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BLUE (best linear unbiased estimator) in practice

  1. Sep 20, 2013 #1
    1. The problem statement, all variables and given/known data
    Let the linear model be [itex]Y_{i}=\alpha + X_{i}\beta + \varepsilon [/itex]. Let the assumptions of the linear model hold. Suppose that the fixed values of X in a data are as follows: [itex] X_{1} - 1, X_{2} - 2, X_{3} - 3, X_{4} - 4 [/itex]. An econometrician proposes the following estimator to estimate the slope of the linear relation between Y and X:

    [itex]\beta^{*}=\frac{Y_{4}+2Y_{3} - 2Y_{2} - Y_{1}}{5} [/itex]

    i) Is this estimator unbiased?
    ii) Derive the sampling variance of this estimator
    iii) Derive the least square estimator of β and its sampling variance
    iv) Compare the sampling variance of β* with the sampling variance of the least square estimator

    3. The attempt at a solution
    i) So far I have shown that β* is unbiased by plugging in for Y(i) (In the sense that [itex] Y_{1} = \alpha + X_{1}\beta + \varepsilon [/itex] and so on, and then plugged in the values for X. I then get β*=β, hence I guess that E(β*)=E(β)=β. Thus, the estimator is unbiased.

    ii) Here I have tried to set up something like this, but I do not think its right...
    [itex] var(\beta^{*}) = \frac{1}{5}[Var(Y_{4}) + 2^{2}Var(Y_{3}) - 2^{2}Var(Y_{2}) - Var(Y_{1}) = \frac{1}{5}[\sigma^{2} + 4\sigma^{2} - 4\sigma^{2} - \sigma^{2} [/itex]

    I think then I used the assumption that the variance is always constant in a linear regression model..

    iii) Here I first wanted to plug in the data into the general OLS-formula for the estimator. However, I believe I do it wrongly here as well... Using this formula:
    [itex] \hat{\beta} = \frac{\sum(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum(x_{i}-\bar{x})^{2}} [/itex]

    Calculating the variance first I find (the mean of the x's is 2.5):
    [itex] Var(x) = (1-2.5)^{2} + (2-2.5)^{2} + (3-2.5)^{2} + (4-2.5)^{2} = 2.25 +0.25 + 0.25 + 2.25 =5 [/itex]
    So far so good. Then I calculate the following:
    [itex] \sum(x_{i} - \bar{x})(y_{i}-\bar{y}) = (1-2.5)(Y_{1} - \bar{Y}) + (2-2.5)(Y_{2}-\bar{Y}) + (3-2.5)(Y_{3} - \bar{Y}) + (4-2.5)(Y_{t} - \bar{Y}) = 1.5Y_{4} + 0.5Y_{3} - 0.5 Y_{2} - 1.5 Y_{1} [/itex].

    Hence the OLS estimator becomes:

    [itex] \hat{\beta} = \frac{1.5Y_{4} + 0.5Y_{3} - 0.5 Y_{2} - 1.5 Y_{1}}{5} [/itex]

    Does this sounds right?

    I am still unsure here about how to calculate the sample variance.
     
  2. jcsd
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