The most common and most likely answer to this crossword clue is the 12 letter word STUBBORNNESS
I've seen this crossword clue in The Times
S
T
U
B
B
O
R
N
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 = stubbornness.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.
Crossword Answer definitions
STUBBORNNESS noun- The state of being stubborn.