If you’ve read and compared the Matthean and Lucan birth narratives and if you’ve read much secondary literature, you know that from the perspective of historical inquiry there are problems. I commend the view of Luke Timothy Johnson in The Historical Jesus: Five Views on matters of the gospel tradition and historical research.
With the various problems the birth narratives present to us, it is reassuring to consider the common elements in Matthew and Luke’s accounts, which suggest a tradition that pre-dated both of them. Fitzmeyer gives a suprisingly detailed list of the doubly attested traditions of Yeshua’s birth and some of these elements may surprise you:
SOURCE: Joseph A. Fitzmeyer, The Gospel According to Luke (Anchor Bible, 307).
(1) Yeshua’s birth is related to the reign of Herod (Luke 1:5; Matt 2:1).
(2) Mary is a virgin engaged to Joseph but they have not yet lived together (Luke 1:27, 34; 2:5; Matt 1:18).
(3) Joseph is of the house of David (Luke 1:27; 2:4; Matt 1:16, 20).
(4) An angel announces the birth of Yeshua (Luke 1:28-30; Matt 1:20-21).
(5) Yeshua is recognized as the Son of David (Luke 1:32; Matt 1:1).
(6) His conception takes place through the Holy Spirit (Luke 1:35; Matt 1:18, 20).
(7) Joseph is not involved in the conception (Luke 1:34; Matt 1:18-25).
(8) The name Yeshua is imposed by heaven prior to the birth (Luke 1:31; Matt 1:21).
(9) The angel identifies Yeshua as Savior (Luke 2:11; Matt 1:21).
(10) Yeshua is born after Joseph and Mary come to live together (Luke 2:4-7; Matt 1:24-25).
(11) Yeshua is born in Bethlehem (Luke 2:4-7; Matt 2:1).
(12) Yeshua, Joseph, and Mary settle in Nazareth (Luke 2:39, 51; Matt 2:22-23).