The instrumental variables strategy is commonly employed in empirical research. For correct inference using this econometric technique, the instruments must be perfectly exogenous and relevant. In fact, the standard t-ratio test statistic used in this context yields unreliable and often inaccurate results even when there is only a slight violation of the exclusion restriction. It is crucial to realize that to make reliable inferences on the structural parameters we need to know the true correlation between the structural error and the instruments. The main innovation in this paper is to identify an appropriate test in this context: a joint null hypothesis of the structural parameters with the correlation between the instruments and the structural error term. Since correlation cannot be estimated, we propose a test statistic involving a grid search over correlation values. To address inference under violations of exogeneity, significant contributions have been made in the recent literature by assuming some degree of non-exogeneity. We introduce a new approach by deriving a modified t-statistic that corrects for the bias associated with non-exogeneity of the instrument. A key advantage of our approach over that of the previous literature is that we do not need to make any assumptions about the degree of violation of exogeneity either as possible values or prior distributions. In particular, our method is not a form of sensitivity analysis. Since our modified test statistic is continuous and monotonic in correlation it is easy to conduct inference by a simple grid search. Even though the joint null may seem to be limiting in interpreting rejection, we can still make accurate inferences on the structural parameters because of a feature of the grid search over correlation values. The procedure for calculating the modified coefficients and statistics is illustrated with two empirical examples.
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