Irish as Patty's pig

Where does the term "Irish as Patty's pig derive? Why is the pig referred to as Irish?

I am guessing that it is "Paddy's pig."

"Paddy is a slang term in British English for an Irish person. It is a diminutive of the Irish forename Padraig, meaning 'Patrick,' and although it is also in use as an ordinary forename its use as slang is now considered offensive or contemptuous by many people."