Bounds in continuous instrumental variable models

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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 non-parametric 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 non-parametric 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 non-parametric assumptions in a unified framework.