predictpiar function

Predictions for a Restricted Periodic Autoregressive Model

Predictions for a Restricted Periodic Autoregressive Model

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.

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,

Φ0yt=Ψ+Φ1YT1+...+ΦPyTP+ϵT, \Phi_0 y_t = \Psi + \Phi_1 Y_{T-1} + ... + \Phi_P y_{T-P} + \epsilon_T ,

where the Φi,i=1,2,...,P\Phi_i, i=1,2,...,P are s×ss \times s matrices containing the c("phiis\n\\phi_{is}\n", "parameters. parameters."), the one-step-ahead forecasts for the year T+1T+1 is straightforward,

yt=Φ01Ψ+Φ01Φ1YT1+...+Φ01ΦPyTP+Φ01+ϵT. 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

σ2Φ01(Φ01), \sigma^2 \Phi_0^{-1} (\Phi_0^{-1})^{'},

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

σ2Φ01(Φ01)+(h1)(ΓΦ01)(ΓΦ01), \sigma^2 \Phi_0^{-1} (\Phi_0^{-1})^{'} + (h-1) (\Gamma \Phi_0^{-1}) (\Gamma \Phi_0^{-1})^{'},

where Γ=Φ01Φ1\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.

See Also

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

Returns

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).

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)