Valid Tests when Instrumental Variables do not Perfectly Satisfy the Exclusion Restriction

There is a growing consensus that it is difficult to pick instruments that perfectly satisfy the exclusion restriction. Drawing on results from Berkowitz, Caner, and Fang (2012, Journal of Econometrics 166: 255–266), we provide in this article a nontechnical summary of how valid inferences can be made when instrumental variables come close to satisfying the exclusion restriction. Although the Anderson–Rubin (1949, Annals of Mathematical Statistics 20: 46–63) test statistic is robust to weak identification, it assumes that the instruments are perfectly orthogonal to the structural error term and is therefore oversized under mild violations of the orthogonality condition. The fractionally resampled Anderson–Rubin (FAR) test is a modification of the Anderson–Rubin test that accounts for violations of the orthogonality condition. We show that in small samples, the size of the resampling block of the FAR test can be modified to obtain valid critical values and analyze its size and power properties. We focus on power and not on size-adjusted power because the FAR test uses only one critical value in its application. We also describe user-written commands to implement the Anderson–Rubin and FAR tests in Stata.