I'm taking a course on time series currently and have been slightly confused about the interplay between unit roots and stationarity in a question I've been attempting to answer. The question set up is a GARCH($m,s$) model of the form $$y_t=\mu + \varepsilon_t$$ $$\varepsilon_t = \nu_t \sigma_t \quad \quad \nu_t\sim N(0,\sigma^2)$$ $$\sigma_t^2 = \alpha_0 + \sum_{i=1}^m \varepsilon_{t-i}^2 +\sum_{j=1}^s \sigma_{t-j}^2$$ and is about when the variance process $\{\sigma_t^2\}$ is wide-sense stationary. Therefore, I think the question is effectively asking for a replication of Theorem 1 of Bollerslev (1986). I could simply replicate the proof given there, but I think I have a different solution but I'm not sure if it is valid. Note, I am not asking for a solution to the question, I am just trying to put my main question into context.
I've given my attempt below for reference:
We first express the $\epsilon_t^2$ process as an ARMA($m,s$) process. We define $\mathscr{F}_{t-1}$ as the information set available at time $t-1$. Since $\mathbb{E}[\nu_t^2]=1$ we can use the tower rule to calcuate, $\mathbb{E}[\epsilon_t^2 | \mathscr{F}_{t-1}]=\mathbb{E}[\sigma_t^2 \mathbb{E}[\nu_t^2|\sigma_t^2]|\mathscr{F}_{t-1}]=\sigma_t^2$ so can convert our expression in the GARCH model for $\sigma_t^2$ by moving to $\mathbb{E}[\epsilon_t^2 | \mathscr{F}_{t-1}]$ to the other side and adding $\epsilon_t^2$ to both sides to obtain $$\epsilon_t^2 = \alpha_0+\sum_{i=1}^m \alpha_i \epsilon_{t-i}^2 +\sum_{j=1}^s \beta_j \sigma^2_{t-j}+w_t$$ where $w_t=\varepsilon_t^2-\mathbb{E}[\varepsilon_t^2 | \mathscr{F}_{t-1}]$ is by definition the innovation. To get rid of the $\sigma^2$'s we can add and subtract $\sum_{j=1}^s \beta_j \epsilon_{t-j}^2$ to obtain the following, $$\epsilon_t^2 = \alpha_0+\sum_{i=1}^m \alpha_i \epsilon_{t-i}^2 +\sum_{j=1}^s \beta_j \epsilon_{t-j}^2 -\sum_{j=1}^s \beta_j (\epsilon_{t-j}^2-\sigma^2_{t-j})+w_t$$
Since $\sigma_{t-j}^2 = \mathbb{E}[\epsilon_{t-j}^2]$ the terms $\epsilon_{t-j}^2 - \sigma_{t-j}^2$ are lagged innovations so we can rewrite the above expression as $$\epsilon_t^2 = \alpha_0+\sum_{i=1}^m \alpha_i \epsilon_{t-i}^2 +\sum_{j=1}^s \beta_j \epsilon_{t-j}^2 -\sum_{j=1}^s \beta_j w_{t-j}+w_t$$
Thus, $\epsilon_t$ follows an ARMA($m,s$) process. This process is stationary if the polynomial $$\phi(z) = 1 - \sum^m_{i=1} \alpha_i z^i - \sum_{j=1}^s \beta_j z^j$$ has all of its roots outside the unit circle. More precisely, the process is stationary iff it does not have a unit root i.e. $\phi(1)>0$ since all coefficients are greater than or equal to 0. So $$\phi(1) = 1 - \sum^m_{i=1} \alpha_i - \sum_{j=1}^s \beta_j>0$$ thus, the process is stationary iff $$\sum^m_{i=1} \alpha_i +\sum_{j=1}^s \beta_j <1$$
My question is whether the statement that the absence of a unit root implies stationarity is permissible? I understand that unit root tests are used to check whether a process is stationary. However, I don't know whether that encompasses all forms of stationarity? That is, does the absence of a unit root guarantee the process will, at the very least, be wide sense stationary? Or do there exist wide sense stationary series which have a unit root? And , if it does imply wide sense stationarity, does the absence of a unit root also imply stronger forms of stationarity as well?
