Instrumental variables (IV) regression is a method for making causal inferences about the effect of a treatment based on an observational study in which there are unmeasured confounding variables. The method requires one or more valid IVs; a valid IV is a variable that is independent of unmeasured confounding variables and has no direct effect on the outcome. Often there is uncertainty about the validity of the proposed IVs. When a researcher proposes more than one IV, the validity of the IVs 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 validity of the proposed IVs, we argue that a sensitivity analysis is needed. A sensitivity analysis examines the impact of plausible amounts of invalidity of the proposed IVs on inferences for the parameters of interest. We develop a method of sensitivity analysis for IV regression with overidentifying restrictions 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 validity of the proposed IVs 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 methods using a study of food demand among rural households in the Philippines.