For a simple autoregressive model satisfying ${p_t = \phi_0 + \phi_1 p_{t-1} +\epsilon_t}$, when you want to test whether the series has a unit root (non-stationary) why is the alternative hypothesis for the test ${|\phi_1| < 1}$, instead of ${|\phi_1| \neq1}$?
The test assumes the error terms are ${Normal(\mu=0, \sigma)}$. Is it because if ${|\phi_1|}$ were ${> 1}$ the time series would explode to infinity rapidly, so we can safely ignore that possiblity?
The $AR(1)$ equation $p_t = \phi p_{t-1} + \epsilon_t$ admits a stationary solution if and only if $|\phi| \neq 1$ (see e.g. Brockwell and Davis Time Series, Proposition 3.5.1).
However, in the case $|\phi| > 1$, the stationary solution is future-dependent. Thus, in practice we restrict ourselves to the so-called causal case, i.e. when $|\phi| < 1$. Thus, in this setting, the alternative to non-stationarity (i.e. $|\phi| = 1$) is $|\phi| < 1$.