# Method for Building the Matrices for the Multivariate Representation of a PAR Model

### Description

This method provides the relevant matrices for the multivariate representation of a PAR or PIAR model
fitted by the functions `fit.ar.par`

, and `fit.piar`

.

### Details

In a quarterly time series, the periodic autoregressive model of order *p* less or equal to 4,

* y_t = ψ_s + φ_{1s} y_{t-1} + φ_{2s} y_{t-2} + ... + φ_{ps} y_{t-p} + ε_t ,*

with *s=1,2,3,4*, can be written as a multivariate model as follows,

*Φ_0 y_t = Ψ + Φ_1 Y_{T-1} + ε_T ,*

where *Φ_0* and *Φ_1* are *S \times S* matrices containing the *φ_{is}
parameters.*

*Φ_0 =*

1 | 0 | 0 | 0 |

-φ_{12} | 1 | 0 | 0 |

-φ_{23} | -φ_{13} | 1 | 0 |

-φ_{34} | -φ_{24} | -φ_{14} | 1 |

*Φ_1 =*

φ_{41} | φ_{31} | φ_{21} | φ_{11} |

0 | φ_{42} | φ_{32} | φ_{22} |

0 | 0 | φ_{43} | φ_{33} |

0 | 0 | 0 | φ_{44} |

The periodically integrated model of order 2,

* y_t - α_s y_{t-1} = μ_s + β_s (y_{t-1} - α_{s-1} y_{t-2}) + ε_t,*

with *s=1,2,3,4*, can be written as a multivariate model as follows,

*Φ_0 y_t = Ψ + Φ_1 Y_{T-1} + ε_T ,*

where the matrix *Φ_0* and *Φ_1* are defined below

*Φ_0 =*

1 | 0 | 0 | 0 |

-α_2 | 1 | 0 | 0 |

0 | -α_3 | 1 | 0 |

0 | 0 | -α_4 | 1 |

*Φ_1 =*

0 | 0 | 0 | α_1 |

0 | 0 | 0 | 0 |

0 | 0 | 0 | 0 |

0 | 0 | 0 | 0 |

The *Φ_0* and *Φ_1* matrices can be used to compute the impact of accumulation of the
shocks *ε_t*. The impact matrix is defined as *Γ Φ_0^{-1}*, where *Γ* is
*Φ_0^{-1} Φ_0*.

That row in which the values of the impact matrix are the highest, entails that the corresponding season undergoes more severe impacts from the accumulation of all shocks. Hence, it is more likely to display fluctuations in the stochastic trend. Put in other words, the impact matrix allow the practitioner to get an idea about how the stochastic trend and the seasonal fluctuations are related.

### Methods

- object = "fit.partsm".
Provides a list object containing the estimated matrices

*Phi0*,*Phi1*, the eigen values of*Phi0^{-1} \%*\% Phi1*, and the time-varing impact of accumulation of shocks calculated as*Phi0^{-1} \%*\% Phi1 \%*\% Phi0^{-1}*. See details below.- object = "fit.piartsm".
Provides the same list as in the latter case. See details below.

### Author(s)

Javier Lopez-de-Lacalle javlacalle@yahoo.es.

### See Also

`fit.partsm-class`

, and `fit.piartsm-class`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 | ```
## Load data and select the deterministic components.
data("gergnp")
lgergnp <- log(gergnp, base=exp(1))
detcomp <- list(regular=c(0,0,0), seasonal=c(1,0), regvar=0)
## Multivariate representation of a PAR(2) model with sesonal intercepts.
out.par <- fit.ar.par(wts=lgergnp, type="PAR", detcomp=detcomp, p=2)
PAR.MVrepr(out.par)
## Multivariate representation of a PIAR(2) model with sesonal intercepts.
out.piar <- fit.piar(wts=lgergnp, detcomp=detcomp, p=2)
PAR.MVrepr(out.piar)
``` |