Factual error: [Special Edition] The Statue of Liberty is incorrectly shown to be facing the Narrows bridge.
Revealing mistake: When the crane falls down in front of the rig and then over the cliff into the abyss it's as if the rig was already just a few yards away from the abyss but when the umbilical pulls the rig again (after Bud says to grab onto something) suddenly there's enough space for the rig to be dragged at least a few hundred yards as if the rig never moved from its touchdown position earlier in the film.
Revealing mistake: Every time the aliens appear anything electrical shuts down but somehow Lindsay's camera is able to take pictures of the small NTI before the electricity powers back on.
Answer: Most of it was television news coverage about the increasing tensions between the US and USSR as it was playing out earlier in the movie. Additional archival footage from previous years' wars implied that the NTIs had been on Earth for quite a long time.