Tail behavior of a threshold autoregressive stochastic volatility model (Q2488465)

In the threshold autoregressive stochastic volatility (TARSV) model the observed returns \(Y_t\) are modelled as \(Y_t=\sigma\exp(\alpha_t/2)\varepsilon_t\), where \(\alpha_t=\varphi_1\alpha_{t-1}+z^{(1)}_t\) if \(Y_{t-1}\leq 0\) and \(\alpha_t=\varphi_2\alpha_{t-1}+z^{(2)}_t\) if \(Y_{t-1}>0\), \(\varepsilon_i\) are i.i.d. with \(E\varepsilon_i=0\), Var \(\varepsilon_i=1\), and \(z_i^{(k)}\) are i.i.d. heavy tailed. The asymptotic expansion of the tail for the stationary distribution of \(\alpha_t\) is obtained.