The Copula Information Criterion and Its Implications for the Maximum Pseudo-Likelihood Estimator

This chapter surveys the asymptotic theory of estimation of a copula from a frequentistic perspective and presents the problems involved in frequentistic model selection among several candidate copulae when using the maximum pseudo-likelihood estimator (MPLE). Frequentistic copula model selection has recently been addressed through the development of the copula information criterion (CIC) — a model selection formula which extends the maximum likelihood-based Akaike information criterion (AIC) to the MPLE. We present the developments leading to the CIC with a focus on its implications, while deferring proofs of underlying limit theorems to the original CIC paper.The CIC is in fact two different formulae, one for misspecified copula models and another for correctly specified copula models, paralleling the Takeuchi information criterion and the Akaike information criterion respectively.These formulae show that there does not exist (in a certain technical sense) an AIC formula for MPL estimation when the parametric copula has extreme behavior near the edge of the unit cube. This means that one cannot make first-order bias-correction terms of a desired part of the attained Kullback-Leibler divergence between the MPL-estimated copula and the data-generating copula in a class of copulae which has received much attention in econometrics. This may be seen as a demarcation of which types of copulae that should be estimated with the MPLE. Interestingly, the main motivating factor for using the MPLE is also the reason for the non-existence of a general MPLE-based AIC formula. A further conclusion is that the CIC provides a counterexample to the often acclaimed intrinsic connection between the AIC and Occam's Razor.