[Paper Review] About Kendall's regression

Alexis Derumigny, Jean‐David Fermanian|arXiv (Cornell University)|Feb 21, 2018

Financial Risk and Volatility Modeling9 references3 citations

TL;DR

This paper proposes a penalized regression model for conditional Kendall's tau, estimating dependence between two variables given covariates using a flexible basis expansion of transformed covariates. It establishes non-asymptotic bounds with explicit constants, proves consistency and asymptotic normality of a two-step estimator, and demonstrates oracle properties and model selection consistency under sparsity.

ABSTRACT

Conditional Kendall's tau is a measure of dependence between two random variables, conditionally on some covariates. We assume a regression-type relationship between conditional Kendall's tau and some covariates, in a parametric setting with a large number of transformations of a small number of regressors. This model may be sparse, and the underlying parameter is estimated through a penalized criterion. We prove non-asymptotic bounds with explicit constants that hold with high probabilities. We derive the consistency of a two-step estimator, its asymptotic law and some oracle properties. Some simulations and applications to real data conclude the paper.

Motivation & Objective

• To model the dependence between two random variables as a function of covariates using conditional Kendall’s tau.
• To address the challenge of high-dimensional, nonparametric estimation of conditional dependence by introducing a parametric, basis-expansion regression framework.
• To enable inference on the influence of covariates on dependence, including testing the 'simplifying assumption' in copula models.
• To develop a computationally efficient alternative to kernel smoothing, which scales poorly with sample size.
• To establish theoretical guarantees, including non-asymptotic bounds, consistency, asymptotic normality, and oracle properties for the estimator.

Proposed method

• Model the transformed conditional Kendall’s tau as a linear combination of basis functions of the covariates: $ \Lambda(\tau_{1,2|\mathbf{Z}=\mathbf{z}}) = \mathbf{\psi}(\mathbf{z})^T \boldsymbol{\beta}^* $, where $ \Lambda $ is a link function such as the Fisher transform.
• Use a penalized criterion (e.g., Lasso-type) to estimate the high-dimensional parameter vector $ \boldsymbol{\beta} $, promoting sparsity and model selection.
• Employ kernel-based estimation for the conditional Kendall’s tau at each covariate value, with theoretical bounds derived under regularity conditions.
• Establish non-asymptotic concentration inequalities for the estimator using moment and entropy conditions on the kernel and design density.
• Derive asymptotic normality of the two-step estimator under regularity conditions on bandwidth, kernel, and densities.
• Use a two-step estimation procedure: first estimate the conditional Kendall’s tau via kernel smoothing, then regress the transformed tau on basis functions of covariates with penalization.

Experimental results

Research questions

• RQ1 Can a flexible, parametric regression model effectively represent the functional relationship between covariates and conditional Kendall’s tau?
• RQ2 What theoretical guarantees (consistency, asymptotic normality, oracle properties) can be established for a penalized two-step estimator of conditional dependence?
• RQ3 How can the 'simplifying assumption' in pair-copula models be tested using the proposed framework?
• RQ4 What are the non-asymptotic error bounds for the estimated conditional Kendall’s tau with explicit constants?
• RQ5 How does the penalized regression approach compare to nonparametric kernel smoothing in terms of computational cost and statistical efficiency?

Key findings

• Non-asymptotic bounds with explicit constants are derived for the estimation error of the conditional Kendall’s tau, holding with high probability.
• The two-step penalized estimator is consistent and asymptotically normal under regularity conditions on bandwidth, kernel, and densities.
• The estimator achieves oracle properties, meaning it selects the correct model with high probability and estimates non-zero coefficients as if the true model were known.
• The method enables testing the 'simplifying assumption' in copula models by assessing whether covariates influence dependence.
• Simulations and real data applications confirm the method's ability to recover sparse, non-linear dependence structures with good finite-sample performance.
• The approach significantly reduces computational cost compared to kernel smoothing, especially for large datasets, due to the parametric regression framework.