# Linear regression and assumptions about response variable

Wikipedia states:

Ordinary linear regression predicts the expected value of a given unknown quantity (the response variable, a random variable) as a linear combination of a set of observed values (predictors). This implies that a constant change in a predictor leads to a constant change in the response variable (i.e. a linear-response model). This is appropriate when the response variable has a normal distribution.

So, Wikipedia makes an assumption about the response variable, namely that it is normally distributed. However, in other sources and here in stack exchange the normality is required for the error terms. If they are not normally distributed we should go for some generalized linear model.

What is the Wikipedia article referring to or is it wrong?

The Wikipedia statement

This is appropriate when the response variable has a normal distribution.

is wrong.

OLS does NOT have assumptions on response variable. But has assumptions on residual (See Gauss–Markov theorem). Also see this post for details.

Why linear regression has assumption on residual but generalized linear model has assumptions on response?

I am stealing @Cliff AB 's example here. The following distribution on $y$ and residual does not violate OLS assumption! Related posts:

What is a complete list of the usual assumptions for linear regression?

How does linear regression use the normal distribution?

What if residuals are normally distributed, but y is not?

• Do you mean OLS makes assumptions on the unobserved errors instead of residuals? Jul 8 '20 at 3:42
• It makes sense that the assumption is on the residuals, not the response variable, but I can't find the mention of this assumption anywhere (in fact, the Wikipedia article states explicitly that the "errors do not need to be normal")... I'm really stumped. Jan 27 at 20:25
• Actually, from what I gather, the errors being normally distributed also imply the response variables being normally distributed, conditional on the predictors (but not centrally normally distributed). This is the benefit of generalized linear models, which allow a conditional distribution which is not normal. However, the response variables themselves (not conditioned on the predictors), aren't normally distributed. Jan 27 at 20:52