Single Index Model - Least Squares Regression

Hi all,

I am slightly confused with a section of the ActEd notes, in chapter 6, on page 10, that has stumped me for some time now.

I am referring to the derivation of the Beta_i (hat) and Alpha_i (hat) coefficients of the single-index model.

From CT3, one would expect the formula for alpha_i (hat), to be:

alpha_i (hat) = E[R_i] - Beta_i (hat) * E[R_M]

However, in this case, for some reason we have

alpha_i (hat) = E[E_it] - Beta_i (hat) * E[E_Mt]

where E_it = E[R_it], etc.

I really do not understand how we get to this expression?

I guess it is a question of why the expected return on security i is equal to the average of all the expected returns of the security i at each time t.

Any help would be greatly appreciated.

Thanks.

 

Hi,

Sorry for the confusion. What you've highlighted is indeed an error, which appears to have been sitting unnoticed in the Course Notes for many years.

As you correctly suggest, the equation for estimating beta is correct, whilst the equation for estimating alpha should be:

alpha_i (hat) = E[R_i] - Beta_i (hat) * E[R_M]

ie we should taking means of the actual security returns and market returns, and we should not be taking means of means.

Apologies for this error, which I'll get corrected in the Course Notes, and thanks for drawing this to our attention.

Graham

 

Graham,

Thanks very much for the reply, much appreciated.

However, I still don't agree that beta_i is correct in the course notes either, or perhaps it is just a notation issue.

Essentially, in the formula for beta_i in the course notes, I think what is written as E_it should be the average of R_it over all times, and hence have no dependence on t (again, including E_it in the expression for beta_i, that sums over t, is taking an average of an average).

Please let me know if you disagree with this.

Thanks again.

 

Hi again,

Yes you're right. "E" is the mean of the "Rt" and as such doesn't depend on t. So, it should just be Ei and EM in the beta formula.

I'm currently updating the Course Notes for the next session and will be amending the beta equation by replacing the "E"'s by "R bars" (for the mean of R), which should include t subscripts.

I'll also add the corrected equations for alpha and beta in the CT8 Corrections document on the ActEd website.

Thanks for taking the time to highlight this.

Graham

 

Sorry, but in my post just now I missed out a "not". The middle sentence should say:

I'm currently updating the Course Notes for the next session and will be amending the beta equation by replacing the "E"'s by "R bars" (for the mean of R), which should not include t subscripts.

Anyway, I've attached the amended page showing what the equations should look like.

Graham

 

Thanks Graham, much appreciated.
The attached agrees exactly with what I had written in 'my' version of the notes.
Best,
Luke