in that the term endogeneity means a lot of things!
As a general specification test the Ramsey Reset test can be used - it is easy to do
https://en.wikipedia.org/wiki/Ramsey_RESET_test
as explained here ( YHat - teh predicted repose - from the original model is squared and included additionally in a revised model - if an F test shows significant improvement in fit then the model is likely to be mis-specified\).