predictpiar: Predictions for a Restricted Periodic Autoregressive Model

In partsm: Periodic Autoregressive Time Series Models

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

wts	a univariate time series object.
p	the order of the PAR model. At present first and second order are considered.
hpred	number of out-of-sample observations to forecast. It must be a multiple of 4.

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 [e-mail].

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)
  

partsm documentation built on June 8, 2025, 10:33 a.m.