His enclosure should be big enough for him to let off steam. He should not be doing so outside his enclosure. It's not safe to let him run around outside for the reasons I gave earlier.I typically do leave him in his enclosure during the day and only take him out to feed him or to take him out to blow off some steam and let him run around.
He is never outside, I live in New Jersey USA and its the dead of winter currently. When he is out of his enclosure he never leaves my eyesight, and he is enclosed in a baby pen which I am always watching.