We consider thermal fluctuations in a system with many degrees of freedom described by macroscopic equations and discuss the question how by elimination of variables one can arrive at a contracted stochastic equation for a subsystem. It is shown that for the validity of the fluctuation-dissipation theorem in the contracted description it is necessary to formulate the linear macroscopic equations in such a way that the equal-time thermal fluctuations of the subsystem are statistically independent of the variables which are eliminated. If this is not the case then a preliminary transformation of variables is required. As special cases we discuss the derivation of the Langevin equation for a Brownian particle immersed in a fluid and for a particle embedded in a harmonic lattice.
