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As I understand the inclusion of the GARCH term, $\sigma^2$, in a GARCH model allows for an infinite number of time series terms, $\epsilon^2$, to influence the conditional variance. Is this the case? How does this characteristic enable the GARCH model to be more parsimonious than the ARCH model?

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  • $\begingroup$ @RichardHardy I edited the other question. Thank you $\endgroup$ – Alvaro GJ Sep 1 '17 at 9:17
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    $\begingroup$ Parsimony is related to the description length of the model, not to its implication. GARCH term makes a better summarization of the past with a single term, compared to many terms of ARCH. Here term roughly means an algebraic description. $\endgroup$ – Cagdas Ozgenc Sep 6 '17 at 8:06
  • $\begingroup$ @CagdasOzgenc, good point. I have updated my answer to explicitly incorporate that. $\endgroup$ – Richard Hardy Sep 6 '17 at 11:47
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As I understand the inclusion of the GARCH term, $\sigma^2$, in a GARCH model allows for an infinite number of time series terms, $\epsilon^2$, to influence the conditional variance. Is this the case?

Yes, it is. A GARCH model can be expressed (under some regularity condition) as an infinite-order ARCH model, thus making the conditional variance $\sigma_t^2$ depend on an infinite number of past squared innovations $\varepsilon_{t-i}^2$ for $i=1,2,\dots$.

How does this characteristic enable the GARCH model to be more parsimonious than the ARCH model?

Citing the original paper introducing the GARCH model (Bollerslev, 1986),

Common to most of the [ARCH model] applications however, is the introduction of a rather arbitrary linear declining lag structure in the conditional variance equation to take account of the long memory typically found in empirical work, since estimating a totally free lag distribution often will lead to violation of the non-negativity constraints. <...> In this light it seems of immediate practical interest to extend the ARCH class of models to allow for both a longer memory and a more flexible lag structure.

(emphasis is mine).

Under long memory, the influence of past innovations dies out gradually and slowly, and any finite-order ARCH model fails to capture this efficiently. On the other hand, the GARCH model excels at this. How? See the answer to your first questions above. Thus GARCH is more parsimonious as it uses just a couple of (or a few) parameters to achieve what the ARCH model would need an infinite number of parameters for. The argument is also very similar (essentially the same) to how an ARMA model is more parsimonious than an AR or an MA model.

References:

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  • $\begingroup$ I get that the $\epsilon$ is the residual of a time-series regression of $Y$ (for simplicity say I use an ARMA model). I am still unclear though, about what the $\sigma$ is (especially in practice). $\endgroup$ – Alvaro GJ Sep 11 '17 at 15:22
  • $\begingroup$ @AlvaroGJ, $\sigma_t$ is the conditional variance of $\varepsilon_t$ when conditioned on past values of $\varepsilon$ and past values of $\sigma$. $\endgroup$ – Richard Hardy Sep 11 '17 at 15:24
  • $\begingroup$ Right, as in a GARCH (1,1) would be: $\sigma^2_t=\gamma+\epsilon^2_{t-1}+\sigma^2_{t-1}$ but what is the $\sigma$ term? Or in other words, how does one go about estimating that term? $\endgroup$ – Alvaro GJ Sep 11 '17 at 15:26
  • $\begingroup$ Oh, sorry, $\sigma_t$ is the square root of the conditional variance, i.e. it is conditional standard deviation. It is latent, of course. But it plays a role in the likelihood of the data and so the parameter can be selected so as to maximize the likelihood. The algorithms to do that is the topic of your other question. $\endgroup$ – Richard Hardy Sep 11 '17 at 15:34
  • $\begingroup$ So $var(Y_t|Y_{t-p},\epsilon_{t-q})$? $\endgroup$ – Alvaro GJ Sep 11 '17 at 15:56

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