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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 $t\approx400$.

Generated using

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')

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 $t\approx400$.

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')

In finance, GARCH (generalized autoregressive conditional heteroskedasticity) effects are widely cited here: stock returns themselves are uncorrelated with their own past if stock markets are efficient (else, you could easily and profitably predict where prices are going), but their squares 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 $t\approx400$.

Generated using

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')

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 $t\approx400$.

Generated using

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')

In finance, ARCH effects are widely cited here: stock returns themselves are uncorrelated if stock markets are efficient, but their squares are not, as there is time dependence in the variances, which cluster in time, with periods of high variance in volatile times.

In finance, GARCH (generalized autoregressive conditional heteroskedasticity) effects are widely cited here: stock returns themselves are uncorrelated with their own past if stock markets are efficient (else, you could easily and profitably predict where prices are going), but their squares 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 $t\approx400$.

Generated using

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')