Why Do DML Estimates Sometimes Fail? Uncovering the Role of the Condition Number

Double Machine Learning (DML) blends machine‑learning models with econometric inference. In theory it should give reliable confidence intervals even with complex nuisance functions. In practice, however, intervals can be wildly off when the orthogonal score is ill‑conditioned.

The authors propose a simple diagnostic, kappa_DML = 1/|J_theta|, that measures how well‑posed the problem is. A small kappa_DML (below 1) means stable estimates and correct coverage. A large kappa_DML inflates both variance and bias, leading to intervals that are too narrow or coverage as low as 40% for a nominal 95% level.

The key result is a non‑asymptotic bound linking coverage error to two terms: n^{-1/2} (the usual sample‑size term) and sqrt(n)·r_n (the nuisance‑estimation error). The condition number amplifies both. Monte‑Carlo simulations reveal three regimes: well‑conditioned, moderately ill‑conditioned, and severely ill‑conditioned. Only the first yields short, trustworthy intervals.

The practical recommendation is to compute kappa_DML and report it alongside DML estimates, just like condition‑number checks in linear algebra or weak‑instrument diagnostics in IV regressions. This lets researchers quickly gauge whether their inference is solid or potentially misleading.

In the partially linear regression (PLR) model examined, the parameter of interest θ appears in an orthogonal moment equation: ψ(W;θ,η)= (Y‑Xθ‑g(Z))·(D‑m(Z)). Here η aggregates the nuisance functions g and m, estimated with flexible methods (Lasso, Random Forest, Neural Nets). The information matrix J_θ = ∂E[ψ]/∂θ is a scalar, whose absolute inverse defines kappa_DML.

A non‑asymptotic Berry‑Esseen theorem shows that the Kolmogorov distance between the DML t‑statistic distribution and the standard normal is bounded by C·(n^{-1/2}+√n·r_n). The term r_n is the combined nuisance‑estimation error, typically O_p(s_n·a_n) where s_n is sparsity and a_n the convergence rate of the machine‑learning methods. When kappa_DML grows, the effective variance becomes kappa_DML^2·σ^2/n and the additional bias is of order kappa_DML·r_n.

This linear dependence yields three scenarios: (i) kappa_DML = o_p(√n) and kappa_DML·r_n →0 guarantee confidence intervals of length O_p(kappa_DML/√n) with correct coverage; (ii) if kappa_DML is of order √n, variance inflates and coverage deteriorates; (iii) for kappa_DML≫√n, bias dominates and intervals become useless.

Simulations reveal that, even with state‑of‑the‑art learners, collinearity between D and Z or poor overlap (propensity scores near 0 or 1) dramatically increase kappa_DML. In high‑dimensional settings (p>n), L1 regularization or random forests can reduce r_n but do not directly affect kappa_DML, which depends on the design matrix geometry.

To implement the diagnostic, compute J_θ̂ = (1/n)∑_i ∂ψ_i/∂θ̂ and set kappa_DML = 1/|J_θ̂|. A practical threshold is 1: values above suggest revisiting model specification, increasing overlap, or reducing dimensionality. This mirrors condition‑number checks in OLS regressions or weak‑instrument tests in IV, offering an interpretable, lightweight measure of DML inference reliability.