The pourpose of this paper is to propose the Stock Market (SM) volatility estimation method based on the Higher Order Cumulant (HOC) function, and to apply it to the cases when stock market returns have a non Gaussiandistribution and/or when a distribution of SM innovations is unknown. 

The HOC functions of the third and fourth order are used not only as a means for non Gaussian model testing but also as sufficient statistics, which is indispensable in estimating the AR and MA parameters of the squared SM returns. 

The empirical analysis is based on the daily closing values of the SMI, DJIA, SP500, DAX, FTSE100, NASDAQ and BSE indexes. The time horizon includes the period between March 30, 2010 and February 6, 2013. ARMA parameter estimation is performed by using the well known GARCH algorithm from Eviews, as well as the estimation algorithm based on higher order cumulant (HOC) functions, which is introduced in this paper. 

Ultimately, the Hinich portmanteau statistics are used to test the adequacy of ARMA-GARCH and ARMA –HOC models. 

The research outcome demonstrates that ARMA-HOC model produces independent innovations and captures the model dynamics while the ARMA -GARCH model fails to do it. All data are taken from Bloomberg.