I read on the wiki about Abilities https://pwo-wiki.info/index.php/Abilities
Moody - Raises a random stat two stages and lowers another one stage after each turn is complete working, but not in game.
Hope you review
Is it not possible to select this ability via the Playerdex for the applicable Pokémon, then?
EDIT: I have researched and it appears Moody in official games has only ever been available as a Hidden Ability. I'm not sure PWO currently implements the ability to select Hidden Abilities; though I will enquire with @Bluerise on this.