A single-index ARCH-M model is studied. Estimators of parametric and nonparametric components for a single-index model are proposed by profile likelihood method. All estimators are shown to be consistent and asymptotically normal. Wald statistic and generalized likelihood ratio statistic to test ARCH effect and goodness-of-fit are proposed, based on asymptotic properties of estimators. Simulations show that proposed estimators and test statistics in this paper perform well.