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We present a study of the dielectric response of (GaAs)p/(AlAs)p (001) superlattices in a wide range of barrier and well widths. We applied density functional theory and a semiempirical method to obtain the superlattice band structures. These were then used as a starting point to evaluate the optical spectra and macroscopic dielectric constants using time-dependent density functional theory. In this context, we investigated the role of crystal local field effects in determining the anisotropy of the dielectric constants. Furthermore, we calculated absorption spectra including the strong continuum excitonic effect through the use of an appropriate model exchange-correlation kernel. We analyzed in detail the complementarity of the ab initio and semiempirical approaches and we compared the successes and limitations of the different approximation schemes.