Linear Regression: Expected Value & Variance of Predicted Values

[twoflower]

Homework Statement

Consider model of linear regression:

$$Y_i = \beta_0 + x_i \beta_1 + \epsilon_i$$

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

$$\widehat{Y} = HY$$

Homework Equations

$$H = X\left(X^T X\right)^{-1}X^T Y$$

The Attempt at a Solution

My attempt is

$$E \widehat{Y} = \beta_0 + X\beta_1 = (\beta_0 + \beta_1, \beta_0 + 2\beta_1, \beta_0 + 3\beta_1, \beta_0 + 4\beta_1, \beta_0 + 5\beta_1)$$

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

$$\widehat Y = X \left(X' X)^{-1}X' Y$$

you can find

$$E[\widehat Y] = E[X \left(X' X\right)^{-1} X' Y]$$

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.