I mentioned last Thursday that it may be possible to find an unbiased estimate of a parameter using IV methods if two endogenous variables have a known relation in the bias they produce in individual IV estimates. Here are the moment conditions corresponding to the expectation and variance assumptions given last time for an error u and endogenous instruments v and w:
E(v^2*u^2)-E(vu)^2 = a^2*(E(w^2*u^2)-E(wu)^2)
where a is an unknown constant.
What is happening is, these two equations allow a reduction by one in the number of parameters in u to be estimated, since we have two new equations but have added only one extra parameter, a. The reduction in the number of parameters to be estimated offsets the contribution of the unknown bias parameter to the number of parameters, and means that the system is as identified as it would be without the bias.
I think this representation makes the direction of further generalisations clear.