  We consider the selection of prediction models for Markovian time series. For
this purpose, we study the theoretical properties of the hold-out method. In
the econometrics literature, the hold-out method is called out-of-sample and is
the main method to select a suitable time series model. This method consists of
estimating models on a learning set and picking up the model with minimal
empirical error on a validation set of future observations. Hold-out estimates
are well studied in the independent case, but, as far as we know, this is not
the case when the validation set is not independent of the learning set. In
this paper, assuming uniform ergodicity of the Markov chain, we state
generalization bounds and oracle inequalities for such method; in particular,
we show that the out-of-sample selection method is adaptative to noise
condition.
