Not monotonically 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 monotonically correlated with each other yet are still not independent, most of the conventional copulas are not up to the task. Examples include (inversed) U-shaped relationships and heteroskedasticity. To fill this gap, this article 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 non monotonic 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.