We extend results of [Liu et al. Proc. Roy. Soc. Lond. A 469, 20130240, 2013] on artificial magneto-electric coupling and magnetism in moderate contrast dielectric layered media via high-order homogenization to the three-dimensional setting. For this, we consider asymptotic expansions in the vector Maxwell system. We then illustrate effect of magneto-electric coupling with numerical simulations for a layered system alternating positively and negatively refracting index. We unveil hyperbolic behaviour and magneto-electric coupling and derive a sharper estimate for the anisotropic effective medium in [Ramakrishna et al., J. Mod. Optics, 50, 1419-1430, 2003]. We also revisit the effective medium description of a PT-symmetric layered structure in [Novitsky, Shalin, Novitsky, Phys. Rev. A 99, 043812, 2019], adding higher-order approximation orders.