So what were the dates of the Roman census that year?
Does anyone today know? :-)
What we do know, thanks to the Bible, is that this is what got Joseph and Mary to Bethlehem, so to be at the place where the Messiah of Israel was to be born.
I am sure that someone could eventually pin it down. But, the point of the matter is that no one can definitively tell you when this baby, born to an out of wedlock mother and his carpenter father, was exactly.
Think about it in our context. A baby is born behind a garage in a shed, because its homeless parents cannot afford rent.
Two thousand years from now will they be able to find a birth certificate? They would know that it was born during the reign of Obama. That will narrow it down to four years.....