Instrumental variables (IV) regression is a method for overcoming the problem of unobserved confounding in estimating causal relationships from observational data. The ability of IV regression to overcome unobserved confounding rests on the proposed instruments satisfying an assumption called the exclusion restriction. Often there is a degree of uncertainty about whether the proposed instruments satisfy the exclusion restriction. When a researcher assumes more restrictions than are needed for identification, the validity of the exclusion restriction can be tested via the "overidentifying restrictions test.'' Although the overidentifying restrictions test does provide some information, the test has no power versus certain alternatives and can have low power versus many alternatives due to its omnibus nature. To fully address uncertainty about the exclusion restriction, we argue that a sensitivity analysis is needed. A sensitivity analysis examines the impact of plausible violations of the exclusion restriction on inferences for the parameters of interest. We develop a method of sensitivity analysis for IV regression that makes full use of the information provided by the overidentifying restrictions test, but provides more information than the test by exploring sensitivity to violations of the exclusion restriction in directions for which the test has low power. Our sensitivity analysis uses interpretable parameters that can be discussed with subject matter experts. We illustrate our methodology using a study of food demand among rural households in the Philippines.