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 (completely data driven) ensuring that the selected model verifies the efficiency property but in the non asymptotic framework. We present an oracle inequality with a leading constant equal to one. Moreover, we also 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 $1-c_0/n^3$ with $c_0>0$. The proposed data driven criteria are based on the minimization of the penalized criterion akin to the Mallows’s $C_p$. Monte Carlo experiments are performed to highlight the obtained results.