  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.
