No.
I took a look at the Economist style guide capitals section here and found no mention of "its."
A search of the Economist website turned up articles with the following titles (bold for emphasis):
Multinationals: China loses its allure
Kenya and its Somalis: Go back home!
Turkey and its neighbours: A reset?
So although the style guide doesn't specifically address the possessive pronoun "its," they consistently leave it lower-case.
