[George Ford]

The IV (independent variables, not instrumental variables) should be correlated with the DV (and instr variables too, though only through the effect on the endogenous variable).

But, the X should not be a different measure of the Y (e.g., regressing income on a categorical income variable or some component of income).

Multicollinearity is another issue, but that doesn't affect the coef (just the standard errors).

[Nae Khar]

Thanks for all your advice! It was very helpful.