Consistency of instruments across specifications

When one has multiple specifications for testing an empirical model using instrumental variable or GMM estimation, then the best instruments for each specification may vary according to exogeneity and correlation criteria. Should a fixed instrument set be used across all specifications, or should the instruments be the best for each specification?

Changing the instrument set risks changing the estimation bias from it, unless the instrument sets all have the same correlation with the error term, which is uncommon in economic applications. Similarly, the uncertainty levels will probably not be directly comparable across specifications.

There are several reasons why correlation with an error term could arise. One is that the instrument set is correlated with some of the variables in the specification, while another is that some of the instruments should have been included in the specification. If the instrumental set is correlated with some of the non-instrumented variables, and these variables do not change across specifications, then the correlation across specifications will change if the set varies between them. In this case, it may be best to use the same set across all specifications. If the non-instrumented variables change across specifications, then the correlation is likely to change across specifications even if the same set is used, so it may be best to use the optimal set for each individual specification.

I think the above comments may vary slightly between instrumental and GMM system estimation due to the way instruments are allocated under the GMM system, but the idea is the same.