Partial identification approaches have seen a sharp increase in interest in econometrics due to improved flexibility and robustness compared to point-identification approaches . However, formidable computational requirements of existing approaches often offset these undeniable advantages---particularly in general instrumental variable models with continuous variables . This article introduces a computationally tractable method for estimating bounds on functionals of counterfactual distributions in continuous instrumental variable models . Its potential applications include  randomized trials with imperfect compliance, the evaluation of social programs and, more generally, simultaneous equations models. The method does not require functional form restrictions a priori, but can incorporate parametric or nonparametric assumptions into the estimation process. It proceeds by solving an infinite dimensional program on the paths of a system of counterfactual stochastic processes in order to obtain the counterfactual bounds. A novel "sampling of paths"- approach provides the practical solution concept and probabilistic approximation guarantees . As a demonstration of its capabilities , the method provides informative nonparametric bounds on household expenditures under the sole assumption that expenditure is continuous , showing that partial identification approaches can yield informative bounds under minimal assumptions. Moreover, it shows that additional monotonicity assumptions lead to considerably tighter bounds, which constitutes a novel assessment of the identificatory strength of such nonparametric assumptions in a unified framework .
Partial identification approaches are a flexible and robust alternative to standard point-identification approaches in general instrumental variable models . However, this flexibility comes at the cost of a ``curse of cardinality'': the number of restrictions on the identified set grows exponentially with the number of points in the support of the endogenous treatment . This article proposes a novel path-sampling approach to this challenge. It is designed for partially identifying causal effects of interest in the most complex models with continuous endogenous treatments. A stochastic process representation allows to seamlessly incorporate assumptions on individual behavior into the model. Some potential applications include dose-response estimation in randomized trials with imperfect compliance, the evaluation of social programs , welfare estimation in demand models, and continuous choice models . As a demonstration , the method provides informative nonparametric bounds on household expenditures under the assumption that expenditure is continuous . The mathematical contribution is an approach to approximately solving infinite dimensional linear programs on path spaces via sampling .
R
have seen a sharp increase in interest in econometrics due to improved flexibility and robustness compared to
are a flexible and robust alternative to standard
D
. However, formidable computational requirements of existing approaches often offset these undeniable advantages---particularly
R
with continuous variables
. However, this flexibility comes at the cost of a ``curse of cardinality'': the number of restrictions on the identified set grows exponentially with the number of points in the support of the endogenous treatment
R
introduces a computationally tractable method for estimating bounds on functionals of counterfactual distributions in continuous instrumental variable models . Its
proposes a novel path-sampling approach to this challenge. It is designed for partially identifying causal effects of interest in the most complex models with continuous endogenous treatments. A stochastic process representation allows to seamlessly incorporate assumptions on individual behavior into the model. Some
A
dose-response estimation in
R
and, more generally, simultaneous equations models. The method does not require functional form restrictions a priori, but can incorporate parametric or nonparametric assumptions into the estimation process. It proceeds by solving an infinite dimensional program on the paths of a system of counterfactual stochastic processes in order to obtain the counterfactual bounds. A novel "sampling of paths"- approach provides the practical solution concept and probabilistic approximation guarantees
, welfare estimation in demand models, and continuous choice models
D
of its capabilities
D
sole
R
, showing that partial identification approaches can yield informative bounds under minimal assumptions. Moreover, it shows that additional monotonicity assumptions lead to considerably tighter bounds, which constitutes a novel assessment of the identificatory strength of such nonparametric assumptions in a unified framework
. The mathematical contribution is an approach to approximately solving infinite dimensional linear programs on path spaces via sampling