I believe that if there had been any kind of missile launch at all, it would have illuminated the sky, and would have been especially visible to aircraft flying above. The attention of the pilots in question would have been immediately diverted toward the source of such an event.
I think a more likely scenario is that the aircraft was leaking fuel, which ignited and left a visible trail which eyewitnesses mistook for a missile or other object streaking up toward the airplane.
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I believe TWA 800 was brought down by a terrorist bomb, either brought on board by a suicide passenger or planted there at the airport. I also believe it was very deliberately covered up by the Clinton Administration which did not want to alarm the public in an election year or invite scrutiny of its previous coverup, namely the Oklahoma City bombing. And guess who was behind both of these incidents? Saddam Hussein. Now do you want to guess again what Sandy Bergeer stuffed into his pants?
The streak must have been descending and most likely was fire in the falling wreckage.
No group ever claimed credit for a bombing, so far as I know.