We investigate the effective parameters and quasi-static resonances for periodic arrays of dielectric spheres. The effective medium model based on the homogenization of normal modes for spherical particles is used to determine the effective permittivity ${ϵ}_{\mathrm{eff}}$ and permeability ${\mu }_{\mathrm{eff}}$ in the quasi-static regime. Major features of ${ϵ}_{\mathrm{eff}}$ and ${\mu }_{\mathrm{eff}}$ are characterized by the Lorentz-type anomalous dispersion around the frequencies pertaining to the leading-order electric and magnetic resonances, respectively. In particular, the anomalous dispersion is depicted by a resonance function associated with the spherical cavity. The underlying mechanism of quasi-static resonance is illustrated with the localized and dipole-like field patterns at the resonant frequencies. A comparison with the effective parameters for periodic arrays of dielectric circular cylinders is also discussed.