GFCI No Equipment Ground not testing
When equipment ground is not present, GFCI devices should test properly. 

However, if equipment ground is not present, external GFCI testers there or at downline outlets  should not test successfully.  Their built-in 3-lamp tester should only display one yellow light.  If it displays two but doesn't trip the GFCI, that means a bootleg ground, which makes the GFCI's protection worthless.