When the word denotes 'cause' it takes the 3rd case-affix.,
विद्यया यशः 'by learning there is produced fame.' धनेन कुलम् 'by wealth, family;' कन्यया शोकः 'by daughter there is grief.'
The word हेतु here is used in its popular sense and not the grammatical hetu 1.4.55. Any thing capable of accomplishing a desired object is called hetu.,