GARCH Model in Time Series Analysis
Model1
The GARCH model is a generalization of the ARCH model, a time series analysis method for identifying heteroscedasticity. $$ (1 - \beta{1} B - \cdots - \beta_{p} B^p) \sigma_{t | t-1}^2 = \omega + (\alpha_{1} B + \cdots + \alpha_{q} B^q) r_{t}^{2} $$
Derivation
Let us begin the derivation with the simplest $ARCH(1)$ model.
2 Given the return $\left\{ r_{t} \right\}$ of the time series data $\left\{ p_{t} \right\}$, the statement that the data has an ARCH effect at lag $1$—that is, autoregressive conditional heteroscedasticity—can be expressed as a formula as follows. $$ r_{t} = \sigma_{t | t-1} \varepsilon_{t} $$
$$ \begin{align} \sigma_{t | t-1}^2 = \omega + \alpha r_{t-1}^{2} \end{align} $$ Here $\alpha$ and $\omega$ are as-yet-unknown coefficients, and $\varepsilon_{t}$ is an iid process innovation that need not be assumed to be white noise but is assumed to have mean $0$ and variance $1$. $\sigma_{t | t-1}^2$ is called the conditional Volatility of $p_{t}$, and by the following formula expansion, the square of the return $r_{t}^2$ becomes an unbiased estimator of $\sigma_{t | t-1}^2$. $$ \begin{align*} E \left( r_{t}^2 | r_{t-j} , j = 1,2, \cdots \right) =& E \left( \sigma_{t | t-1}^2 \varepsilon_{t}^2 | r_{t-j} , j = 1,2, \cdots \right) \\ =& \sigma_{t | t-1}^2 E \left( \varepsilon_{t}^2 | r_{t-j} , j = 1,2, \cdots \right) \\ =& \sigma_{t | t-1}^2 E \left( \varepsilon_{t}^2 \right) \\ =& \sigma_{t | t-1}^2 \end{align*} $$ That $r_{t}^{2}$ is an unbiased estimator of $\sigma_{t | t-1}^2$ means that, setting $\eta_{t} := r_{t}^{2} - \sigma_{t | t-1}^2$, by substituting $\eta_{t}$ into $(1)$ we can obtain an autocorrelation model $AR(1)$ for $\left\{ r_{t}^{2} \right\}$. $$ \left( r_{t}^{2} \right) = \omega + \alpha \left( r_{t-1}^{2} \right) + \eta_{t} $$ Here, assuming that $r_{t}$ has some constant population variance $\sigma^2$ and taking the expectation of both sides, $$ \begin{align} \sigma^2 = \omega + \alpha \sigma^2 \end{align} $$ is obtained. One should not be confused by $\sigma$ suddenly appearing here: the subscript-free $\sigma$ is not the population variance of the original time series data $p_{t}$, but the population variance of its return $r_{t}$. Since $E (r_{t}) = E (\sigma_{t | t-1} \varepsilon_{t} ) = 0$, $$ \begin{align*} \sigma =& \text{var} (r_{t}) \\ =& E(r_{t}^{2}) - E(r_{t})^2 \\ =& E(r_{t}^{2}) \end{align*} $$ is how it is computed. At least in the $ARCH(1)$ model, it is natural for $p_{t}$ to have heteroscedasticity. Since, by $(2)$, $\displaystyle \omega = \sigma^2 \left( 1 - \alpha \right)$, unraveling $(1)$ with respect to $\sigma_{t | t-1}^2$ gives $$ \begin{align*} \sigma_{t | t-1}^2 =& \omega + \sigma r_{t}^{2} \\ =& (1 - \alpha) \sigma^2 + \alpha r_{t}^{2} \end{align*} $$ that is, $\sigma_{t | t-1}^2$ appears as a weighted average of $\sigma^2$ and $r_{t}$; the closer $\alpha$ is to $1$, the more strongly it is affected by the previous return $r_{t}$, and the closer it is to $0$, the more it means there is no ARCH effect. In that case, the analysis, without worrying about $\omega$, ultimately reduces to the problem of estimating the coefficient $\sigma$ in the $AR(1)$ model. Simply put, it is a rehash of the ARMA model.
Generalization
The generalization of the ARCH model can be done in the same manner. $$ \sigma_{t | t-1}^2 = \omega + \beta_{1} \sigma_{t-1| t-2}^2 + \cdots + \beta_{p}\sigma_{t -p | t-p-1}^2 $$ The above will be the autoregressive model $AR(p)$ of $\sigma_{t | t-1}^2$, and $$ \sigma_{t | t-1}^2 = \omega + \alpha_{1} r_{t-1}^2 + \cdots + \alpha_{q} r_{t-q}^2 $$ the above will be the moving average model $MA(q)$ of $\sigma_{t | t-1}^2$. In this way, the ARMA model $ARMA(p,q)$ for $\sigma_{t | t-1}^2$ may be called the generalized ARCH model $GARCH(p,q)$. Using the backshift operator $B$, we obtain the following much simpler formula. $$ (1 - \beta{1} B - \cdots - \beta_{p} B^p) \sigma_{t | t-1}^2 = \omega + (\alpha_{1} B + \cdots + \alpha_{q} B^q) r_{t}^{2} $$ Conceptually, if the GARCH model is no different from the ARMA model, then the method for finding the orders $p,q$ is also no different, and likewise the method of finding them using the EACF can be used as is.
Meanwhile, we can now define volatility clustering a little more elegantly. Instead of the vague explanation ‘a phenomenon in which the variance grows larger and smaller’, we can say ‘when the data follows a GARCH model of large order, we say there is volatility clustering’.