  The reduction of high-dimensional systems to effective models on a smaller
set of variables is an essential task in many areas of science. For stochastic
dynamics governed by diffusion processes, a general procedure to find effective
equations is the conditioning approach. In this paper, we are interested in the
spectrum of the generator of the resulting effective dynamics, and how it
compares to the spectrum of the full generator. We prove a new relative error
bound in terms of the eigenfunction approximation error for reversible systems.
We also present numerical examples indicating that if Kramers--Moyal (KM) type
approximations are used to compute the spectrum of the reduced generator, it
seems largely insensitive to the time window used for the KM estimators. We
analyze the implications of these observations for systems driven by
underdamped Langevin dynamics, and show how meaningful effective dynamics can
be defined in this setting.
