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Abstract: Modelling the dependence relations among a large number of quantitative variables has broad applications in various fields. Dependence models with copulas are widely used in multivariate applications when the classical assumption of Gaussian-distributed variables does not hold, and tail inferences are needed. For a large number of highly correlated variables, the dependence relation can be explained using several latent variables; these methods are known as factor models.
In Gaussian factor models (1-factor, bi-factor, oblique factor) and their factor copula counterparts, we propose a way to estimate the latent variables with proxies. We show the proxies, which are defined as the conditional expectation of the latent factors given the observed variables, are consistent (under weak regularity conditions) as the number of observed variables linked to each latent variable increases. The proxies can help to select the bivariate linking copulas in factor copulas and to estimate the copula parameters. With estimated "latent variables", existing copula methods can be applied in modelling the dependence relations of a large number of variables to yield a parsimonious dependence structure. Examples of dependence graphs will be used for illustration.