Approximate Likelihood with Proxy Variables for Parameter Estimation in High-Dimensional Factor Copula Models

Factor copula models involve latent variables that explain much of the dependence in the observed variables. Their log-likelihoods can involve one-dimensional or multi-dimensional integration. For the one-factor copula with weak residual dependence and for the oblique factor copula model, we show that, under some mild assumptions, proxy variables that are unweighted averages computed from the observed variables can be used for the latent variables when the dimension is large. Then alternative log-likelihoods without integrals can be used for parameter estimation. The proxy variables can help to select appropriate linking copulas in some factor copula models and to perform numerically faster maximum likelihood estimation of parameters. Simulation studies show that parameter estimates obtained using the proxy variable approach are close to those obtained using the maximum likelihood approach. The proxy variable approach is used to analyze a financial data set of stock returns in a single sector.