fGarch package

The main function in this package is the garchFit() function.

> garchFit(formula = ~ garch(1, 1), data = dem2gbp, 
+          cond.dist = c("norm", "snorm", "ged", "sged",
+                        "std", "sstd", "snig", "QMLE"), 
+          include.mean = TRUE, trace = TRUE, ...)
> # additional arguments removed

The garchFit() function fits an so called ARMA - GARCH model to the data where the residuals of an ARMA process are modelled with a GARCH approach. $$\begin{align*} x_t &= \mu + \sum\limits_{i = 1}^p a_i x_{t-i} + \sum \limits_{j = 1}^q b_j \epsilon_{t-j} + \epsilon_t \\ \\ \epsilon_t &= \sigma_t z_t \\ \\ z_t &\sim D_{\nu}(0,1) \\ \\ \sigma_t^2 &= \omega + \sum\limits_{i = 1}^m \alpha_i \epsilon_{t-i}^2 + \sum\limits_{j = 1}^n \beta_j \sigma_{t-j}^2, \end{align*}$$

where \(D_{\nu}(0,1)\) is the probability density function of the innovations with zero mean and unit variance, \(\nu\) represents the optional parameter of the distribution if choosing a distribution other than normal.

By default, the orders \(p\) and \(q\) of the ARMA model are set to zero but \(\mu\) is included. The GARCH orders \(m\) and \(n\) are set to 1 and the innovations \(z\) are assumed to be normal,

$$\begin{align*} x_t &= \mu + \epsilon_t \\ \\ \epsilon_t &= \sigma_t z_t \\ \\ z_t &\sim N(0,1) \\ \\ \sigma_t^2 &= \omega + \alpha_1 \epsilon_{t-1}^2 + \beta_1 \sigma_{t-1}^2. \end{align*}$$