I am currently reviewing in-depth materials from Svensson (2007) and trying to retrieve equation (2.6) myself. To put it simply, I have such equation:
$$\pi_{t+1} = \gamma \pi_{t} + \alpha x_{t} + \varepsilon_{t}$$
and a loss function got under the hypothesis that $$\mathbb{E}(\pi_{t}) = \mathbb{E}(x_{t}) = 0$$
Which is written as:
$$\mathcal{L}_{t} = \mathbb{E}[\pi_{t}^{2}] + \lambda\mathbb{E}[x_{t}^{2}] = \mathbb{V}[\pi_{t}] + \lambda \mathbb{V}[x_{t}]$$
I also know that we can state:
$$x_{t} = -f\pi_{t}$$
And thus the first equation rewrites as:
$$\pi_{t+1} = (\gamma - \alpha f)\pi_{t} + \varepsilon_{t}$$
My question is: how can we find that equation (2.6) ?
$$\mathbb{V}(\pi_{t}) = \frac{1}{1 - (\gamma - \alpha f)^{2}}\mathbb{V}(\varepsilon_{t})$$
Is there something I am missing on Variance fundamentals when it comes to a "2 times" problem ?