An excavator is an excellent tool for this. Dozers almost as effective. But, unless there's a reason to actually remove the stump, I'll cut it off as low to the ground as possible, drill holes with an auger bit and start putting used motor oil on it repeatedly just filling the holes for a year of so and then build a fire with other burnable product and set it ablaze. Rarely do I have to repeat the process with a second fire. It will leave a void that needs filling but, I usually just filled with manure from the farm animals. Of course, one must be aware of current environmental conditions before setting any fire.
If the stump is located where a foundation must go, it must be physically removed and it never hurts to compact the soil once that has been accomplished.
Rams