  In the extraction of $\alpha_s$ from hadronic tau decay data several moments
of the spectral functions have been employed. Furthermore, different
renormalization group improvement (RGI) frameworks have been advocated, leading
to conflicting values of $\alpha_s$. Recently, we performed a systematic study
of the perturbative behavior of these moments in the context of the two
main-stream RGI frameworks: Fixed Order Perturbation Theory (FOPT) and Contour
Improved Perturbation Theory (CIPT). The yet unknown higher order coefficients
of the perturbative series were modelled using the available knowledge of the
renormalon singularities of the QCD Adler function. We were able to show that
within these RGI frameworks some of the commonly employed moments should be
avoided due to their poor perturbative behavior. Furthermore, under reasonable
assumptions about the higher order behavior of the perturbative series FOPT
provides the preferred RGI framework.
