We often see statements like “linear regression makes the assumption that the data is normally distributed”, “Data has no or little multicollinearity”, or other such blunders (you know who you are..).
Let’s set the whole thing straight.
Linear regression assumes nothing about your data
It has to be said. Linear regression does not even assume linearity for that matter, I argue. It is simply an estimator, a function. We don’t need to ask anything from a function.
Consider that linear regression has an additional somewhat esoteric, geometric interpretation. When we perform a linear regression you simply find the best possible, closest possible, linear projection we can. A linear combination in your X space that is as close as possible in a Euclidean sense (squared distance) to some other vector y.
That is IT! a simple geometric relation. No assumptions needed whatsoever.
You don’t ask anything from the average when you use it as an estimate for the mean do you? So why do that when you use regression? We only need to ask more if we do something more.