  Moment approximation methods are gaining increasing attention for their use
in the approximation of the stochastic kinetics of chemical reaction systems.
In this paper we derive a general moment expansion method for any type of
propensities and which allows expansion up to any number of moments. For some
chemical reaction systems, more than two moments are necessary to describe the
dynamic properties of the system, which the linear noise approximation (LNA) is
unable to provide. Moreover, also for systems for which the mean does not have
a strong dependence on higher order moments, moment approximation methods give
information about higher order moments of the underlying probability
distribution. We demonstrate the method using a dimerisation reaction,
Michaelis-Menten kinetics and a model of an oscillating p53 system. We show
that for the dimerisation reaction and Michaelis-Menten enzyme kinetics system
higher order moments have limited influence on the estimation of the mean,
while for the p53 system, the solution for the mean can require several moments
to converge to the average obtained from many stochastic simulations. We also
find that agreement between lower order moments does not guarantee that higher
moments will agree. Compared to stochastic simulations our approach is
numerically highly efficient at capturing the behaviour of stochastic systems
in terms of the average and higher moments, and we provide expressions for the
computational cost for different system sizes and orders of approximation. {We
show how the moment expansion method can be employed to efficiently {quantify
parameter sensitivity}.} Finally we investigate the effects of using too few
moments on parameter estimation, and provide guidance on how to estimate if the
distribution can be accurately approximated using only a few moments.
