Beta - Cov(x,y)/Var(x) or SC(x,y)/SS(x)

nosit

New Member
#1
Hello everyone,

Recently I am trying to understand formulas more in depth instead of just knowing them by heart.

When deriving the least squares estimators, which is Y=B0 +B1X, the B1 equals:

Σ (Xi - Xbar) (Yi-Ybar)/Σ (Xi - Xbar)^2

The thing is that this is apparently not Cov(x,y)/Var(x) because numerator and denominator are not being divided by "n-1" or simply by "n".

Therefore, why it is said Beta equals Cov(x,y)/Var(x) and not Sum of Cross products / Sum of Squares, aka SC(x,y)/SS(x)?

Thank you in advance.
 
#5
Thank you for the reply.
But when I start solving for B1, it starts in this way:

/B1 Σ (Yi - (B0 + B1.Xi))=0

So there is no "n" there, could you please explain me how do I add an "n" in a way that it makes sense?
 

Dason

Ambassador to the humans
#6
It's easier to add it at the end. Multiply by 1 which doesn't change anything so it's perfectly acceptable. Keep in mind that n/n = 1
 

katxt

Active Member
#7
Start with Σ (Xi - Xbar) (Yi-Ybar)/Σ (Xi - Xbar)^2 written as a fraction
Divide top and bottom by n,
It turns into Cov(x,y)/Var(x)