# Counting Defiers

## TL;DR

This paper develops methods to count different types of individuals, including defiers, in instrumental variable models, relaxing the usual monotonicity assumption, and applies it to a fertility study with virtual randomization.

## Contribution

It introduces estimators for counts of defiers, always takers, compliers, and never takers allowing for defiers in binary instrument models, especially useful in natural experiments.

## Key findings

- Allowing for defiers changes the estimated responsiveness of fertility to sex composition.
- The model provides new insights into treatment effects when defiers are present.
- Empirical results show increased fertility responsiveness when defiers are considered.

## Abstract

The LATE monotonicity assumption of Imbens and Angrist (1994) precludes "defiers," individuals whose treatment always runs counter to the instrument, in the terminology of Balke and Pearl (1993) and Angrist et al. (1996). I allow for defiers in a model with a binary instrument and a binary treatment. The model is explicit about the randomization process that gives rise to the instrument. I use the model to develop estimators of the counts of defiers, always takers, compliers, and never takers. I propose separate versions of the estimators for contexts in which the parameter of the randomization process is unspecified, which I intend for use with natural experiments with virtual random assignment. I present an empirical application that revisits Angrist and Evans (1998), which examines the impact of virtual random assignment of the sex of the first two children on subsequent fertility. I find that subsequent fertility is much more responsive to the sex mix of the first two children when defiers are allowed.