Reducing the Bias Hidden Inside Economic Estimates

When preliminary estimates distort the answer

Suppose researchers want to measure how hiring discrimination varies across firms. They first estimate each firm’s response to applicant characteristics, then combine those estimates to calculate an average or a measure of dispersion. With few applications per firm, the individual estimates are noisy. Simply squaring and averaging estimated coefficients can produce a biased estimate of their true second moment.

This is an illustration of the problem addressed by Stéphane Bonhomme, Koen Jochmans, Whitney K. Newey and Martin Weidner in Higher-Order Neyman Orthogonality in Moment-Condition Models. This article concerns version 3 of their arXiv preprint, revised on 5 November 2026. Its central contribution is theoretical; its numerical evidence comes from simulations, not a new empirical study of employers.

The authors work with moment conditions: restrictions requiring certain quantities to average to zero in the population. These provide a common framework for methods including linear regression and instrumental variables. One set of conditions determines background, or “nuisance,” parameters; another identifies the quantity researchers actually want.

Making the estimating equation less sensitive

A standard protection is Neyman orthogonality. At the true parameter values, a small change in a nuisance parameter has no first-order effect on the expected estimating equation. That reduces the impact of preliminary estimation errors, but the remaining bias can still matter when those estimates are inaccurate.

The paper extends this protection to a chosen higher order. Under its assumptions, first-order orthogonality makes the local disturbance quadratic in the combined nuisance error; second-order orthogonality makes it cubic. More generally, order q makes the disturbance of order q plus one. These are local mathematical bounds, not promises that an estimator will be unbiased in a finite dataset.

Importantly, the protection covers both the original nuisance parameters and an additional parameter representing a left inverse of the Jacobian—the matrix describing how the nuisance moment conditions change when their parameters change. It also covers interactions between errors in these two estimates.

Combining approximation with independent observations

The construction joins two ideas. First, the authors approximate the target using a polynomial expansion. Second, they estimate products in that polynomial using separate, independent observations. Independence allows the expectation of a product to equal the product of expectations.

These “copies” mean independent observations, not duplicated records. In implementation, the estimating equation is averaged over combinations of distinct observations, producing a U-statistic. Preliminary estimates are obtained from held-out data. Cross-fitting swaps the estimation and evaluation roles of sample subsets and averages the results.

The authors provide closed-form formulas when the nuisance moment conditions are affine—linear plus a constant—and extend them to nonlinear conditions. For the nonlinear case, rooted trees organize the additional derivative terms and their coefficients.

A distinctive feature is that increasing the orthogonality order does not require adding more nuisance parameters. The authors also show how transformed moment conditions can reduce the additional Jacobian-inverse nuisance to a single scalar, even with more nuisance conditions than unknowns. That reduction has a cost: more independent observations enter each estimating term, with potentially greater sampling variability.

What the simulations show

The simulation exercise considers 1,000 firms, between 20 and 100 applications per firm, and 1,000 simulation repetitions. Binary callbacks depend on applicant race and gender through a stylized model inspired by earlier hiring-discrimination research. The authors explicitly do not calibrate it to that study’s empirical findings.

For the average race coefficient, the unregularized estimators are all close to unbiased, while orthogonalization adds some sampling variability. The more revealing comparison concerns the average squared coefficient, which helps describe dispersion across firms.

Without regularization of the Jacobian-inverse estimate, first-order correction has greater absolute bias than the plug-in estimator. Second-order correction has lower bias than the plug-in estimator only from 60 applications per firm onward.

Regularization, which shrinks preliminary estimates using a hierarchical model, changes the picture. Combined with it, second-order correction has lower bias than the plug-in approach throughout the simulated application counts. First-order correction still leaves substantial bias. The findings support a combination of higher-order correction and accurate preliminary estimation, not correction alone.

Where the result stops

The theory requires independence, sufficient smoothness and a full-column-rank nuisance Jacobian. Nuisance parameters are finite-dimensional, although the general framework permits their dimension to grow with sample size. The appendix develops asymptotic inference for grouped data under additional assumptions; the authors identify other settings as requiring further work.

They also leave the choice of orthogonality order unresolved and identify nonparametric nuisance components as an extension. The contribution is therefore a general construction for reducing a specific source of estimation bias—not evidence that higher-order methods always improve economic measurement.