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

[nosit]

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.

 

[Deleted member 4993]

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.

You are "trying to understand formulas more in depth instead of just knowing them by heart" - excellent endeavor.

You say:

B1 = Σ (Xi - Xbar) (Yi-Ybar)/Σ (Xi - Xbar)^2

Where did you get that equation? Did you go through the derivation of the equation?

 

[nosit]

Actually I saw in this video:

So it triggered me doubt as it ended up showing SC(x,y)/SS(x)

 

[nosit]

Some people told me that I need to stick a "n" in denominator and numerator if I understood well. But I am not sure when should I put this n's. Does anyone know?