Discriminating between (in)valid external instruments and (in)valid exclusion restrictions

In models estimated by (generalized) method of moments a test on coefficient restrictions can either be based on a Wald statistic or on the difference between evaluated criterion functions. From their correspondence it easily follows that the statistic used for testing instrument validity, the Sargan-Hansen overidentifying restrictions (OR) statistic, is equivalent to an exclusion restrictions test statistic for a nonunique group of regressor variables. We prove that asymptotically this is the case too for incremental OR tests. However, we also demonstrate that, despite this equivalence of test statistics, one can nevertheless distinguish between either the (in)validity of some additional instruments or the (un)tenability of particular exclusion restrictions. This, however, requires to be explicit about the adopted maintained hypothesis. It also
highlights that recent warnings in the literature that overidentifying restrictions tests may mislead practitioners should not be directed towards the test, but to practitioners who do not realize that inference based on such tests is unavoidably conditional on the validity of particular just-identifying statistically untestable assumptions.