Version:	1.1-4
Date:	2025-04-21
Title:	Periodic Autoregressive Time Series Models
Imports:	methods
Depends:	R (≥ 2.14.0)
Description:	Basic functions to fit and predict periodic autoregressive time series models. These models are discussed in the book P.H. Franses (1996) "Periodicity and Stochastic Trends in Economic Time Series", Oxford University Press. Data set analyzed in that book is also provided. NOTE: the package was orphaned during several years. It is now only maintained, but no major enhancements are expected, and the maintainer cannot provide any support.
License:	GPL-2
URL:	https://github.com/MatthieuStigler/partsm
NeedsCompilation:	no
Packaged:	2025-04-21 17:19:07 UTC; matifou
Author:	Javier Lopez-de-Lacalle [aut, cph], Matthieu Stigler [cre]
Maintainer:	Matthieu Stigler <Matthieu.Stigler@gmail.com>
Repository:	CRAN
Date/Publication:	2025-04-21 17:30:01 UTC

Test for the Significance of the p+1 Autoregressive Parameters in an AR(p) or PAR(p) Model

Description

Test for the significance of prospective autoregressive parameters of order p+1 in an AR(p) or PAR(p) model. It is performed as an F-statistic that sets the parameters of order p+1 equal to zero.

Usage

    Fnextp.test (wts, detcomp, p, type)
  

Arguments

Value

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

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

See Also

fit.ar.par, and Ftest.partsm-class.

Examples

    ## Test the significance of a second order lag in a PAR model for the Real GNP in Germany.
    ## Including seasonal intercepts.
    data("gergnp")
    lgergnp <- log(gergnp, base=exp(1))
    detcomp <- list(regular=c(0,0,0), seasonal=c(1,0), regvar=0)
    out <- Fnextp.test(wts=lgergnp, detcomp=detcomp, p=1, type="PAR")
  

Test for Periodic Variation in the Autoregressive Parameters

Description

Test for periodic variation in the autoregressive parameters.

Usage

    Fpar.test (wts, detcomp, p)
  

Arguments

Details

An F-test based on the residual sum of squares is performed to test for periodic variation in the autoregressive parameters.

On the basis of the following model,

y_t = \phi_{1s} y_{t-1} + ... + \phi_{ps} y_{t-p} + \epsilon_t,

for s=1,...,S, where S is the periodicity of the time series,

the null-hypothesis is non-periodicity: \phi_{is}=\phi_i, for s=1,...,S and i=1,2,...,p. When the null hypothesis is imposed an AR(p) is estimated, whereas the alternative is a PAR(p) model.

The F-statistic when four seasonal intercepts are included follows an F((S-1)*p, n-(S+S*p)) distribution, where S is the periodicity of the series and n the number of observations.

Value

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

Author(s)

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

See Also

fit.ar.par, and Ftest.partsm-class.

Examples

    ## Test for periodicity in a second order PAR model for
    ## the logarithms of the Real GNP in Germany time series.
    data("gergnp")
    lgergnp <- log(gergnp, base=exp(1))
    detcomp <- list(regular=c(0,0,0), seasonal=c(1,0), regvar=0)
    out <- Fpar.test(wts=lgergnp, detcomp=detcomp, p=2)
  

Test for a Parameter Restriction in a PAR Model.

Description

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.

Usage

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

Arguments

Details

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 = \mu_s + \alpha_s y_{t-1} + \beta_s (y_{t-1} - \alpha_{s-1} y_{t-2}) + \epsilon_t,

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

• H0: \alpha_s = 1, for s=1,...S-1,

• H0: \alpha_s = -1, for s=1,...S-1.

For S=4, if the hypothesis \alpha_1*\alpha_2*\alpha_3*\alpha_4=1 cannot be rejected (see LRurpar.test), the null hypotheses above entails that either \alpha_4=1 or \alpha_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.

Value

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

Author(s)

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

See Also

Ftest.partsm-class, and LRurpar.test.

Examples

    ## 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")
  

Test for Seasonal Heteroskedasticity

Description

Test for seasonal heteroskedasticity.

Usage

    Fsh.test (res, s)
  

Arguments

Details

On the basis of the following model,

x_t^2 = \omega_0 + \omega_1 D_{1,t} + ... + \omega_{S-1} D_{S-1,t} + \epsilon_t,

where S is the periodicity of the data, the null hypothesis of

the null-hypothesis \omega_{i}=0, for i=1,...,S-1 is tested.

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

Value

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

Author(s)

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

See Also

Ftest.partsm-class.

Examples

    ## Fsh test for the residuals of the first differences
    ## of the logarithms of the Real GNP in Germany
    ## on an AR(4) model with seasonal intercepts.
    data("gergnp")
    lgergnp <- log(gergnp, base=exp(1))
    wts <- ts(c(NA, diff(gergnp, lag=1)), frequency=4, start=start(lgergnp))

    detcomp=list(regular=c(0,0,0), seasonal=c(1,0), regvar=0)
    ar4 <- fit.ar.par(wts=lgergnp, type="AR", p=4, detcomp=detcomp)
    out <- Fsh.test(res=residuals(ar4@lm.ar), s=frequency(wts))
  

Ftest.partsm Class

Description

This class contains the information provided by the F-tests available in the package 'partsm'.

Slots

test.label:

Object of class "character": A label to identify the test that the information is related to.

test.name:

Object of class "character": A one-line description of the test.

p:

Object of class "numeric": The order of the AR or PAR model.

Fstat:

Object of class "numeric": The F-statistic.

df:

Object of class "numeric": The freedom degrees.

pval:

Object of class "numeric": The p-value.

pvl:

Object of class "character": A symbol indicating the significance of the F-statistic according to usual codes, i.e. Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1.

h0md:

Object of class "lm": The summary of the model fitted for the null hypothesis.

hamd:

Object of class "ANY": The summary of the model fitted for the alternative hypothesis.

Methods

show:

This method reports the F-test statistic, the null and the alternative hypotheses entailed in the procedure, as well as the freedom degrees, the p-value and the codified p-value.

summary:

In addition to the information reported by show, a summary of the fitted models for the null and the alternative hypotheses is also added.

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

See Also

Fnextp.test, Fpar.test, Fsh.test, and Fpari.piar.test.

LRur.partsm Class

Description

This class contains the information provided by LRurpar.test.

Slots

test.label:

Object of class "character": A label to identify the test.

test.name:

Object of class "character": A one-line description of the test.

p:

Object of class "numeric": The lag order parameter of the model.

LR:

Object of class "numeric": The LR statistic.

LRtau:

Object of class "numeric": The one side test statistic.

h0nls:

Object of class "matrix": The estimated coefficients of the non-linear PIAR model.

halm:

Object of class "lm": The estimated PAR model for the alternative hypotheses.

Methods

show:

Shows the LR statistics and a one-side test constructed as sign(g(\hat{\alpha}) - 1) * LR^{1/2}, where g(\hat{\alpha}) is the product of the periodic differencing filter parameters estimated under the alternative.

summary:

Displays the same output as show but a summary of the null and the alternative hypotheses is also displayed.

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

See Also

LRurpar.test.

Likelihood Ratio Test for a Single Unit Root in a PAR(p) Model

Description

This function performs the likelihood ratio test for a single unit root in a PAR(p) model up to order 2.

Usage

    LRurpar.test (wts, detcomp, p)
  

Arguments

Details

In a quarterly time series, the PAR(1) model, y_t = \alpha_{s,1} y_{t-1} + \epsilon_t with \epsilon_t ID(0,1), contains a unit root if g(\alpha) = \Pi_{s=1}^4 \alpha_{s,1} = 1. To test this hypothesis, a likelihood ratio test, LR, is built as the logarithm of the ratio between the residual sum of squares in the unrestricted and the restricted model, weighted by the number of observations.

The unrestricted PAR model is estimated by OLS, whereas the model in which the null hypothesis is imposed, i.e. \Pi_{s=1}^4 \alpha_{s,1} = 1, is estimated by nonlinear least squares.

The critical values are reported in Osterwald-Lenum (1992), table 1.1 (for the case where p-r=1).

In this version, PAR models up to order 2 with seasonal intercepts are considered, since the function fit.piar does not allow for higher orders.

Value

An object of class LRur.partsm-class containing the LR statistics and a one-side test constructed as sign(g(\hat{\alpha}) - 1) * LR^{1/2}, where g(\hat{\alpha}) is the product of the periodic differencing filter parameters estimated under the alternative.

Author(s)

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

References

H.P. Boswijk and P.H. Franses (1996), Unit roots in periodic autoregressions. Journal of Time series Analysis, 17, pp. 221-245.

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

Osterwald-Lenum, M. (1992), A Note with Quantiles of the Asymptotic Distribution of the Maximum Likelihood Cointegration Rank Test Statistics: Four Cases. Oxford Bulletin of Economics and Statistics, 54, pp.461-472.

See Also

fit.ar.par, fit.piar, and LRur.partsm-class.

Examples

    ## Test for a single unit root in a PAR(2) model 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 <- LRurpar.test(wts=lgergnp, detcomp=detcomp, p=2)
  

MVPAR Class

Description

This class contains the matrices for the multivariate representation of an object of class fit.partsm-class.

Slots

Phi0:

Object of class "matrix": Matrix for the multivariate representation. See details in PAR.MVrepr-methods.

Phi1:

Object of class "matrix": Matrix for the multivariate representation. See details in PAR.MVrepr-methods.

Gamma.eigenvalues:

Object of class "numeric": Eigenvalues of the matrix Phi0^{-1} \%*\% Phi1.

tvias:

Object of class "matrix": Time-varying impact of accumulation of shocks calculated as Phi0^{-1} \%*\% Phi1 \%*\% Phi0^{-1}.

Methods

show

signature(object = "MVPAR"): Shows the information contained in the slots.

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

See Also

PAR.MVrepr-methods, and fit.partsm-class.

MVPIAR Class

Description

This class contains the matrices for the multivariate representation of an object of class fit.piartsm-class.

Slots

Phi0:

Object of class "matrix": Matrix for the multivariate representation. See details in PAR.MVrepr-methods.

Phi1:

Object of class "matrix": Matrix for the multivariate representation. See details in PAR.MVrepr-methods.

Gamma.eigenvalues:

Object of class "numeric": Eigenvalues of the matrix Phi0^{-1} \%*\% Phi1.

tvias:

Object of class "matrix": Time-varying impact of accumulation of shocks calculated as Phi0^{-1} \%*\% Phi1 \%*\% Phi0^{-1}.

Omega0:

Object of class "ANY": Matrix for internal use.

Omega1:

Object of class "ANY": Matrix for internal use.

Pi0:

Object of class "ANY": Matrix for internal use.

Pi1:

Object of class "ANY": Matrix for internal use.

Methods

show

signature(object = "MVPIAR"): Shows the information contained in the slots.

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

See Also

PAR.MVrepr-methods, and fit.piartsm-class.

Multivariate representation of a PAR model

Description

The function PAR.MVrepr applied on the out.par object of class fit.partsm, shows the multivariate representation as well as some complementary information of a PAR process.

Usage

    PAR.MVrepr(object)
  

Arguments

Details

For more information, see the vignette, section 2.

Value

A list with different objects

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

Examples

    ## Models for the the logarithms of the Real GNP in Germany.
    data("gergnp")
    lgergnp <- log(gergnp, base=exp(1))

    ## Fit an PAR model
    detcomp <- list(regular=c(0,0,0), seasonal=c(1,0), regvar=0)
    out.par <- fit.ar.par(wts=lgergnp, type="PAR", detcomp=detcomp, p=2)

    ## Show the matrix representation: 
    out.MV <- PAR.MVrepr(out.par)
    out.MV
  

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

Description

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.

Details

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} + \epsilon_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} + \epsilon_T ,

where \Phi_0 and \Phi_1 are S \times S matrices containing the \phi_{is} parameters.

\Phi_0 =

\Phi_1 =

The periodically integrated model of order 2,

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

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

\Phi_0 y_t = \Psi + \Phi_1 Y_{T-1} + \epsilon_T ,

where the matrix \Phi_0 and \Phi_1 are defined below

\Phi_0 =

\Phi_1 =

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

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.

Author(s)

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

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)
  

Autocorrelation function for several transformations of the original data

Description

This function is based on the acf function and extends it by allowing for some transformations of the data before computing the autocovariance or autocorrelation function.

Usage

      acf.ext1 (wts, transf.type, perdiff.coeffs, type, lag.max, showcat, plot)
   

Arguments

Details

The implemented transformations are the following:

• "orig": Original series.

• "fdiff": First differences of the original series.

• "sdiff": Seasonal differences of the original series.

• "fsdiff": Fisrt and seasonal differences of the original series.

• "fdiffsd": Residuals of the first differences on four seasonal dummy variables.

• "perdiff": Periodic differences of the original series.

• "perdiffsd": Residuals of the periodic differences on four seasonal dummy variables.

Value

Lags at which the acf is computed, estimates of the acf, and p-values for the significance of the acf at each lag.

Author(s)

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

See Also

acf.

Examples

      ## Logarithms of the Real GNP in Germany
      data("gergnp")
      lgergnp <- log(gergnp, base=exp(1))

      out <- acf.ext1(wts=lgergnp, transf.type="orig",
                      type="correlation", lag.max=12, showcat=TRUE, plot=FALSE)

      out <- acf.ext1(wts=lgergnp, transf.type="perdiffsd", 
                      perdiff.coeff = c(1.004, 0.981, 1.047, 0.969),
                      type="correlation", lag.max=12, showcat=TRUE, plot=FALSE)
   

Unemployment in Canada (1960.1-1987.4)

Description

OECD Main Economic Indicators: Unemployment in Canada. Sample: 1960.1 - 1987.4. Remark: measured in 1000 persons. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("canun")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

Unemployment in Canada. (1960.1-1987.4). Seasonally Adjusted

Description

OECD Main Economic Indicators: Unemployment in Canada. Sample: 1960.1 - 1987.4. Remark: Seasonally Adjusted. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("canunsa")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time canunsas (Oxford University Press, 1996).

Fit an Autoregressive or Periodic Autoregressive Model

Description

This function fits either an autoregressive (AR) or a periodic autoregressive (PAR) model and extract the estimates for the autoregressive or periodic autoregressive coefficients.

Usage

    fit.ar.par (wts, type, detcomp, p)
  

Arguments

Details

If type is "AR" the following model is estimated by ordinary least squares:

y_t = \phi_{1} y_{t-1} + \phi_{2} y_{t-2} + ... + \phi_{p} y_{t-p} + \epsilon_t.

If type is "PAR", the following model is estimated by ordinary least squares:

y_t = \alpha_{1s} y_{t-1} + \alpha_{2s} y_{t-2} + ... + \alpha_{ps} y_{t-p} + \epsilon_t,

for s=1,...,S, where S is the periodicity of the time series.
Deterministic components can be added to models above. Be careful when defining the detcomp argument. To include an intercept and seasonal intercepts, or a regular trend with seasonal trends, will cause multicollinearity problems.

Value

A fit.partsm-class class object reporting the estimates of the autoregressive or periodic autoregressive coefficients. See fit.partsm-class to check further information available from this class via the methods show and summary.

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

See Also

fit.piartsm-class, fit.partsm-class, and PAR.MVrepr-methods.

Examples

    ## Models for the the logarithms of the Real GNP in Germany.
    data("gergnp")
    lgergnp <- log(gergnp, base=exp(1))

    ## Fit an AR(4) model with intercept and seasonal dummies.
    detcomp <- list(regular=c(1,0,c(1,2,3)), seasonal=c(0,0), regvar=0)
    out.ar <- fit.ar.par(wts=lgergnp, type="AR", detcomp=detcomp, p=4)

    ## Fit a PAR(2) model with seasonal intercepts.
    detcomp <- list(regular=c(0,0,0), seasonal=c(1,0), regvar=0)
    out.par <- fit.ar.par(wts=lgergnp, type="PAR", detcomp=detcomp, p=2)
  

fit.partsm Class

Description

This class contains information on the autoregressive or periodic autoregressive parameters estimated by fit.ar.par.

Slots

type:

Object of class "character": The type of the fitted model, an autoregressive model, "AR", or a periodic autoregressive model, "PAR".

p:

Object of class "numeric": The lag order parameter of the model.

lm.ar:

Object of class "ANY": The summary of a fitted AR model. When an AR type model is selected, it is of class "lm", otherwise the slot is empty.

lm.par:

Object of class "ANY": The summary of a fitted PAR model. When a PAR type model is selected, it is of class "lm", otherwise the slot is empty.

ar.coeffs:

Object of class "ANY": The autoregressive parameters estimates. When a PAR type model is selected, it is of class "matrix", otherwise the slot is empty.

par.coeffs:

Object of class "ANY": The periodic autoregressive parameters estimates. When a PAR type model is selected, it is of class "matrix", otherwise the slot is empty.

Methods

show:

This method reports the autoregressive or periodic autoregressive estimates, depending whether the model is an AR model or a PAR model.

summary:

In addition to the information reported by show, a summary of the fitted model is also added.

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

See Also

fit.ar.par.

Fit a Periodically Integrated Autoregressive Model.

Description

Fit a periodically integrated periodic autoregressive model.

Usage

    fit.piar (wts, detcomp, p, initvalues=NULL)
  

Arguments

Details

The following equation is estimated by non-linear least squares

y_t = \alpha_s y_{t-1} + \beta_s (y_{t-1} - \alpha_{s-1} y_{t-2}) + \epsilon_t,

under the restriction \Pi_{i=1}^{S} \alpha_i = 1 for s=1,...,S, where S denotes the number of seasons. Regressors defined in detcomp can also be included. Obviously, for a first order PIAR process \beta parameters are equal to zero.

Value

An object of class fit.piartsm-class containing the estimated coefficients in the restricted non-linear model, the residuals, and the periodic autoregressive coefficients. On the basis of the estimated alpha parameters, the periodically differenced data are also computed. See fit.piartsm-class for methods that display this information.

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

See Also

nls, fit.ar.par, and fit.piartsm-class.

Examples

    ## Fit a PIAR(2) model 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 <- fit.piar(wts=lgergnp, detcomp=detcomp, p=2, initvalues=NULL)
  

fit.piartsm Class

Description

This class contains information on the periodic autoregressive parameters estimated by fit.piar.

Slots

p:

Object of class "numeric": The order of the PIAR model.

nls.parameters:

Object of class "matrix": Estimated coefficients of the non-linear PIAR model.

nls.res:

Object of class "numeric": Residuals of the non-linear PIAR model.

par.coeffs:

Object of class "matrix": Periodic autoregressive parameters estimates.

pdiff.data:

Object of class "ts": Periodically differenced data.

Methods

show:

Reports the periodic autoregressive coefficients estimates.

summary:

Like show, the periodically differenced data are also displayed.

plot:

Plot the the periodically differenced data, as well as the seasonal paths of the transformed data.

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

See Also

fit.piar.

Real GNP in Germany (1960.1-1990.4)

Description

Real GNP in Germany. Sample: 1960.1 - 1990.4. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("gergnp")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

J. Wolters (1992): Persistence and Seasonality in Output and Employment of the Federal Republic of Germany, Recherches Economiques de Louvain, 58 pp.421-39.

Real GNP in Germany (1960.1-1990.4). Seasonally Adjusted

Description

Real GNP in Germany. Sample: 1960.1 - 1990.4. Remark: Seasonally Adjusted. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("gergnpsa")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

J. Wolters (1992): Persistence and Seasonality in Output and Employment of the Federal Republic of Germany, Recherches Economiques de Louvain, 58 pp.421-39.

Graphical Representation of the Periodically Differenced Data

Description

On the basis of the estimated parameters in a PIAR model, this function displays the periodically differenced data, as well as two different representations of the seasonal paths for the transformed data.

Usage

    plotpdiff (x)
  

Arguments

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

See Also

fit.piartsm-class, and fit.piar.

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)

    ## Fit a PIAR(2) model with seasonal intercepts.
    out.piar <- fit.piar(wts=lgergnp, detcomp=detcomp, p=2)
    plotpdiff(out.piar)
  

Plot of the Out-of-Sample Forecasts in a PIAR Model

Description

This function displays a plot of the predictions and the corresponding 95 per cent confidence intervals based on a PIAR model. In this version, this function is implemented for quarterly observed data, PIAR models up to order 2 are considered, and seasonal intercepts are included by default.

Usage

    plotpredpiar (x)
  

Arguments

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

See Also

pred.piartsm-class, and predictpiar.

Examples

    ## Load data and select the deterministic components.
    data("gergnp")
    lgergnp <- log(gergnp, base=exp(1))

    ## Fit a PIAR(2) model with seasonal intercepts.
    out.pred <- predictpiar(wts=lgergnp, p=2, hpred=24)
    plotpredpiar(out.pred)
  

pred.piartsm Class

Description

This class contains the information provided by predictpiar.

Slots

wts:

Object of class "wts": The observed time series.

hpred:

Object of class "numeric": The number of forecasts.

p:

Object of class "numeric": The lag order parameter of the PIAR model.

fcast:

Object of class "ts": The out-of-sample forecasts.

fse:

Object of class "ts": The forecast standard errors.

ucb:

Object of class "ts": The upper 95 per cent confidence bound.

lcb:

Object of class "ts": The lower 95 per cent confidence bound.

Methods

show:

Shows out-of-sample forecasts and the corresponding standard errors, as well as the 95 per cent confidence intervals.

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

See Also

predictpiar.

Predictions for a Restricted Periodic Autoregressive Model

Description

This function performs predictions for a restricted periodic autoregressive model. This version considers PIAR models up to order 2 with seasonal intercepts. It is implemented for quarterly observed data.

Usage

    predictpiar (wts, p, hpred)
  

Arguments

Details

Upon the multivariate representation,

\Phi_0 y_t = \Psi + \Phi_1 Y_{T-1} + ... + \Phi_P y_{T-P} + \epsilon_T ,

where the \Phi_i, i=1,2,...,P are s \times s matrices containing the \phi_{is} parameters., the one-step-ahead forecasts for the year T+1 is straightforward,

y_t = \Phi_0^{-1} \Psi + \Phi_0^{-1} \Phi_1 Y_{T-1} + ... + \Phi_0^{-1} \Phi_P y_{T-P} + \Phi_0^{-1} + \epsilon_T .

Multi-step-ahead forecasts are obtained recursively.

The prediction errors variances for the one-step-ahead forecast are the diagonal elements of

\sigma^2 \Phi_0^{-1} (\Phi_0^{-1})^{'},

whereas for h=2,3,... years ahead forecasts it becomes

\sigma^2 \Phi_0^{-1} (\Phi_0^{-1})^{'} + (h-1) (\Gamma \Phi_0^{-1}) (\Gamma \Phi_0^{-1})^{'},

where \Gamma = \Phi_0^{-1} \Phi_1.

This version considers PIAR models up to order 2 for quarterly observed data. By default, seasonal intercepts are included in the model as deterministic components.

The number of observations to forecast, hpred must be a multiple of 4.

Value

An object of class pred.piartsm-class containing the forecasts and the corresponding standard errors, as well as the 95 per cent confidence intervals.

Author(s)

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

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

See Also

fit.piar, PAR.MVrepr-methods, and pred.piartsm-class.

Examples

    ## 24 step-ahead forecasts in a PIAR(2) model for the
    ## logarithms of the Real GNP in Germany.
    data("gergnp")
    lgergnp <- log(gergnp, base=exp(1))
    pred.out <- predictpiar(wts=lgergnp, p=2, hpred=24)
  

Methods for Function 'show' in Package 'partsm'

Description

This method shows the information provided by functions implemented in package 'partsm'.

Methods

object = "fit.partsm".

Shows the estimates of the autoregressive or periodic autoregressive coefficients.

object = "Ftest.partsm".

Shows the F-test statistic, the null and the alternative hypotheses entailed in the procedure, as well as the freedom degrees, the p-value and a symbol indicating the significance of the F-statistic according to usual codes, i.e. Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1.

object = "fit.piartsm".

Shows the estimated periodic autoregressive coefficients in the restricted non-linear PIAR.

object = "LRur.partsm".

Shows the LR statistics and a one-side test constructed as sign(g(\hat{\alpha}) - 1) * LR^{1/2}, where g(\hat{\alpha}) is the product of the periodic differencing filter parameters estimated under the alternative.

object = "pred.piartsm".

Shows out-of-sample forecasts and the corresponding standard errors, as well as the 95 per cent confidence intervals.

Author(s)

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

See Also

fit.partsm-class, Ftest.partsm-class, and summary.

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)

    ## Fit an AR(4) model with intercept and seasonal dummies.
    dcar <- list(regular=c(1,0,c(1,2,3)), seasonal=c(0,0), regvar=0)
    out.ar <- fit.ar.par(wts=lgergnp, type="AR", detcomp=dcar, p=4)
    show(out.ar)

    ## Fit a PAR(2) model with seasonal intercepts.
    out.par <- fit.ar.par(wts=lgergnp, type="PAR", detcomp=detcomp, p=2)
    show(out.par)

    ## Fnextp.test
    Fnextp.out <- Fnextp.test(wts=lgergnp, detcomp=detcomp, p=1, type="PAR")
    show(Fnextp.out)

    ## Fpar.test
    Fpar.out <- Fpar.test(wts=lgergnp, detcomp=detcomp, p=2)
    show(Fpar.out)

    ## Fsh.test
    ar4 <- fit.ar.par(wts=lgergnp, type="AR", p=4, detcomp=detcomp)
    Fsh.out <- Fsh.test(res=residuals(ar4@lm.ar), s=frequency(lgergnp))
    show(Fsh.out)

    ## Fit a PIAR(2) model with seasonal intercepts.
    out.piar <- fit.piar(wts=lgergnp, detcomp=detcomp, p=2)
    show(out.piar)

    ## Fpari.piar.test
    Fpari1.out <- Fpari.piar.test(wts=lgergnp, detcomp=detcomp, p=2, type="PARI1")
    show(Fpari1.out)

    ## Fit a PIAR(2) model with seasonal intercepts.
    out.piar <- fit.piar(wts=lgergnp, detcomp=detcomp, p=2)
    show(out.piar)

    ## Test for a single unit root in a PAR(2) model with seasonal intercepts.
    out.LR <- LRurpar.test(wts=lgergnp, detcomp=detcomp, p=2)
    show(out.LR)

    ## 24 step-ahead forecasts in a PIAR(2) model.
    pred.out <- predictpiar(wts=lgergnp, p=2, hpred=24)
    show(pred.out)
  

Methods for Function 'summary' in Package 'partsm'

Description

This method summarises the information provided by functions implemented in package 'partsm'.

Methods

object = "fit.partsm".

Displays the estimates of the autoregressive or periodic autoregressive coefficients, and a summary of the fitted model is also added.

object = "Ftest.partsm".

Shows a summary of the information provided by the F-test statistics in package 'partsm'. The null and the alternative hypotheses entailed in the procedure, as well as the freedom degrees, the p-value and a symbol indicating the significance of the F-statistic according to usual codes, i.e. Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1.

In addition, a summary of the fitted models for the null and the alternative hypotheses is also added.

object = "fit.piartsm".

Displays the estimated periodic autoregressive coefficients in the restricted non-linear PIAR, the other coefficients entailed in the selected model, and the periodically differenced data.

object = "fit.LRur.partsm".

Reports the the LR statistics and a one-side test, and a summary of the fitted models under the null and the alternative hypotheses.

Author(s)

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

See Also

fit.partsm-class, Ftest.partsm-class, and show.

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)

    ## Fit an AR(4) model with intercept and seasonal dummies.
    dcar <- list(regular=c(1,0,c(1,2,3)), seasonal=c(0,0), regvar=0)
    out.ar <- fit.ar.par(wts=lgergnp, type="AR", detcomp=dcar, p=4)
    summary(out.ar)

    ## Fit a PAR(2) model with seasonal intercepts.
    detcomp <- list(regular=c(0,0,0), seasonal=c(1,0), regvar=0)
    out.par <- fit.ar.par(wts=lgergnp, type="PAR", detcomp=detcomp, p=2)
    summary(out.par)

    ## Fnextp.test
    Fnextp.out <- Fnextp.test(wts=lgergnp, detcomp=detcomp, p=1, type="PAR")
    summary(Fnextp.out)

    ## Fpar.test
    Fpar.out <- Fpar.test(wts=lgergnp, detcomp=detcomp, p=2)
    summary(Fpar.out)

    ## Fsh.test
    ar4 <- fit.ar.par(wts=lgergnp, type="AR", p=4, detcomp=detcomp)
    Fsh.out <- Fsh.test(res=residuals(ar4@lm.ar), s=frequency(lgergnp))
    summary(Fsh.out)

    ## Fit a PIAR(2) model.
    out.piar <- fit.piar(wts=lgergnp, detcomp=detcomp, p=2)
    summary(out.piar)

    ## Fpari.piar.test
    Fpari1.out <- Fpari.piar.test(wts=lgergnp, detcomp=detcomp, p=2, type="PARI1")
    summary(Fpari1.out)

    ## Fit a PIAR(2) model with seasonal intercepts.
    out.piar <- fit.piar(wts=lgergnp, detcomp=detcomp, p=2)
    summary(out.piar)
    
    ## Test for a single unit root in a PAR(2) model with seasonal intercepts.
    out.LR <- LRurpar.test(wts=lgergnp, detcomp=detcomp, p=2)
    summary(out.LR)
  

Real per Capita Disposable Income in Sweden (1963.1-1988.1)

Description

Real per capita disposable income in Sweden. Sample: 1963.1 - 1988.1. Remark: measured in logs. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("swdipc")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

B. Assarson (1991): 'The Stochastic Behaviour of Durables and Nondurables Consumption in Sweden', Working paper 1991:21, Department of Economics, Uppsala University.

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

Real per Capita non-durables Consumption in Sweden (1963.1 - 1988.1)

Description

Real per capita non-durables consumption in Sweden. Sample: 1963.1 - 1988.1. Remark: measured in logs. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("swndcpc")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

B. Assarson (1991): 'The Stochastic Behaviour of Durables and Nondurables Consumption in Sweden', Working paper 1991:21, Department of Economics, Uppsala University.

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

United Kingdom Total Consumption (1955.1-1988.4)

Description

United Kingdom total consumption. Sample: 1955.1 - 1988.4. Remark: at 1985 prices. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("ukcons")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

D.R. Osborn (1990): A Survey of Seasonality in UK Macroeconomic Variables, International Journal of Forecasting, 6, pp. 327-36.

United Kindom Exports of Goods and Services (1955.1-1988.4)

Description

United Kindom exports of goods and services. Sample: 1955.1 - 1988.4. Remark: at 1985 prices. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("ukexp")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

D.R. Osborn (1990): A Survey of Seasonality in UK Macroeconomic Variables, International Journal of Forecasting, 6, pp. 327-36.

United Kingdom Gross Domestic Product (1955.1-1988.4)

Description

United Kingdom gross domestic product. Sample: 1955.1 - 1988.4. Remark: at 1985 prices. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("ukgdp")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

D.R. Osborn (1990): A Survey of Seasonality in UK Macroeconomic Variables, International Journal of Forecasting, 6, pp. 327-36.

United Kindom Imports of Goods and Services (1955.1-1988.4)

Description

United Kindom imports of goods and services. Sample: 1955.1 - 1988.4. Remark: at 1985 prices. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("ukimp")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

D.R. Osborn (1990): A Survey of Seasonality in UK Macroeconomic Variables, International Journal of Forecasting, 6, pp. 327-36.

Real Total Investment in the United Kindom (1955.1-1988.4)

Description

Real total investment in the United Kindom. Sample: 1955.1 - 1988.4. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("ukinvest")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

D.R. Osborn (1990): A Survey of Seasonality in UK Macroeconomic Variables, International Journal of Forecasting, 6, pp. 327-36.

United Kindom non-durables Consumption (1955.1-1988.4)

Description

United Kindom nondurables consumption. Sample: 1955.1 - 1988.4. Remark: at 1985 prices. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("ukndcons")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

D.R. Osborn (1990): A Survey of Seasonality in UK Macroeconomic Variables, International Journal of Forecasting, 6, pp. 327-36.

United Kindom Public Investment (1962.1-1988.4)

Description

United Kindom public investment. Sample: 1962.1 - 1988.4. Remark: at 1985 prices. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("ukpinvest")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

D.R. Osborn (1990): A Survey of Seasonality in UK Macroeconomic Variables, International Journal of Forecasting, 6, pp. 327-36.

United Kindom Workforce (1955.1-1988.4)

Description

United Kindom workforce. Sample: 1955.1 - 1988.4. Remark: Consisting of workforce in employment and unemployment. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("ukwf")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

D.R. Osborn (1990): A Survey of Seasonality in UK Macroeconomic Variables, International Journal of Forecasting, 6, pp. 327-36.

Total Industrial Production Index for the United States (1960.1-1991.4)

Description

OECD Main Economic Indicators: Total industrial production index for the United States. Sample: 1960.1 - 1991.4. Remark: (1985=100). This time series is part of the data set used in the book Franses (1996).

Usage

    data ("usaipi")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).

Total Industrial Production Index for the United States (1960.1-1991.4). Seasonally Adjusted

Description

OECD Main Economic Indicators: Total industrial production index for the United States. Sample: 1960.1 - 1991.4. Remark: (1985=100), Seasonally Adjusted. This time series is part of the data set used in the book Franses (1996).

Usage

    data ("usaipisa")
  

Format

An object of class 'ts' containing the data.

Source

Previously available on P.H. Franses's homepage: https://www.eur.nl/en/people/philip-hans-franses.

References

P.H. Franses: Periodicity and Stochastic Trends in Economic Time Series (Oxford University Press, 1996).