# On Heckits, LATE, and Numerical Equivalence

## TL;DR

This paper demonstrates that certain parametric estimators of LATE, including Heckman's control function approach, are algebraically equivalent to IV estimates, highlighting that differences often relate to target parameters rather than functional form assumptions.

## Contribution

It establishes algebraic equivalence between Heckit's control function estimators and IV, clarifying the source of differences in structural versus IV estimates.

## Key findings

- Heckit's estimator is algebraically equivalent to IV under certain conditions
- Equivalence holds for a semi-parametric family and at interior solutions for MLE
- Differences in estimates often reflect target parameter disagreements

## Abstract

Structural econometric methods are often criticized for being sensitive to functional form assumptions. We study parametric estimators of the local average treatment effect (LATE) derived from a widely used class of latent threshold crossing models and show they yield LATE estimates algebraically equivalent to the instrumental variables (IV) estimator. Our leading example is Heckman's (1979) two-step ("Heckit") control function estimator which, with two-sided non-compliance, can be used to compute estimates of a variety of causal parameters. Equivalence with IV is established for a semi-parametric family of control function estimators and shown to hold at interior solutions for a class of maximum likelihood estimators. Our results suggest differences between structural and IV estimates often stem from disagreements about the target parameter rather than from functional form assumptions per se. In cases where equivalence fails, reporting structural estimates of LATE alongside IV provides a simple means of assessing the credibility of structural extrapolation exercises.