I am working on a time series problem and I really am having trouble understanding many things.
The model we are using is AR(1) where $$y_t = \beta_1y_{t-1}+e_t$$ and the least squares estimator of $\beta_1$ is $$\hat{\beta_1} = \frac{\sum_{t=2}^n y_ty_{t-1}}{\sum_{t=2}^n y_{t-1}^2}$$
My goal is to show that $$Var[\hat{\beta}]= \frac{\sigma^2_e}{\sum_{t=2}^n y_{t-1}^2}$$
I understand that $Var[e_t]=\sigma^2_e$ because it is assumed that $e_t \sim N(0,\sigma^2_e)$.
The problem also mentions that $y_{t-1}$ are fixed.
This really confuses me, because how can $y_{t-1}$ be fixed while $y_t$ is not?
I want to simplify
$$Var\left[\frac{\sum_{t=2}^n y_ty_{t-1}}{\sum_{t=2}^n y_{t-1}^2}\right]$$
but I do not know the distribution of $y_ty_{t-1}$ and $y_{t-1}^2$ so I do not know how to proceed from here.
I would appreciate your help.