I read in many books or notes online that (1)volatile series do not differ significantly from white noise and that (2)their squared values will exhibit correlation. Although I agree with (2), I can't seem to agree with (1). I used the tsdisplay function from forecast package to plot the acf of SP500(MASS) and bmw(evir) data and found that there are a few significant autocorrelation. This isn't so obvious when plotting the acf using the built in acf function as the correlation at lag 0 mask the correlation at subsequent lags.

I also generated a white noise series and although the acf seem the same there were not really any significant autocorrelation.

Can anyone explain this to me?



1 Answer 1


Real data always would have more features than the idealised model. Search for textbook models of SP500, I think you'll find that it would be modelled as ARMA + GARCH, i.e. the spikes in the autocorrelation function will be explained by ARMA part of the model.

The key property of returns data is the (2) property. It should really read that the correlations for squared values are much more prominent relative to the correlations for the original values as opposed for iid white noise process, for which correlations are non-significant for both squared and non-squared values.

  • $\begingroup$ By ARMA+GARCH you mean ARMA for the mean eqn and GARCH for the variance eqn?I read in Statistics and Data Analysis for Financial Engineering that but there was something else about ARMA/GARCH model for errors and I didn't know if both were the same thing. thanks $\endgroup$
    – ankc
    Commented Oct 7, 2013 at 13:34
  • $\begingroup$ Yes, I mean ARMA mean equation and GARCH error. $\endgroup$
    – mpiktas
    Commented Oct 7, 2013 at 15:57

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