A copula with a flexibly dependence structure can capture complexity and heterogeneity in economic and financial time series. Based on the recently proposed single-index copula, we propose a simultaneous variable selection and estimation procedure. This method allows for choosing the most relevant state variables by using a penalized estimation with large sample properties derived. Simulation results demonstrate the good performance of the method in selecting relevant state variables and estimating unknown index coefficients and dependence parameters. We apply the proposed procedure to four states' housing markets in the United States and identify six macroeconomic factors that drive their dependence structure.
