BLUE (best linear unbiased estimator) in practice

[Charlotte87]

Homework Statement

Let the linear model be $Y_{i}=\alpha + X_{i}\beta + \varepsilon$. Let the assumptions of the linear model hold. Suppose that the fixed values of X in a data are as follows: $X_{1} - 1, X_{2} - 2, X_{3} - 3, X_{4} - 4$. An econometrician proposes the following estimator to estimate the slope of the linear relation between Y and X:

$\beta^{*}=\frac{Y_{4}+2Y_{3} - 2Y_{2} - Y_{1}}{5}$

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

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 $Y_{1} = \alpha + X_{1}\beta + \varepsilon$ 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...
$var(\beta^{*}) = \frac{1}{5}[Var(Y_{4}) + 2^{2}Var(Y_{3}) - 2^{2}Var(Y_{2}) - Var(Y_{1}) = \frac{1}{5}[\sigma^{2} + 4\sigma^{2} - 4\sigma^{2} - \sigma^{2}$

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:
$\hat{\beta} = \frac{\sum(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum(x_{i}-\bar{x})^{2}}$

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

Hence the OLS estimator becomes:

$\hat{\beta} = \frac{1.5Y_{4} + 0.5Y_{3} - 0.5 Y_{2} - 1.5 Y_{1}}{5}$

Does this sounds right?

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

 

[lippyka]

I have been trying to use the same formula as in ii) but I do not think that is right. iv) I don't know.