Covariance - what is happening here

jc2008

I found an example of a covariance proof in a book, but I don't understand one of the steps.

The model is [latex]E(y_i) = \beta_0 + \beta_1x_i[/latex]
and I want to find the covariance between [latex]y_i[/latex] and [latex]\hat{\beta_1}[/latex]

So the book writes:
[latex]$Cov$(y_i, \hat{\beta_1}) = \frac{1}{Sx}$Cov$(y_i, Sxy)[/latex], pulling the Sx out of the covariance ([latex]\hat{\beta_1}[/latex] = Sxy/Sx
Then [latex]$Cov$(y_i, \hat{\beta_1}) = \frac{1}{Sx}$Cov$(y_i, \displaystyle\sum_{j}y_j(x_j - \bar{x}))[/latex]
Then [latex]$Cov$(y_i, \hat{\beta_1}) = \frac{1}{Sx}$Cov$(y_i, y_i(x_i - \bar{x}))[/latex]

What did they do to get to the last line from the previous line - i.e. how did they get rid of the summation sign?

Thanks,
Jack.

Colours

Hi Jack,

I don't recall the underlying relationships at play between the variables and formulae (having done some study in this area quite a few moons ago), it looks to me like perhaps the reason why the summation sign can be done away with in the last line is because all the terms of the form Cov(y_i,y_j(x_j-xbar) are zero EXCEPT when j=i, that is the summation of all those terms ranging through j can simply be replaced by the only non-zero one which is Cov(y_i,y_i(x_i-xbar).