  In this paper we introduce an efficient fat-tail measurement framework that
is based on the conditional second moments. We construct a goodness-of-fit
statistic that has a direct interpretation and can be used to assess the impact
of fat-tails on central data conditional dispersion. Next, we show how to use
this framework to construct a powerful normality test. In particular, we
compare our methodology to various popular normality tests, including the
Jarque--Bera test that is based on third and fourth moments, and show that in
many cases our framework outperforms all others, both on simulated and market
stock data. Finally, we derive asymptotic distributions for conditional mean
and variance estimators, and use this to show asymptotic normality of the
proposed test statistic.
