Data-driven model selection for same-realization predictions in autoregressive processes

Abstract

This paper is about the one-step ahead prediction of the future of observations drawn from an infinite-order autoregressive AR( $$\infty $$ ) process. It aims to design penalties (fully data driven) ensuring that the selected model verifies the efficiency property but in the non-asymptotic framework. We show that the excess risk of the selected estimator enjoys the best bias-variance trade-off over the considered collection. To achieve these results, we needed to overcome the dependence difficulties by following a classical approach which consists in restricting to a set where the empirical covariance matrix is equivalent to the theoretical one. We show that this event happens with probability larger than $$1-c_0/n^2$$ with $$c_0>0$$ . The proposed data-driven criteria are based on the minimization of the penalized criterion akin to the Mallows’s $$C_p$$ .