Var(Y_knot_Hat)

#1
Hello all,

I need a little help with this:

Show Var(Y_knot_Hat) = Sigma^2 * (1/n + (x_knot - x_bar)^2 / Sxx)

Start with Y_knot_hat = b_knot + b_1*x_knot

thanks in advance!
 

Dragan

Super Moderator
#4
Hello all,

I need a little help with this:

Show Var(Y_knot_Hat) = Sigma^2 * (1/n + (x_knot - x_bar)^2 / Sxx)

Start with Y_knot_hat = b_knot + b_1*x_knot

thanks in advance!

See J. Johnston, Econometric Methods (3rd Edition) pp. 42-45 for a proof of what you are trying to do.
 
#5
I have no idea how I can get that book.

It was published in '84, and there is a 4th edition out, but I still don't have access to it.

is it a long proof, could you maybe just help me out with the covariance issue of b knot and b1?

thanks again.
 

Dragan

Super Moderator
#6
.........could you maybe just help me out with the covariance issue of b knot and b1?

thanks again.
Briefly:

cov[b0,b1] = E[(b0 - E[b0])(b1 - E[b1])]

= E[(b0 - Beta0)(b1 - Beta1)]

= -Xbar*E[b1 - Beta1]^2

= -Xbar*Var[b1]

= -Xbar*(Sigma^2 / SSx),

where I am making use of b0 = Ybar -b1*Xbar and E[b0] = Ybar - Beta1*Xbar, giving (b0 - E[b0]) = -Xbar*(b1 - Beta1).