partsm (version 1.1-4)

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)

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.

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.

Author

Javier Lopez-de-Lacalle .

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.

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)