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Suppose I have the model $$y_t = \alpha_0 + \alpha_1 y_{t-1} + ... + \alpha_p y_{t-p} + \beta_0 x_t + ... + \beta_q x_{t-q} + \epsilon_t,$$ where $\{x_t\}$ is a stationary process and $\epsilon_t$ has variance $(1+c|y_{t-1}|)^2$, for some constant $c>0$. Suppose the ADL model without the heteroskedasticity would be stationary, what can we say about the stationarity of the ADL model with the heteroskedasticity? I suspect that it should be stationary for sufficiently small $c$, but I'm not certain and I have no clue on how to go about proving it. Any help would be appreciated.

On a related note: if this model turns out to be too odd to analyse like this, what form of heteroskedasticity, other than ARCH, would result in a stationary process?

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