Suppose we have a $0$ mean time serie representing stock index returns about a title, $r$.
I also know it follows an $ARCH(p)$ model with parameters $\omega$ and $\alpha$, specified in the following notation:
$$\sigma_t^2 = \omega + \sum_{i=1}^{p}\alpha_i r_{t-i}^2 .$$
I know that if $r_t$ follow an $ARCH(p)$, than $r_t^2$ follow an autoregressive process of the same order, an $AR(p)$.
The question is: are parameters the same? I mean, can I estimate $ARCH(p)$ parameters modelling $r_t^2$?
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Define
$$ v_t = r_{t}^2 - E_{t-1}[r_{t}^2] = r_{t}^2 - \sigma_{t}^2 $$
Plug this into the ARCH equation
$$ r_t^2 - v_t = w + \sum_{i=1}^p \alpha_i r_{t-i}^2 $$
Rearranging yields the AR(p) model
$$ r_t^2 = w + \sum_{i=1}^p \alpha_i r_{t-i}^2 + v_t $$
Yes, one should be able to estimate the model with least squares. However, there are some drawbacks discussed in Chapter 6 of Francq and Zakoian's book "GARCH Models: Structure, Statistical Inference and Financial Applications" (2010).
answered May 28, 2019 at 16:44
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$\begingroup$ Is $v_t$ i.i.d.? I am just wondering if this is a vanilla AR model or a model with autoregressive lags and a peculiar error term. $\endgroup$ Commented May 29, 2019 at 13:39
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$\begingroup$ Yes, there will be heteroskedasticity in the error term. This is also one of the reasons why the OLS approach is inefficient. $\endgroup$ Commented May 29, 2019 at 19:58
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$\begingroup$ I see. So this is AR w.r.t. the conditional mean only. Is the error term some sort of GARCH, or are the squares of errors not conditionally autocorrelated? $\endgroup$ Commented May 30, 2019 at 7:25