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  1.  
  2.                                    Fascicle II.3 _ Rec. E.506     3
  3.                                  
  4.                                  
  5.                                  
  6.                               ANNEX B
  7.                      (to Recommendation E.506)
  8.                                  
  9.             Example using weighted least squares method
  10.  
  11. B.1  Telex data
  12.  
  13.       The  telex traffic between the following countries  has  been
  14. analyzed:
  15.      _  Germany (D)
  16.      _  Denmark (DNK)
  17.      _  USA (USA)
  18.      _  Finland (FIN)
  19.      _  Norway (NOR)
  20.      _  Sweden (S)
  21.  
  22.       The  data consists of yearly observations from 1973  to  1984
  23. [19].
  24.  
  25. B.2  Forecasting
  26.  
  27.       Before  using  the  weighted least squares  method,  separate
  28. forecasts for the traffic matrix have to be made. In this example a
  29. simple   ARIMA   (0,2,1)   model   with   logarithmic   transformed
  30. observations without explanatory variables is used for forecasting.
  31. It  may  be possible to develop better forecasting models  for  the
  32. telex traffic between the various countries. However the main point
  33. in this example only is to illustrate the use of the weighted least
  34. squares technique.
  35.  
  36.      Forecasts for 1984 based on observations from 1973 to 1983 are
  37. given in Table B_1/E.506.
  38.                                  
  39.                                  
  40.                           TABLE B_1/E.506
  41.    Forecasts for telex traffic between Germany(D), Denmark(DNK),
  42.      USA(USA), Finland(FIN), Norway(NOR) and Sweden(S) in 1984
  43.                                                          
  44.         From                                             
  45.                D   DNK  USA  FIN  NOR    S     Sum    Forecas
  46.                                                         ted
  47.     To                                                  sum
  48.                                                          
  49.                                                          
  50.        D       _   486   12  287  239        28 005   27 788
  51.                     9   630   9    7    523
  52.                                          0
  53.       DNK           _             127        10 831   10 805
  54.               519       165  751   0    195
  55.                6         5               9
  56.       USA     11   131   _        165        17 193   17 009
  57.               103   3        719   7    240
  58.                                          1
  59.       FIN                     _                6496     6458
  60.               265  715  741       489   189
  61.                5                         6
  62.       NOR          125             _           7580     7597
  63.               241   5   182  541        154
  64.                5         1               8
  65.        S           182       179  133    _   12 063   12 053
  66.               482   1   228   8    3                     
  67.                8         3          
  68.                           
  69.                                                          
  70.       Sum     26   997   19  668  714   13               
  71.               197   3   130   8    6    034              
  72.                                          
  73.                                                          
  74.     Forecast  26   996   19  665  711   12               
  75.      ed sum   097   7   353   9    0    914              
  76.                                          
  77.  
  78.       It should be noticed that there is no consistency between row
  79. and  column  sum  forecasts and forecasts of the  elements  in  the
  80. traffic matrix. For instance, the sum of forecasted outgoing  telex
  81. traffic from Germany is 28 005, while the forecasted row sum is  27
  82. 788.
  83.  
  84.      To adjust the forecasts to get consistency and to utilize both
  85. row/column  forecasts  and forecasts of the  traffic  elements  the
  86. weighted least squares method is used.
  87.  
  88. B.3  Adjustment of the traffic matrix forecasts
  89.  
  90.       To  be  able  to use the weighted least squares  method,  the
  91. weights  and  the  separate  forecasts are  needed  as  input.  The
  92. separate forecasts are found in Table B_2/E.506, while the  weights
  93. are based on the mean squared one step ahead forecasting errors.
  94.  
  95.       Let yt be the traffic at time t. The ARIMA (0,2,1) model with
  96. logarithmic transformed data is given by:
  97.  
  98.                 zt = (1 _ B)2 ln yt = (1 _ qB) at
  99.  
  100.  
  101. or
  102.  
  103.                          zt = at _ qat_1
  104.  
  105.  
  106. where
  107.  
  108.      zt = ln yt _ 2 ln yt_1 + ln yt_2
  109.  
  110.      at is white noise,
  111.  
  112.      q  is a parameter,
  113.  
  114.      B  is the backwards shift operator.
  115.  
  116.      The mean squared one step ahead forecasting error of zt is:
  117.  
  118.                     MSQ =  S (zt _ z¢t_1(1))2
  119.  
  120.  
  121. where
  122.  
  123.      t_1(1) is the one step ahead forecast.
  124.  
  125.       The  results  of using the weighted least squares  method  is
  126. found  in  Table  B_3/E.506  and show that  the  factors  in  Table
  127. B_1/E.506  have been adjusted. In this example only  minor  changes
  128. have been performed because of the high conformity in the forecasts
  129. of row/column sums and traffic elements.
  130.                                  
  131.                                  
  132.                           TABLE B_2/E.506
  133.    Inverse weights as mean as squared one step ahead forecasting
  134.                               errors
  135.     of telex traffic (100_4) between Germany(D), Denmark(DNK),
  136.      USA(USA), Finland(FIN), Norway(NOR) and Sweden(S) in 1984
  137.                                                          
  138.        From                                              
  139.               D     DNK    USA    FIN    NOR     S      Sum
  140.     To                                                   
  141.                                                          
  142.                                                          
  143.        D       _                                        7.77
  144.                    28.72  13.18  11.40   8.29  44.61
  145.       DNK             _                                10.61
  146.              5.91         43.14  18.28  39.99  18.40
  147.       USA                    _                         21.27
  148.             23.76  39.19         42.07  50.72  51.55
  149.       FIN                           _                  17.46
  150.             23.05  12.15  99.08         34.41  19.96
  151.       NOR                                  _           20.56
  152.             21.47  40.16  132.5  24.64         17.15
  153.                             7
  154.        S                                          _     6.48
  155.              6.38  12.95  28.60  28.08   8.76            
  156.                                            
  157.                                                          
  158.       Sum                                                
  159.              6.15   3.85  14.27   9.55  12.94   8.53     
  160.                                                   
  161.  
  162.  
  163.                                  
  164.                                  
  165.                           TABLE B_3/E.506
  166.  Adjusted telex forecasts using the weighted least squares method
  167.                                                          
  168.        From                                              
  169.               D     DNK    USA    FIN    NOR     S      Sum
  170.     To                                                   
  171.                                                          
  172.                                                          
  173.        D       _    4850    12    2858   2383          27 865
  174.                            684                  5090
  175.       DNK             _            750   1257          10 825
  176.              5185          1674                 1959
  177.       USA     11    1321     _     717   1644          17 090
  178.              001                                2407
  179.       FIN            715            _     487           6471
  180.              2633          745                  1891
  181.       NOR           1258           540     _            7617
  182.              2402          1870                 1547
  183.        S            1817          1788   1331     _    12 066
  184.              4823          2307                          
  185.                              
  186.                                                          
  187.       Sum     26    9961    19    6653   7102    12      
  188.              044           280                  894      
  189.                                                   
  190.  
  191.  
  192.  
  193.  
  194.