Which expression is correct: "he ran out the house" or "he ran out of the house"?
"He ran out of the house" is correct.
In the sentence "He ran out the house", 'out' is an adverb modifying the verb 'ran'. The subject 'he' carried out an action 'ran' ON the object 'house'. In effect, you are saying "he ran the house" and you are just modifying how he ran it.
By using 'out of' it becomes a preposition and links the action of the verb 'ran' in space and time, to the object 'house'.