  The Bogoliubov-Born-Green-Kirkwood-Yvon or Time-Dependent Density Matrix
(TDDM) hierarchy of equations for higher density matrices is truncated at the
three body level in approximating the three body correlation function by a
quadratic form of two body ones, closing the equations in this way. The
procedure is discussed in detail and it is shown in non-trivial model cases
that the approximate inclusion of three body correlation functions is very
important to obtain precise results. A small amplitude approximation of this
time dependent nonlinear equation for the two body correlation function is
performed (STDDM*-b) and it is shown that the one body sector of this
generalised non-linear second RPA equation is equivalent to the Self-Consistent
RPA (SCRPA) approach which had been derived previously by different techniques.
It is discussed in which way SCRPA also contains the three body correlations.
TDDM and SCRPA are tested versus exactly solvable model cases.
