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Linear regression

  1. Sep 17, 2008 #1
    1. The problem statement, all variables and given/known data

    Consider model of linear regression:

    [tex]
    Y_i = \beta_0 + x_i \beta_1 + \epsilon_i
    [/tex]

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

    [tex]
    \widehat{Y} = HY
    [/tex]

    2. Relevant equations

    [tex]
    H = X\left(X^T X\right)^{-1}X^T Y
    [/tex]


    3. The attempt at a solution

    My attempt is

    [tex]
    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)
    [/tex]

    Is it correct?

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

    Thank you.
     
  2. jcsd
  3. Sep 18, 2008 #2

    statdad

    User Avatar
    Homework Helper

    "Find expected value and variance of predicted values"

    Since you know that

    [tex]
    \widehat Y = X \left(X' X)^{-1}X' Y
    [/tex]

    you can find

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

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

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