Linear Regression Model with one variable and no constant

#1
Hi, I've been given a problem sheet and I'd like some help as to start answering the questions,

Given the model; Yi=β1Xi + ui

a) show that this model is linear

b) prove β1 is conditionally unbiased

For part a) all I think I should out the question in the form, β1=ΣaiYi, where ai=Xi / ΣXi^2 . Is this all that would be needed?

For part b) I've been unable to show E(β1)=β1, I come to a point that E(β1) has β1 on the right hand side, but I can't see a away to separate them to find the conditions. I've done this by using V(Xi) to find E(Xi)=... and Cov(Xi,Yi) to find E(XiYi) =...

So they could be used as, β1 = ΣXiYi / ΣXi

Any comments on this part b) would be greatly appreciated, I'd include my methodology but I couldn't find a way for the text box to accept my formulas, and I don't have a scanner.

I've been unable to find a thread that goes over the proofs I'm looking for.
 

Dragan

Super Moderator
#2
Hi, I've been given a problem sheet and I'd like some help as to start answering the questions,

Given the model; Yi=β1Xi + ui

a) show that this model is linear

b) prove β1 is conditionally unbiased

For part a) all I think I should out the question in the form, β1=ΣaiYi, where ai=Xi / ΣXi^2 . Is this all that would be needed?

For part b) I've been unable to show E(β1)=β1, I come to a point that E(β1) has β1 on the right hand side, but I can't see a away to separate them to find the conditions. I've done this by using V(Xi) to find E(Xi)=... and Cov(Xi,Yi) to find E(XiYi) =...

So they could be used as, β1 = ΣXiYi / ΣXi

Any comments on this part b) would be greatly appreciated, I'd include my methodology but I couldn't find a way for the text box to accept my formulas, and I don't have a scanner.

I've been unable to find a thread that goes over the proofs I'm looking for.
Hint: In terms of part (b), you need to show that you cannot simultaneously have: Sum[u_i * X_i] = Sum[u_i] = 0.

When you impose the additional condition of Sum[u_i] = 0 then the estimate of β1 (β1_hat) becomes:

β1_hat = Mean value of Y / Mean value of X

which is not an unbiased estimate of β1.