Despite the recent introduction of novel solution methods for Dynamic Stochastic General Equilibrium (DSGE), perturbation methods are still among the most popular and widely used solution techniques for DSGE models. Unfortunately, nonlinear perturbation solutions produce paths with stochastic properties that invalidate the econometric analysis. This paper proposes a correction that renders the econometric analysis valid and sound. The proposed correction is simple to implement in existing software packages such as Dynare, it does not add any significant computational effort and, as a result, does not impact computational times. The corrected solution retains the same approximation properties as standard higher-order perturbation methods and, in contrast to those methods, generates stable sample paths that are stationary, geometrically ergodic and absolutely regular. Additionally, moments are shown to be bounded. Transformed perturbation solutions are an alternative to the pruning method as proposed in Kim et al. (2008). The advantages of our approach are that, unlike pruning, we do not need to sacrifice accuracy around the steady-state by ignoring higher-order effects, and furthermore, we also deliver a policy function. Moreover, the newly proposed solution is always more accurate globally than standard perturbation methods. We demonstrate the superior accuracy of our method in a range of simple examples.