Let ($\epsilon_t - \alpha$)~i.i.d Bernoulli [(1-$\alpha) -\alpha$)]. Define the regression of ($\epsilon_t - \alpha$) on $r^2_{t-1}$ without a constant, and where $r^2_{t-1}$ is a return series.
Given $\epsilon_t - \alpha = a* r^2_{t-1} + \mu_t$
Is the $\hat{\alpha}$ unbiased and consistent?
My solution:
I personally defined $\hat{\alpha_{OLS}}$ as: $\sum^T_{t=1} (r^2_{t-1} * r^{2'}_{t-1})^{-1} * \sum^T_{t=1} (r^2_{t-1}*(\epsilon_t-\alpha)$ where the usual yt = ($\epsilon_t - \alpha$).
Now, to check if it's correct, i computed the expected value and the result, since the E($\epsilon_t - \alpha) =0 $, is zero, implying that the OLS estimator is not unbiased.
Moreover, also checking the consistency using the LLN leads to zero.
Do you think is it correct?