A non-perturbative general relativistic approach to global astrometry was developed by de Felice et al. (1998) to handle satellite astrometry data in a genuine relativistic framework. In this contribution, the framework above has been further exploited to account for stellar motions and parallax. Because of the relevance that accurate knowledge (to 10−5 or better) of the relativistic parameter γ has to fundamental physics, a Parametrized Post-Newtonian (PPN) model has also been implemented, which allows the direct estimation of γ along with the astrometric parameters. These models have been tested on end-to-end simulations of the mission GAIA. The results show that, within the limitation of the simulation and the assumptions of the adopted model, measurements accurate to 100 μarcsec of large arcs among stars repeated over a few years can be modelled to establish a dense reference frame with a precision of a few tens of μarcseconds. Moreover, our experiments indicate that γ can be estimated to better than 10−6.