Conditional Density Estimation with Neural Networks: Best Practices and Benchmarks

TL;DR

This paper presents best practices for neural network-based conditional density estimation in finance, introducing regularization and normalization techniques that improve accuracy and robustness over existing methods.

Contribution

It introduces noise regularization and data normalization schemes for neural conditional density estimators, enhancing their performance and stability in financial applications.

Findings

Outperforms popular semi- and non-parametric estimators on benchmarks.

Effective in estimating higher moments, quantiles, and non-linear transformations.

Demonstrates superior performance on simulated and Euro Stoxx 50 data.

Abstract

Given a set of empirical observations, conditional density estimation aims to capture the statistical relationship between a conditional variable $\mathbf{x}$ and a dependent variable $\mathbf{y}$ by modeling their conditional probability $p(\mathbf{y}|\mathbf{x})$. The paper develops best practices for conditional density estimation for finance applications with neural networks, grounded on mathematical insights and empirical evaluations. In particular, we introduce a noise regularization and data normalization scheme, alleviating problems with over-fitting, initialization and hyper-parameter sensitivity of such estimators. We compare our proposed methodology with popular semi- and non-parametric density estimators, underpin its effectiveness in various benchmarks on simulated and Euro Stoxx 50 data and show its superior performance. Our methodology allows to obtain high-quality estimators for statistical expectations of higher moments, quantiles and non-linear return transformations, with very little assumptions about the return dynamic.