  We study statistical inference for small-noise-perturbed multiscale dynamical
systems under the assumption that we observe a single time series from the slow
process only. We construct estimators for both averaging and homogenization
regimes, based on an appropriate misspecified model motivated by a second-order
stochastic Taylor expansion of the slow process with respect to a function of
the time-scale separation parameter. In the case of a fixed number of
observations, we establish consistency, asymptotic normality, and asymptotic
statistical efficiency of a minimum contrast estimator (MCE), the limiting
variance having been identified explicitly; we furthermore establish
consistency and asymptotic normality of a simplified minimum constrast
estimator (SMCE), which is however not in general efficient. These results are
then extended to the case of high-frequency observations under a condition
restricting the rate at which the number of observations may grow vis-\`a-vis
the separation of scales. Numerical simulations illustrate the theoretical
results.
