I. Howard Marshall offers explanation of census and dating of Jesus's birth.
I. Howard Marshall, The Gospel of Luke (Grand Rapids, MI: Eerdmans, 1978), 100, 102, 104
After the conquest of Egypt (30 BC) it is 'practically certain' that a census was held in 10-9 BC (F.F. Bruce, NBD, 203; cf Res Gestae 2:8), and that this was repeated every 14 years. About the same time a census, which took 40 years to complete, was commenced in Gaul. This latter census was a prima descriptio, a kind of 'Domesday Book' reckoning, to list for the first time all who were liable to taxation and what their resources were. . .In any case, Luke's statement can be regarded as a sufficiently accurate description of the emperor's intention that the whole empire should pay taxes. . .
These considerations show that the character of the census described by Luke is far from impossible and hence many recent writers are prepared to admit that Luke's description of a census reflects historical reality. The major difficulty that remains is the date. According to Lk 1:5 and Mt. 1-2 Jesus was born during the reign of Herod the Great (ob. 4 BC), but Quirinius was governor of Syria after his death in AD 6-9, Josephus refers to the census carried out at this time, and describes the riots which accompanied it under Judaas the Galilean (cf. Acts 5:37). Josephus' narrative gives the impression that this was the first Roman census in the country. Hence it has been concluded that Luke has wrongly associated the birth of Jesus which took place before Herod's death with the census which in fact happened some ten years later. . . In attempting to come to a conclusion on the matter it seems certain that Luke intended to record an enrolment as resulting from a general policy of Augustus during taxation. It is not impossible that this enrolment was carried out by Quirinius acting in a special capacity before the death of Herod but this theory remains speculative. It is also not impossible that Luke has recorded the first stages of an enrolment by another governor of Syria and completed by Quirinius, although the impression we get from Josephus is of a new action being commenced in AD 6. Historically, the solution least open to difficulty is that Herod was forced to carry out some kind of enrolment in his own realm under Roman pressure, perhaps in association with the oath of loyalty (Hauck 37), and that this was regarded by Luke as part of the general fiscal measures of Augustus.