# 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 oberved data.

### Usage

1 2 | ```
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

1 2 3 4 5 6 |