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This is how I would define it, if anyone has any objections please let me know!

AR(m)-ARCH(m) time series is an ARCH(m) process in which the variance at time t is conditional on the previous m times such that:

$Var(a_t|z_{t-1},...,z_{t-m})=\sigma^2_t=\omega+\alpha_1a_{t-1}^2+...+\alpha_ma_{t-m}^2$.

ie , the mean is modeled by AR(m) and the variances as ARCH(m)

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  • $\begingroup$ What is $a_t$, what is $z_t$? For the AR($p$)-ARCH($m$) model you need an equation for the conditional mean and one for the conditional variance; the conditional mean equation seems to be missing in your definition. Also, do you intentionally have the same lag order for the AR and the ARCH parts? In practice, they could very well be different. $\endgroup$
    – Richard Hardy
    Commented Mar 31, 2016 at 8:34
  • $\begingroup$ At least your statement about AR($m$)-ARCH($m$), the mean is modeled by AR($m$) and the variances as ARCH($m$), is correct (although I would use variances in singular instead, since you already use mean in singular). $\endgroup$
    – Richard Hardy
    Commented Mar 31, 2016 at 11:19
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    $\begingroup$ I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? $\endgroup$
    – Richard Hardy
    Commented Feb 13, 2017 at 18:27

1 Answer 1

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Here are the equations defining an AR($p$)-ARCH($m$) model for a time series variable $x_t$:

$$ \begin{aligned} x_t &= \varphi_0 + \varphi_1 x_{t-1} + \dotsc + \varphi_p x_{t-p} + \varepsilon_t, \\ \varepsilon_t &= \sigma_t z_t, \\ z_t &\sim i.i.N(0,1), \\ \sigma_t^2 & = \alpha_0 + \alpha_1 \varepsilon_{t-1}^2 + \dotsc + \alpha_m \varepsilon_{t-m}^2. \end{aligned} $$

For the special case of $p=m$ just replace $p$ with $m$ in the top equation.

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