Although subgrid-scale models of similarity type are insufficiently dissipative for practical applications to large-eddy simulation, in recently published a priori analyses, they perform remarkably well in the sense of correlating highly against exact residual stresses. Here, Taylor-series expansions of residual stress are exploited to explain the observed behavior and “success” of similarity models. Specifically, the first few terms of the exact residual stress ${\tau }_{\mathrm{kl}}$ are obtained in (general) terms of the Taylor coefficients of the grid filter. Also, by expansion of the test filter, a similar expression results for the resolved turbulent stress tensor ${\mathcal{L}}_{\mathrm{kl}}$ in terms of the Taylor coefficients of both the grid and test filters. Comparison of the expansions for ${\tau }_{\mathrm{kl}}$ and ${\mathcal{L}}_{\mathrm{kl}}$ yields the grid- and test-filter dependent value of the constant $c_L$ in the scale-similarity model of Liu et al. [J. Fluid Mech. 275, 83 (1994)]. Until recently, little attention has been given to issues related to the convergence of such expansions. To this end, we re-express the convergence criterion of Vasilyev et al. [J. Comput. Phys. 146, 82 (1998)] in terms of the transfer function and the cutoff wave number of the filter. As a rule of thumb, the less dissipative the filter (e.g., the higher the cutoff), the faster the rate of convergence.