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I have a simple question about the GARCH model.

We know that the $\alpha$ and $\beta$ parameters of the models are fitted for the local volatility of each time $t$ as follows:

$$\sigma_t^2= \alpha_0 + \sum_{i=1}^q \alpha_i \varepsilon_{t-i} + \sum_{i=1}^p \beta_i \sigma_{t-i}^2$$

with $\varepsilon_t=\sigma_t z_t$ and $z_t \sim N(0,1)$

However, the wikipedia article says that the process $y$ is behaving as follows:

$$y_t=a_0 + \sum_{i=1}^q a_i y_{t-i} + \varepsilon_t$$

I just wanted to make sure that here we assume that $y_t$ is the "original" time series values, and hence that $\varepsilon_t$ was the "return" of the time series at time $t$. Is that correct?

In a financial application, would $y_t$ be the price or the return at time $t$?

EDIT:

As the answers indicates that $y_t$ models the returns, I'm a bit surprised because usually you use the maximum log-likelihood:

$$\log L = -\frac{1}{2} \sum_{i=1}^n \left(\log (2 \pi) + \log (\sigma_{i}^2) + \frac{y_i^2}{\sigma_i^2} \right)$$

But this is only true if $y_i \sim N(0,\sigma_i^2)$

Now clearly with the setup presented above $\text{Var}(y_i) = \sigma_i^2$, but if $a \neq 0 ~ \forall i$, then $E[y_i] = a_0 + \sum_{i=1}^q a_i y_{t-i} \neq 0$

Is it because the log-likelihood is computed assuming $a=0 \forall i$?

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This is pretty common notation:

  • $y_t$ is the return at $t$.

  • $\varepsilon_t$ is residual from modeling the returns as an $AR(q)$ process as shown in the equations.

Taken together, you have an $AR(q)-GARCH(p,q)$ model there (with slight abuse of notation as we have $q$ twice).

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  • $\begingroup$ Ok but then, when you perform a fitting from the GARCH model, you should have 3 parameters fitted: $a, \alpha, \beta$ but I never see the $a$ in the output after at fitting. Why is that? $\endgroup$ – SRKX May 1 '12 at 6:53
  • $\begingroup$ For a financial time series, you would say that $y_t$ is the return, not a price? $\endgroup$ – SRKX May 1 '12 at 7:01
  • $\begingroup$ Was my question mislead by the log-likelihood computation included in the edit then? $\endgroup$ – SRKX May 1 '12 at 19:49
  • $\begingroup$ Can someone answer @SRKX question about why some implementations (e.g., fGarch in R), don't give us the fit for the $a$ parameter? (However, EViews does for some reason). $\endgroup$ – Jase Dec 9 '12 at 3:08
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You want to model returns. The blog post http://www.portfolioprobe.com/2011/01/12/the-number-1-novice-quant-mistake/ explains why you want returns rather than prices in the context of regression. The same sort of logic applies to garch -- except maybe even more so.

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  • $\begingroup$ Ok, so $y_t$ in GARCH model is supposed to be the price series right? $\endgroup$ – SRKX May 1 '12 at 12:13
  • $\begingroup$ No, the returns. $\endgroup$ – Patrick Burns May 1 '12 at 17:09
  • $\begingroup$ Could you please have a look at my edit? $\endgroup$ – SRKX May 1 '12 at 19:48
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    $\begingroup$ Your log likelihood formula is assuming the mean of the returns is zero. The more general formula would replace y_i with y_i - \mu. $\endgroup$ – Patrick Burns May 2 '12 at 9:02

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