Balanced model truncation has been considered by many authors, since it is a simple and, nevertheless, efficient model reduction technique. In many cases the approximation error may be bounded by a function of the neglected singular values. In this paper the performance of balanced truncation of state space models for ARMA processes is analysed, where the goodness of fit is measured by the asymptotic Gaussian likelihood function. It is shown that locally, i.e. close to the set of lower order systems, minimum phase balanced truncation and stochastically balanced truncation give almost optimal results.