# AR-ARCH conditional variance

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....

• What do you think about my answer? I see you have neither upvoted nor accepted it, nor asked for further clarification. Commented May 2, 2020 at 6:35

since $$\mathbb{E}(\sigma_t^2|x_{t-2})=0$$
Also, the variance of $$\epsilon$$ in an ARCH model should be 1, not $$\sigma^2_{\epsilon}$$.