The most common and most likely answer to this crossword clue is the 14 letter word PIG-HEADEDNESS
I've seen this crossword clue in The Guardian
P
I
G
-
H
E
A
D
E
D
N
E
S
S
Buddy explains!
The straight part
We know that "obstinacy" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case obstinacy = pig-headedness.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.