ABC for copula estimation
Clara Grazian and Brunero Liseo (di Roma) have just arXived a note on a method merging copulas, ABC, and empirical likelihood. The approach is rather hybrid and thus not completely Bayesian, but this must be seen as a consequence of an ill-posed problem. Indeed, as in many econometric models, the model there is not fully defined: the marginals of iid observations are represented as being from well-known parametric families (and are thus well-estimated by Bayesian tools), while the joint distribution remains uncertain and hence so does the associated copula. The approach in the paper is to proceed stepwise, i.e., to estimate correctly each marginal, well correctly enough to transform the data by an estimated cdf, and then only to estimate the copula or some aspect of it based on this transformed data. Like Spearman’s ρ. For which an empirical likelihood is computed and aggregated to a prior to make a BCel weight. (If this sounds unclear, each BEel evaluation is based on a random draw from the posterior samples, which transfers some uncertainty in the parameter evaluation into the copula domain. Thanks to Brunero and Clara for clarifying this point for me!)
At this stage of the note, there are two illustrations revolving around Spearman’s ρ. One on simulated data, with better performances than a nonparametric frequentist solution. And another one on a Garch (1,1) model for two financial time-series.
I am quite glad to see an application of our BCel approach in another domain although I feel a tiny bit uncertain about the degree of arbitrariness in the approach, from the estimated cdf transforms of the marginals to the choice of the moment equations identifying the parameter of interest like Spearman’s ρ. Especially if one uses a parametric copula which moments are equally well-known. While I see the practical gain in analysing each component separately, the object created by the estimated cdf transforms may have a very different correlation structure from the true cdf transforms. Maybe there exist consistency conditions on the estimated cdfs… Maybe other notions of orthogonality or independence could be brought into the picture to validate further the two-step solution…