= Stationary versus causal solution of a two-sided AR(1) equation
{title2=$|\phi|\ne1\text{ stationary};\quad|\phi|<1\text{ causal}$}
With nondegenerate white noise, a two-sided AR(1) equation has a unique weakly stationary solution exactly when $|\phi|\ne1$. The solution is causal for $|\phi|<1$ and <anticausal> for $|\phi|>1$. At $\phi=\pm1$, an n-term noise sum has variance $n\sigma^2$, while its difference-of-stationary-values representation has variance at most four times the stationary variance, a contradiction.
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