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

TL;DR

This paper introduces the normal mode copula, a new statistical tool designed to model dependencies between variables that are not linearly correlated but still dependent, especially in complex relationships like heteroskedasticity.

Contribution

It characterizes the properties of the normal mode copula and demonstrates its superior performance over traditional copulas in modeling complex dependencies.

Findings

Normal mode copula is asymmetric and nonmonotonic under certain conditions.

It outperforms conventional copulas in modeling U.S. House vote share and campaign expenditure.

Provides a new approach for dependency modeling beyond linear correlation.

Abstract

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.