  We discuss the relevance of information contained in cross-correlations among
different degrees of freedom, which is crucial in non-equilibrium systems. In
particular we consider a stochastic system where two degrees of freedom $X_1$
and $X_2$ - in contact with two different thermostats - are coupled together.
The production of entropy and the violation of equilibrium
fluctuation-dissipation theorem (FDT) are both related to the cross-correlation
between $X_1$ and $X_2$. Information about such cross-correlation may be lost
when single-variable reduced models, for $X_1$, are considered. Two different
procedures are typically applied: (a) one totally ignores the coupling with
$X_2$; (b) one models the effect of $X_2$ as an average memory effect,
obtaining a generalized Langevin equation. In case (a) discrepancies between
the system and the model appear both in entropy production and linear response;
the latter can be exploited to define effective temperatures, but those are
meaningful only when time-scales are well separated. In case (b) linear
response of the model well reproduces that of the system; however the loss of
information is reflected in a loss of entropy production. When only linear
forces are present, such a reduction is dramatic and makes the average entropy
production vanish, posing problems in interpreting FDT violations.
