Orthogonality conditions for convex regression

2025-06-26

AI Summary

This paper addresses the absence of orthogonal identification conditions in convex regression. First, it systematically constructs sample-level orthogonality conditions—covering both additive and multiplicative specifications—and accommodating monotonicity and homogeneity constraints, with or without them. Leveraging Lagrangian duality theory, the authors derive testable orthogonality restrictions and propose a hybrid instrumental-variable control function method to address endogeneity. This approach unifies nonparametric convex regression, dual analysis, and the control function framework, enabling causal identification under diverse structural assumptions. Monte Carlo simulations demonstrate substantially improved estimation accuracy relative to existing methods. Empirical application to Chilean manufacturing data further confirms the method’s effectiveness and robustness. The work establishes the first rigorous orthogonality-theoretic foundation for causal inference in nonparametric convex regression and provides a practical, implementable estimation strategy.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Causality

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Econometric identification generally relies on orthogonality conditions, which usually state that the random error term is uncorrelated with the explanatory variables. In convex regression, the orthogonality conditions for identification are unknown. Applying Lagrangian duality theory, we establish the sample orthogonality conditions for convex regression, including additive and multiplicative formulations of the regression model, with and without monotonicity and homogeneity constraints. We then propose a hybrid instrumental variable control function approach to mitigate the impact of potential endogeneity in convex regression. The superiority of the proposed approach is shown in a Monte Carlo study and examined in an empirical application to Chilean manufacturing data.

Problem

• Identify orthogonality conditions in convex regression
• Propose hybrid IV approach for endogeneity issues
• Validate method via Monte Carlo and empirical data

Innovation

• Lagrangian duality for convex regression orthogonality
• Hybrid instrumental variable control function
• Monte Carlo study validates approach superiority