In the Old Testament, the original Hebrew refers to a male goat as a hairy goat (as opposed to the female goat). The scape goat onto whom the sins of the whole people were put on the Day of Atonement was always a male goat. It was driven out into the desert to die. Thus, I opine, that to run around like a hairy goat is to run around like the scapegoat in the desert, trying not to be killed by wild beasts.