The Proper Chronology for Agrippa's Reception of his Kingdom and the Calculation of his Regnal Years on His Coinage

There has been so much debate about when Pilate was dismissed from Judea, I think Sir Winston Smith (A Dictionary of the Bible) gets it basically right when he says:

In AD 37, Pilate having been recalled to Rome Jerusalem was visited by Vitellius, the prefect of Syria, at the time of the Passover. Vitellius conferred two great benefits on the city. He remitted the duties levied on produce, and he allowed the Jews again to have the free custody of the high-priest's vestments. He removed Caiaphas from the high-priesthood, and gave it to Jonathan son of Annas. He then departed, apparently leaving a Roman officer in charge of the Antonia (Ant. xviii. 4, § 3). Vitellius was again at Jerusalem this year, probably in the autumn, with Herod the tetrarch (xviii. 5, § 3); while there he again changed the high-priest, substituting for Jonathan, Theophilus his brother. The news of the death of Tiberius and the accession of Caligula reached Jerusalem at this time. Marcellus was appointed procurator by the new emperor.(p. 1620)

It was in this year that the Emperor made Agrippa the king of Judea but he only seized his physically set foot to claim his new kingdom in the late summer of 38 CE.

Not a single person has properly solved the riddle of the dating of coinage for Agrippa's reign because of this confusion. Schwartz attempts to fill in some of the gaps but ultimately fails writing:

Antipas and Herodias were exiled in the summer of 39, before Gaius departed for the north, as is virtually universally agreed on the basis of the conjunction of Roman, Josephan and numismatic evidence. At that time, Agrippa was not in Rome, so he sent a freedman, Fortunatus, with a letter containing his accusations. It seems, therefore, that Agrippa was still busy with his kingdom's affairs. According to Antip, Caligula gave Antipas' territories and property to Agrippa upon taking them away from the tetrarch (BJ2.183; Ant. 18.252). However, various scholars have noted a difficulty with this, namely, Ant. 19.351 states that Agrippa received Antipas' territories during the fourth year of his own monarchy. If Agrippa's reign began in the spring (Nisan) of 37, then his fourth year began in the spring of 40, more than a half a year after Antipas was unseated!

However, we must emphasize the premise: "If Agrippa's reign began in the spring of 37." It seems, in fact, as especially A. Stein has argued, that Agrippa counted his years from the autumn (Tishri) of 36. For some of his coins bear Gaius' name and portrait and are dated to Agrippa's fifth year: given Gaius' death in January 41 , this is impossible if Agrippa counted from Nisan 37. (We ignore such such desperate assumptions as that the coins had been minted months in advance and somehow got into circulation.) Rather, we should assume that Agrippa counted his regnal years from the autumn new year preceding his first enthronement, ie, Tishri (ca. September/October) 36. In fact, there is no good reason for the usual assumption that the Herodians counted their years from the spring, and several recent scholars have argued that the autumnal era was the usual one.(Agrippa I: the Last King of Judea p. 57, 58)

But Schwartz does not let the facts get in the way of a good solution - Gaius's reign only began in March of 37. The only solution that makes any sense is that Agrippa simply used the Roman calendar which began in January to calculate his regnal years. This is what all other kings did. There is no reason to think that this client king would have bucked convention.