# On the construction of confidence intervals for ratios of expectations

## TL;DR

This paper develops a generalized delta method and bootstrap techniques for constructing confidence intervals for ratios of expectations, especially in small samples or when the denominator approaches zero, with practical guidelines for reliability.

## Contribution

It introduces a generalized delta method and bootstrap consistency results for ratios of expectations, addressing limitations of traditional methods in small samples and near-zero denominators.

## Key findings

- Nonasymptotic confidence intervals are possible but not at all confidence levels.
- A new index helps assess the reliability of delta method-based intervals.
- Simulations and an application demonstrate practical usefulness.

## Abstract

In econometrics, many parameters of interest can be written as ratios of expectations. The main approach to construct confidence intervals for such parameters is the delta method. However, this asymptotic procedure yields intervals that may not be relevant for small sample sizes or, more generally, in a sequence-of-model framework that allows the expectation in the denominator to decrease to $0$ with the sample size. In this setting, we prove a generalization of the delta method for ratios of expectations and the consistency of the nonparametric percentile bootstrap. We also investigate finite-sample inference and show a partial impossibility result: nonasymptotic uniform confidence intervals can be built for ratios of expectations but not at every level. Based on this, we propose an easy-to-compute index to appraise the reliability of the intervals based on the delta method. Simulations and an application illustrate our results and the practical usefulness of our rule of thumb.