Unit root ($|z|=1,\Phi (z)=0$)

A unit root is a zero of an autoregressive polynomial on the unit circle. A zero at $1$ is removed by first differencing. A conjugate pair $e^{\pm i\omega }$ corresponds to the real factor $1-2\cos \omega B+B^2$. Nonzero white-noise excitation at an uncancelled unit root prevents a finite-variance stationary solution.

A large, slowly damping oscillatory sample autocorrelation function suggests autoregressive zeros near a conjugate pair on the unit circle. For period $s$, the pair is near $e^{\pm 2\pi i/s}$, with minimal real filter $1-2\cos (2\pi /s)B+B^2$. Such a plot is evidence, not proof: a finite sample cannot distinguish a near-unit stationary model or a random sinusoid solely from the shape.