Instrumental variables are widely used for the identification of the causal effect of one random variable on another under unobserved confounding. The distribution of the observable variables for a discrete instrumental variable model satisfies certain inequalities but no conditional independence relations. Such models are usually tested by checking whether the relative frequency estimators of the parameters satisfy the constraints. This ignores sampling uncertainty in the data. Using the observable constraints for the instrumental variable model, a likelihood analysis is conducted. A significance test for its validity is developed, and a bootstrap algorithm for computing confidence intervals for the causal effect is proposed. Applications are given to illustrate the advantage of the suggested approach.