The main question is: an AR(1)/ARCH(1) process has the variance of the ARCH(1)? I've tried to compute the unconditional variance of an AR(1)/ARCH(1) model, so an AR(1) in which the noise is modelled as an ARCH(1). I've computed the variance of the ARCH(1) as (w/1-α), then I've used the ordinary formula to obtain the variance of an AR(1), so sigma^2/(1-φ^2) , substituting sigma^2 with the variance of the ARCH(1). Is this the right procedure?
My computation is: Variance of arch = 1/1-0.35 = 1.538 AR(1) variance = 1.538/(1-0.5814)=3.193 This is my solution, but it is not the solution of the book (according to me is wrong!) that says that that AR(1) unconditional variance of the AR(1) is equal to 1/1-0-35=1.53