Instrument selection when there are no good instruments

I mentioned last Thursday that my paper on technology transfers could be questioned on the result variability in the last few tables and the selection of foreign direct investment lags as instruments in the GMM estimations. The issues are more related than I had anticipated when preparing the paper.

Instrumental variables may be weakly correlated with their instrumented variables and/or may be highly correlated with the error term. Both are problems, but I had favoured weakly correlated, low error correlated instruments if no highly correlated, error correlation free instruments were available. However, recent econometrics literature indicates that quite severe bias problems and non-standard distributions of estimates can arise from using weak instruments.

So if no highly variable correlated, error uncorrelated instruments are available across the full set of specifications, I would use three different sets of instruments and mention the results in a robustness testing section of a research paper. The sets are:
1) the weakly correlated, error uncorrelated instruments
2) the strongly correlated, error correlated instruments
3) a set consisting of the optimal instruments for each individual specification, even if they change across the full complement of specifications

None of the sets are perfect - the third suffers from variable biases across specifications. But if they give similar results, probably qualitatively rather than quantitatively, then we can be more confident than when using the weakly correlated variables alone.