Job Market Paper:
This article introduces two results for instrumental variable models with a continuous endogenous variable, allowing for the most general unobserved heterogeneity. The first result is a proof of a generalization of Pearl's conjecture (Pearl 1995), showing that the exclusion restriction of an instrument cannot be tested in this setting without making structural assumptions. The proof is constructive and opens the door to potentially reestablish testability under weak assumptions. The second and main result is an approach for estimating sharp bounds on any possible causal effect, making no or minimal structural assumptions on the model besides the exclusion restriction. The key for this is to consider the instrumental variable model as two dependent stochastic processes and to construct an infinite dimensional linear program on their paths, the solution to which provides the counterfactual bounds. This framework can in principle encompass every structural assumption made in instrumental variable models and is the natural generalization of the complier-, defier-, never taker-, always taker distinction to the continuous setting. To showcase the estimation procedure, we obtain bounds on distributional causal effects of expenditures on leisure and food, using the 1996 UK Family Expenditure Survey. We find that food is a necessity while leisure is a luxury good, thereby corroborating the predictions from economic theory thereby corroborating the predictions from economic theory while introducing only minimal assumptions during the estimation process.
Econometrics, Causal Inference, Applied Microeconomics, Machine Learning, Finance