Limits To (Machine) Learning

TL;DR

This paper introduces the Limits-to-Learning Gap (LLG), a fundamental lower bound on the discrepancy between empirical and true data-generating processes in machine learning, especially relevant for financial data.

Contribution

It characterizes the LLG, derives bounds for it, and demonstrates its significance in explaining underestimation of predictability and excess volatility in financial models.

Findings

LLGs are large in financial variables, indicating underestimation of predictability.

LLG-based bounds refine classic financial bounds like Hansen-Jagannathan.

The LLG explains excess volatility in financial markets.

Abstract

Machine learning (ML) methods are highly flexible, but their ability to approximate the true data-generating process is fundamentally constrained by finite samples. We characterize a universal lower bound, the Limits-to-Learning Gap (LLG), quantifying the unavoidable discrepancy between a model's empirical fit and the population benchmark. Recovering the true population $R^2$, therefore, requires correcting observed predictive performance by this bound. Using a broad set of variables, including excess returns, yields, credit spreads, and valuation ratios, we find that the implied LLGs are large. This indicates that standard ML approaches can substantially understate true predictability in financial data. We also derive LLG-based refinements to the classic Hansen and Jagannathan (1991) bounds, analyze implications for parameter learning in general-equilibrium settings, and show that the LLG provides a natural mechanism for generating excess volatility.