Autoregressive Integratded Moving Average (ARIMA) Model

TO DO

Definition

The AR process is defined by

\[\Phi\left(B\right)y_t=\epsilon_t\]

where:

\[\Phi\left(B\right)=1+\varphi_1 B + \cdots + \varphi_p B^p\]

is an auto-regressive polynomial.

It conforms to the general SSF by setting the state vector

\[\alpha_t= \begin{pmatrix} y_{t-nlags} \\ \vdots \\ y_{t-1} \\ \hline y_{t} \\ y_{t+1|t} \\ \vdots \\ y_{t+h|t} \end{pmatrix}\]

of size $r0=max(p,h+1)$, where $h$ is the forecast horizon desired by the user, and $nlags=0$ by default. If independently of the AR order the user specifies a number of lagged values required ( $nlags$), then the size of the state vector will be $r=r0+nlags$.

The transition matrices are

\[T_t = \begin{pmatrix} 0 &1 & 0 & \cdots & 0 \\0& 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & 1\\ -\varphi_r & \cdots & \cdots & \cdots &-\varphi_1 \end{pmatrix}\]

with $\varphi_{j}=0$ for $j>p$

\[S_t = \begin{pmatrix} 0 \\ \vdots \\ 0\\ \hline 1 \\ \psi_1 \\ \vdots\\ \psi_s \end{pmatrix}\] \[V_t = \sigma_{ar}^{2} S S'\]

Depending on the estimation algorithm used, it may be preferrable to slightly modify the model and set $\sigma_{ar}^2=1$

ar


This R function is used to define an autoregressive model that belongs to the JD+ class JD3_SsfItem

output

Object of the JD+ class JD3_SsfItem:

usage

ar(name, ar, fixedar, variance, fixedvariance, nlags, zeroinit)

Argument Definition Default Remarks
name string with the name of the component ex. "trend", do not forget the commmas
ar double array of AR coefficients, i.e. $\varphi_1$, $\ldots$, $\varphi_p$ The autoregressive order is determined by the size of the array, e.g. of your write c(1.5, -0.6), you are implicitely fixing the order of your AR(p) model to $p=2$ with $\varphi_{1}=1.5$ and $\varphi_{2}=-0.6$
fixedar logical that triggers estimation of the AR coefficients $\varphi_1$, $\ldots$, $\varphi_p$ (FALSE) or fixes them (TRUE) to a pre-specified values above, set by the parameter ar FALSE
variance double for the value of $\sigma^2_{ar}$ 1 variance of the innovation is actually $\sigma^2\sigma^2_{ar}$
fixedvariance logical that triggers estimation of $\sigma^2_{ar}$ (FALSE) or fixes it (TRUE) to a pre-specified value set by the parameter variance FALSE
nlags integer specifying how many lags of the state variable are needed 0 Note that the number of lags desired are independent from the order of the AR model. You may have an order $p=2$ and and do not need to specify any of its lags in the measurement equation (in this case, the default nlags=0 would be sufficient)
nfcasts integer specifying how many forecasts of the state variable are needed 0 Note that the resulting state space representation stacks the lagged values of the state together with the forecasts, as shown in the state-space representation above

Examples of use

jd3_ssf_ar2("cycle", c(1.5,-0.4), fixedar=FALSE, variance=1, fixedvariance=TRUE, nlags=4, nfcasts=4) // 
jd3_ssf_ar2("cycle", c(1.5,-0.4), fixedar=FALSE, variance=1, fixedvariance=FALSE, nlags=0, , nfcasts=0) // default
jd3_ssf_ar2("cycle", c(1.5,-0.4))