
Simple Linear Regression
Hey,
Im having trouble with a question finding an estimate for $\displaystyle \beta_1$ as well as showing its unbiased and finding its variance. There is a set of data with the points
$\displaystyle (1,Y_1) , (2,Y_2) , (3,Y_3) , (4,Y_4), (5,Y_5) $
We want to fit a simple linear regression of the model:
$\displaystyle Y_i = \beta_0 + \beta_1i + \epsilon_i $
Where: $\displaystyle \epsilon_i ~ N(0,\sigma^2) $
The question asks me to find an estimate for $\displaystyle \beta_1$ by joining each set of points and writing the slope of each line formed. Thus, using the average of all the individual slopes to calculate $\displaystyle \beta_1$
$\displaystyle \hat\beta_1 = \frac{\sum\gamma_i}{4}$
Where: $\displaystyle \gamma_i$ is the slope between the points $\displaystyle (i, Y_i) \mbox{and} (i+1,Y_{i+1})$
My attempt:
The slope of any $\displaystyle \gamma_i $ is:
$\displaystyle \gamma_i = \frac{Y_{i+1}  Y_i}{i+1i} = Y_{i+1}  Y_i $
Thus:
$\displaystyle \sum\gamma_i = (Y_2  Y_1) + (Y_3  Y_2) + (Y_4  Y_3) + (Y_5  Y_4) = (Y_5  Y_1)
$
Which leads me to the estimate $\displaystyle \hat\beta_1 = \frac{Y_5  Y_1}{4}$
Can someone tell me if I am on the right track, and how can I determine the bias of this estimator? Aren't the values just constant...? Thanks