The Validity of Instruments Revisited

Valid instrumental variables must be relevant and exogenous. However, in practice it is difficult to find instruments that are exogenous in that they satisfy the knife-edged orthogonality condition and at the same time are strongly correlated with the endogenous regressors. In this paper we show how a mild violation of the exogeneity assumption affects the limit of the Anderson-Rubin test (1949). This test statistic is frequently used in economics due to the fact that it is robust to identification problems. However, when there is mild violation of exogeneity the test is oversized and with larger samples the problem gets worse. In order to correct this problem, we introduce the fractionally resampled Anderson-Rubin test (FAR) that is derived by modifying the resampling technique introduced by Wu(1990). We show the FAR test does not overreject the null hypothesis when we use half of the sample without replacement as the block size from the original sample. As a novel scheme, we treat the block size as a random variable and prove that this choice recovers the limit of the Anderson-Rubin (1949) test. We also prove that this optimal choice of block size converges in probability to 1/2. Simulations show that in finite samples the FAR is conservative; thus, we propose a range of block size choices that has very good size and power when there are possible violations of the exogeneity assumption.