For the linear regression $y_t = Bx_t+e_t$ where we have the assumptions: $E(e_t)=0$, $E(e_t^2) = \sigma^2$, $E(e_t e_s)= 0$ for $s\neq t $ a ), suppose $x_t = \frac{1}{t}$ for all t. Is $\hat {B}$ unbiased an consistent or not?
b)Suppose $x_t = \frac{1}{2}$ for all t. Show whether $\hat{B}$ is unbiased or not. Show whether $\hat{B}$ is consistent for $\beta$ or not.
For point a I state that the estimator is unbiased because $$E(\hat{B}) = \beta + E\left(\sum_{t=1}^{T} \frac{1}{t}e_t\right) = \beta + \sum_{t=1}^{T} \frac{1}{t}E(e_t) = \beta$$ then I checked if the variance is equal to $0$ to verify consistency and : \begin{align*} Var(\hat{\beta}) &= Var\left(\frac{\sum_{t=1}^{T} \frac{1}{t}e_t}{\sum_{t=1}^{T} \frac{1}{t}^2}\right) \\ &= \frac{1}{\left(\sum_{t=1}^{T} \frac{1}{t^2}^2\right)} \cdot \sum_{t=1}^{T} \left(\frac{1}{t}\right)^2 Var(e_t) \\ &= \sigma^2 \cdot \frac{1}{\sum_{t=1}^{T} \frac{1}{t}^2} \end{align*}
Since the denominator is finite then the variance does not converge to $0$ because $\sum_{t=1}^T \frac{1}{t^2}\lt \infty$ then the $\hat{B}$ is not consistent Is my attempt correct?
for point B I state that the estimator is unbiased for the same reason of point. How can I show if the $\hat{B}$ is consistent or not in this case?
Any help would be really appreciated