# Var(Y_knot_Hat)

#### floriandotti

##### New Member
Hello all,

I need a little help with this:

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

#### floriandotti

##### New Member
I've gotten to

var(y_knot_Hat) = sigma^2 ( 1/n + (x_knot + x_bar^2)/Sxx)

... how does (x_knot + x_bar^2)/Sxx) = (x_knot-x_bar)^2 / Sxx

#### floriandotti

##### New Member
it has something to do with Cov(b_knot,b_1) = -sigma^2 * x_bar / Sxx

#### Dragan

##### Super Moderator
Hello all,

I need a little help with this:

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

See J. Johnston, Econometric Methods (3rd Edition) pp. 42-45 for a proof of what you are trying to do.

#### floriandotti

##### New Member
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
.........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).