PAR.MVrepr-methods {partsm} | R Documentation |
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
.
In a quarterly time series, the periodic autoregressive model of order p less or equal to 4,
y_t = psi_s + phi_{1s} y_{t-1} + phi_{2s} y_{t-2} + ... + phi_{ps} y_{t-p} + ε_t ,
with s=1,2,3,4, can be written as a multivariate model as follows,
Phi_0 y_t = Psi + Phi_1 Y_{T-1} + ε_T ,
where Phi_0 and Phi_1 are S times S matrices containing the phi_{is} parameters.
Phi_0 =
1 | 0 | 0 | 0 |
-phi_{12} | 1 | 0 | 0 |
-phi_{23} | -phi_{13} | 1 | 0 |
-phi_{34} | -phi_{24} | -phi_{14} | 1 |
Phi_1 =
phi_{41} | phi_{31} | phi_{21} | phi_{11} |
0 | phi_{42} | phi_{32} | phi_{22} |
0 | 0 | phi_{43} | phi_{33} |
0 | 0 | 0 | phi_{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,
Phi_0 y_t = Psi + Phi_1 Y_{T-1} + ε_T ,
where the matrix Phi_0 and Phi_1 are defined below
Phi_0 =
1 | 0 | 0 | 0 |
-α_2 | 1 | 0 | 0 |
0 | -α_3 | 1 | 0 |
0 | 0 | -α_4 | 1 |
Phi_1 =
0 | 0 | 0 | α_1 |
0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 |
The Phi_0 and Phi_1 matrices can be used to compute the impact of accumulation of the shocks ε_t. The impact matrix is defined as Γ Phi_0^{-1}, where Γ is Phi_0^{-1} Phi_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.
Javier López-de-Lacalle javlacalle@yahoo.es.
fit.partsm-class
, and fit.piartsm-class
.
## 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)