-P_{t-1})/P_{t-1}$$, with $$P_t$$ the price at time $$t$$, themselves are uncorrelated with their own past $$r_{t-1}$$ if stock markets are efficient (else, you could easily and profitably predict where prices are going), but their squares $$r_t^2$$ and $$r_{t-1}^2$$ 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 $$tapprox400$$.

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') ``

Rate this post