If there is no ice or snow on the roof, then it is probably caused by condensation forming in the attic.
Heat from inside the house works it's way into the attic and the extream cold air outside causes the moisture in the warm air to form conensation on the underside of the roof.
Condensation runs down the roof rafters and drops near the outside walls.
If you have a A-shaped roof, there should be signs of the water in the front and back walls.
There are no signs of water on the third floor ceiling?
It sounds like the water is running down the inside of the walls and when it hits the first floor it's being diverted or blocked and is traveling across the ceiling and escaping through the lowest part, the light fixtures. This is not a safe situation and you should not have any power going to the fixtures. (it could cause a fire!)
I think you need to gain access to this attic area and see how you can better vent it and also see if there is any insulation on the attic floor.
If you had snow or ice on the roof, I would think that you might have ice dams.
Ice dams are caused when the snow on the roof melts under the snow because of the heat in the house. The water runs under the snow and hits the ice that has formed at the edge of the roof and then the water backs up and goes under the shingles and gets in through the seams in the underlayment.