Linear Regression: Expected Value & Variance of Predicted Values

[twoflower]

Homework Statement

Consider model of linear regression:

<br /> Y_i = \beta_0 + x_i \beta_1 + \epsilon_i<br />

i = 1, ..., 5, where \epsilon_i \sim \mathcal{N}(0, \sigma^2) are independent. Find expected value and variance of predicted values \widehat{Y}_i considering that observations are obtained in points 1, 2, 3, 4, 5 (ie. x_i = i for i = 1, ..., 5) and \sigma^2 = 1. Hint: remember that

<br /> \widehat{Y} = HY<br />

Homework Equations

<br /> H = X\left(X^T X\right)^{-1}X^T Y<br />

The Attempt at a Solution

My attempt is

<br /> E \widehat{Y} = \beta_0 + X\beta_1 = (\beta_0 + \beta_1, \beta_0 + 2\beta_1, \beta_0 + 3\beta_1, \beta_0 + 4\beta_1, \beta_0 + 5\beta_1)<br />

Is it correct?

Anyway, even if it is, how do I find the variance and how do I use the hint? :)

Thank you.

 

[statdad]

"Find expected value and variance of predicted values"

Since you know that

<br /> \widehat Y = X \left(X&#039; X)^{-1}X&#039; Y<br />

you can find

<br /> E[\widehat Y] = E[X \left(X&#039; X\right)^{-1} X&#039; Y]<br />

Use the properties of expected value and the expected value of Y (unless I'm totally missing something, I don't see how the form of the x_i applies here).

As far as finding the variance of \widehat Y, you can use the hint. Write out the
matrix X (first column consists of ones, for the intercept, and you know the values of x to use in the second column), and use the matrix formulas for the covariance matrix in regression to find the variances. Because the x values are consecutive integers, a little algebra in the matrix multiplication will give nice forms for the entries.