3  Available models

Note: This chapter is under construction.

To be fair, this chapter can only give a condensed overview of what kinds of models there are. Time series come with many different characteristics, including

all of which can be but do not necessarily have to be part of an observed time series. Hence, it is only natural that there is an extensive amount of different models and approaches, usually specialized for specific purposes and situations. Moreover, the quantity to forecast also crucially restricts the pool of suitable models. Do you want to forecast the time-varying (conditional) variation of an observed series? Then models such as ARIMA are not sensible, which are for forecasting (conditional) means.

3.1 SARIMAX

The SARIMAX (seasonal autoregressive integrated moving average with exogenous regressors) model is a potent generalization of the ARIMA model. We assume \(u_t\), \(t\in\mathbb{Z}\), to be iid random variables with mean zero and finite variance \(\sigma_u^2\). Then the SARIMAX process \(\{Y_t\}_{t\in\mathbb{Z}}\) is defined through \[\begin{equation} \Phi(B^s)\phi(B)(1-B)^d (1-B^s)^D(Y_t-\mathbf{x}_t^{'}\mathbf{\beta})=\Theta(B^s)\theta(B)u_t, \end{equation}\] where \(B\) is the so-called backshift operator, so that for any time series variable \(Z_t\) we have \(BZ_t = Z_{t-1}\) and \(B^sZ_t=Z_{t-s}\). \(s\) is a positive integer reflecting the seasonal period. Moreover, \(\Phi(B^s)=1-\sum_{i=1}^{P}\Phi_iB^{is}\), \(\Theta(B^s)=1+\sum_{j=1}^{Q}\Theta_jB^{js}\), \(\phi(B)=1-\sum_{l=1}^{p}\phi_lB^{l}\), and \(\theta(B)=1+\sum_{k=1}^{q}\theta_kB^{k}\) are polynomials without common roots. \(\Phi_1,\dots,\Phi_P\), \(\Theta_1,\dots, \Theta_Q\), \(\phi_1,\dots,\phi_p\), and \(\theta_1,\dots,\theta_q\) are all real-valued coefficients. \(d,D \in \mathbb{N}_0\) are the orders of differencing. \(\mathbf{x}_t = (X_{1,t} \hspace{2mm} X_{2,t} \hspace{2mm} \dots \hspace{2mm} X_{m,t})^{'}\) is a vector with values of \(m\) exogenous regressors at time \(t\), and \(\beta=(\beta_1 \hspace{2mm} \dots \hspace{2mm} \beta_m)^{'}\) is a corresponding vector of coefficients. A short-hand for such a process is \(\text{SARIMAX}(p, d, q)(P,D,Q)\), where the terms in the brackets are the model orders.

This model nests the standard \(\text{ARIMAX}(p, d, q)\) for \(P=Q=D=0\): \[\begin{equation} \phi(B)(1-B)^d (Y_t - \mathbf{x}_t^{'}\mathbf{\beta}) = \theta(B)u_t. \end{equation}\] Note that one can include a linear time trend by setting the \(r\)th regressor, \(r\in\{1,\dots,m\}\), to \(X_{r,t}=t\), which corresponds to a constant term in the differenced series.

A basic \(\text{ARMA}(p, q)\) model (with non-zero mean \(\mu \in \mathbb{R}\)) is then simply given by \[\begin{equation} \phi(B) (Y_t - \mu) = \theta(B)u_t. \end{equation}\] This model is without differencing.

3.2 FARIMA

Sometimes an observed time series returns very slowly to its long-term mean over time. In such cases, an ARMA model may not be able to capture this persistence effectively, while a full differencing step as with an ARIMA with \(d=1\) would drop too much useful information. In such a scenario, we are very likely dealing with a case of long-range dependence, also called long memory. A natural extension of an ARMA is then a fractionally integrated ARMA (FARIMA), which reads \[\begin{equation} \phi(B)(1-B)^d (Y_t - \mu) = \theta(B)u_t. \end{equation}\] While this formula is similar to that of an ARIMA, here we assume that \(d\in(0, 0.5)\). Therefore, \((1-B)^d\) does not represent a number of full integrations but a fractional integration.