Sometimes in choosing instruments for an instrumental variable estimation one has a choice of two instrument sets. The first is highly correlated with the variables to be instrumented, but exogeneity tests indicate that it is correlated with the error term. The second is weakly correlated with the instrumented variables but exogeneity tests indicate that it is not correlated with the error term. Which to choose, if no better sets can be found?
The correlation with an error term means that the coefficient estimates produced by the instruments will be biased; the weak correlation with the instrumented variables will increase the uncertainty in the estimates. If the weakly correlated instruments produce coefficient estimates with an acceptable standard error, then it is probably best to go with them. Otherwise, using the error correlated instruments may be best, since the biased confidence interval will probably lie within the other confidence interval, if the error term correlation is not colossal.
Similar considerations apply when choosing GMM system instruments.
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