"Yours" is usually a possessive pronoun with an implicit noun. What is the implicit noun in the case of "yours sincerely"?
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The "yours" in "Yours sincerely" is a possessive pronoun. It is a shortening of the phrase "I am yours sincerely" so the implicit noun is "I," who is presumably the writer of the letter.
Now you are asking "I am your what sincerely," and here is an interesting article from 1900 that would suggest "friend," although given the traditional valediction of "Your humble and obedient servant," I think "servant" would suffice as well.
What I think is particularly interesting is that the noun has been omitted, perhaps specifically to make this closing ambiguous, while simultaneously emphasizing that whatever I am with regard to you, it is true and without fault.
"Yours sincerely" is a valediction, and the "yours" is a shortening of "your servant"
Edit: Converting the link in my comment:
"Yours" doesn't just stand for "your servant". It can for anything from "your friend" to "your benefactor". "Yours" is just a shortened form for any of these.
I thought to quote these 2 posts from Quora, which enlarges on the existing answers:
Concerning the grammar, User 'Mike Mendis' 's answer: