Regression Degrees of Freedom

CFA Level II Quant
#1

My question comes from page 294 in the CFA II quant book, reading 9, example 17 (Performance Evaluation: The Dreyfus Appreciation Fund)

If degrees of freedom = number of observations minus estimated observations, in this example 60

And residual degrees of freedom = 58 because you are estimating the slope coefficient and intercept coefficient

Why does the regression degrees of freedom = 1? This suggests that every regression coefficient and independent variable is an estimate except 1 of them. Which one is it? Furthermore, it seems that the degrees of freedom for the regression is the number of independent variables, which is a different logic than that for the residual degrees of freedom. Can anybody clarify this for me? Thank you.

#2

If degrees of freedom = number of observations minus estimated observations, in this example 60

This is wrong

DF = number of observations - number of estimated slopes… (not estimated observations)

Why does the regression degrees of freedom = 1?

The DF of the regression is n - k, where k is the number of slopes estimated. If you have 1 independent variable only you would have in this example 59 DF for the regression (Mistaken) and would be the b1 coefficient (the only slope in simple linear regression). The b0 coefficient is not a slope, it is an intercept !

Regards

#3

I believe that you’re mistaken, Harrogath.

There are three degrees-of-freedom numbers associated with regression. If you have n observations and k independent variables, then the:

  • Regression degrees of freedom = k
  • Residual degrees of freedom = n − (k + 1)
  • Total degrees of freedom = n − 1
#4

Yes, you right, my bad. Sorry!

#5

S2000magician, would you kindly explain why the Regression degrees of freedom = k? Thank you.

#6

Because you have k independent variables, you have k (slope) coefficients that you estimate from the data; those k coefficients represent the k degrees of freedom for the regression: the coefficients are free to take on any values whatsoever.

The intercept is considered separately from the slopes: note that for the residual degrees of freedom you lose the k degrees of freedom for the regression (the slopes) and one more degree of freedom for the intercept.

Any deeper explanation would have to get into the algebra of vector spaces and affine spaces; I hope that it’s sufficient to leave it at this.