Convert PIAR coefficients to PAR coefficients

Usage

pi1ar2par(picoef, parcoef)
piar2par(picoef, parcoef)

Arguments

picoef: coefficients of the periodic integration filter, a numeric vector or matrix, see Details.
parcoef: coefficients of the periodically correlated part of the model.

Details

These functions expand periodic filters represented in multiplicative form. The non-periodic analogue of the operation of these functions is representing a multiplicative filter like \((1-B)(1-aB)\), where \(B\) is the backward shift operator, by the single filter \(1 - (1+a)B + aB^2\), which is just a product of the polynomials, \(1-B\) and \(1 - aB\).

In the non-periodic case however this operation is not, in general, equivalent to multiplication of the corresponding polynomials. It is also not commutative.

pi1ar2par converts PIAR(1) model coefficients specified as a set of coefficients corresponding to a periodic unit root and PAR coefficients to coefficients for a single filter.

piar2par does the same but admits higher order periodic integration.

picoef is a matrix, specifying one or more first order periodic unit root filters. Each column contains the coefficients of one filter. If there is only one filter, its coefficients can be given as a numeric vector.

The filters are applied from right to left, in the sense that first the PAR filter is applied to the time series, then the filter specified by the last column and so on.

Value

a matrix, whose i-th row contains the coefficients for the i-th season.

Author

Georgi N. Boshnakov

Examples

## Lina's example
parcoef    <- rbind(c(0.5, -0.06), c(0.6, -0.08),
                    c(0.7, -0.1),  c(0.2, 0.15) )
picoef1    <- c(0.8, 1.25, 2, 0.5)
parcoef2   <- pi1ar2par(picoef1, parcoef)

picoef2    <- c(4, 0.25, 5, 0.2)
coefper2I2 <- pi1ar2par(picoef2, parcoef2)