It is ungrammatical; how can you have something you do not have? I do not (don't) have a dog is correct; you do not have what you do not have.
Oh no...i always thought it was a correct way to say that!!
By the way...I have a related question. The differences between
I'm running out of sth...I have nearly finished something [flour/milk/butter/patience]
I'm out of sth... I have completely used up something and have none left
I'm without sth... I do not have something
are they interchangeable?
You may say either. The first is more British in usage, the second is more American. Both are correct. You could say either anytime and anywhere.