pepsi: moving average expansion of a periodic autoregression
In pear: Package for Periodic Autoregression Analysis

Description Usage Arguments Details Value Side Effects References See Also Examples

A periodic autoregression can be represented as an infinite periodic moving average process. This function calculates the coefficients in this expansion. These coefficients are needed in various time series computations such as in computing the variances of forecasts, variances of residual autocorrelations and theoretical autocovariances of a periodic autoregression. The function pepsi is used by pear to calculate the estimated standard deviations of the residual autocorrelations in a fitted periodic autoregression.

1	pepsi(phi, lag.max)

`phi`	matrix with (i,j)-entry phi[i, j] where phi[i,j] is the autoregressive coefficient for period i at lag j. Here i=1,...,p and j=1,...,m where m is highest ar order specified.
`lag.max`	maximum number of lags to calculate in the moving average expansion.

The moving average expansion for a periodic autoregressive is defined in equation (1.4) of McLeod (1994) and the algorithm implements the recursion given in equation (1.5).

matrix with (i,j)-entry psi[i, j] where psi[i,j] is the autoregressive coefficient for period i at lag j. Here i=1,...,p and j=1,...,lag.max.

none

McLeod, A.I. (1994), "Diagnostic Checking of Periodic Autoregression" Journal of Time Series Analysis, Vol. 15, No. 2, pp.221–233.

pear

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data(Fraser)
pear.out <- pear(log(Fraser), ic="bic")
pepsi(pear.out$phi,lag.max=20)

           lags
periods         lag 1      lag 2       lag 3        lag 4       lag 5
  period 1  0.6633120  0.4942948  0.37289514  0.432195529  0.07022041
  period 2  0.7616731  0.5052269  0.37649104  0.284024192  0.32919171
  period 3  0.8127845  0.6190761  0.41064063  0.306006089  0.23085047
  period 4  0.7652456  0.6219798  0.47374527  0.314240938  0.23416982
  period 5  0.2018468  0.1544623  0.12554460  0.095623945  0.06342851
  period 6  0.2609638 -0.2076482  0.03951781  0.032119461  0.02446453
  period 7  0.7691441  0.0298545 -0.19419975  0.004002795  0.00325341
  period 8  0.7508529  0.5775140  0.02241633 -0.145815441  0.00300551
  period 9  0.7505846  0.5635786  0.43347313  0.016825355 -0.10944682
  period 10 1.1590270  0.1883114  0.50544298  0.388758466  0.01508975
  period 11 0.7543983  0.8743680  0.14206180  0.381305314  0.29327872
  period 12 0.7451919  0.5621715  0.65157197  0.105863311  0.28414565
           lags
periods            lag 6        lag 7        lag 8         lag 9        lag 10
  period 1   0.188477229  0.144966141  0.005626893 -0.0366022316  0.0007544357
  period 2   0.053484996  0.143558034  0.110416809  0.0042858527 -0.0278789350
  period 3   0.267561922  0.043471776  0.116681748  0.0897450732  0.0034834747
  period 4   0.176657307  0.204750587  0.033266586  0.0892901954  0.0686770235
  period 5   0.047266418  0.035657704  0.041328242  0.0067147525  0.0180229362
  period 6   0.016227617  0.012092689  0.009122704  0.0105734602  0.0017179092
  period 7   0.002478035  0.001643710  0.001224880  0.0009240471  0.0010709955
  period 8   0.002442832  0.001860639  0.001234185  0.0009197044  0.0006938234
  period 9   0.002255890  0.001833552  0.001396567  0.0009263599  0.0006903159
  period 10 -0.098156902  0.002023185  0.001644413  0.0012525052  0.0008308018
  period 11  0.011383678 -0.074049398  0.001526287  0.0012405424  0.0009448878
  period 12  0.218548939  0.008483025 -0.055181015  0.0011373768  0.0009244422
           lags
periods            lag 11        lag 12        lag 13        lag 14
  period 1   0.0006131937  0.0004670531  3.098020e-04  2.308619e-04
  period 2   0.0005746334  0.0004670531  3.557418e-04  2.359678e-04
  period 3  -0.0226595667  0.0004670531  3.796135e-04  2.891414e-04
  period 4   0.0026657137 -0.0173401340  3.574103e-04  2.904976e-04
  period 5   0.0138622344  0.0005380657 -3.500050e-03  7.214212e-05
  period 6   0.0046110067  0.0035465284  1.376593e-04 -8.954564e-04
  period 7   0.0001740086  0.0004670531  3.592311e-04  1.394364e-05
  period 8   0.0008041600  0.0001306549  3.506882e-04  2.697297e-04
  period 9   0.0005207732  0.0006035901  9.806753e-05  2.632211e-04
  period 10  0.0006191068  0.0004670531  5.413272e-04  8.795143e-05
  period 11  0.0006267554  0.0004670531  3.523441e-04  4.083763e-04
  period 12  0.0007041228  0.0004670531  3.480442e-04  2.625640e-04
           lags
periods            lag 15        lag 16        lag 17        lag 18
  period 1   1.741618e-04  2.018583e-04  3.279666e-05  8.802888e-05
  period 2   1.758413e-04  1.326544e-04  1.537500e-04  2.498033e-05
  period 3   1.917910e-04  1.429211e-04  1.078194e-04  1.249656e-04
  period 4   2.212642e-04  1.467672e-04  1.093697e-04  8.250834e-05
  period 5   5.863600e-05  4.466146e-05  2.962448e-05  2.207593e-05
  period 6   1.845691e-05  1.500149e-05  1.142623e-05  7.579159e-06
  period 7  -9.070160e-05  1.869518e-06  1.519515e-06  1.157374e-06
  period 8   1.046962e-05 -6.810356e-05  1.403733e-06  1.140932e-06
  period 9   2.024550e-04  7.858334e-06 -5.111748e-05  1.053620e-06
  period 10  2.360687e-04  1.815709e-04  7.047712e-06 -4.584449e-05
  period 11  6.635041e-05  1.780898e-04  1.369767e-04  5.316782e-06
  period 12  3.043187e-04  4.944379e-05  1.327111e-04  1.020740e-04
           lags
periods            lag 19        lag 20
  period 1   6.770689e-05  2.628058e-06
  period 2   6.704923e-05  5.157051e-05
  period 3   2.030363e-05  5.449657e-05
  period 4   9.562940e-05  1.553726e-05
  period 5   1.665404e-05  1.930248e-05
  period 6   5.647928e-06  4.260787e-06
  period 7   7.677000e-07  5.720838e-07
  period 8   8.690174e-07  5.764297e-07
  period 9   8.563662e-07  6.522711e-07
  period 10  9.449347e-07  7.680283e-07
  period 11 -3.458500e-05  7.128571e-07
  period 12  3.962023e-06 -2.577246e-05