Not Linearly Correlated, But Dependent: A Family of Normal Mode Copulas

When scholars study joint distributions of multiple variables, copulas are useful. However, if the variables are not linearly correlated with each other yet are still not independent, most of conventional copulas are not up to the task. Examples include (inversed) U-shaped relationships and heteroskedasticity. To fill this gap, this manuscript sheds new light on a little-known copula, which I call the "normal mode copula." I characterize the copula's properties and show that the copula is asymmetric and nonmonotonic under certain conditions. I also apply the copula to a dataset about U.S. House vote share and campaign expenditure to demonstrate that the normal mode copula has better performance than other conventional copulas.