Why does downsampling increase the phase angle of poles in autoregressive models?
Decimating a signal $x[n]$ by an integer factor $M$ results in the following $\mathcal{Z}$-transform:

$$Y(z)=\frac{1}{M}\sum_{k=0}^{M-1}X\left(e^{-j2\pi k/M}z^{1/M}\right)\tag{1}$$

If you assume that $x[n]$ was filtered by an anti-aliasing filter before decimation, the transfer function of the decimated filter can be approximated by the term for $k=0$ in (1):

$$Y(z)\approx\frac{1}{M}X(z^{1/M})\tag{2}$$

So if $z_{\infty}$ is a pole of $X(z)$ then $z_{\infty}^{M}$ is a pole of $Y(z)$. Writing the pole in polar coordinates $z_{\infty}=re^{j\phi}$ gives

$$z_{\infty}^M=r^Me^{jM\phi}$$

So the angle of the pole is increased by a factor $M$.