  Generalized Langevin equations (GLEs) can be systematically derived via
dimensional reduction from high-dimensional microscopic systems. For linear
models the derivation can either be based on projection operator techniques
such as the Mori-Zwanzig (MZ) formalism or by "integrating out" the bath
degrees of freedom. Based on exact analytical results we show that both routes
can lead to fundamentally different GLEs and that the origin of these
differences is based inherently on the non-equilibrium nature of the
microscopic stochastic model. The most important conceptional difference
between the two routes is that the MZ result intrinsically fulfills the
generalized second fluctuation-dissipation theorem while the integration result
can lead to its violation. We supplement our theoretical findings with
numerical and simulation results for two popular non-equilibrium systems:
Time-delayed feedback control and the active Ornstein-Uhlenbeck process.
