The interpretation of Luke chapter 2, i.e., "aspect of the writer" + "context" = a discrepancy in the interpretation or no discrepancy, i.e., we are shooting for an inspired interpretation, making sure we are the same page with presenting a Bible discrepancy with a 'complete comparison.' Regarding "the date of the birth of Jesus" is not the intent of Luke. A glance at Lk.3:1, 2 shows how Luke proceeds when he is fixing an important date, i.e., It is at once apparent that, s far as Luke is concerned, the important thing to be dated most carefully is not the birth or death of Jesus, but the beginning of the work of the forerunner of Jesus. Thus, right from the start we have one side of the comparison, incomplete (false premise) hence not a valid discrepancy, i.e., either side invalid, makes both invalid; however let's continue and see what we come up with, curious?
Working with Luke because from past scrutinizing of Gospel's birth story, remember he was the clos
est. Cutting through the chase working backwards then forward to attempt to arrive at a closer date than 6 B.C. won't happen. Herod died in the spring of 4 B.C., and that Jesus was born a year or two before his death (Matt.2). Quirinius functioned in Syria during 7 B.C. and after that date, in a governing capacity. Also we know that Varus, who was afterword so thoroughly defeated by the Germans, was governor of Syria and administered its regular affairs in the year 6 B.C.
Although Luke and Matthew not concerned about the date of the birth of Jesus, both result in about 6 B.C.