# Gauss-Markov assumptions for Multiple linear regression

Are the Gauss-Markov assumptions the same for Simple Linear Regression and Multiple Linear Regression? I cant seem to find the answer for this and my literature seem to suggest that they have different formulas.

(Literature: An introduction to Econometrics - James H. Stock, Mark W. Watson.)

These are the Gauss-Markov assumptions used in the Simple linear regression chapter:

According to My book, these below here are the Gauss Markov assumptions for Multiple Linear Regression, and you can note that the second assumption is written in matrix form.

• Hi: ii) in the matrix case covers 2 and 3 in the OLS case. iii) in the matrix case really doesn't apply to the OLS case because it's unlikely that that you wouldn't have full column rank in the OLS case because there are only 2 columns and the first is a column of 1's. So, in short, yes, the assumptions are the same. Oct 23, 2020 at 14:46
• @mlofton thanks for the clarification. As i understand it now, the Gauss Markov assumption below are only valid for multiple linear regression with Matrix cases, and otherwise the standard Gauss Markov assumption (the uppert most) are valid whether its linear or multiple regression. Oct 23, 2020 at 15:08
• Hi: I'm sorry that I wasn't clear. What I should have said is that the Gauss Markov assumption has to hold whether it's OLS or multiple regression. I mistakenly answered by explaining why they look slightly different in what you described. Basically, think of it as one set of assumptions that always need to hold and think of OLS as just a special case of multiple regression. But the point is that the assumptions are needed in OLS also. Oct 24, 2020 at 12:55

Simple linear regression is a special case of multiple linear regression that only has one feature ($$x$$ variable). Consequently, any theorem that applies to multiple linear regression must apply to simple linear regression, so, yes, the Gauss-Markov assumptions are the same.
It then becomes an issue of how to translate the multiple linear regression notation into simple linear regression. The answer is that your model matrix just has a column of $$1$$s for the intercept and then one column for your lone feature. Depending on how you want to express the assumptions, you might want to write the matrix as $$n\times 2$$, or you might just want to say that you have feature $$X_1$$ and that’s all.