>>11006933Research has been done, but in a generalized from. The question was how competence in X influences once self-evaluation of X. The result is the Dunning-Kruger-Effect. It predicts that incompetent people overshoot and competent people undershoot their self-assessment. Explanation: Incompetent people lack the skills to acurately measure their level of competency (because those are also the skills needed to be competent) and default to a favorable self-assessment.
This should generalize to IQ, I would bet that there is research on it.
Also keep in mind that this does not tell you anything about a given individual; we are in psychologyland here. That means that many effects just appear when you look at millions of people. (I.e. if you were to make a business abusing the Duning-Kruger-Effect, it won't work on 200 people, you would have to way increase your reach to make a reliable profit.)