Simple Linear Regression

Mar 2008

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:
\(\displaystyle \gamma_i = \frac{Y_{i+1} - Y_i}{i+1-i} = Y_{i+1} - Y_i \)


\(\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