<p>I don't believe $w&gt;0$ 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.</p>

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

CC BY-SA 3.0
2015-05-07T12:08:55.953Z
2015-05-07T12:08:55.953Z
BKay
<p>I have a time series model:</p>

<p>$$y_t = \sigma_t \epsilon_t$$</p>

<p>$$\sigma_t = w + \gamma|y_{t-1}|$$</p>

<p>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.</p>

<p>a) What restrictions on the parameters w and $\gamma$ seem sensible? </p>

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

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

<hr>

<p>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:</p>

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

<p>Is this right or have I done some hocus pocus?</p>

<p>For (c) I'm stuck:</p>

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

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

CC BY-SA 3.0
2016-08-23T17:03:22.133Z
This would better fit [Cross Validated](http://stats.stackexchange.com).
Richard Hardy
<finance><time-series>
ARCH Model - Expectation of Absolute Value
Brian
# ARCH Model - Expectation of Absolute Value

(Asked by: username_0 on 2015-05-07)

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.

**Tags:** finance, time-series

> **Comments:**
> **username_1**: This would better fit [Cross Validated](http://stats.stackexchange.com).  

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## 1 answers

### Answer by username_2 on 2015-05-07. Score: 1

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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