# Parameters in Autoregressive representation of an ARCH model

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 $r_t^2$?

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).

• 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. – Richard Hardy May 29 '19 at 13:39
• Yes, there will be heteroskedasticity in the error term. This is also one of the reasons why the OLS approach is inefficient. – Johan Stax Jakobsen May 29 '19 at 19:58
• 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? – Richard Hardy May 30 '19 at 7:25