Efficient and consistent model selection procedures for time series

This paper studies the problem of model selection in a large class of causal time series models that includes ARMA or AR (8) processes as well as GARCH or ARCH (8), APARCH, ARMA-GARCH - and many other processes. First, we study the asymptotic behavior of the ideal penalty that minimizes the risk defined from a quasi-likelihood estimation among a finite family of models containing the true model. We then establish general conditions on the penalty term to obtain properties of consistency and efficiency. In particular, we prove that consistent model selection criteria outperform the classical AIC criterion in terms of efficiency. Finally, we derive the usual BIC criterion from a Bayesian approach and, retaining all second-order terms of the Laplace approximation, a data-driven criterion, which we call KC'. Monte Carlo experiments illustrate the asymptotic results obtained and show that the KC' criterion performs better than the AIC and BIC criteria in terms of consistency and efficiency.