The Validity of Instruments Revisited

Valid instrumental variables must be relevant and exogenous. However, in practice it is difficult to find instruments that perfectly satisfy the 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 (1949) test. The Anderson-Rubin (AR) test statistic is frequently used because it is robust to identification problems. However, when there is a mild violation of exogeneity the AR test is oversized and with larger samples the problem gets worse. In order to correct this problem, we introduce the fractionally resampled Anderson-Rubin (FAR) test that is derived by modifying the resampling technique of Wu (1990). Our technical innovation is to treat the block size as a random variable. We prove that this choice recovers the limit of the AR test under a mild violation of exogeneity. We also prove that the optimal of block size converges in probability to one half. Simulations show that in finite samples the FAR is conservative; thus, we propose block sizes in the range of one quarter to one third that have good finite sample properties.