Fpari.piar.test: Test for a Parameter Restriction in a PAR Model.

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.


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



a univariate time series object.


a vector indicating the deterministic components included in the auxiliar regression. See the corresponding item in fit.ar.par.


the order of the initial AR or PAR model. In this version PAR models up to order 2 with seasonal intercepts are considered.


a character string indicating which restriction should be tested. "PARI1" inidicates 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.

See Also

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.
    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")

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