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 $fsim 50%$ (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.
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