PAR.MVrepr-method function

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

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

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. methods

Methods

  • object = "fit.partsm".: Provides a list object containing the estimated matrices Phi0Phi0, Phi1Phi1, the eigen values of Phi01%%Phi1Phi0^{-1} \%*\% Phi1, and the time-varying impact of accumulation of shocks calculated as Phi01%%Phi1%%Phi01Phi0^{-1} \%*\% Phi1 \%*\% Phi0^{-1}. See details below.

  • object = "fit.piartsm".: Provides the same list as in the latter case. See details below.

Details

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

yt=ψs+ϕ1syt1+ϕ2syt2+...+ϕpsytp+ϵt, y_t = \psi_s + \phi_{1s} y_{t-1} + \phi_{2s} y_{t-2} + ... + \phi_{ps} y_{t-p} + \epsilon_t ,

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

Φ0yt=Ψ+Φ1YT1+ϵT, \Phi_0 y_t = \Psi + \Phi_1 Y_{T-1} + \epsilon_T ,

where Φ0\Phi_0 and Φ1\Phi_1 are S×SS \times S matrices containing the c("phiis\n\\phi_{is}\n", "parameters. parameters.")

Φ0=\Phi_0 =

11000000
ϕ12-\phi_{12}110000
ϕ23-\phi_{23}ϕ13-\phi_{13}1100
ϕ34-\phi_{34}ϕ24-\phi_{24}ϕ14-\phi_{14}11

Φ1=\Phi_1 =

ϕ41\phi_{41}ϕ31\phi_{31}ϕ21\phi_{21}ϕ11\phi_{11}
00ϕ42\phi_{42}ϕ32\phi_{32}ϕ22\phi_{22}
0000ϕ43\phi_{43}ϕ33\phi_{33}
000000ϕ44\phi_{44}

The periodically integrated model of order 2,

ytαsyt1=μs+βs(yt1αs1yt2)+ϵt, y_t - \alpha_s y_{t-1} = \mu_s + \beta_s (y_{t-1} - \alpha_{s-1} y_{t-2}) + \epsilon_t,

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

Φ0yt=Ψ+Φ1YT1+ϵT, \Phi_0 y_t = \Psi + \Phi_1 Y_{T-1} + \epsilon_T ,

where the matrix Φ0\Phi_0 and Φ1\Phi_1 are defined below

Φ0=\Phi_0 =

11000000
α2-\alpha_2110000
00α3-\alpha_31100
0000α4-\alpha_411

Φ1=\Phi_1 =

000000α1\alpha_1
00000000
00000000
00000000

The Φ0\Phi_0 and Φ1\Phi_1 matrices can be used to compute the impact of accumulation of the shocks ϵt\epsilon_t. The impact matrix is defined as ΓΦ01\Gamma \Phi_0^{-1}, where Γ\Gamma is Φ01Φ0\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.

See Also

fit.partsm-class, and fit.piartsm-class.

Examples

## 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)

Author(s)

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