Affiliations:

John K. Dagsvik, Statistics Norway, Research Department;

Mariachiara Fortuna, freelance statistician, Turin;

Sigmund Hov Moen, Westerdals Oslo School of Arts, Communication and Technology.

Corresponding author:

John K. Dagsvik, E-mail: john.dagsvik@ssb.no

Mariachiara Fortuna, E-mail: mariachiara.fortuna@vanlog.it (reference for code and analysis)

Table C1. Estimation results when using the characteristic function estimator and the Whittle method. Monthly data.

Weather station \(H_c\) \(H_w\) \(SE(H_w)\)
Argentina, Buenos Aires 0.785 0.713 0.016
Australia, Adelaide 0.696 0.662 0.016
Australia, Alice Springs 0.700 0.683 0.017
Australia, Cap Otway 0.803 0.744 0.016
Austria, Kremsmunster 0.655 0.651 0.016
Austria, Vienna 0.684 0.659 0.015
Belgium, Uccle 0.660 0.643 0.014
Canada, Winnipeg 0.654 0.660 0.016
Croatia, Zagreb 0.654 0.650 0.015
Czech Republic, Prague 0.684 0.670 0.012
Denmark, Copenhagen 0.755 0.758 0.013
Denmark, Vestervig 0.725 0.763 0.016
Egypt, Alexandria 0.773 0.798 0.018
France, Nantes 0.643 0.643 0.015
France, Paris 0.733 0.672 0.012
Germany, Berlin 0.664 0.662 0.012
Germany, Hohenpeissenberg 0.617 0.605 0.012
Germany, Karlsruhe 0.642 0.629 0.016
Greece, Athens 0.682 0.698 0.015
Greenland, Illulisat 0.738 0.725 0.016
Greenland, Ivittuut 0.782 0.751 0.020
Hungary, Budapest 0.627 0.645 0.012
Iceland, Djupivogur 0.762 0.736 0.016
Iceland, Reykjavik 0.731 0.711 0.016
India, Agra 0.731 0.753 0.019
India, Allahabad 0.699 0.694 0.017
India, Bombay 0.783 0.788 0.017
India, Indore 0.734 0.709 0.017
India, Madras 0.751 0.753 0.017
India, Nagpur 0.697 0.708 0.017
Israel, Jerusalem 0.685 0.699 0.016
Italy, Bologna 0.702 0.698 0.014
Italy, Milan 0.724 0.709 0.012
Japan, Hiroshima 0.731 0.733 0.017
Japan, Nagasaki 0.738 0.715 0.017
Japan, Tokyo 0.795 0.744 0.016
Kazakhstan, Kazalinsk 0.609 0.655 0.018
Luxembourg, Luxembourg 0.675 0.658 0.014
New Zealand, Wellington 0.774 0.752 0.017
Norway, Andoya 0.723 0.725 0.016
Norway, Bergen 0.678 0.687 0.015
Norway, Bodo 0.680 0.698 0.016
Norway, Dombas 0.633 0.664 0.015
Norway, Karasjok 0.655 0.679 0.016
Norway, Mandal 0.682 0.724 0.016
Norway, Oksoy Lighthouse 0.719 0.771 0.016
Norway, Ona 0.711 0.749 0.016
Norway, Oslo 0.693 0.724 0.014
Norway, Roros 0.667 0.695 0.016
Norway, Tromso 0.670 0.690 0.016
Norway, Utsira 0.735 0.768 0.016
Norway, Vardo 0.765 0.751 0.016
Pakistan, Lahore 0.649 0.693 0.016
Portugal, Lisbon 0.769 0.710 0.017
Romania, Sulina 0.653 0.699 0.017
Russia, Archangelsk 0.675 0.661 0.016
Russia, Sort 0.640 0.694 0.018
Russia, St Petersburg 0.697 0.696 0.017
Spain, Gibraltar 0.773 0.765 0.015
Sweden, Bromma 0.694 0.736 0.012
Sweden, Stockholm 0.681 0.721 0.012
Sweden, Tullinge 0.672 0.727 0.012
Sweden, Uppsala 0.690 0.718 0.011
Switzerland, Basel 0.625 0.622 0.012
Switzerland, Geneva 0.693 0.667 0.012
UK, Aberdeen 0.691 0.704 0.017
UK, Belfast 0.650 0.665 0.016
UK, Cambridge 0.678 0.673 0.016
UK, Durham 0.698 0.686 0.015
UK, Edinbourg 0.644 0.670 0.013
UK, London 0.721 0.686 0.015
UK, Plymouth 0.624 0.676 0.017
USA, Atlanta 0.632 0.641 0.016
USA, Bismarck 0.655 0.640 0.016
USA, Boise 0.654 0.656 0.016
USA, Boston 0.693 0.670 0.016
USA, Chattanooga 0.637 0.647 0.016
USA, Cincinatti 0.656 0.645 0.016
USA, Columbus 0.629 0.631 0.016
USA, Concord 0.687 0.662 0.016
USA, Des Moines 0.626 0.632 0.016
USA, Detroit 0.659 0.653 0.016
USA, Dodge City 0.626 0.611 0.016
USA, Fargo 0.656 0.655 0.016
USA, Galveston 0.662 0.688 0.017
USA, Indianapolis 0.611 0.622 0.016
USA, Jacksonville 0.608 0.651 0.016
USA, Knoxville 0.624 0.630 0.016
USA, Las Vegas 0.643 0.647 0.017
USA, Madison 0.641 0.647 0.016
USA, Marquette 0.688 0.686 0.017
USA, Milwaukee 0.689 0.676 0.016
USA, Mobile 0.617 0.651 0.016
USA, Nashville 0.581 0.603 0.016
USA, New Orleans 0.696 0.695 0.017
USA, New York 0.745 0.699 0.014

Due to the fact that the monthly time series are quite long, the estimates of the Hurst parameter are quite precise. From Table C1 we note that the difference between the characteristic function estimates and the Whittle estimates of the Hurst parameter are only significantly different in a few cases.

Table C2. Estimates and test statistics based on annual data

Weather station \(H_c\) \(Q(H_c)\) \(H_w\) \(SE(H_w)\) \(Q(H_w)\)
Argentina, Buenos Aires 0.951 4.960 0.938 0.055 -0.222
Australia, Adelaide 0.882 2.511 0.781 0.058 0.336
Australia, Alice Springs 0.647 -0.040 0.708 0.058 0.062
Australia, Cap Otway 0.905 0.222 0.869 0.059 0.157
Austria, Kremsmunster 0.728 -0.679 0.782 0.058 0.273
Austria, Vienna 0.811 -0.527 0.806 0.055 -0.270
Belgium, Uccle 0.740 -0.086 0.739 0.050 0.011
Canada, Winnipeg 0.713 0.346 0.728 0.058 0.174
Croatia, Zagreb 0.723 0.888 0.780 0.055 -0.144
Czech Republic, Prague 0.745 0.442 0.716 0.043 0.012
Denmark, Copenhagen 0.817 0.092 0.753 0.045 0.007
Denmark, Vestervig 0.699 0.093 0.733 0.056 0.008
Egypt, Alexandria 0.882 0.224 0.862 0.064 0.010
France, Nantes 0.738 -0.494 0.720 0.052 0.120
France, Paris 0.873 0.574 0.802 0.042 -0.010
Germany, Berlin 0.726 -0.053 0.712 0.041 -0.041
Germany, Hohenpeissenberg 0.701 1.053 0.684 0.043 -0.338
Germany, Karlsruhe 0.728 0.209 0.819 0.059 0.172
Greece, Athens 0.754 0.863 0.788 0.054 0.094
Greenland, Illulisat 0.806 0.839 0.805 0.057 0.000
Greenland, Ivittuut 0.797 -0.202 0.804 0.071 -0.273
Hungary, Budapest 0.682 0.288 0.663 0.043 -0.115
Iceland, Djupivogur 0.852 0.084 0.841 0.058 0.332
Iceland, Reykjavik 0.889 0.996 0.885 0.057 0.146
India, Agra 0.802 -0.281 0.844 0.066 0.184
India, Allahabad 0.706 -1.135 0.807 0.059 -0.839
India, Bombay 0.793 -0.072 0.887 0.059 0.087
India, Indore 0.820 -0.303 0.899 0.059 -0.534
India, Madras 0.775 -0.511 0.906 0.059 -0.328
India, Nagpur 0.610 -0.424 0.727 0.058 -0.158
Israel, Jerusalem 0.702 0.022 0.654 0.057 -0.060
Italy, Bologna 0.819 0.602 0.845 0.048 -0.723
Italy, Milan 0.851 -1.014 0.826 0.043 -0.281
Japan, Hiroshima 0.798 -0.165 0.738 0.059 -0.267
Japan, Nagasaki 0.823 0.026 0.761 0.058 0.014
Japan, Tokyo 0.926 -0.235 0.851 0.058 -0.086
Kazakhstan, Kazalinsk 0.611 -0.250 0.563 0.061 -0.216
Luxembourg, Luxembourg 0.815 -1.068 0.825 0.051 1.062
New Zealand, Wellington 0.810 -0.618 0.919 0.060 -0.296
Norway, Andoya 0.773 0.049 0.761 0.055 -0.013
Norway, Bergen 0.783 -0.377 0.717 0.053 0.206
Norway, Bodo 0.700 -0.289 0.682 0.054 -0.130
Norway, Dombas 0.679 3.476 0.637 0.053 3.275
Norway, Karasjok 0.655 -0.386 0.652 0.056 0.627
Norway, Mandal 0.620 -0.499 0.625 0.054 0.368
Norway, Oksoy Lighthouse 0.666 -0.126 0.672 0.055 0.156
Norway, Ona 0.674 -0.139 0.702 0.056 0.030
Norway, Oslo 0.692 0.464 0.699 0.047 -0.130
Norway, Roros 0.727 -0.333 0.688 0.055 -0.229
Norway, Tromso 0.616 -0.318 0.641 0.054 0.005
Norway, Utsira 0.738 -0.094 0.753 0.055 -0.040
Norway, Vardo 0.724 -0.053 0.770 0.054 -0.017
Pakistan, Lahore 0.659 0.047 0.743 0.057 0.020
Portugal, Lisbon 0.933 0.211 0.931 0.060 -0.221
Romania, Sulina 0.591 0.121 0.631 0.057 -0.121
Russia, Archangelsk 0.707 1.187 0.746 0.058 -0.369
Russia, Sort 0.594 0.722 0.581 0.062 0.431
Russia, St Petersburg 0.670 -0.161 0.706 0.058 7.045
Spain, Gibraltar 0.787 -0.075 0.855 0.056 0.283
Sweden, Bromma 0.688 0.202 0.690 0.041 -0.043
Sweden, Stockholm 0.614 5.121 0.632 0.041 -0.940
Sweden, Tullinge 0.624 1.523 0.622 0.041 -0.312
Sweden, Uppsala 0.715 1.135 0.710 0.040 -0.291
Switzerland, Basel 0.664 0.558 0.720 0.042 -0.753
Switzerland, Geneva 0.845 -0.537 0.818 0.042 0.563
UK, Aberdeen 0.771 0.268 0.767 0.058 -0.245
UK, Belfast 0.707 -0.576 0.727 0.058 0.252
UK, Cambridge 0.773 -1.596 0.781 0.056 4.131
UK, Durham 0.771 -0.656 0.761 0.052 3.554
UK, Edinbourg 0.605 -1.430 0.626 0.045 2.282
UK, London 0.798 -0.543 0.809 0.053 1.394
UK, Plymouth 0.559 0.268 0.671 0.058 1.633
USA, Atlanta 0.766 2.043 0.725 0.058 1.774
USA, Bismarck 0.749 1.135 0.761 0.058 0.631
USA, Boise 0.725 -0.017 0.698 0.057 -0.281
USA, Boston 0.728 0.389 0.724 0.058 0.541
USA, Chattanooga 0.744 0.715 0.695 0.057 0.684
USA, Cincinatti 0.758 -0.108 0.718 0.058 0.685
USA, Columbus 0.705 -1.449 0.702 0.057 0.248
USA, Concord 0.790 0.012 0.729 0.058 -0.255
USA, Des Moines 0.621 0.042 0.623 0.056 -0.247
USA, Detroit 0.707 1.846 0.663 0.057 -1.948
USA, Dodge City 0.648 0.123 0.715 0.058 -0.191
USA, Fargo 0.738 2.083 0.725 0.058 0.948
USA, Galveston 0.674 -0.566 0.666 0.057 -0.104
USA, Indianapolis 0.667 1.495 0.658 0.057 -0.726
USA, Jacksonville 0.664 0.212 0.618 0.056 -0.423
USA, Knoxville 0.744 -0.773 0.680 0.057 0.267
USA, Las Vegas 0.707 -0.310 0.694 0.060 -0.100
USA, Madison 0.673 -0.336 0.682 0.057 -0.347
USA, Marquette 0.694 -0.026 0.716 0.058 -0.167
USA, Milwaukee 0.683 -1.088 0.755 0.058 -0.247
USA, Mobile 0.678 3.408 0.672 0.057 0.667
USA, Nashville 0.609 -0.240 0.625 0.057 0.360
USA, New Orleans 0.861 4.034 0.812 0.058 -0.570
USA, New York 0.907 4.772 0.843 0.049 2.002

From the results in Table C2 we note that the estimates of the Hurst parameter based on annual data are, on average, higher than the corresponding estimates based on monthly data. Furthermore, we see that data from 9 weather stations reject the FGN hypothesis when using the characteristic function estimate of the Hurst parameter whereas data from 6 weather stations reject the FGN when using the Whittle estimate of the Hurst parameter.

Table C3. Estimates and test statistics based on Moberg et al. (2005) time series

Parameters and statistics Value
\(\mu\) -0.354
\(\sigma\) 0.220
\(\mu_c\) -0.354
\(\sigma_c\) 0.051
\(H_c\) 0.917
\(H_w\) 0.990
\(SE(H_w)\) 0.015
\(Q(H_c)\) -11.205
\(Q(H_w)\) 104.220

The results of Table C3 show that the FGN model is rejected for the Moberg data when the respective estimated Hurst parameters are used.

Table C4. Chi-square statistics based on the data of Moberg et al. (2009)

H Q(H)
0.92 -10.595
0.93 -8.332
0.94 -5.274
0.95 -0.946
0.96 5.599
0.97 16.575
0.98 38.621

The results of Table C4 shows that the power of the Q test is high (conditional on the FGN model). In particular, when H = 0.95 then Q(H) \(\in\) (-1.96, 1.96) whereas when H equals 0.94 or 0.96 (or further away from 0.95) then Q(H) \(\notin\) (-1.96, 1.96) which means rejection of FGN.

Table C5. Stationarity test. Moberg data

Test statistic Test result Test criterion
Significance level: 0.05 3.397 no rejection 5.494
Significance level: 0.1 3.430 no rejection 5.350

Table C6. Stationarity test. Annual data

Weather station Test statistic Test result Test criterion
Argentina,Buenos Aires 3.086 no rejection 5.865
Australia,Adelaide 3.282 no rejection 5.878
Australia,Alice Springs 3.947 no rejection 5.802
Australia,Cap Otway 3.719 no rejection 5.788
Austria,Kremsmunster 3.229 no rejection 5.742
Austria,Vienna 2.722 no rejection 5.747
Belgium,Uccle 3.905 no rejection 5.941
Canada,Winnipeg 3.593 no rejection 5.628
Croatia,Zagreb 4.149 no rejection 5.813
Czech Republic,Prague 4.066 no rejection 5.971
Denmark,Copenhagen 3.912 no rejection 5.957
Denmark,Vestervig 3.926 no rejection 5.864
Egypt,Alexandria 5.144 no rejection 5.493
France,Nantes 3.805 no rejection 5.855
France,Paris 4.139 no rejection 5.952
Germany,Berlin 4.309 no rejection 5.976
Germany,Hohenpeissenberg 3.038 no rejection 5.987
Germany,Karlsruhe 2.600 no rejection 5.717
Greece,Athens 3.658 no rejection 5.581
Greenland,Illulisat 3.716 no rejection 5.865
Greenland,Ivittuut 3.042 no rejection 5.740
Hungary,Budapest 3.673 no rejection 5.941
Iceland,Djupivogur 6.377 rejection 5.849
Iceland,Reykjavik 3.259 no rejection 5.717
India,Agra 4.363 no rejection 5.828
India,Allahabad 2.495 no rejection 5.822
India,Bombay 4.361 no rejection 5.878
India,Indore 2.763 no rejection 5.892
India,Madras 4.714 no rejection 5.718
India,Nagpur 3.669 no rejection 5.869
Israel,Jerusalem 3.665 no rejection 5.821
Italy,Bologna 4.363 no rejection 5.907
Italy,Milan 4.002 no rejection 5.935
Japan,Hiroshima 2.940 no rejection 5.333
Japan,Nagasaki 2.960 no rejection 5.637
Japan,Tokyo 2.550 no rejection 5.569
Kazakhstan,Kazalinsk 3.289 no rejection 5.751
Luxembourg,Luxembourg 3.368 no rejection 5.902
New Zealand,Wellington 3.390 no rejection 5.606
Norway,Andoya 4.493 no rejection 5.914
Norway,Bergen 2.312 no rejection 5.861
Norway,Bodo 4.287 no rejection 5.896
Norway,Dombas 5.485 no rejection 5.867
Norway,Karasjok 3.920 no rejection 5.821
Norway,Mandal 3.929 no rejection 5.924
Norway,Oksoy Lighthouse 4.187 no rejection 5.873
Norway,Ona 3.902 no rejection 5.894
Norway,Oslo 3.286 no rejection 5.928
Norway,Roros 3.352 no rejection 5.832
Norway,Tromso 3.834 no rejection 5.871
Norway,Utsira 2.863 no rejection 5.868
Norway,Vardo 2.742 no rejection 5.867
Pakistan,Lahore 2.830 no rejection 5.889
Portugal,Lisbon 4.938 no rejection 5.784
Romania,Sulina 2.957 no rejection 5.396
Russia,Archangelsk 3.720 no rejection 5.683
Russia,Sort 3.222 no rejection 5.437
Russia,St Petersburg 3.695 no rejection 5.852
Spain,Gibraltar 5.704 no rejection 5.867
Sweden,Bromma 3.244 no rejection 5.963
Sweden,Stockholm 3.083 no rejection 5.955
Sweden,Tullinge 3.508 no rejection 5.950
Sweden,Uppsala 3.182 no rejection 5.920
Switzerland,Basel 4.411 no rejection 5.961
Switzerland,Geneva 4.266 no rejection 5.955
UK,Aberdeen 2.114 no rejection 5.818
UK,Belfast 2.766 no rejection 5.846
UK,Cambridge 2.815 no rejection 5.863
UK,Durham 2.867 no rejection 5.930
UK,Edinbourg 3.406 no rejection 5.914
UK,London 4.037 no rejection 5.909
UK,Plymouth 4.379 no rejection 5.852
USA,Atlanta 3.985 no rejection 5.897
USA,Bismarck 3.510 no rejection 5.635
USA,Boise 3.839 no rejection 5.567
USA,Boston 3.373 no rejection 5.828
USA,Chattanooga 3.835 no rejection 5.863
USA,Cincinatti 4.886 no rejection 5.860
USA,Columbus 3.413 no rejection 5.831
USA,Concord 2.573 no rejection 5.911
USA,Des Moines 2.397 no rejection 5.799
USA,Detroit 3.542 no rejection 5.841
USA,Dodge City 2.791 no rejection 5.799
USA,Fargo 2.176 no rejection 5.584
USA,Galveston 2.841 no rejection 5.879
USA,Indianapolis 3.422 no rejection 5.825
USA,Jacksonville 3.743 no rejection 5.800
USA,Knoxville 2.886 no rejection 5.866
USA,Las Vegas 2.441 no rejection 5.746
USA,Madison 2.792 no rejection 5.842
USA,Marquette 3.198 no rejection 5.843
USA,Milwaukee 2.621 no rejection 5.819
USA,Mobile 2.741 no rejection 5.887
USA,Nashville 3.836 no rejection 5.858
USA,New Orleans 2.708 no rejection 5.859
USA,New York 4.448 no rejection 5.617

From Table C6 we note that only in one case (Djupivogur, Iceland) do the data reject the stationarity hypothesis.

Table C7. Stationarity test. Monthly data

Weather station Test statistic Test result Test criterion
Argentina,Buenos Aires 3.086 no rejection 5.865
Australia,Adelaide 3.282 no rejection 5.878
Australia,Alice Springs 3.947 no rejection 5.802
Australia,Cap Otway 3.719 no rejection 5.788
Austria,Kremsmunster 3.229 no rejection 5.742
Austria,Vienna 2.722 no rejection 5.747
Belgium,Uccle 3.905 no rejection 5.941
Canada,Winnipeg 3.593 no rejection 5.628
Croatia,Zagreb 4.149 no rejection 5.813
Czech Republic,Prague 4.066 no rejection 5.971
Denmark,Copenhagen 3.912 no rejection 5.957
Denmark,Vestervig 3.926 no rejection 5.864
Egypt,Alexandria 5.144 no rejection 5.493
France,Nantes 3.805 no rejection 5.855
France,Paris 4.139 no rejection 5.952
Germany,Berlin 4.309 no rejection 5.976
Germany,Hohenpeissenberg 3.038 no rejection 5.987
Germany,Karlsruhe 2.600 no rejection 5.717
Greece,Athens 3.658 no rejection 5.581
Greenland,Illulisat 3.716 no rejection 5.865
Greenland,Ivittuut 3.042 no rejection 5.740
Hungary,Budapest 3.673 no rejection 5.941
Iceland,Djupivogur 6.377 rejection 5.849
Iceland,Reykjavik 3.259 no rejection 5.717
India,Agra 4.363 no rejection 5.828
India,Allahabad 2.495 no rejection 5.822
India,Bombay 4.361 no rejection 5.878
India,Indore 2.763 no rejection 5.892
India,Madras 4.714 no rejection 5.718
India,Nagpur 3.669 no rejection 5.869
Israel,Jerusalem 3.665 no rejection 5.821
Italy,Bologna 4.363 no rejection 5.907
Italy,Milan 4.002 no rejection 5.935
Japan,Hiroshima 2.940 no rejection 5.333
Japan,Nagasaki 2.960 no rejection 5.637
Japan,Tokyo 2.550 no rejection 5.569
Kazakhstan,Kazalinsk 3.289 no rejection 5.751
Luxembourg,Luxembourg 3.368 no rejection 5.902
New Zealand,Wellington 3.390 no rejection 5.606
Norway,Andoya 4.493 no rejection 5.914
Norway,Bergen 2.312 no rejection 5.861
Norway,Bodo 4.287 no rejection 5.896
Norway,Dombas 5.485 no rejection 5.867
Norway,Karasjok 3.920 no rejection 5.821
Norway,Mandal 3.929 no rejection 5.924
Norway,Oksoy Lighthouse 4.187 no rejection 5.873
Norway,Ona 3.902 no rejection 5.894
Norway,Oslo 3.286 no rejection 5.928
Norway,Roros 3.352 no rejection 5.832
Norway,Tromso 3.834 no rejection 5.871
Norway,Utsira 2.863 no rejection 5.868
Norway,Vardo 2.742 no rejection 5.867
Pakistan,Lahore 2.830 no rejection 5.889
Portugal,Lisbon 4.938 no rejection 5.784
Romania,Sulina 2.957 no rejection 5.396
Russia,Archangelsk 3.720 no rejection 5.683
Russia,Sort 3.222 no rejection 5.437
Russia,St Petersburg 3.695 no rejection 5.852
Spain,Gibraltar 5.704 no rejection 5.867
Sweden,Bromma 3.244 no rejection 5.963
Sweden,Stockholm 3.083 no rejection 5.955
Sweden,Tullinge 3.508 no rejection 5.950
Sweden,Uppsala 3.182 no rejection 5.920
Switzerland,Basel 4.411 no rejection 5.961
Switzerland,Geneva 4.266 no rejection 5.955
UK,Aberdeen 2.114 no rejection 5.818
UK,Belfast 2.766 no rejection 5.846
UK,Cambridge 2.815 no rejection 5.863
UK,Durham 2.867 no rejection 5.930
UK,Edinbourg 3.406 no rejection 5.914
UK,London 4.037 no rejection 5.909
UK,Plymouth 4.379 no rejection 5.852
USA,Atlanta 3.985 no rejection 5.897
USA,Bismarck 3.510 no rejection 5.635
USA,Boise 3.839 no rejection 5.567
USA,Boston 3.373 no rejection 5.828
USA,Chattanooga 3.835 no rejection 5.863
USA,Cincinatti 4.886 no rejection 5.860
USA,Columbus 3.413 no rejection 5.831
USA,Concord 2.573 no rejection 5.911
USA,Des Moines 2.397 no rejection 5.799
USA,Detroit 3.542 no rejection 5.841
USA,Dodge City 2.791 no rejection 5.799
USA,Fargo 2.176 no rejection 5.584
USA,Galveston 2.841 no rejection 5.879
USA,Indianapolis 3.422 no rejection 5.825
USA,Jacksonville 3.743 no rejection 5.800
USA,Knoxville 2.886 no rejection 5.866
USA,Las Vegas 2.441 no rejection 5.746
USA,Madison 2.792 no rejection 5.842
USA,Marquette 3.198 no rejection 5.843
USA,Milwaukee 2.621 no rejection 5.819
USA,Mobile 2.741 no rejection 5.887
USA,Nashville 3.836 no rejection 5.858
USA,New Orleans 2.708 no rejection 5.859
USA,New York 4.448 no rejection 5.617

Table C7 shows that stationarity (based on the default option of Cho’s test) is rejected for data from 14 weather stations when monthly time series are used.

Table C8. Estimation of H using the Wavelet Lifting estimator. Monthly data

Weather station \(H_{wav}\) \(Q(H_{wav})\)
Argentina, Buenos Aires 0.622 -5.299
Australia, Adelaide 0.622 -1.906
Australia, Alice Springs 0.643 -1.490
Australia, Cap Otway 0.696 -4.644
Austria, Kremsmunster 0.602 -1.606
Austria, Vienna 0.610 -2.172
Belgium, Uccle 0.595 -1.695
Canada, Winnipeg 0.615 -1.349
Croatia, Zagreb 0.609 -1.449
Czech Republic, Prague 0.625 -2.273
Denmark, Copenhagen 0.694 -4.464
Denmark, Vestervig 0.699 -2.392
Egypt, Alexandria 0.740 -3.498
France, Nantes 0.602 -1.312
France, Paris 0.603 -4.348
Germany, Berlin 0.621 -1.907
Germany, Hohenpeissenberg 0.567 -1.001
Germany, Karlsruhe 0.569 -1.383
Greece, Athens 0.657 -1.622
Greenland, Illulisat 0.664 -2.943
Greenland, Ivittuut 0.695 -2.970
Hungary, Budapest 0.606 -1.260
Iceland, Djupivogur 0.668 -4.356
Iceland, Reykjavik 0.644 -3.351
India, Agra 0.697 -2.155
India, Allahabad 0.644 -1.767
India, Bombay 0.715 -4.660
India, Indore 0.642 -2.954
India, Madras 0.688 -3.555
India, Nagpur 0.665 -1.588
Israel, Jerusalem 0.652 -1.589
Italy, Bologna 0.642 -3.037
Italy, Milan 0.642 -4.870
Japan, Hiroshima 0.665 -2.826
Japan, Nagasaki 0.661 -2.904
Japan, Tokyo 0.664 -5.614
Kazakhstan, Kazalinsk 0.619 -0.592
Luxembourg, Luxembourg 0.606 -2.034
New Zealand, Wellington 0.703 -3.066
Norway, Andoya 0.669 -2.437
Norway, Bergen 0.635 -1.900
Norway, Bodo 0.651 -1.500
Norway, Dombas 0.618 -1.215
Norway, Karasjok 0.634 -1.152
Norway, Mandal 0.680 -0.963
Norway, Oksoy Lighthouse 0.713 -1.528
Norway, Ona 0.680 -2.199
Norway, Oslo 0.671 -2.014
Norway, Roros 0.641 -1.650
Norway, Tromso 0.648 -1.162
Norway, Utsira 0.703 -2.522
Norway, Vardo 0.694 -3.376
Pakistan, Lahore 0.649 -1.011
Portugal, Lisbon 0.627 -4.469
Romania, Sulina 0.658 -0.876
Russia, Archangelsk 0.610 -1.618
Russia, Sort 0.658 -0.460
Russia, St Petersburg 0.643 -1.817
Spain, Gibraltar 0.707 -4.289
Sweden, Bromma 0.687 -2.110
Sweden, Stockholm 0.673 -1.558
Sweden, Tullinge 0.679 -1.350
Sweden, Uppsala 0.665 -2.545
Switzerland, Basel 0.585 -1.229
Switzerland, Geneva 0.606 -3.239
UK, Aberdeen 0.654 -1.924
UK, Belfast 0.621 -1.307
UK, Cambridge 0.616 -2.031
UK, Durham 0.625 -2.584
UK, Edinbourg 0.633 -0.943
UK, London 0.614 -3.479
UK, Plymouth 0.644 -0.635
USA, Atlanta 0.607 -0.909
USA, Bismarck 0.598 -1.234
USA, Boise 0.616 -1.300
USA, Boston 0.617 -1.920
USA, Chattanooga 0.617 -0.855
USA, Cincinatti 0.614 -1.090
USA, Columbus 0.601 -0.866
USA, Concord 0.608 -1.973
USA, Des Moines 0.604 -0.734
USA, Detroit 0.614 -1.196
USA, Dodge City 0.582 -0.776
USA, Fargo 0.608 -1.420
USA, Galveston 0.646 -1.334
USA, Indianapolis 0.594 -0.666
USA, Jacksonville 0.621 -0.723
USA, Knoxville 0.592 -0.926
USA, Las Vegas 0.616 -0.933
USA, Madison 0.609 -1.009
USA, Marquette 0.630 -2.118
USA, Milwaukee 0.625 -1.860
USA, Mobile 0.624 -0.638
USA, Nashville 0.589 -0.382
USA, New Orleans 0.644 -2.132
USA, New York 0.629 -4.211

Figure C1. Comparison between the Wavelet Lifting and the Whittle estimator

Wavelet Lifting vs Whittle estimates of H, with 95% confidence bands