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 Phi0, Phi1, the eigen values of Phi0−1%∗%Phi1, and the time-varying 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.
Details
In a quarterly time series, the periodic autoregressive model of order p less or equal to 4,
yt=ψs+ϕ1syt−1+ϕ2syt−2+...+ϕpsyt−p+ϵt,
with s=1,2,3,4, can be written as a multivariate model as follows,
Φ0yt=Ψ+Φ1YT−1+ϵT,
where Φ0 and Φ1 are S×S matrices containing the c("phiis\n", "parameters.")
Φ0=
1
0
0
0
−ϕ12
1
0
0
−ϕ23
−ϕ13
1
0
−ϕ34
−ϕ24
−ϕ14
1
Φ1=
ϕ41
ϕ31
ϕ21
ϕ11
0
ϕ42
ϕ32
ϕ22
0
0
ϕ43
ϕ33
0
0
0
ϕ44
The periodically integrated model of order 2,
yt−αsyt−1=μs+βs(yt−1−αs−1yt−2)+ϵt,
with s=1,2,3,4, can be written as a multivariate model as follows,
Φ0yt=Ψ+Φ1YT−1+ϵT,
where the matrix Φ0 and Φ1 are defined below
Φ0=
1
0
0
0
−α2
1
0
0
0
−α3
1
0
0
0
−α4
1
Φ1=
0
0
0
α1
0
0
0
0
0
0
0
0
0
0
0
0
The Φ0 and Φ1 matrices can be used to compute the impact of accumulation of the
shocks ϵt. The impact matrix is defined as ΓΦ0−1, where Γ is
Φ0−1Φ0.
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)