An automatic finite-sample robustness metric: Can dropping a little data make a big difference?

We propose a method to assess the sensitivity of statistical analyses to the removal of a small fraction of the data. Manually checking the influence of all possible small subsets is computationally infeasible, so we provide an approximation to find the most influential subset. Our metric, the "Approximate Maximum Influence Perturbation," 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 the approximation performance and, 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 economics papers can be overturned by removing less than 1% of the sample.