# pear: fit a periodic autoregression model In pear: Package for Periodic Autoregression Analysis

## Description

Periodic time series models of any order, say m[j], for the j-th period, j=1,...,p can be fit to a periodic time series of period p. A generalization of the Yule-Walker method is used so that when p=1 the results from this function will be equivalent to the Splus function ar.yw(). If desired the aic or bic criterion can be used to select the model orders. Otherwise the user can select based on the partial autocorrelation function as suggested in Box and Jenkins (1976) and McLeod (1994).

## Usage

 `1` ```pear(z, m, ic="none") ```

## Arguments

 `z` a univariate time series object. Note that the period of z is given by attr(z, "tsp")[3]. Additional information about the time series can be provided in a title string by setting attr(z, "title") to the desired string. This title will then be displayed on the plot. Abbreviations for the periods may be provided in attr(z, "abb"). For example, to use the standard monthly abbreviations: attr(z, "abb")<-month.abb. These abbreviations will be used to aid one in interpreting the output. `m` If ic="none" then m is a required argument. In this case m specifies the order of the periodic autoregression to be fitted. Typically m is specified as a vector of length p where p is the period and m[k], k=1,...,p indicates the order for the k-th period. For convenience, if all periods are the same order then m can just be that scalar value. When ic="aic" or ic="bic" then the argument m is ignored. `ic` The default ic="none" means the model orders are supplied. Otherwise if ic="aic" or ic="bic" the automatic criteria aic or bic are used.

## Details

Let z[t] be a period time series with period p and let m[j] denote the order of the autoregressive model for the j-th period, j=1,...,p. The parameters of this model can be estimated using the Yule-Walker type equations given in McLeod eq (3.1) and (3.2). The covariance matrix of the autoregressive parameters is obtained by replacing the theoretical autocovariances in eq (3.3, note addendum correction) with their sample values.

## Value

a list with the following named components: model.orders vector of length p, indicating the fitted ar order for each period k, k=1,...,p phi matrix of dimension p by m where m = max(model.orders). The (i,j) entry is phi[i,j] which is the autoregression coefficient for period i and lag j. se.phi matrix of standard deviations for the estimated phi's. For those phi's set to 0, the corresponding se.phi's are also set to 0. resvar vector of length p residuals time series object of length(z) portmanteau.test list: portmanteau test at various lags The named components of this list are: QM = matrix of portmanteau statistics for each period and lag QM.df = corresponding df of QM QM.sl = corresponding sl of QM residual.acf residual autocorrelation matrix residual.acf.sd estimated standard errors of the residual autocorrelations cov list with p components: cov[[i]] is the estimated covariance matrix for the parameters of period i

none

## References

Hipel, K.W. and McLeod, A.I. (1994) "Time Series Modelling of Water Resources and Environmental Systems" Elsevier, Amsterdam ISBN 0-444-89270-2. (1013 pages).

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

McLeod, A.I. (1995), Errata (see file errata.tex included with these files)

peacf, pepacf, ar.yw

## Examples

 ```1 2``` ```data(Fraser) pear(log(Fraser), ic="bic") ```

### Example output

```\$model.orders
period 1  period 2  period 3  period 4  period 5  period 6  period 7  period 8
1         1         1         1         1         3         2         1
period 9 period 10 period 11 period 12
1         3         1         1

\$phi
lag 1      lag 2     lag 3
period 1  0.6633120  0.0000000 0.0000000
period 2  0.7616731  0.0000000 0.0000000
period 3  0.8127845  0.0000000 0.0000000
period 4  0.7652456  0.0000000 0.0000000
period 5  0.2018468  0.0000000 0.0000000
period 6  0.2609638 -0.2603229 0.1984197
period 7  0.7691441 -0.1708642 0.0000000
period 8  0.7508529  0.0000000 0.0000000
period 9  0.7505846  0.0000000 0.0000000
period 10 1.1590270 -0.6816364 0.3640488
period 11 0.7543983  0.0000000 0.0000000
period 12 0.7451919  0.0000000 0.0000000

\$se.phi
lag 1      lag 2      lag 3
period 1  0.06411463 0.00000000 0.00000000
period 2  0.06830949 0.00000000 0.00000000
period 3  0.07801672 0.00000000 0.00000000
period 4  0.12172331 0.00000000 0.00000000
period 5  0.06715109 0.00000000 0.00000000
period 6  0.07954896 0.05963304 0.07706646
period 7  0.10493248 0.08051855 0.00000000
period 8  0.06854167 0.00000000 0.00000000
period 9  0.08905648 0.00000000 0.00000000
period 10 0.13656207 0.20895763 0.16588797
period 11 0.09123781 0.00000000 0.00000000
period 12 0.07113319 0.00000000 0.00000000

\$resvar
period 1   period 2   period 3   period 4   period 5   period 6   period 7
0.02947049 0.02544491 0.03173720 0.08818641 0.04713156 0.02345282 0.02533583
period 8   period 9  period 10  period 11  period 12
0.01581397 0.02495953 0.03599596 0.05159159 0.03847035

\$residuals
Jan           Feb           Mar           Apr           May
1912                              0.000000e+00  5.721555e-02  1.175709e-01
1913 -3.258406e-01  2.596907e-01 -2.226714e-01 -2.222533e-01 -4.688577e-01
1914  4.067548e-01 -1.784681e-01  1.570446e-01  1.811963e-01  1.343034e-01
1915 -1.744470e-02 -9.860352e-02  8.031318e-02  4.490414e-01 -2.267148e-01
1916 -1.003728e-01  5.067634e-01 -1.414051e-01 -1.992890e-01 -1.556473e-01
1917 -8.176069e-02 -1.092491e-01 -1.724108e-01 -3.430667e-01  1.980510e-02
1918  3.538601e-01 -1.567799e-01 -8.208499e-02  9.487146e-02  6.224578e-02
1919 -1.308004e-01 -1.323231e-01 -2.009289e-01  2.062348e-01 -1.250024e-01
1920  6.614437e-02 -9.087964e-02 -2.252497e-01 -4.585874e-01 -4.922424e-02
1921 -3.350202e-02  4.293999e-02 -4.596902e-02 -3.071827e-02  1.193670e-01
1922 -1.458264e-01 -2.183531e-01 -1.495018e-01 -3.973890e-01 -3.460873e-02
1923 -4.821564e-02 -2.211147e-01 -1.092823e-01  8.305990e-02  4.383948e-02
1924 -2.769485e-01  4.168267e-01 -1.950578e-01 -5.142389e-01  3.606164e-01
1925 -1.230843e-01 -4.329207e-02 -9.079959e-02  4.364792e-01  3.309091e-01
1926  1.778208e-02 -2.199408e-01  1.235637e-01  4.088300e-01 -9.218828e-02
1927 -1.596624e-03 -1.766224e-01 -1.241188e-01 -3.199559e-01 -2.395240e-01
1928  1.838263e-01 -1.861212e-01  5.125989e-02 -3.677473e-02  3.170613e-01
1929 -9.055998e-02 -1.028659e-01 -5.384942e-02 -5.050318e-01 -3.710180e-01
1930  2.831683e-01 -1.068893e-01 -2.106693e-01  3.045650e-01 -3.257996e-01
1931  1.152278e-02 -4.857992e-02  5.291106e-02 -3.028069e-01 -3.057653e-02
1932 -8.086112e-02  8.860597e-02  2.041928e-01  1.665643e-01  8.595320e-02
1933 -9.829030e-02 -1.757170e-01 -1.024656e-01 -1.123653e-01  1.205825e-01
1934  9.888489e-02  1.844503e-01  2.834700e-02  4.108331e-01  1.658607e-01
1935  1.706912e-01  3.790624e-01 -2.141793e-01 -5.033792e-01  2.426439e-02
1936 -1.499829e-01 -4.709321e-02 -5.685531e-02  1.981667e-01  2.353384e-01
1937 -5.305823e-01 -1.738699e-01  1.291833e-01  1.501383e-03 -1.648938e-01
1938 -4.675644e-02 -2.106905e-01  7.750340e-02 -4.234799e-02 -2.210396e-01
1939  3.333940e-02 -2.064353e-01  1.683528e-01  3.776290e-01  1.331839e-01
1940 -3.272108e-01 -7.667727e-02  1.143320e-01  2.409685e-01 -5.393414e-02
1941  1.609808e-04  2.520528e-02  7.966475e-02  3.582103e-01 -3.745282e-01
1942 -2.105621e-01 -1.423219e-01 -1.407671e-01 -2.398250e-02 -2.552821e-01
1943 -4.405427e-02  2.563151e-03 -1.469047e-01  4.634041e-01 -4.738079e-01
1944 -2.183619e-02 -8.320794e-02 -2.921771e-02  1.956457e-01 -3.630967e-01
1945 -1.698236e-02 -8.739679e-02 -1.418986e-01 -4.004576e-01 -2.785992e-02
1946 -8.839638e-02 -1.133921e-01  7.855637e-02 -1.785497e-01  3.233297e-01
1947 -1.139764e-01 -1.611775e-02  1.462319e-01  3.479846e-01  8.894095e-02
1948  7.518810e-02 -2.326357e-01 -9.972051e-02 -3.176611e-01  4.243752e-01
1949 -1.184598e-01 -3.133870e-02  4.284637e-02  4.120314e-01  2.370064e-01
1950 -2.864985e-01 -1.599167e-02 -5.282667e-02 -3.755754e-01 -2.621055e-02
1951 -1.591443e-01 -5.538402e-02 -1.477331e-01  1.836240e-01  2.356696e-01
1952 -4.855685e-03 -5.467704e-02 -1.381822e-01  2.903710e-01  1.664290e-01
1953  5.194858e-02  6.656803e-02 -1.502567e-01 -3.414065e-01  1.565815e-01
1954 -1.588315e-01  1.653662e-01 -1.426590e-01 -6.693839e-01 -4.614976e-03
1955 -1.399330e-01 -5.993195e-02 -1.691378e-01 -1.600370e-01 -2.530214e-01
1956 -6.693086e-02 -1.099230e-01 -5.780330e-02  2.638718e-01  1.890978e-01
1957  3.256565e-01  1.014507e-01 -1.299792e-01 -1.702829e-01  5.556314e-01
1958  1.557201e-01  5.806317e-02  1.576609e-01  8.294779e-02  2.122073e-01
1959  6.401319e-02 -1.684218e-01  3.073768e-03 -3.505305e-02  2.343356e-02
1960 -6.637268e-02  1.129964e-02  1.271617e-02  2.927019e-01 -1.696833e-01
1961  1.668581e-01 -8.044133e-03  5.493180e-02  1.027905e-01  4.585702e-02
1962  2.110816e-01  4.557641e-01 -4.371250e-01  3.939627e-01 -2.135676e-01
1963  3.826947e-02  3.808115e-01  4.740467e-02 -1.697578e-01 -1.281356e-01
1964  7.592362e-02  2.562593e-02 -6.675663e-02 -1.396830e-01 -1.583550e-01
1965  8.885925e-02  1.072119e-01  6.978780e-02  3.970724e-02  4.108524e-02
1966 -8.178084e-02  3.492787e-02  5.506327e-02  3.611153e-01  2.329127e-02
1967  2.532721e-02  8.048379e-02 -5.098134e-02 -3.171424e-01  1.693669e-01
1968  4.253393e-01  1.508643e-01  3.633916e-01 -5.334193e-01  7.865000e-02
1969 -9.929665e-03  6.059335e-02 -3.533855e-02  2.927670e-01  6.606432e-03
1970 -9.545309e-02  1.005926e-01  1.112506e-02 -2.606569e-01 -2.417743e-01
1971  1.385822e-01  2.470989e-01 -1.361118e-01 -6.551537e-02  2.614472e-01
1972 -3.752375e-02  1.390318e-01  5.919686e-01 -2.046261e-01  2.526680e-01
1973  1.385134e-01 -6.146881e-02  3.833664e-02 -7.366117e-02  4.115909e-02
1974  6.048914e-02  3.454127e-02  4.513380e-02  2.632911e-01  1.417339e-01
1975 -5.750201e-02  2.258386e-02 -4.350287e-02 -3.064058e-01 -1.097118e-01
1976  2.562484e-02  2.797923e-02 -4.199140e-02  2.571136e-01  3.260860e-01
1977  1.097080e-01  1.413908e-01  1.503646e-02  1.246058e-01 -8.442757e-02
1978  8.338447e-03 -3.973398e-02  2.024498e-01  7.430409e-02 -2.237706e-01
1979 -7.385170e-02 -5.988406e-02  2.829998e-01 -3.480068e-01  7.822072e-02
1980 -1.925261e-01 -4.482000e-02  6.006344e-02  1.987021e-01  5.739316e-02
1981  3.494699e-01  1.758681e-02  4.795265e-03 -2.057843e-01  1.031290e-02
1982 -1.295020e-01  1.726459e-01 -4.691558e-02 -4.965449e-01  2.153010e-01
1983  5.702008e-02  9.658362e-02  2.951005e-01 -1.127614e-01 -1.920073e-01
1984  3.893655e-01 -4.876059e-03  2.014093e-01 -4.182497e-02 -5.502572e-01
1985 -4.719049e-02 -2.057526e-02  1.221452e-01  2.610693e-01  5.919898e-02
1986  1.419888e-01  4.394951e-02  4.802117e-01 -8.802753e-02 -2.833737e-01
1987  2.186597e-01  4.059202e-02  3.702811e-01 -1.288066e-02  5.142783e-03
1988 -1.700258e-01  3.127982e-02  1.905478e-01  4.055424e-01  5.641129e-02
1989 -1.750462e-02 -1.101529e-01 -1.414288e-02  2.332095e-01  2.996384e-03
1990  1.173560e-01 -2.201329e-01  1.249404e-01  5.211622e-01 -6.401760e-02
Jun           Jul           Aug           Sep           Oct
1912 -1.202146e-01 -2.803713e-02  2.233733e-01 -1.752565e-01  2.622513e-02
1913  2.690574e-01 -6.057507e-02  4.087512e-01  1.126472e-01 -3.561219e-01
1914  2.898698e-02  1.902480e-01 -7.338789e-02 -1.376321e-01  1.335855e-01
1915 -2.540459e-01  2.082256e-01  1.186783e-01 -3.696631e-01  6.025071e-02
1916 -5.011390e-02  1.756759e-01 -1.037972e-02 -8.524152e-02 -1.291617e-01
1917  7.355925e-02  5.703068e-02 -7.108466e-02 -1.423159e-01  2.040347e-01
1918  1.373858e-01  1.557848e-01  2.826005e-02 -6.676824e-02 -1.771807e-01
1919  2.704738e-02  2.662001e-01  1.029624e-01 -1.805426e-01 -2.002980e-01
1920 -2.881611e-02  5.026379e-01  2.099146e-02 -4.542309e-02  3.330792e-01
1921  2.590800e-01 -1.487811e-02  4.594751e-02  2.076856e-03  2.582504e-01
1922  9.972273e-02 -1.542273e-01 -1.022915e-02  7.840095e-02  1.194174e-01
1923  1.399698e-01 -5.111860e-02 -3.117602e-02  6.575656e-02 -4.621102e-01
1924 -2.153841e-01 -1.769803e-02  1.653629e-01  2.007175e-01 -1.848261e-01
1925  2.382071e-02 -6.676082e-02 -5.231161e-02  1.166957e-01 -5.576425e-01
1926 -3.524365e-01  1.166507e-01 -1.571206e-01 -1.427060e-01  5.993417e-02
1927  7.093142e-02 -3.119283e-02 -2.467673e-02  4.437430e-02  3.600082e-01
1928 -1.838322e-02  6.251838e-02 -1.269100e-01 -2.158108e-01 -1.223446e-01
1929 -1.029336e-01 -2.774244e-01  4.181095e-02 -9.285818e-02 -9.624925e-02
1930  5.077680e-02  1.702640e-02 -9.922748e-02  1.050590e-02 -2.547496e-01
1931 -1.784442e-01  3.565084e-02 -1.357435e-01  5.496140e-02  2.604526e-02
1932  4.011842e-02 -5.530825e-02  1.557837e-01 -7.485458e-02 -1.661912e-02
1933  8.944173e-02  2.850927e-01  1.060835e-01 -9.671990e-02  2.432258e-03
1934 -2.748280e-02 -4.643543e-02  1.020571e-01  1.109698e-01  1.811410e-01
1935 -1.101294e-01  1.373210e-01 -1.008135e-01 -4.584641e-02 -1.796651e-01
1936  2.273690e-02 -8.447749e-02 -8.750893e-02  3.603283e-02  1.168324e-01
1937  2.453657e-02 -8.001239e-02 -4.317106e-02  1.153736e-01 -2.106405e-01
1938 -1.592959e-01 -1.452031e-01 -1.832604e-01  1.594965e-01 -3.193678e-01
1939 -3.935457e-02  9.193839e-02 -3.174929e-03 -4.654706e-02  1.747356e-01
1940 -1.031797e-01  1.757107e-02  4.945885e-02  4.060456e-02 -1.007948e-03
1941 -2.425885e-01 -5.117404e-02 -2.948433e-02  3.174737e-01  3.993207e-01
1942 -6.703813e-02 -7.693244e-02 -7.262046e-04 -1.394020e-01  5.373566e-02
1943  4.612135e-02  1.327843e-01 -4.248265e-02 -1.467874e-01 -9.447438e-02
1944 -1.039380e-01 -2.699738e-01  2.159153e-01  1.204964e-01  2.068312e-01
1945 -9.958986e-02 -2.470953e-01 -1.168399e-01 -6.988819e-02 -1.050313e-01
1946 -3.600342e-02 -3.705854e-02 -6.837567e-02 -1.999755e-01 -3.409423e-01
1947  8.900721e-02 -9.381864e-03 -8.268189e-02 -3.400417e-02  1.598462e-01
1948  2.684652e-01 -3.393429e-01  3.201265e-01  1.834571e-01  4.354448e-02
1949 -9.006497e-02 -1.431996e-01  1.996554e-01 -9.776569e-02 -6.817774e-03
1950  1.854538e-01 -1.399461e-02 -1.978122e-01 -6.389357e-02 -1.582617e-01
1951 -1.550082e-01  7.148529e-02 -1.774093e-01 -2.361388e-01 -7.276339e-02
1952  9.561457e-03  1.154318e-01 -1.215320e-01 -1.627866e-01 -1.799888e-02
1953 -2.403374e-01 -2.666028e-03 -1.149180e-01  1.473896e-01 -3.756548e-02
1954 -4.277458e-02  1.785317e-01  3.783343e-02  1.562748e-01 -4.101380e-02
1955  1.746120e-01  2.147978e-01 -4.171627e-02 -2.594388e-01 -3.549588e-02
1956  1.277255e-01 -1.467581e-01 -1.566598e-01 -2.229309e-02  1.035079e-01
1957 -2.009198e-01 -7.514927e-02  1.049863e-01  1.073800e-01 -2.156733e-01
1958  1.503477e-02 -2.302792e-01 -7.240577e-02  1.464299e-01  3.693583e-01
1959  6.104185e-02  9.937913e-02 -7.689098e-02  5.087359e-01 -5.210581e-02
1960  1.049451e-01  2.345201e-01 -1.358140e-01  1.928290e-01  7.017317e-02
1961  1.186213e-01 -2.797219e-01 -7.213172e-03  2.601062e-01  1.714685e-01
1962  1.443756e-01  8.554226e-02  1.658879e-01 -4.269605e-02  2.551728e-01
1963 -6.234988e-02 -2.947931e-02  5.925672e-02  1.846672e-01  1.370089e-02
1964  3.965558e-01  3.307166e-02  9.390725e-02  8.705444e-02  3.144259e-01
1965  5.239422e-02 -1.102085e-01  9.234569e-02 -7.598318e-02  3.113498e-01
1966  8.112159e-02  8.877706e-02  3.065916e-02 -2.806065e-05  1.897854e-01
1967  2.466878e-01 -7.069238e-02 -1.124010e-01 -8.848577e-03  3.402421e-02
1968 -3.277706e-02  1.968847e-01 -1.344176e-01  2.197119e-01 -1.093057e-01
1969 -4.965164e-03 -2.138298e-01  1.056325e-01  2.319791e-01  1.400778e-01
1970 -5.158920e-02 -3.692680e-01 -9.286558e-02  1.760737e-02 -1.912210e-02
1971  8.638631e-03 -9.380319e-04  2.157335e-03 -8.883692e-03  7.239712e-03
1972  3.118920e-01  7.743355e-03 -5.803339e-02 -1.509193e-01  7.233543e-02
1973 -1.414996e-01  7.950575e-03 -1.116547e-01 -2.004636e-01  4.028870e-01
1974  2.092732e-01  2.016992e-01 -1.562025e-02 -1.255940e-01  2.026512e-02
1975 -4.475951e-02  8.734588e-02 -1.026225e-01 -2.417311e-03  9.090727e-03
1976  9.214029e-03  3.753047e-01  3.545975e-01  1.847864e-01 -1.597822e-01
1977 -1.649375e-01  3.249929e-02  1.257900e-01 -2.783384e-02 -5.601774e-02
1978 -1.217191e-01 -6.758387e-02 -2.562966e-02  2.260764e-01 -5.410701e-02
1979 -1.841141e-01 -4.253111e-02 -2.059831e-01 -2.813471e-02 -1.464405e-01
1980 -3.137598e-01 -2.703491e-02 -2.035204e-02  2.920754e-01 -7.364804e-02
1981 -1.648135e-01 -2.280137e-02  1.165201e-01 -1.218628e-01 -2.703769e-02
1982  6.637926e-02  1.295684e-01  1.154062e-01  1.889398e-01 -1.727285e-01
1983 -1.233023e-01  3.869405e-02  1.940881e-02 -5.106174e-02 -1.657570e-01
1984  3.744296e-02  1.675092e-01 -6.311491e-02  1.611361e-01  4.348891e-02
1985  1.044547e-01 -1.890907e-01 -1.133199e-01 -1.442031e-02  1.749806e-01
1986  2.297531e-01 -1.990856e-01 -7.227606e-02 -1.535200e-01 -4.306682e-02
1987 -1.928576e-01 -1.471645e-01  5.978790e-02 -2.131642e-01 -2.232250e-01
1988 -8.362957e-02 -6.479436e-02  6.164953e-03 -1.028996e-01  2.821337e-02
1989 -8.534829e-02 -1.992696e-01  1.365019e-01 -1.685848e-01  2.297495e-02
1990  3.549551e-01 -2.924441e-02 -1.506332e-01 -2.353447e-01  2.259642e-02
Nov           Dec
1912 -2.023753e-01 -8.371192e-02
1913 -3.563444e-01 -8.110854e-02
1914  1.139835e-01 -1.114747e-01
1915  5.381637e-02 -4.329405e-01
1916 -2.645437e-01 -2.380060e-01
1917  1.631487e-01  2.027358e-01
1918 -1.520926e-01 -6.843971e-02
1919  6.797975e-02  1.743626e-01
1920 -2.222222e-01 -4.729552e-02
1921 -2.644246e-02  4.648648e-02
1922 -2.981411e-01  1.886132e-01
1923 -1.948645e-01  1.065701e-01
1924 -1.736052e-01  2.386236e-01
1925 -1.507200e-01  8.469582e-01
1926  1.888699e-02 -7.084517e-02
1927 -2.001024e-01 -1.460653e-01
1928 -2.885730e-01 -2.709626e-01
1929 -1.747114e-01  1.140587e-02
1930 -6.123198e-02 -5.212326e-02
1931 -1.215488e-01 -1.452958e-01
1932  2.465383e-01  1.636122e-01
1933  5.480525e-01 -3.243086e-03
1934  2.689330e-01 -1.194561e-01
1935 -5.697012e-02  9.610555e-02
1936  1.604938e-01  6.201187e-02
1937  2.431485e-01 -1.784854e-01
1938 -2.771129e-01  7.396836e-03
1939  2.183005e-01  5.816706e-01
1940 -2.050157e-01 -8.081577e-03
1941 -6.325106e-02  1.599149e-01
1942 -1.667850e-01  1.096391e-01
1943 -2.364824e-01 -1.094085e-01
1944 -3.279357e-02 -1.049793e-01
1945 -1.408098e-01 -5.111273e-02
1946 -2.809509e-01  7.737759e-02
1947 -5.690200e-02 -4.013909e-02
1948 -1.518133e-01 -3.095895e-01
1949  3.526678e-01 -2.698931e-02
1950  1.533514e-01  1.762746e-01
1951 -6.839538e-04 -7.012491e-02
1952 -3.478542e-01 -1.774476e-01
1953  2.418168e-01 -6.858480e-02
1954  3.427314e-01  9.720156e-02
1955 -6.328121e-02 -7.980323e-02
1956 -6.844017e-02  6.350043e-03
1957  2.409651e-01  2.428767e-01
1958 -2.457013e-01  1.854303e-01
1959  1.750241e-01  7.820218e-02
1960  2.120967e-02 -8.181141e-02
1961 -1.468061e-01 -2.700201e-01
1962  3.628659e-01  6.089297e-02
1963 -8.611982e-02  1.311099e-01
1964 -9.233234e-02 -8.129846e-02
1965  1.029483e-01 -9.087624e-02
1966 -7.199190e-02  2.649409e-01
1967  2.180664e-01 -6.796753e-02
1968  2.145288e-01 -8.866777e-02
1969  6.706357e-02  1.046292e-01
1970 -2.408416e-01  6.118401e-03
1971  1.819947e-02 -1.902634e-01
1972 -3.428856e-02 -6.931781e-02
1973 -9.893930e-03 -4.450927e-03
1974 -1.812127e-01  2.082858e-02
1975  4.150293e-01 -1.499411e-02
1976 -4.629606e-02  1.868597e-01
1977 -8.649529e-02  9.016773e-02
1978 -3.220429e-02 -1.027890e-01
1979 -2.899158e-01  2.473983e-01
1980 -3.740064e-03  5.105995e-01
1981  4.071851e-01 -1.338013e-01
1982 -1.861787e-01  5.009710e-02
1983  4.873755e-01 -4.322056e-01
1984 -3.341521e-01 -1.514237e-01
1985 -2.481359e-01 -2.144634e-01
1986  6.220532e-02 -8.177442e-02
1987  2.358583e-01 -1.187045e-01
1988  1.197035e-01 -1.311440e-01
1989  5.399037e-01  3.786655e-02
1990  4.909906e-01 -1.496410e-01

\$portmanteau.test
\$portmanteau.test\$QM
[,1]      [,2]      [,3]     [,4]
[1,] 1.2385593  7.662516  8.557323 12.67232
[2,] 6.7037426 10.314739 19.925393 29.65042
[3,] 4.4260039  6.286687 12.741855 22.10372
[4,] 7.0814729  7.953657 12.343296 16.23929
[5,] 1.2756624  3.519195 15.541614 20.53343
[6,] 0.3345657 14.774391 18.576333 23.05116
[7,] 2.7661078 10.266698 14.730405 17.24386
[8,] 4.5565922 11.572028 14.379916 17.55193
[9,] 4.7421034  7.700167 13.178117 14.25953
[10,] 1.8503846  6.016364 13.748339 19.26425
[11,] 2.7374340  6.952487  8.354099 16.61374
[12,] 7.5349809 15.032765 26.068905 35.15483

\$portmanteau.test\$QM.df
[,1] [,2] [,3] [,4]
[1,]    4    9   14   19
[2,]    4    9   14   19
[3,]    4    9   14   19
[4,]    4    9   14   19
[5,]    4    9   14   19
[6,]    2    7   12   17
[7,]    3    8   13   18
[8,]    4    9   14   19
[9,]    4    9   14   19
[10,]    2    7   12   17
[11,]    4    9   14   19
[12,]    4    9   14   19

\$portmanteau.test\$QM.sl
[,1]       [,2]       [,3]       [,4]
[1,] 0.8717102 0.56848115 0.85832122 0.85491505
[2,] 0.1523971 0.32561132 0.13251121 0.05641935
[3,] 0.3514119 0.71091781 0.54694342 0.27912456
[4,] 0.1316449 0.53882312 0.57875437 0.64125736
[5,] 0.8654910 0.94012295 0.34216070 0.36314375
[6,] 0.8459603 0.03900261 0.09927870 0.14759244
[7,] 0.4291096 0.24680395 0.32448403 0.50640302
[8,] 0.3358898 0.23852513 0.42180755 0.55252930
[9,] 0.3147971 0.56462153 0.51254894 0.76835716
[10,] 0.3964552 0.53783990 0.31706680 0.31357700
[11,] 0.6026802 0.64206631 0.87007477 0.61601782
[12,] 0.1101766 0.09003991 0.02536967 0.01337899

\$residual.acf
lags
periods             lag 1         lag 2        lag 3        lag 4        lag 5
period 1  -9.074425e-02  1.273701e-02  0.041466695  0.073403977  0.013077003
period 2   5.430577e-02 -2.458664e-01  0.127661923  0.070487889  0.014692426
period 3  -1.163953e-01  1.594117e-01 -0.043457572  0.060937812 -0.106053752
period 4   1.258787e-01 -2.221958e-01  0.023543959  0.149194025 -0.036154937
period 5  -3.926900e-02  2.157657e-02  0.010352620  0.070126997  0.094866588
period 6   7.061298e-17 -1.621368e-16 -0.051544675 -0.039689166  0.001710566
period 7   5.096794e-03 -1.134987e-02  0.009973078 -0.169991576  0.076567798
period 8  -2.522742e-02  6.219444e-02  0.038789593  0.060271993 -0.219172227
period 9   1.128500e-01 -1.514618e-01 -0.031995535  0.004317709 -0.152671045
period 10 -2.019760e-16  3.606214e-16 -0.032783424  0.118472296 -0.091171005
period 11  8.528105e-02 -1.052427e-01  0.092714190  0.076690084 -0.042718697
period 12 -2.769031e-02 -1.679117e-01  0.257622467 -0.004649663  0.005239010
lags
periods           lag 6       lag 7        lag 8       lag 9       lag 10
period 1  -0.07997898  0.20316707 -0.169840857 -0.06646055  0.019134506
period 2   0.16695454  0.06222316 -0.013248934  0.08808317  0.077598889
period 3   0.05277002 -0.05003146 -0.081047044  0.08558261  0.063827793
period 4  -0.09922215  0.01735967 -0.001327322 -0.02710302 -0.004222671
period 5  -0.02491033 -0.03054336  0.001776585 -0.14466338 -0.074535888
period 6  -0.16244332  0.09084480  0.355089220 -0.05793567  0.127997625
period 7   0.11959298  0.10288826  0.207407104 -0.11327220 -0.114833043
period 8   0.25759068 -0.06312977 -0.083522088  0.01676992  0.104755702
period 9  -0.04550089  0.16848808 -0.023851623  0.07264836  0.032517331
period 10  0.14312125  0.04976958 -0.017740522  0.16666508 -0.040742566
period 11  0.12251533  0.11091475  0.105492581  0.02978174  0.118437230
period 12 -0.10946470  0.24630913  0.051528421 -0.13967105  0.009734803
lags
periods          lag 11       lag 12       lag 13       lag 14       lag 15
period 1   0.09808334 -0.028347279 -0.008849588 -0.026495795 -0.009408534
period 2  -0.07525990  0.192862120  0.250270912 -0.007721045  0.122179835
period 3  -0.05501715  0.239858813  0.082921108  0.106376562 -0.021520244
period 4  -0.01933725 -0.160597593  0.093705556 -0.134642835 -0.033671927
period 5   0.15656141  0.199839163 -0.239089776 -0.158536443  0.050002317
period 6   0.02501159 -0.002585942 -0.040052012 -0.157217219 -0.143397355
period 7   0.04015840  0.183130160 -0.037297456  0.012261713 -0.135083520
period 8  -0.03157999  0.127517652  0.103748448  0.009790310  0.081016478
period 9   0.00355698  0.226035553 -0.084950125  0.052275606 -0.078204518
period 10 -0.01543402  0.057475849  0.008101748 -0.041028067  0.302092614
period 11  0.03868042  0.059960660 -0.050321882  0.092970491  0.034045071
period 12 -0.21956514  0.161023994 -0.103066245 -0.134065241  0.187482676
lags
periods           lag 16       lag 17      lag 18       lag 19       lag 20
period 1  -0.108914086  0.008288044  0.17558821  0.056728707  0.073643519
period 2   0.217806193  0.115035044 -0.09743203  0.007045642  0.226423696
period 3  -0.107091220 -0.065376609  0.13094330  0.078637840 -0.276463119
period 4  -0.007679860  0.153199803 -0.10252187  0.118237053  0.006926512
period 5  -0.005615818 -0.072931798  0.03825222 -0.096617663 -0.213165434
period 6   0.186012735  0.052686998 -0.08031297  0.049760738  0.096879508
period 7  -0.152161409 -0.012084065 -0.01359687  0.061578180 -0.063526946
period 8   0.019116873  0.039695453 -0.04352545  0.171589208  0.079269504
period 9   0.066163208  0.022958996 -0.01608694  0.085179857  0.033112301
period 10  0.094002988 -0.090349447  0.14782214  0.129534191  0.115359281
period 11 -0.110827346  0.170414619  0.22727146  0.004984024  0.101133507
period 12 -0.001607617 -0.134820673 -0.18064329  0.183606985 -0.170391233

\$residual.acf.sd
lags
periods          lag 1       lag 2      lag 3      lag 4     lag 5     lag 6
period 1  0.08633936 0.093938065 0.10598928 0.10650102 0.1131185 0.1119606
period 2  0.08611677 0.098572324 0.10251300 0.10909916 0.1093871 0.1131645
period 3  0.08819503 0.096846801 0.10380263 0.10609313 0.1100061 0.1101794
period 4  0.08559165 0.099166597 0.10373310 0.10755605 0.1088418 0.1110672
period 5  0.06496980 0.104307688 0.10824256 0.10966048 0.1108818 0.1112994
period 6  0.00766863 0.004428359 0.08549513 0.09912249 0.1037048 0.1075404
period 7  0.04965632 0.034367519 0.09742581 0.11168745 0.1120745 0.1122172
period 8  0.07163479 0.092393197 0.11245332 0.10802872 0.1125081 0.1125084
period 9  0.08736839 0.090107961 0.10085991 0.11247534 0.1098291 0.1125084
period 10 0.01652553 0.012832852 0.07118343 0.09220197 0.1124528 0.1079895
period 11 0.08276627 0.085136741 0.11210596 0.10776766 0.1099378 0.1125011
period 12 0.07663214 0.099818618 0.10073928 0.11232208 0.1103346 0.1113234
lags
periods         lag 7     lag 8     lag 9    lag 10    lag 11    lag 12
period 1  0.1125356 0.1132256 0.1130622 0.1132277 0.1132277 0.1132277
period 2  0.1124965 0.1128279 0.1132265 0.1131320 0.1132277 0.1132277
period 3  0.1124702 0.1120629 0.1122648 0.1125081 0.1124504 0.1125088
period 4  0.1111666 0.1124865 0.1122509 0.1123677 0.1125084 0.1124750
period 5  0.1120301 0.1120630 0.1125013 0.1124229 0.1124617 0.1125086
period 6  0.1088303 0.1110627 0.1111624 0.1124864 0.1122501 0.1123672
period 7  0.1123414 0.1123842 0.1124593 0.1124627 0.1125080 0.1124999
period 8  0.1125086 0.1125087 0.1125087 0.1125087 0.1125088 0.1125088
period 9  0.1125086 0.1125086 0.1125087 0.1125087 0.1125088 0.1125088
period 10 0.1125081 0.1125084 0.1125085 0.1125087 0.1125087 0.1125087
period 11 0.1118979 0.1125087 0.1125087 0.1125088 0.1125088 0.1125088
period 12 0.1125052 0.1122258 0.1125087 0.1125088 0.1125088 0.1125088
lags
periods        lag 13    lag 14    lag 15    lag 16    lag 17    lag 18
period 1  0.1132277 0.1132277 0.1132277 0.1132277 0.1132277 0.1132277
period 2  0.1132277 0.1132277 0.1132277 0.1132277 0.1132277 0.1132277
period 3  0.1125088 0.1125088 0.1125088 0.1125088 0.1125088 0.1125088
period 4  0.1125088 0.1125088 0.1125088 0.1125088 0.1125088 0.1125088
period 5  0.1124975 0.1125088 0.1125088 0.1125088 0.1125088 0.1125088
period 6  0.1125084 0.1124749 0.1125088 0.1125088 0.1125088 0.1125088
period 7  0.1125039 0.1125088 0.1125076 0.1125088 0.1125088 0.1125088
period 8  0.1125088 0.1125088 0.1125088 0.1125088 0.1125088 0.1125088
period 9  0.1125088 0.1125088 0.1125088 0.1125088 0.1125088 0.1125088
period 10 0.1125088 0.1125088 0.1125088 0.1125088 0.1125088 0.1125088
period 11 0.1125088 0.1125088 0.1125088 0.1125088 0.1125088 0.1125088
period 12 0.1125088 0.1125088 0.1125088 0.1125088 0.1125088 0.1125088
lags
periods        lag 19    lag 20
period 1  0.1132277 0.1132277
period 2  0.1132277 0.1132277
period 3  0.1125088 0.1125088
period 4  0.1125088 0.1125088
period 5  0.1125088 0.1125088
period 6  0.1125088 0.1125088
period 7  0.1125088 0.1125088
period 8  0.1125088 0.1125088
period 9  0.1125088 0.1125088
period 10 0.1125088 0.1125088
period 11 0.1125088 0.1125088
period 12 0.1125088 0.1125088

\$cov
\$cov\$`period 1`
[,1]
[1,] 0.004110686

\$cov\$`period 2`
[,1]
[1,] 0.004666186

\$cov\$`period 3`
[,1]
[1,] 0.006086609

\$cov\$`period 4`
[,1]
[1,] 0.01481656

\$cov\$`period 5`
[,1]
[1,] 0.004509269

\$cov\$`period 6`
[,1]         [,2]          [,3]
[1,]  0.0063280375 -0.001095628 -0.0004168932
[2,] -0.0010956277  0.003556099 -0.0025039456
[3,] -0.0004168932 -0.002503946  0.0059392389

\$cov\$`period 7`
[,1]         [,2]
[1,]  0.011010825 -0.002037675
[2,] -0.002037675  0.006483237

\$cov\$`period 8`
[,1]
[1,] 0.004697961

\$cov\$`period 9`
[,1]
[1,] 0.007931057

\$cov\$`period 10`
[,1]        [,2]         [,3]
[1,]  0.018649200 -0.01664179  0.003292144
[2,] -0.016641792  0.04366329 -0.024571970
[3,]  0.003292144 -0.02457197  0.027518820

\$cov\$`period 11`
[,1]
[1,] 0.008324338

\$cov\$`period 12`
[,1]
[1,] 0.005059931
```

pear documentation built on May 29, 2017, 11:40 p.m.