F statistic

[brian.field]

I feel terrible asking what I assume is an elementary question, but alas, this is why I am here.

David - I do not believe that you address the F statistic's representation as ESS/k / RSS/n-k-1 explicitly anywhere in the S and W videos. Is there some reason this was not covered? Is it equivalent to the expression utilizing the t^2's that you did cover in the lecture? Is the ESS version a heteroskedastic version, i.e., a robust version? Is it always appropriate? Etc.

Thanks!

Brian

 

[David Harper CFA FRM]

Hi Brian,

It's a smart question

The reason I didn't include F = (ESS/df)/(RSS/df) is that I don't think S&W show it. Inexplicably, as that was the more familiar (and intuitive) formula before S&W replaced previous, better econometric readings (I think S&W on F-stat is *weak* and confusing)).

F = (ESS/df)/(RSS/df) is the F stat for the so-called "overall" regression F-stat; i.e., the test of the joint null that all regressors (independent variables) are equal to zero. This is the basic F-stat. We want to note that this is a special case of a restricted regression: the test of joint null that all independents = 0 is equivalent to restricting all of the regression coefficients (i.e., q = number of independent variables). Again, the (common) overall regression F-stat is a special case of a restricted regression where restrictions (q) is set equal to number of independents (which is equal to ESS df).

So this overall F-stat is a special case of S&W's homoskedastic F-stat given by 7.14:
F = [(R^2 - R^2 restricted)/q] / [(1-R^2)/(n - k unrestricted - 1), but the special case where q = k = the number of regressors in the unrestricted regression), such that:
F = [(R^2 - 0)/k] / [(1-R^2)/(n - k - 1)] = [(R^2/k)]/[(1-R^2)/(n-k-1)].

So, as far as I am concerned, there is one general F-stat and the difference is the number of restrictions. I'm not aware that the FRM has ever gone beyond the "overall" regression F-stat. (given this, the t^2 should only be equivalent when the unrestricted regression happens to have two independent variables: in which case, the overall F-stat is a test of the joint null that two regressors are zero). I hope that explains.

 

[brian.field]

Ahhh....that makes sense. I appreciate the prompt (and thorough) response. Thanks again.

Brian