In this paper we derive closed-form formulas of feedback capacity and nonfeedback achievable rates, for Additive Gaussian Noise (AGN) channels driven by nonstationary autoregressive moving average (ARMA) noise (with unstable one poles and zeros), based on time-invariant feedback codes and channel input distributions. From the analysis and simulations follows the surprising observations, (i) the use of time-invariant channel input distributions gives rise to multiple regimes of capacity that depend on the parameters of the ARMA noise, which may or may not use feedback, (ii) the more unstable the pole (resp. zero) of the ARMA noise the higher (resp. lower) the feedback capacity, (iii) certain conditions, known as detectability and stabilizability are necessary and sufficient to ensure the feedback capacity formulas and nonfeedback achievable rates {\it are independent of the initial state of the ARMA noise}. Another surprizing observation is that Kim's \cite{kim2010} characterization of feedback capacity which is developed for stable ARMA noise, if applied to the unstable ARMA noise, gives a lower value of feedback capacity compared to our feedback capacity formula.