General Hannan and Quinn Criterion for Common Time Series

A common solution in model selection is to choose the model, minimizing a penalized based criterion which is the sum of two terms: the first one is the empirical risk (least squares, likelihood) that measures the goodness of fit and the second one is an increasing function of the complexity which aims to penalize large models and control the bias. Therefore a challenging task when designing a penalized criterion is the specification of the penalty term. Considering leading model selection criteria (BIC, AIC, Cp, HQ to name a few), one can see that the penalty term is a product of the model dimension with a sequence which is specific to the criteria. Indeed, a criterion is designed according to the goal one would like to achieve. The classical properties for model selection criteria include consistency, efficiency (oracle inequality, asymptotic optimality), adaptative in the minimax sense.