"Oh, well -- I was at Hogwarts meself but I -- er -- got expelled, ter tell yeh the truth. In me third year. They snapped me wand in half an' everything. But Dumbledore let me stay on as gamekeeper. Great man, Dumbledore."
-- Hagrid (PS4)
Hagrid carries a pink umbrella which conceals his broken wand. which was snapped in half when he was expelled from Hogwarts in june, 1943. The wand/umbrella seems to work just fine. He used it to start a fire, put a pig-tail jinx on Dudley (with a “flash of violet light” and a “sound like a firecracker”), and to speed up a Muggle rowboat (PS4, PS5).
Hagrid's wand was "Oak, sixteen inches, rather bendy," and he swore to Mr. Ollivander that although he still had the pieces he never used them, but Harry noticed he clutched the pink umbrella as he said this (PS5).
From the Web
A very interesting theory about the pink umbrella from the SuperCarlinBrothers (YouTube video)