Behaviour of skewness, kurtosis and normality tests in long memory data

We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque-Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither root n-consistent nor asymptotically normal. The normalizations needed to obtain the limiting distributions depend on the long memory parameter d. A direct consequence is that if data are long memory then testing normality with the Jarque-Bera test by using the chi-squared critical values is not valid. Therefore, statistical inference based on skewness, kurtosis, and the Jarque-Bera normality test, needs a rescaling of the corresponding statistics and computing new critical values of their nonstandard limiting distributions.