# Fat Tails and Volatility clustering relationship

Let's say I have a return time series that after a proper ARMA modelling exhibits fat-tallness from QQ-plot.

Can this be a consequence of volatility clustering so that by applying a GARCH model I can obtain normally distributed residuals? When is this the case (if any)?

In other words: what's the relationship between significant autocorrelation in squared residuals and leptokurtosis?

• Short answer: the relationship is there. Volatility clustering produces heavier tails. This should be described in some of the early ARCH-GARCH literature. (Since I do not have a reference, I am only posting this as a comment.) Apr 29, 2018 at 18:15
• @Richard Hardy Thank you for your comment! So if I model a GARCH and the variables results statistically significant but QQ-plot does not "improve" what can I conclude? Apr 29, 2018 at 18:22
• Engle's original ARCH paper (p. 992) says The first-order ARCH process generates data with fatter tails than the normal density. Bollerslev's original GARCH paper (p. 313) says Hence the GARCH(1,1) process is leptokurtic (heavily tailed), a property the process shares with the ARCH(q) process; cf. Milhoj (1984). And Tsay "Analysis of Financial Time Series" (2010, 3rd edition, p. 133) says Consequently, similar to ARCH models, the tail distribution of a GARCH(1,1) process is heavier than that of a normal distribution. Apr 29, 2018 at 18:23
• You should be looking at the QQ plot of standardized residuals from the GARCH model, not raw residuals. They should match the assumed distribution (you can assume nonnormal distributions for standardized residuals). Apr 29, 2018 at 18:26
• @Richard Hardy Great you're right! Just to be sure, so the same (standardise the residuals before) would apply if I want to investigate whether a t-distribution fit better in my Garch model, right? Thank you very much for the help. Apr 29, 2018 at 18:38

The relationship is there; volatility clustering produces heavier tails. Engle's original ARCH paper (p. 992) says

The first-order ARCH process generates data with fatter tails than the normal density.

Bollerslev's original GARCH paper (p. 313) says

Hence the GARCH(1,1) process is leptokurtic (heavily tailed), a property the process shares with the ARCH(q) process; cf. Milhoj (1984).

And Tsay's "Analysis of Financial Time Series" (2010, 3rd edition, p. 133) says

Consequently, similar to ARCH models, the tail distribution of a GARCH(1,1) process is heavier than that of a normal distribution.

after giving a proof of one special case.

References