Specify the model for the conditional mean in a dual model, where the conditional mean is modelled through an ARMA or a FARIMA model and the conditional standard deviations through a GARCH-type model simultaneously.
mean_spec(orders = c(0, 0), long_memo = FALSE, include_mean = TRUE)An object of class "mean_spec" is returned.
a two-element numeric vector with the model orders; the first element is the autoregressive order \(p^{*}\), while the second element is the moving-average order \(q^{*}\).
a logical value that indicates whether the long-memory version of the model should be considered or not.
a logical value indicating whether or
not to include the constant unconditional mean in the estimation
procedure; for include_mean = FALSE, the unconditional
mean of the series is fixed to zero and not being estimated.
Let \(\left\{y_t\right\}\), with \(t \in \mathbb{Z}\) as the time index, be a theoretical time series that follows $$\beta(B)(1- B)^{D}(y_t - \mu)=\alpha(B)r_t,$$ where \(\beta(B) = 1 - \sum_{i=1}^{p^{*}}\beta_i B^{i}\) and \(\alpha(B) = 1 + \sum_{j=1}^{q^{*}}\alpha_j B^{j}\) are the AR- and MA-polynomials of orders \(p^{*}\) and \(q^{*}\), respectively, with real coefficients \(\beta_i\), \(i=1,\dots,p^{*}\), and \(\alpha_j\), \(j=1,\dots,q^{*}\). \(B\) is the backshift operator. \(\beta(B)\) and \(\alpha(B)\) are commonly assumed to be without common roots and to have roots outside of the unit circle. Furthermore, \(\mu\) is a real-valued coefficient representing the unconditional mean in \(\left\{y_t\right\}\). \(D \in [0, 0.5)\) is the fractional differencing parameter. \(\left\{r_t\right\}\) is a zero-mean (weak) white noise process, for example a member of the GARCH-models (with mean set to zero) presented in this package (see the descriptions in fEGarch_spec, fiaparch, figarch, etc.).
The for \(D=0\), which can be achieved through long_memo = FALSE,
the formulas above describe an autoregressive moving-average (ARMA) model.
For \(D \in (0, 0.5)\), they describe a fractionally integrated ARMA (FARIMA)
model.
mean_spec()
mean_spec(orders = c(1, 1))
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