Dear Mike and Kit,
Thanks much for your response. Your first paragraph speaks directly to the
problem I was facing. Regarding your subsequent remarks: The tests for
instrument strength/weakness are all (or nearly all) fine, and I can
conduct them with the just-identifying instruments. In other words, the
correlation or lack thereof between Z and X can well be captured with
the model I have. The problem was with testing for instrument orthogonality (validity). Conceptually, I would have thought that it
should be
possible to test whether the excluded instruments are in fact exogenous
and orthogonal to the error term, even if the model is just-identified.
But both your and Kit's comments clarified to a certain extent why this
may not be possible after all.
Thanks also for the literature. Two of them actually concern the
weak instrument problem which I believe I do have a handle on (the other
two are intro to econometrics texts; I have worked a lot with the
graduate Wooldridge, though not with the intro Wooldridge).
Kit, thanks for reminding me that I can use nonlinear transformations of
my instruments as additional instruments to make the model over-identified
and thus be able to run the battery of instrument validity tests. This
option was in my mind at some point but it seemed to have escaped, so
thanks for bringing it back!
Jennifer
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