No; it only copies stat changes from things like Bulk Up, Intimidate, Parting Shot, etc. It does not copy things from abilities like Defeatist or Flash Fire.
I just remember it as, "it copies everything that the game displays '[Pokemon]'s [stat] [fell/rose]!'" for. This means that speed increases Speed Boost gives you can be copied through the use of Psych Up; it depends on the form of boost the ability gives you.