Nobody knows the exact date of the messiyahs birth. however, most experts agree that it most likely was not december 25th. Remember, miriam and yoseph were required to take part in the roman census. It is unlikely that the roman government would require its citizens to go to bethlehem in the winter. It would be too much of a burden.
Secondly, scripture says that shepherds were abiding in the fields tending their flocks at the time of the messiyahs birth. It was not the tradition of shepherds to do this in winter. The latest one would usually find shepherds in the fields in Israel would be in late october.
