<p>Yes they are equivalent it is just different notation.</p>
<p>For example, in Hamilton Time Series analysis the ARCH is given on pp 658 as:</p>
<p><span class="math-container">$$E(u^2_t | u^2_{t-1} ...  u^2_{t-m}) = \zeta + \alpha_1 u^2_{t-1}+ ...+  \alpha_m u^2_{t-m}$$</span></p>
<p>which for only one lag collapses to:</p>
<p><span class="math-container">$$E(u^2_t | u^2_{t-1} ) = \zeta + \alpha_1 u^2_{t-1}$$</span></p>
<p>Just because <span class="math-container">$u$</span> is used instead of <span class="math-container">$\epsilon$</span> or <span class="math-container">$\sigma$</span> that does not make it different model (in addition here also the information set is just described directly instead of putting <span class="math-container">$I$</span> there but it is still an ARCH model).</p>
<p>However, this being said usually <span class="math-container">$\sigma$</span> is reserved to be the parameter of interest in ARCH model. For example, following Verbeek (2008) a guide to modern econometrics 4th ed pp 325 by definition:</p>
<p><span class="math-container">$$\sigma^2_t \equiv E \{ \epsilon_t^2 | \mathcal{I}_{t-1} \}= \omega +\alpha \epsilon^2_{t-1}.$$</span></p>
<p>It is useful to use notation which does not use the same letter for expected variance as in GARCH model you would also want to include lags of expected/predicted variance (e.g.  <span class="math-container">$\sigma^2_t = \omega +\alpha \epsilon^2_{t-1} + \beta \sigma^2_{t-1}$</span>), but since in question you say this is for ARCH I don't think it makes any difference.</p>

CC BY-SA 4.0
2020-12-06T12:21:08.267Z
$\sigma_t^2$ is not the dependent variable but a parameter of interest. The dependent variable is $x_t$ modelled as $x_t=\mu_t+u_t$ where $\mu_t$ is the conditional mean and $u_t$ is the residual. [This](https://stats.stackexchange.com/questions/41509/what-is-the-difference-between-garch-and-arma/231512#231512) might be relevant.
Richard Hardy
CC BY-SA 4.0
2020-12-06T12:25:22.550Z
@RichardHardy thank you for pointing that out I edited my answer
1muflon1
CC BY-SA 4.0
2020-11-29T20:38:33.153Z
2020-12-06T12:25:05.350Z
2020-12-06T12:25:05.350Z
1muflon1
<p>It is the same in practice, they are both variances some people will write it as <span class="math-container">$\sigma^2$</span> some as <span class="math-container">$\epsilon^2$</span>. Variance in error tells you what is the variance in the variable you predict.</p>

CC BY-SA 4.0
2020-11-29T20:07:00.083Z
This is a theoretical exercise so, they are the same in practice but also in theory?
GregorSilvei
CC BY-SA 4.0
2020-11-30T01:15:46.673Z
they are the same also in theory
csilvia
CC BY-SA 4.0
2020-11-29T19:10:26.393Z
2020-11-29T19:10:26.393Z
csilvia
<p>In my problem set about ARCH models I'm given that</p>
<p><span class="math-container">$\epsilon^2_{t}=\alpha\epsilon^2_{t-1}+v_{t}$</span></p>
<p>But then I'm asked to calculate
<span class="math-container">$E(\sigma^2_{t+n}|I_{t-1})$</span>.</p>
<p>So is the same to calculate <span class="math-container">$E(\epsilon^2_{t+n}|I_{t-1})$</span>, isn't it?</p>
<p>Thanks in advance</p>

<econometrics><autoregressive><heteroskedasticity><variance>
Arch Model and $\sigma$
GregorSilvei
# Arch Model and $\sigma$

(Asked by: username_0 on 2020-11-29)

In my problem set about ARCH models I'm given that

$\epsilon^2\_{t}=\alpha\epsilon^2\_{t-1}+v\_{t}$

But then I'm asked to calculate
$E(\sigma^2\_{t+n}|I\_{t-1})$.

So is the same to calculate $E(\epsilon^2\_{t+n}|I\_{t-1})$, isn't it?

Thanks in advance

**Tags:** econometrics, autoregressive, heteroskedasticity, variance

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## 2 answers

### Answer by username_1 on 2020-11-29. Score: 0 | Accepted answer

Yes they are equivalent it is just different notation.

For example, in Hamilton Time Series analysis the ARCH is given on pp 658 as:

$$E(u^2\_t | u^2\_{t-1} ... u^2\_{t-m}) = \zeta + \alpha\_1 u^2\_{t-1}+ ...+ \alpha\_m u^2\_{t-m}$$

which for only one lag collapses to:

$$E(u^2\_t | u^2\_{t-1} ) = \zeta + \alpha\_1 u^2\_{t-1}$$

Just because $u$ is used instead of $\epsilon$ or $\sigma$ that does not make it different model (in addition here also the information set is just described directly instead of putting $I$ there but it is still an ARCH model).

However, this being said usually $\sigma$ is reserved to be the parameter of interest in ARCH model. For example, following Verbeek (2008) a guide to modern econometrics 4th ed pp 325 by definition:

$$\sigma^2\_t \equiv E \{ \epsilon\_t^2 | \mathcal{I}\_{t-1} \}= \omega +\alpha \epsilon^2\_{t-1}.$$

It is useful to use notation which does not use the same letter for expected variance as in GARCH model you would also want to include lags of expected/predicted variance (e.g. $\sigma^2\_t = \omega +\alpha \epsilon^2\_{t-1} + \beta \sigma^2\_{t-1}$), but since in question you say this is for ARCH I don't think it makes any difference.

> **Comments:**
> **username_2**: $\sigma_t^2$ is not the dependent variable but a parameter of interest. The dependent variable is $x_t$ modelled as $x_t=\mu_t+u_t$ where $\mu_t$ is the conditional mean and $u_t$ is the residual. [This](https://stats.stackexchange.com/questions/41509/what-is-the-difference-between-garch-and-arma/231512#231512) might be relevant.  
> **username_1**: @RichardHardy thank you for pointing that out I edited my answer  

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### Answer by username_3 on 2020-11-29. Score: 0

It is the same in practice, they are both variances some people will write it as $\sigma^2$ some as $\epsilon^2$. Variance in error tells you what is the variance in the variable you predict.

> **Comments:**
> **username_0**: This is a theoretical exercise so, they are the same in practice but also in theory?  
> **username_3**: they are the same also in theory  

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