I am running an -ivprobit- model with one endogenous regressor and one instrument. I am trying out two different instruments, let's call them A and B. If I instrument using A, I get the p-value for Wald's test of exogeniety as 0.04 (indicating that using -ivprobit- is appropriate due to the presence of endogeniety), but if I use instrument B, I get a p-value of 0.96 suggesting that there is no endogeniety.

This doesn't sound correct to me. I have the same endogenous variable--according to me, simply changing the instrument shouldn't change the Wald exogeniety test results.

Could someone explain how to interpret my results and if/where am I going wrong?