Solving Models of Economic Dynamics with Ridgeless Kernel Regressions

TL;DR

This paper introduces a ridgeless kernel approach for solving economic dynamic models formulated as differential-algebraic equations, overcoming limitations of traditional shooting methods and handling multiple steady states effectively.

Contribution

It develops a novel kernel-based method that recovers solutions satisfying boundary conditions without direct enforcement, with theoretical guarantees and practical demonstrations.

Findings

Kernel solutions satisfy asymptotic boundary conditions asymptotically.

Method converges to the correct solution in canonical models.

Handles models with multiple steady states effectively.

Abstract

This paper proposes a ridgeless kernel method for solving infinite-horizon, deterministic, continuous-time models in economic dynamics, formulated as systems of differential-algebraic equations with asymptotic boundary conditions (e.g., transversality). Traditional shooting methods enforce the asymptotic boundary conditions by targeting a known steady state -- which is numerically unstable, hard to tune, and unable to address cases with steady-state multiplicity. Instead, our approach solves the underdetermined problem without imposing the asymptotic boundary condition, using regularization to select the unique solution fulfilling transversality among admissible trajectories. In particular, ridgeless kernel methods recover this path by selecting the minimum norm solution, coinciding with the non-explosive trajectory. We provide theoretical guarantees showing that kernel solutions satisfy asymptotic boundary conditions without imposing them directly, and we establish a consistency result ensuring convergence within the solution concept of differential-algebraic equations. Finally, we illustrate the method in canonical models and demonstrate its ability to handle problems with multiple steady states.