Piece-wise linear isotonic regression

2026-05-14

AI Summary

This study addresses the limitation of traditional isotonic regression, whose step-function outputs lack economically interpretable marginal quantities such as shadow prices or elasticities. To overcome this, the authors propose a piecewise-linear smoothing framework that preserves monotonicity while incorporating conditional convexity to formulate a bilevel optimization model. This approach transforms isotonic regression estimates into continuous piecewise-linear functions, thereby recovering marginal effects with meaningful economic interpretation. The method is applicable to both univariate and multivariate settings and consistently achieves substantial reductions in mean squared error, regardless of whether the underlying function is convex. Its practical utility is demonstrated through an empirical application to the analysis of agglomeration economies in Finnish municipalities, confirming both its effectiveness and real-world relevance.

Abstract

Isotonic regression provides a flexible, tuning-free approach to estimating monotonic functions without imposing global curvature constraints, yet the estimated regression function is inherently a step function. This paper addresses a key limitation of such estimators: their inability to provide meaningful marginal properties, such as shadow prices or elasticities. We propose a novel piece-wise linear smoothing framework that recovers meaningful marginal estimates even in non-convex settings. Building on the concept of conditional convexity originally developed in deterministic frontier analysis, we formulate the smoothing process as a bilevel optimization problem that fits a continuous, monotonic, piece-wise linear function to the initial isotonic regression predictions. Monte Carlo simulations demonstrate that the proposed approach can significantly improve estimation precision, reducing mean squared error in both convex and non-convex settings for univariate and multivariate data. We apply this approach to analyze agglomeration economies in Finnish municipalities, illustrating its practical value.

Problem

• isotonic regression
• marginal properties
• shadow prices
• elasticities
• step function

Innovation

• piece-wise linear isotonic regression
• bilevel optimization
• conditional convexity
• marginal estimation
• monotonic function estimation