Predictions for a Restricted Periodic Autoregressive Model

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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 oberved data.

Usage

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    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,

Φ_0 y_t = Ψ + Φ_1 Y_{T-1} + ... + Φ_P y_{T-P} + ε_T ,

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

y_t = Φ_0^{-1} Ψ + Φ_0^{-1} Φ_1 Y_{T-1} + ... + Φ_0^{-1} Φ_P y_{T-P} + Φ_0^{-1} + ε_T .

Multi-step-ahead forecasts are obtained recursively.

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

σ^2 Φ_0^{-1} (Φ_0^{-1})^{'},

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

σ^2 Φ_0^{-1} (Φ_0^{-1})^{'} + (h-1) (Γ Φ_0^{-1}) (Γ Φ_0^{-1})^{'},

where Γ = Φ_0^{-1} Φ_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

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    ## 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)
  

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