Test if estimator is unbiased

Question

I'm having problems with the following question for my econometrics homework.

Is $\ \ \hat \beta_2 = (y_n - y_1)/(n - 1)\ $ an unbiased estimator of $\beta_2$

for $\ \ y_t\ =\ \beta_1\ +\ \beta_2 \ x_t\ +\ e_t,\ \ t = 1, \dots, n$,

where $x_t = t$ is a trend variable, and $\ (e_1,\dots,e_n)\sim\text{iid}(\text{mean } 0,\text{ variance }\alpha^2)$?

I know that if the estimator is unbiased then $E(\hat\beta_2) = \beta_2$, but I'm not sure how to test what is given.

Answer 1

From the given information, $$\hat \beta_2 = \frac{y_n - y_1}{n-1} = \frac{(\beta_1 + \beta_2 n + e_n)-(\beta_1 + \beta_2 1 + e_1)}{n-1} = \frac{\beta_2 (n-1) + (e_n-e_1)}{n-1} = \beta_2 + \frac{e_n-e_1}{n-1}$$ so $$E(\hat \beta_2) = E \left ( \frac{y_n - y_1}{n-1} \right ) = \beta_2 + E \left ( \frac{e_n-e_1}{n-1} \right ) = \beta_2 + \frac{E(e_n - e_1)}{n-1} = \beta_2$$ because $E(e_n) = E(e_1) = 0$.

Therefore, $\hat \beta_2$ is unbiased for $\beta_2$.