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+Φ1YT−1+...+ΦPyT−P+ϵT,
where the Φi,i=1,2,...,P are s×s matrices containing the c("phiis\n", "parameters."), the one-step-ahead forecasts for the year T+1 is straightforward,
Multi-step-ahead forecasts are obtained recursively.
The prediction errors variances for the one-step-ahead forecast are the diagonal elements of
σ2Φ0−1(Φ0−1)′,
whereas for h=2,3,... years ahead forecasts it becomes
σ2Φ0−1(Φ0−1)′+(h−1)(ΓΦ0−1)(ΓΦ0−1)′,
where Γ=Φ0−1Φ1.
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.
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)