Fpari.piar.test: Test for a Parameter Restriction in a PAR Model.
In partsm: Periodic Autoregressive Time Series Models

Description Usage Arguments Details Value Author(s) See Also Examples

This function performs a test for a parameter restriction in a PAR model. Two restrictions can be considered and entail that the process contain either the unit root 1 or the seasonal unit root -1. In this version PAR models up to order 2 can be considered.

1 2	Fpari.piar.test (wts, detcomp, p, type)

`wts`	a univariate time series object.
`detcomp`	a vector indicating the deterministic components included in the auxiliary regression. See the corresponding item in `fit.ar.par`.
`p`	the order of the initial AR or PAR model. In this version PAR models up to order 2 with seasonal intercepts are considered.
`type`	a character string indicating which restriction should be tested. `"PARI1"` indicates that the unit root is tested whereas `"PARI-1"` test for the unit root -1.

On the basis of the following PAR model (in this version PAR models up to order 2 are considered and seasonal intercepts are included default),

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

for s=1,...,S, two different hypotheses can be tested:

H0: α_s = 1, for s=1,...S-1,
H0: α_s = -1, for s=1,...S-1.

For S=4, if the hypothesis α_1*α_2*α_3*α_4=1 cannot be rejected (see LRurpar.test), the null hypotheses above entails that either α_4=1 or α_4=-1.

When the first H0 is not rejected, the PAR model contains the unit root 1, and the periodic difference filter is just the first order difference, (1-L), where L is the lag operator.

When the second H0 is not rejected, the PAR model contains the unit root -1, and the periodic difference filter is simplified as (1+L).

In both null hypotheses it is said that the data behave as a PAR model for an integrated series, known as PARI. If those null hypotheses are rejected, the corresponding model is called a periodically integrated autoregressive model, PIAR.

The asymptotic distribution of the F-statistic is F(S-1, n-k), where n is the number of observations and k the number of regressors.

In this version PAR models up to order 2 can be considered.

An object of class Ftest.partsm-class containing the F-test statistic, the freedom degrees an the corresponding p-value.

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

Ftest.partsm-class, and LRurpar.test.

    ## Test for the unit root 1 in a PAR(2) with seasonal intercepts for
    ## the logarithms of the Real GNP in Germany.
    data("gergnp")
    lgergnp <- log(gergnp, base=exp(1))
    detcomp <- list(regular=c(0,0,0), seasonal=c(1,0), regvar=0)
    out <- Fpari.piar.test(wts=lgergnp, detcomp=detcomp, p=2, type="PARI1")