Did Christian Bale sing by himself in "I'm Not There"?

He sang in "Newsies" (1992), but he's not in "Empire of the Sun" (1987). So how about "in I'm Not There" (2007)? Did he sing "The Lonesome Death of Hattie Carroll" and "Pressing On" by himself, or did he just lip-synched? Thank you for your answer.