I am struggling with the calculation of PACF (partial autocorrelated functions) in $ARMA(p,q)$. The lecture notes I have gives an example of calculating PACF of $AR(1)$. For $$y_t=a_0+a_1y_{t-1}+\varepsilon_t,$$ what the note does is to redefine $$y_t^*=y_t-E(y_t)$$ and obtain $y_t^*=\phi_{11}y_{t-1}^*+e_t$ where $\phi_{11}$ is the first PACF value. Then it uses Yule-Walker equations to find PACF. However, I am not too sure the general way to do it. For example, if for $AR(2)$, do we have $y_t^*=\phi_{11}y_{t-1}^*+e_t$ or $y_t^*=\phi_{11}y_{t-1}^*+\phi_{12}y_{t-2}^*+e_t$? In fact, I was even not sure what that $e_t$ should eqaul...
I am hoping that someone could tell me the general way for calculating this, or provide some simple but more general examples, say $AR(2)$, to elucidate the progress of the calculation. Your help is much appreciated!