Did Christ know who He was as a youth? Did Jesus Christ know as a baby that He was the Son of God?

When He worked carpentry with Joseph, His earthly father, did He know then Who or What He was? Did He sense in anyway prior to being a young prodigy (learning and eventually teaching in the temple) that He was the Anointed One?

Do the Scriptures say anything about this issue? Do any church traditions or the writings of the Early Church Fathers offer any perspective?
Luke 2 tells the story of Jesus as a boy in the Temple.  His parents had taken Him to a feast there and had left Him behind on their return trip. When they returned to Jerusalem to find Him, Jesus asks His mother a question--"Did you not know that I must be in my Father's house?"

Now, the Temple was not where Joseph lived, but was the House of God.  So, from that very early age, Jesus understood who He was and that His Father was not Joseph, but God the Father.


  Now his parents went to Jerusalem every year at the Feast of the
  Passover. And when he was twelve years old, they went up according to
  custom. And when the feast was ended, as they were returning, the boy
  Jesus stayed behind in Jerusalem. His parents did not know it, but
  supposing him to be in the group they went a day's journey, but then
  they began to search for him among their relatives and acquaintances,
  and when they did not find him, they returned to Jerusalem, searching
  for him. After three days they found him in the temple, sitting among
  the teachers, listening to them and asking them questions. And all who
  heard him were amazed at his understanding and his answers. And when
  his parents saw him, they were astonished. And his mother said to him,
  “Son, why have you treated us so? Behold, your father and I have been
  searching for you in great distress.” And he said to them, “Why were
  you looking for me? Did you not know that I must be in my Father's
  house?” And they did not understand the saying that he spoke to them. 
  And he went down with them and came to Nazareth and was submissive to
  them. And his mother treasured up all these things in her heart.  Luke 2:41-51 ESV