I am currently studying how to $\bf{identify}$ the parameter $B_1$ in a simple univariate regression model where we have $Y=B_0+B_1X+\epsilon$ with the usual assumption of $X$ being exogenous, homoscedasticity etc. I understand how to do this. My notes give the asymptotic distribution of $\hat{B_1}$ as follows:
$$\sqrt{n}[\hat{B_1}-B_1] \approx N(0,\frac{\sigma^2}{Var(X)})$$
The text doesn't really explain how it came to this asymptotic distribution, can someone please explain in simple terms? I understand that we are almost "standardizing" $\hat{B_1}$ by subtracting the true $B_1$ parameter and multiplying by $\sqrt{n}$ but I don't understand how the variance of the asymptotic distribution came about.
I also know that $B_1$ in this case is:
$$B_1=\frac{Cov(Y,X)}{Var(X)}$$
Can someone please help? I'd greatly appreciate it! This is not a homework problem, just a part of my lecture notes that do not go in depth so I am confused.
Many thanks!
