Many modern estimation methods in econometrics approximate an objective function, for instance, through simulation or discretization. These approximations typically affect both bias and variance of the resulting estimator. We provide a higher-order expansion of such "approximate" estimators that...

Multivalued treatment models have only been studied so far under restrictive assumptions: ordered choice, or more recently unordered monotonicity. We show how marginal treatment effects can be identified in a more general class of models. Our results rely on two main assumptions: treatment...

Many econometric models used in applied work integrate over unobserved heterogeneity. We show that a class of these models that includes many random coefficients demand systems can be approximated by a "small-o" expansion that yields a straightforward 2SLS estimator. We study in detail the...

Multivalued treatment models have typically been studied under restrictive assumptions: ordered choice, and more recently unordered monotonicity. We show how treatment effects can be identified in a more general class of models that allows for multidimensional unobserved heterogeneity. Our...

Multivalued treatments are commonplace in applications. We explore the use of discrete-valued instruments to control for selection bias in this setting. Our discussion revolves around the concept of targeting: which instruments target which treatments. It allows us to establish conditions under...