1. The Synoptic Problem and the binary time series 1

2. The synoptic problem is concerned with hypotheses about the relationships between the synoptic gospels of Mark (Mk), Matthew (Mt) and Luke (Lk). The two-source or two-document hypothesis: Matthew and Luke had two sources in common, Mark and a hypothetical “Q”, both of which Matthew and Luke used independently of each other. The assumption of Markan priority: the gospel of Mark was the first to be written. The authors of the gospels of Matthew and Luke used the text of Mark as a basis for their own gospels, but making alterations, omissions and additions. The assumption of Markan priority, which we make here, is implicit in the two source hypothesis, but does not imply it. 2

3. 3 The two-source hypothesis Mt LkMk Q Q is the hypothetical source that accounts for the large quantity of Mt-Lk double tradition material that is absent from Mk.

4. 4 The Farrer theory Mk Mt Lk

5. 5 The three-source theory Mt LkMk Q

6. 6 The Mark-Luke model Mk Lk Mt

7. 7

8. The question of independence In the standard form of the two-source hypothesis, it is assumed that Matthew and Luke were independent in their use of Mark, in the sense of not collaborating or neither having the other as a source. Although this might suggest that they were statistically independent in the choice of the words that they retained unchanged from Mark, this is not necessarily the case. We might expect the criteria that Matthew and Luke used in choosing whether to retain, alter or omit words from Mark to have some similarities. The result would be that there would be some departures from statistical independence. However, departures from independence are more easily explained if we assume either (i) that Luke used Matthew as well as Mark or (ii) that Matthew used Luke as well as Mark. 8

9. Construct a word by word transcription of Farmer's colour-coded Greek text of Mark into a bivariate binary time series of length 11078, which is the number of words in the text that was used by Farmer. Verbal agreements are coded 1 and non-agreements 0. The subscript t of the time series refers to the position of the word in the text of Mark. The first component X t of the bivariate time series is constructed by writing 1 if the word in position t is present unchanged in Matthew and 0 otherwise. The second component Y t is constructed by writing 1 if the word is present unchanged in Luke and 0 otherwise. A covariate binary time series Z t is constructed by writing 1 if the word is part of the direct speech of Jesus or John or the divine voice and 0 otherwise. 9

10. chapterversewordtXtXt YtYt ZtZt 14110629000 14111630111 14112631111 1421632110 1422633000 1423634010 1424635010 1425636110 1426637110 1427638110 1428639000 1429640100 1431641000 1432642000 10