This paper explores the single-item newsvendor problem under a novel setting which combines temporal dependence and tractable robust optimization. First, the demand is modeled as a time series which follows an autoregressive process AR(p), p>0. Second, a robust approach to maximize the worst-case revenue is proposed: a robust distribution-free autoregressive method for the newsvendor problem, which copes with non-stationary time series, is formulated. A closed-form expression for the optimal solution is found for p=1; for the remaining values of p, the problem is expressed as a nonlinear convex optimization program, to be solved numerically. The optimal solution under the robust method is compared with those obtained under three versions of the classic approach, in which either the demand distribution is unknown, and autocorrelation is neglected, or it is assumed to follow an AR(p) process with normal error terms. Numerical experiments show that our proposal usually outperforms the previous benchmarks, not only with regard to robustness, but also in terms of the average revenue. Extensions to multiperiod and multiproduct models are also discussed.