[Paper Review] A path-sampling method to partially identify causal effects in instrumental variable models

Florian Gunsilius|arXiv (Cornell University)|Oct 21, 2019

Advanced Causal Inference Techniques10 references4 citations

TL;DR

This paper introduces a path-sampling method to overcome the 'curse of cardinality' in partial identification of causal effects within general instrumental variable models, especially with continuous endogenous variables. By modeling counterfactual outcomes as stochastic processes and solving an infinite-dimensional linear program via path sampling, the method enables robust, nonparametric bounds on causal effects with probabilistic approximation guarantees.

ABSTRACT

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.

Motivation & Objective

• To address the 'curse of cardinality' in partial identification of causal effects, where the number of constraints grows exponentially with endogenous variable support size.
• To extend linear programming approaches for partial identification to high-cardinality and continuous endogenous variables, where existing methods fail.
• To develop a computationally feasible method for estimating informative nonparametric bounds on causal effects under minimal structural assumptions.
• To incorporate behavioral assumptions (e.g., monotonicity) naturally into the model via path restrictions in a stochastic process framework.
• To provide a general, flexible, and theoretically grounded computational framework for infinite-dimensional optimization in counterfactual models.

Proposed method

• Represent the instrumental variable model as a system of stochastic processes indexed by unobservable heterogeneity, using potential outcomes notation.
• Formulate the causal identification problem as an infinite-dimensional linear program on path spaces, where the objective is to find an optimal probability measure on paths.
• Approximate the infinite-dimensional program by sampling a finite set of paths, reducing it to a semi-infinite program.
• Use concentration inequalities (e.g., Vapnik, van der Vaart & Wellner) to derive probabilistic guarantees on the approximation quality of the sampled program.
• Implement a data-driven approach to select the penalty parameter λ by monitoring convergence and stability of solution paths during optimization.
• Extend the framework to include nonparametric restrictions such as monotonicity, convexity, and martingale properties via path filtering.

Experimental results

Research questions

• RQ1 Can a computationally feasible method be developed to partially identify causal effects in general instrumental variable models with continuous endogenous variables?
• RQ2 How can the 'curse of cardinality'—where constraints grow exponentially with support size—be overcome in partial identification frameworks?
• RQ3 To what extent can behavioral assumptions (e.g., monotonicity) be embedded into the model through path restrictions in a stochastic process representation?
• RQ4 Can path-sampling provide informative, nonparametric bounds on causal effects in real-world data, such as household expenditure, under minimal assumptions?
• RQ5 What theoretical guarantees can be established for the approximation error when replacing an infinite-dimensional linear program with a sampled finite-dimensional counterpart?

Key findings

• The path-sampling method successfully produces informative nonparametric bounds on household expenditures using the 1995/1996 UK family expenditure survey, validating its empirical relevance.
• The method recovers well-established economic facts—such as food being a necessity good and leisure a luxury good—demonstrating robustness and empirical credibility.
• Monotonicity assumptions significantly sharpen the identified bounds, indicating strong identificatory content and improved clarity in results.
• The method provides probabilistic approximation guarantees for the sampled program, ensuring convergence to the true infinite-dimensional solution with high probability under sufficient path sampling.
• The approach outperforms alternative sampling methods by operating on paths rather than inequalities, offering greater flexibility and theoretical grounding.
• The current implementation supports univariate settings and monotonicity constraints, with extensions to multivariate and richer nonparametric restrictions possible via sparsity or factor models.