I am in the process of working through some problem sets. I have studied some time series, but my knowledge of ARCH models is pretty basic. I am given the following information:
$Y_t = a_0 + a_1 Y_{t-1} + \epsilon_t$
where $\epsilon_t | I_{t-1} \sim N(0,h_t)$ and $h_t = \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \alpha_2 \epsilon_{t-2}^2$
I solved for the following,
Conditional variance:
$E(Y_t | I_{t-1}) = a_0 + a_1 Y_{t-1}= \mu$
then
$Var(Y_t)=E[(Y_t- \mu)^2] = E(E_t^2) = ht = \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \alpha_2 \epsilon_{t-2}^2$
Unconditional Variance:
At this point I think we can create a new series for $Y_t$ since we are not conditioning, so I wrote,
$Y_t = a_0 + a_1 (a_0 + a_1 Y_{t-2} + \epsilon_{t-1}) + \epsilon_t$, repeat this infinitely many times and get
$Y_t = \frac{a_0}{1-a_1} + \sum_{j=0}^\infty a_1^j \epsilon_{t-j}$
$E(Y_t) = \frac{a_0}{1-a_1}$ and
$Var(Y_t) = E[(Y_t - \mu )^2] = \sum_{j=0}^\infty a_1^{2j} \epsilon_{t-j}^2 = (\alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \alpha_2 \epsilon_{t-2}^2) + a_1^2 (\alpha_0 + \alpha_1 \epsilon_{t-2}^2 + \alpha_2 \epsilon_{t-3}^2) + a_1^4(\alpha_0 + \alpha_1 \epsilon_{t-3}^2 + \alpha_2 \epsilon_{t-4}^2) + \dots$
I am not sure if this cleans up nicely, but what does this say for the variance? it seems like the unconditional is just going to blow up infinitely (given I could have made a mistake in the derivation of this question). I am also not sure if I should state some stability conditions in this question similar to AR(1) processes (e.g., $\alpha_1 < 1$ and $\alpha_2 < 1$), since these are variances do those conditions apply?
If I was to conclude a statement, is it acceptable to stay the model captures volatility clustering via 2 period lags?