An Automatic Finite-Sample Robustness Metric: When Can Dropping a Little Data Make a Big Difference? (2011.14999v5)

Abstract: Study samples often differ from the target populations of inference and policy decisions in non-random ways. Researchers typically believe that such departures from random sampling -- due to changes in the population over time and space, or difficulties in sampling truly randomly -- are small, and their corresponding impact on the inference should be small as well. We might therefore be concerned if the conclusions of our studies are excessively sensitive to a very small proportion of our sample data. We propose a method to assess the sensitivity of applied econometric conclusions to the removal of a small fraction of the sample. Manually checking the influence of all possible small subsets is computationally infeasible, so we use an approximation to find the most influential subset. Our metric, the "Approximate Maximum Influence Perturbation," is based on the classical influence function, and is automatically computable for common methods including (but not limited to) OLS, IV, MLE, GMM, and variational Bayes. We provide finite-sample error bounds on approximation performance. At minimal extra cost, we provide an exact finite-sample lower bound on sensitivity. We find that sensitivity is driven by a signal-to-noise ratio in the inference problem, is not reflected in standard errors, does not disappear asymptotically, and is not due to misspecification. While some empirical applications are robust, results of several influential economics papers can be overturned by removing less than 1% of the sample.

• The paper introduces AMIP, a novel metric derived using Taylor expansion to measure the maximal impact of small data omissions on econometric estimators.
• It shows that higher signal-to-noise ratios and evenly distributed influence scores lead to more robust estimates across models like OLS, IV, GMM, and Bayesian methods.
• Simulations and case studies, including the Oregon Medicaid study, validate that omitting less than 1% of data can significantly shift analytic outcomes.

An Automatic Finite-Sample Robustness Metric: When Can Dropping a Little Data Make a Big Difference?

The paper, authored by Tamara Broderick, Ryan Giordano, and Rachael Meager, explores a critical issue in empirical research: the robustness of econometric conclusions against the removal of small subsets of data. The paper proposes an easily computable metric, the "Approximate Maximum Influence Perturbation" (AMIP), which quantifies how the exclusion of minor portions of a dataset can impact the results of econometric analyses. The authors emphasize the importance of this robustness measure, especially in applied economic research where data imperfections and sampling issues are prevalent.

Major Contributions

1. AMIP Derivation: The paper outlines the derivation of the AMIP using a Taylor series expansion around the empirical influence function. This approach approximates the maximum change in an estimator when small subsets of a sample are omitted. The AMIP relies on influence scores calculated from the classical influence function, facilitating an automated and fast computation process applicable to broad econometric models such as OLS, IV, GMM, and Bayesian methods.
2. Robustness Insights: The paper provides a nuanced understanding of two critical factors impacting robustness: signal-to-noise ratio and data influence shape. It highlights that a higher signal-to-noise ratio often results in more robust estimates. Additionally, the shape of the influence (i.e., the distribution of influence scores across data points) can indicate potential non-robustness when specific proportions of data exert disproportionate influence on outcomes.
3. Simulation and Theory: The authors utilize both theoretical proofs and simulations to establish the validity of their approximation. They demonstrate that the approximation error tends to be minimal for small sample proportions, ensuring practical reliability.
4. Case Studies: To demonstrate the application and practicality of their approach, the paper presents several case studies, including the Oregon Medicaid paper and various microcredit trials across different countries. These examples reveal non-trivial sensitivity in econometric analyses, where significant alterations in results occur after omitting less than 1% of the sample.

Implications and Future Directions

The authors discuss the implications of their findings on the interpretation of empirical results, especially in policy decision contexts. They assert that reliance on classical standard errors alone may overlook substantial sensitivities that could undermine the broader applicability of research conclusions. These findings suggest that introducing AMIP-based sensitivity metrics could lead to more robust and reliable policy recommendations.

The research opens avenues for incorporating robustness checks into standard econometric toolkits. Future developments could include refining the AMIP's precision across more complex models and investigating its applicability in settings with heavily skewed or non-standard data distributions.

Conclusion

This work provides a significant advancement in robustness analysis by offering a computationally feasible method to assess the sensitivity of econometric conclusions to small data perturbations. By identifying cases where conclusions are overly reliant on limited data observations, the AMIP can serve as a vital supplement to traditional methods, encouraging more resilient and generalizable inferences in empirical research.