# State Space model with heteroskedastic disturbance - approximation of error term

We have a state-space model with a heteroskedastic disturbance term modelled according to some time-varying process (e.g. ARCH, GARCH etc.). The disturbance term is modelled according to e.g. $$h_{t+1}=\alpha_0+\alpha_1 {\epsilon_t^*}^2$$. Since term $${\epsilon_t^*}^2$$ is unobserved, we can instead estimate $$h_{t+1}$$ as follows: $$h_{t+1}=\alpha_0+\alpha_1\mathbb{E}[{\epsilon_t^*}^2| I_t]$$ where $$I_t$$ is the all information up to time step $$t$$. This follows Harvey, Ruiz, and Sentana (1992). We can then follow Kim and Nelson $$($$2017$$)$$ $$($$Ch.6, pp.143-144$$)$$ in estimating $$\mathbb{E}[{\epsilon_t^*}^2| I_t]$$ by using the fact that $$\epsilon_t^*=\mathbb{E}[\epsilon_t^*|I_t]+(\epsilon_t^*-\mathbb{E}[\epsilon_t^*|I_t])$$ (1). We can then square the term $$\epsilon_t^*$$ and take the conditional expectation to obtain the expression $$\mathbb{E}[{\epsilon_t^*}^2|I_t]=\mathbb{E}[\epsilon_t^*|I_t]^2+\mathbb{E}[(\epsilon_t^*-\mathbb{E}[\epsilon_t^*|I_t])^2]$$ (2) given again by Kim and Nelson (2017) (Ch.6, pp.143-144).

I'm not sure how to go from (1) to (2). This is what I've done so far. If we square equation (1) we have:

(3) $${\epsilon_t^*}^2=\mathbb{E}[{\epsilon_t^*}|I_t]^2+({\epsilon_t^*}-\mathbb{E}[{\epsilon_t^*}|I_t])^2+2\mathbb{E}[{\epsilon_t^*}|I_t]({\epsilon_t^*}-\mathbb{E}[{\epsilon_t^*}|I_t])$$

If we then take the conditional expectation, we have:

(4) $$\mathbb{E}[{\epsilon_t^*}^2|I_t]=\mathbb{E}[{\epsilon_t^*}|I_t]^2+\mathbb{E}[({\epsilon_t^*}-\mathbb{E}[{\epsilon_t^*}|I_t])^2|I_t]$$

where the cross-product cancels out because:

(5)$$\mathbb{E}\Big[2\mathbb{E}[\epsilon_t^*|I_t]\times(\epsilon_t^*-\mathbb{E}[\epsilon_t^*|I_t])\Big| I_t\Big]=$$

(6)$$\mathbb{E}\big[2\mathbb{E}[\epsilon_t^*|I_t]\epsilon_t^*-2\mathbb{E}[\epsilon_t^*|I_t]^2\big| I_t\big]=$$

(7) $$\mathbb{E}[2\mathbb{E}[\epsilon_t|I_t]\epsilon_t^*|I_t]-\mathbb{E}[2\mathbb{E}[\epsilon_t^*|I_t]^2|I_t]=$$

(8) $$2\mathbb{E}[\epsilon_t^*|I_t]\mathbb{E}[\epsilon_t^*|I_t]-2\mathbb{E}[\epsilon_t^*|I_t]^2=0$$.

In going from (7) to (8), I believe that the conditional expectation of a conditional expectation is just the original conditional expectation, since it equates to a known value as we are in time-step $$t$$ where all information up to $$t$$ is already known.

But if the above is correct, I'm still not sure how/why the conditional term in (4) falls away when comparing the expression to (2) as taken from the mentioned textbook.

Any help is appreciated.

• This is the original poster. I was able to ascertain that the small derivation I stated is indeed correct. So the question is answered. Commented May 1 at 7:38