In this paper we present three theorems which give insight into the regularizing properties of {\minres}. While our theory does not completely characterize the regularizing behavior of the algorithm, it provides a partial explanation of the observed behavior of the method. Unlike traditional attempts to explain the regularizing properties of Krylov subspace methods, our approach focuses on convergence properties of the residual rather than on convergence analysis of the harmonic Ritz values. The import of our analysis is illustrated by two examples. In particular, our theoretical and numerical results support the following important observation: in some circumstances the dimension of the optimal Krylov subspace can be much smaller than the number of the components of the truncated spectral solution that must be computed to attain comparable accuracy. Also cross-referenced as UMIACS-TR-98-62