Consider a strictly stationary ARCH(2) process $X_t = \sigma_tZ_t,$ $t \in \mathbb{Z}$, with iid standard normal noise $(Z_t)$ where $\sigma^2_t=\alpha_0+\alpha_1 X^2_{t-1} +\alpha_2 X^2_{t-2} ,$ $\alpha_i>0,i=0,1,2$.
Assume that $E[\sigma_0^4] < \infty$. It follows that $v_t = X_t^2 - \sigma_t^2$, $t \in \mathbb{Z}$, constitutes white noise. Then $(X_t^2)$ is an AR(2) process with noise $v_t$.
I understand all of this. But then the text goes on to say that $(X_t^2)$ has spectral density, and computes the spectral density using a formula for causal ARMAs. Isn’t it supposed to first justify that $(X_t^2)$ is causal? And I can’t see how to do this. What did I miss?