Autoregressive model of order p
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
\[\alpha_t= \begin{pmatrix} y_t \\ y_{t-1} \\ \vdots \\ y_{t-p+1} \end{pmatrix}\]and the transition matrices
\[T_t = \begin{pmatrix}-\varphi_1 & \cdots & \cdots & -\varphi_p \\ 1 & \cdots & \cdots & 0 \\ \vdots & \ddots & \ddots & \vdots\\ 0 & 0 & 1 & 0 \end{pmatrix}\] \[S_t = \begin{pmatrix} \sigma_{ar} \\ 0 \\ \vdots\\ 0 \end{pmatrix}\] \[V_t = S S'\]Depending on the estimation algorithm used, it may be preferrable to slightly modify the model and set $\sigma_{ar}^2=1$
jd3_ssf_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
jd3_ssf_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) |
| zeroinit | logical determining the initial condition for the state variable, which is equal to zero if TRUE is chosen. The default FALSE triggers the an initialization based on the unconditional mean and variance of the AR(p) process | FALSE |
Examples of use
jd3_ssf_ar("cycle", c(1.5,-0.4), fixedar=FALSE, variance=1, fixedvariance=TRUE, nlags=4, zeroinit=FALSE) //
jd3_ssf_ar("cycle", c(1.5,-0.4), fixedar=FALSE, variance=1, fixedvariance=FALSE, nlags=0, zeroinit=FALSE) // default
jd3_ssf_ar("cycle", c(1.5,-0.4))