  We present an analysis of the results of the OGLE-III microlensing campaign
towards the Large Magellanic Cloud (LMC). We evaluate for all the possible lens
populations along the line of sight the expected microlensing quantities,
number of events and duration. In particular we consider lensing by massive
compact halo objects (MACHOs) in the dark matter haloes of both the Milky Way
(MW) and the LMC, and "self lensing" by stars in the LMC bar and disc, in the
MW disc and in the stellar haloes of both the LMC and the MW. As a result we
find that the self-lensing signal is able to explain the 2 OGLE-III
microlensing candidates. In particular, we estimate the expected MW disc signal
to be almost as large as that from LMC stars and able, by itself, to explain
the observed rate. We evaluate a 95% CL \emph{upper} limit for $f$, the halo
mass fraction in form of MACHOs, in the range 10-20% for $(10^{-2}-0.5)
\mathrm{M}_\odot$, and $f=24%$ for $1 \mathrm{M}_\odot$ (below 10% in this full
range, and in particular below 5% for $(10^{-2}-0.1) \mathrm{M}_\odot$) for the
Bright (All) samples of source stars. Furthermore, we find that these limits do
not rise much even if we assume the observed events \emph{are} MACHOs. For the
All sample we also evaluate a rather significant constraint on $f$ for larger
values of the MACHO mass, in particular $f\sim 50%$ (95% CL) for $100
\mathrm{M}_\odot$, to date the stronger bound coming from microlensing analyses
in this mass range. Finally, we discuss these results in the framework of the
previous observational campaigns towards the LMC, that of the MACHO and the
EROS collaborations, and we present a joint analysis of the OGLE-II and the
OGLE-III campaigns.
