The rumour was cultivated by the Beatles or their label
The Beatles themselves, including Paul, their record label, producer and manager started the rumour.
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The Argument
There was a clear financial motive for the record label to start or encourage the rumour, which led to massive sales of all the Beatles' albums. In November 1969, Capitol Records sales managers reported a significant increase in sales of Beatles catalogue albums, attributed to the rumour stating "this is going to be the biggest month in history in terms of Beatles sales." American record sales were boosted - Abbey Road comfortably outsold all the band's previous albums and both Sergeant Pepper and The Magical Mystery Tour, both of which had been out of the charts, re-entered the Billboard LP chart.
There is more speculation about the role of The Beatles themselves, including Paul, due to their creativity and sense of humour. Hints include a song called "Saint Paul" released by Terry Knight, that seemed to hint at Paul being dead, and was published by MacLen Music - McCartney and Lennon's publishing company.
John Lennon later admitted that he added cryptic lyrics such as "the Walrus was Paul" and backmasking section of "Revolution 9" so a reversed section seemed to say "turn me on, dead man" to boost the rumour. The WKNR radio station reported that the rumour was started by Lennon, who said in a radio interview, that although "insane" it was good publicity for Abbey Road. Lennon’s bitterness over the Beatles break up was reflected in the lyrics to his 1971 song "How do you sleep?, saying “those freaks was right when they said you was dead.”
Other clues continued to be dropped in the 1970’s to maintain public interest in the rumour including Ringo’s solo song 'I'm the Greatest,' written by Lennon features the lyric "Yes I'm Billy Shears..." and in the 1978 Sergeant Pepper movie, Peter Frampton plays a character called Billy Shears.
Counter arguments
Capitol Records have specifically denied having started the rumour. In October 1969 Apple Records refuted the rumours, releasing a statement from Paul saying "I am alive and well."
Premises
[P1] The Beatles, their management and the record label had financial incentives to start the rumour.
[P2] The rumour did boost the sales of their final two albums.
[P3] The Beatles and those around them started the rumour.
Rejecting the premises
[Rejecting P3] The record label has publicly denied the rumours.
Further Reading
Schaffner, Nicholas (1978). The Beatles Forever. New York, NY: McGraw-Hill. ISBN 0-07-055087-5