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This paper studies the model selection problem in a large class of causaltime series models, which includes both the ARMA or AR($\infty$) processes, aswell as the GARCH or ARCH($\infty$), APARCH, ARMA-GARCH and many othersprocesses. To tackle this issue, we consider a penalized contrast based on thequasi-likelihood of the model. We provide sufficient conditions for the penaltyterm to ensure the consistency of the proposed procedure as well as theconsistency and the asymptotic normality of the quasi-maximum likelihoodestimator of the chosen model. It appears from these conditions that theBayesian Information Criterion (BIC) does not always guarantee the consistency.We also propose a tool for diagnosing the goodness-of-fit of the chosen modelbased on the portmanteau Test. Numerical simulations and an illustrativeexample on the FTSE index are performed to highlight the obtained asymptoticresults, including a numerical evidence of the non consistency of the usual BICpenalty for order selection of an AR(p) models with ARCH($\infty$) errors.

Consistent model selection criteria and goodness-of-fit test for affine causal processes