🤖 AI Summary
This study addresses the challenge of non-unique identification of model parameters arising from data insufficiency. To overcome this, it proposes a multi-marginal entropy optimal transport framework that reformulates the partial identification problem as an optimal transport problem over path space, and employs the Sinkhorn iterative algorithm to efficiently solve the resulting infinite-dimensional optimization problem. Theoretically, the paper establishes the convergence of the proposed algorithm and derives the consistency rate of the estimator. Empirically, it demonstrates computationally efficient implementation with precise bound estimation, successfully applying the approach to demand model analysis. Ultimately, this work provides a unified theoretical and computational paradigm for addressing partial identification problems in settings characterized by limited data availability.
📝 Abstract
In many statistical settings, the available data and maintained assumptions do not suffice to uniquely identify the model parameters of interest. In such cases, one can only identify sets which are guaranteed to contain the true parameters. These are often characterized through linear programs that optimize over models compatible with the observed data. These programs can be infinite-dimensional in the optimizer and the number of constraints. We provide a unified way to characterize and solve such optimization problems by phrasing them as optimal transport problems on path spaces. This allows us to regularize the problem with an entropy penalty, recasting it as a multi-marginal entropic optimal transport problem, which can be solved efficiently via Sinkhorn iterations. In addition, it allows us to establish convergence of the regularized value to the sharpest bound, derive consistency rates for a plug-in estimator, and obtain asymptotic distribution for approximate bounds. The method is general and accommodates settings ranging from instrumental variable models with continuous variables to welfare estimation in heterogeneous demand models. We verify the statistical and computational properties in simulations and provide an application to demand estimation.