# Regression coefficients without intercept [duplicate]

Could someone recommend a link or help me out here: where can I find the formula for the regression without an intercept, and how is it deriveed differently than the formula with the intercept? (matrix form)

## 1 Answer

It's the same formula you use in regression with an intercept: $$\hat{\beta}=\left(X^TX\right)^{-1}X^Ty$$. Just make sure there isn't a column of ones for the intercept. It's the same derivation as the formula with an intercept.

Here's a link to some notes that derive regression coefficients: Statistics 512: Applied Linear Models - Chapter 5: Linear Regression in Matrix Form [PDF].

• Thank you. If you don't mind me asking another question - if I'm looking to compare variances of the same regression with and without an intercept by looking at their ratio, is there a difference in formulas?
– user14386
Oct 28, 2012 at 20:59
• Yup. The model without an intercept is actually a special case (nested) of the model with an intercept. Without the intercept, we are actually assuming the intercept is zero. Oct 28, 2012 at 21:03
• But if I need to take a ratio of variances, doesn't that mean their formula will just be the same?
– user14386
Oct 28, 2012 at 21:11
• The residual sums of squares will not be the same unless the regression model with an intercept fits the intercept to zero. Oct 28, 2012 at 21:20
• I apologize. I think I read your initial comment incorrectly. The standard F Test for choosing between two models, one of which is nested in the other, is valid. It's no different because you aren't including an intercept. Oct 28, 2012 at 21:22