This is a very tricky situation because a lot of people do not understand what is the difference in 4:3 vs. 16:9.
There are standard 4:3, 4:3 letterboxed, & 4:3 anamorphic
These are all different ways of displays, and most of the time, the problem lies in the way you display your output.
A true 16:9 footage will display in a 4:3 aspect in a squished mode (4:3 anamorphic).
So technically, when you master it out to a 4:3 aspect, it should be a squished picture.
If you master it out in a 16:9 mode, it will look normal, but with letterboxed on a 4:3 display.
Aspect conversion should be done in a proper way with decidated video processors, but the current trend is to do it the cheap and easy way by rescale. There are proper calculations involved in doing a rescale.
When you convert 16:9 to 4:3 picture, technically u lose picture content & resolution.
When you convert 4:3 to 16:9, you lose horizontal resolution.