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I have a quick question. I found this notion of "ARCH nested in GARCH" in one of the papers I'm reading right now, and I can't quite understand what it means.

if anyone can help, I will be grateful.

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The GARCH ($p$,$q$) model for a time series $(x_{t})_{t \in \mathbb{Z}}$ is defined as \begin{align} x_t &= \sigma_t \epsilon_t \\ \sigma_t^2 &= a_0 + a_1 x_{t-1}^2 + \dots + a_p x_{t-p}^2 + b_1 \sigma_{t-1}^2 + \dotsb + b_q \sigma_{t-q}^2. \end{align}

ARCH nested in GARCH means that ARCH is a restricted version of GARCH, i.e. ARCH model is

\begin{align} x_t &= \sigma_t \epsilon_t \\ \sigma_t^2 &= a'_0 + a'_1 x_{t-1}^2 + \dots + a'_p x_{t-p}^2, \end{align}

with restriction $b_i = 0$ for $i\in[q]$.

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