  We discuss the probabilistic properties of the variation based third and
fourth moments of financial returns as estimators of the actual moments of the
return distributions. The moment variations are defined under non-parametric
assumptions with quadratic variation method but for the computational
tractability, we use a square root stochastic volatility model for the
derivations of moment conditions for estimations. Using the S\&P 500 index high
frequency data, the realized versions of the moment variations is used for the
estimation of a stochastic volatility model. We propose a simple estimation
method of a stochastic volatility model using the sample averages of the
variations and ARMA estimation. In addition, we compare the results with a
generalized method of moments estimation based on the successive relation
between realized moments and their lagged values.
