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I have a time series model:

$$y_t = \sigma_t \epsilon_t$$

$$\sigma_t = w + \gamma|y_{t-1}|$$

Where $\epsilon_t$ are i.i.d normally distributed with mean zero and variance one. Suppose t = $0, ±1, ±2, ±3, \ldots$ and that the process $y_t$ is weakly and strongly stationary.

a) What restrictions on the parameters w and $\gamma$ seem sensible?

b) Calculate $E|y_t|$ assuming that this quantity is finite, noting that $E|\epsilon_{t-1}| = \sqrt{\frac{2}{\pi}}$

(c) Write $|y_t|$ as an AR(1) process and hence calculate the autocorrelation function of absolute returns: $$p_j = \frac{cov(|y_t|, |y_{t-j}|)}{cov(|y_t|, |y_{0}|)}$$

So for (a) I feel like: $\gamma \in [0, 1)$ should be reasonable as we expect periods of high-volatility to cluster. Do we need to assume w > 0 for a positive standard deviation? I think these make a difference on the answer for b:

$$ E|y_t| = E[|\sigma_t \epsilon_t|] = E[|\sigma_t| |\epsilon_t|]$$ $$ \text{as they are independent:} $$ $$ = E[|\sigma_t|] E[|\epsilon_t|] = E[|w + \gamma|y_{t-1}||]\sqrt{\frac{2}{\pi}}$$ $$ \text{as we assumed w and $\gamma$ are positive:}$$ $$ = E[w + \gamma|y_{t-1}|]\sqrt{\frac{2}{\pi}} = \{w + \gamma E[|y_{t-1}|]\}\sqrt{\frac{2}{\pi}} $$ $$ \text{as we have a weakly stationary series:}$$ $$ E|y_t| = \{w + \gamma E[|y_{t}|]\}\sqrt{\frac{2}{\pi}} \implies E|y_t| = \frac{w\sqrt{\frac{2}{\pi}}}{1 - \gamma \sqrt{\frac{2}{\pi}}} $$

Is this right or have I done some hocus pocus?

For (c) I'm stuck:

I get $|y_t| = \{w + \gamma |y_{t-1}|\}|\epsilon_t|$ but can't make anything useful out of the covariance.

Any help is much appreciated. I'm not too familiar with ARCH/GARCH models.

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I don't believe $w>0$ is necessary or sufficient to get a positive standard deviation everywhere. I think you need $\sigma_0 \geq 0$ and $w\geq 0$ with one of the two inequalities strict.

If $\gamma > 1$ then the series $\sigma$ will grow explosively if $w + \gamma \cdot \sigma_{t-1} >0$ or behave explosively harmonic if $w + \gamma \cdot \sigma_{t-1} <0 $. Neither of those seems particularly desirable most for economic situations.

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