Bayesian inference with monotone instrumental variables

Sampling variations complicate the classical inference on the analogue bounds under the monotone instrumental variables assumption, since point estimators are biased and confidence intervals are difficult to construct. From the Bayesian perspective, a solution is offered in this paper. Using a conjugate Dirichlet prior, we derive some analytic results on the posterior distribution of the two bounds of the conditional mean response. The bounds of the unconditional mean response and the average treatment effect can be obtained with Bayesian simulation techniques. Our Bayesian inference is applied to an empirical problem which quantifies the effects of taking extra classes on high school students' test scores. The two MIVs are chosen as the education levels of their fathers and mothers. The empirical results suggest that the MIV assumption in conjunction with the monotone treatment response assumption yield good identification power.