"Arnold raced out of the door": grammatical or not?

Solution 1:

Consider the following three sentences.

1. He raced out the door.
2. He raced out of the doorway.
3. He raced out of the door.

In the first sentence out is a preposition meaning 'through to the outside'. This is entirely unproblematic: he was in a room or building, and he raced out of it through the door(way).

In formal English the second sentence is equally clear: out is an adverb meaning 'moving or appearing to move away from a particular place, especially one that is enclosed', and the sentence means that he raced out of the doorway in which he had been standing. I shouldn’t be at all surprised, however, to find that some people use it synonymously with (1).

Since 'doorway' is one perfectly standard meaning of door, and since it is more than a little difficult to stand in a physical door, (3) is formally synonymous with (2). However, many people, evidently including at least one writer for Murder, She Wrote, use it synonymously with (1). For those people out of has in effect become a compound preposition meaning 'through to the outside' as well as the combination of adverb and preposition found in the formal interpretation of (2). I suspect that this, like in back of for behind, is more common in the US than in the UK. In the US, at least, it’s common enough to qualify as normal English, though there are also speakers like me who would never use it because it’s ungrammatical in their idiolects.

Solution 2:

I think this is a job for Google Ngrams.

Google Ngrams shows that in 1900, out of the door was the expression both in the U.S. and the U.K. But the majority of Americans switched from using out of the door to out the door around 1940, and now out of the door is quite rare. This same switch wasn't made in the U.K. until 1990. So I would guess out of the door sounds funny to many Americans, but both expressions sound perfectly fine to U.K. speakers (except maybe for traditionalists trying futilely to stop the language from evolving).

I believe the choice between these expressions is idiomatic, and trying to analyze which to use with pure reason is not going to yield worthwhile results.

Solution 3:

The NOAD has the following notes about using out of, or out:

The use of out as a preposition (rather than the standard prepositional phrase out of), as in "he threw it out the window," is common in informal contexts, and is standard in American, Australian, and New Zealand English. Traditionalists do not accept it as part of standard British English, however.

The phrase, whatever it uses out of or out, is grammatically correct; in both the cases, the phrase is standard in American English, but the standard British English phrase is the one using out of.