How to draw a vertical line of desired length in equations.

Try enclosing your expression into \left. and \right\vert, as in

\left.\frac{\partial L}{\partial\bm{\beta}}\right\vert_{\hat\beta}=-2\bm{X'y}+2\bm{X'X\hat\beta}=0

which yields:

Well, if you are manually defining the size, as in the example you could just put a \Bigg| at the right place. Probably better would be using \left, \middle and \right.

By the way: You should prefer \[ and \] instead of double $$.

\[ E\left[ F_{m-1}\Big( \text{some big contents} \Big) \middle|\: \mathcal{F}_1 \right] \]