Exegesis:

egeneto de … exēlthen ‘and it happened that … went out,’ cf. on 1.8.

en tais hēmerais ekeinais ‘in those days’ (cf. on 1.39) serves to bring out the temporal relationship between the narratives of ch. 1 and ch. 2 and is best understood as pointing back to en tais hēmerais Hērōdou in 1.5.

exēlthen dogma para Kaisaros Augoustou ‘a decree went out from Caesar Augustus,’ a general expression, not official language.

dogma ‘decree,’ usually of one issued by a person of unquestionable authority.

Kaisar, representing Latin Caesar, originally a name which the first Roman emperor acquired by adoption as the son of Julius Caesar, soon became a title. Since Kaisar Augoustos (here) and Tiberios Kaisar (3.1) are in accordance with the official Roman form of the names of both emperors Kaisar is best understood as a part of the name.

apographesthai pasan tēn oikoumenēn ‘that the whole world should have itself registered,’ indicating the content of the imperial decree in the form of an accusative and infinitive.

apographō ‘to register,’ here either in the middle ‘to have oneself registered,’ or, in the passive ‘to be enrolled,’ preferably the former. Here apographesthai refers to registration for tax purposes (cf. on v. 2, apographē).

oikoumenē scil. gē lit. ‘the inhabited earth,’ hence ‘mankind’; it refers here to the inhabitants of the Roman empire as in Acts 17.6; the equation of oikoumenē to the Roman empire or its inhabitants was not uncommon.

Translation:

In those days, or, ‘in that time/period also’; care should be taken not to suggest a connexion with the day of John’s manifestation to Israel (1.80).

A decree went out from Caesar Augustus, or, ‘came down from C. A.’ (Tagalog), ‘was given by C. A.,’ cf. “a proclamation was made by C. A.” (Phillips); or, ‘C. A. posted a placard (Sranan Tongo), put-out strong word (i.e. order)’ (Ekari); or again, shifting to a possessive, ‘there-was a proclamation of C. A.’ (cf. Javanese, and Bible de Jérusalem). A decree. Where no specific term exists one may use ‘order,’ ‘command.’ Tboli uses a term that points to civil law in contrast to customary law. Caesar Augustus. If taken as part of the proper name (see Exegesis) “Caesar” will have to be transliterated. In some languages the use of a name qualifier or title is required with the proper name; hence e.g. ‘big chief Kaisali Angustu’ (Kituba), ‘King-of-kings Kaisar Augustus’ (cf. Balinese).

That, introducing the contents of the decree, may have to be rendered, ‘saying/commanding/ordering,’ or, ‘(in it) he said/commanded/ordered,’ followed by direct or indirect discourse.

All the world, preferably, ‘the whole kingdom’ (Kannada, Ekari), ‘all his subjects’ (Tae,’ the same in Tboli, lit. ‘all persons holding to-him’), ‘all people in all countries of his empire’ (Kituba), ‘all the people included in the Roman empire’ (Cuyono).

Should be enrolled, preferably, ‘should have itself/themselves enrolled.’ The middle voice is sometimes expressed by a causative, e.g. ‘that all his subjects should cause-to-have-written (i.e. cause someone to write down) their name’ (Tboli), or it may have to be circumscribed, e.g. ‘that in the whole kingdom people should come to those who would register them.’ To enroll, or, ‘to hold a census (lit. listing of souls)’ (Bahasa Indonesia KB), ‘to write names’ (East and Toraja-Sa’dan, Kituba, Tboli), ‘to write names in a book’ (Dan), ‘to make a list of’ (Huanuco Quechua), ‘to record the number of’ (Telugu), ‘to count’ (Ekari).

Quoted with permission from Reiling, J. and Swellengrebel, J.L. A Handbook on the Gospel of Luke. (UBS Handbook Series). New York: UBS, 1971. For this and other handbooks for translators see here . Make sure to also consult the Handbook on the Gospel of Mark for parallel or similar verses.