  This paper details the modeling pipeline and validates the baseline analysis
choices of the DES Year 3 joint analysis of galaxy clustering and weak lensing
(a so-called "3$\times$2pt" analysis). These analysis choices include the
specific combination of cosmological probes, priors on cosmological and
systematics parameters, model parameterizations for systematic effects and
related approximations, and angular scales where the model assumptions are
validated. We run a large number of simulated likelihood analyses using
synthetic data vectors to test the robustness of our baseline analysis. We
demonstrate that the DES Year 3 modeling pipeline, including the calibrated
scale cuts, is sufficiently accurate relative to the constraining power of the
DES Year 3 analyses. Our systematics mitigation strategy accounts for
astrophysical systematics, such as galaxy bias, intrinsic alignments, source
and lens magnification, baryonic effects, and source clustering, as well as for
uncertainties in modeling the matter power spectrum, reduced shear, and
estimator effects. We further demonstrate excellent agreement between two
independently-developed modeling pipelines, and thus rule out any residual
uncertainties due to the numerical implementation.
