AR-ARCH conditional variance

Question

Consider a AR(1)+ARCH(1) model: \begin{align*} &x_t=a_0+a_1x_{t-1}+u_t,\\ &u_t=\sigma_t\epsilon_t,\>\>\>\epsilon_t\sim N(0,\sigma^2_{\epsilon}),\\ &\sigma_t=\sqrt{b_0+b_1\sigma^2_{t-1}}. \end{align*}

I want to calculate $\text{Var}(u_t|x_{t-2})$. I tried to use total law of variance \begin{align*} \text{Var}(u_t|x_{t-2}) = \mathbb{E}(\text{Var}(u_t|x_{t-1},x_{t-2})|x_{t-2}) + \text{Var}(\mathbb{E}(u_t|x_{t-1},x_{t-2})|x_{t-2}). \end{align*} The first part since $u_t$ and $x_{t-1}$ are uncorrelated \begin{align*} \text{Var}(u_t|x_{t-1},x_{t-2}) = \text{Var}(u_t|x_{t-1}) = \sigma^2_t. \end{align*} The second part \begin{align*} \mathbb{E}(u_t|x_{t-1},x_{t-2}) = \mathbb{E}(u_t|x_{t-1}) = 0. \end{align*} Since $\mathbb{E}(\sigma_t^2|x_{t-2})=0$ so $\text{Var}(u_t|x_{t-2}) =0$? It doesn't seem right....

Answer 1

Here is a tip: there is one point that immediately looks suspect:

since $\mathbb{E}(\sigma_t^2|x_{t-2})=0$

The conditional expectation of a nonnegative quantity should not be zero.

Also, the variance of $\epsilon$ in an ARCH model should be 1, not $\sigma^2_{\epsilon}$.