Maybe it's too much too soon for him? If he'd cut his managerial teeth on a smaller club he could have gained a lot of experience before he felt confident enough to move up a notch. Because to me he looks out of his depth at times, like too much is going on at once and he's getting stressed with all the pressure. A lot of his decision making is irrational at best, and that makes me think just what is his plan, what is his thinking - basically, trying to second guess the man is futile because he doesn't seem to know himself what he's thinking.
If they got rid of him now, or perhaps a bit further into the season, he could take a break from the pressure and relax, keep away from the stresses of Prem management, and after a break come back and start again at a mid-table club.
Arsenal would benefit by getting in a more level-headed and experienced manager/coach, and perhaps just aim for a European place or a trophy for a while before aiming for the big prize with all the other juggernaut clubs.
I'm not in any way trying to disrespect the club here, but I do genuinely feel Arteta is not cut out for pressure-cooker management just yet.