‘‘Moving median with trend and seasonality’’ Eleftherios Giovanis* * Current student in Master program of “Economics” in University of Macedonia , Department of Economics, Thessaloniki, Greece * Current student in Master of program “Quality Assurance” in Hellenic Open University, School of Science & Technology , Patra, Greece Abstract In this paper is examined and presented an alternative method in time-series analysis and forecasting. In the specific project is being a concentration of certain ideas that I had as a
student in the third and fourth year of my undergraduate studies in the Economic Science, as I had the unique experience of learning the basic econometrics and the time-series analysis and forecasting. Consequently in this work I am trying to attribute with the simplest presentation and easiest way ,the methodology of moving median method, as it can be proved very easy comprehensible to everyone, because mathematics that are been used can be easy understood . It can
be useful to
financial analysts and can be applied in microeconomics and
macroeconomics. Key words: basic econometrics, moving median, forecasting, trend, seasonality
Electronic copy available at: http://ssrn.com/abstract=969012
1. Introduction
In this paper is not being an effort to prove that moving method is a better method than the other already known methods , neither to replace any other method, neither to propose it as the most optimum, after each method have the advantages and its disadvantages, methods as the moving average, the linear moving average, the simple and double exponential smoothing, the Winter' s model, as well as the ARIMA models. The method of moving median that is proposed in this work with various variants, can in certain cases be proved very useful and reliable, while in other cases can be proved unreliable and disastrous. Also it does not constitute an original method, because very simply it is based, if no in entire methodology, but in very great degree in previous methods and it does not constitute a scientific method based on mathematical theorems, but it is based on empirical content. In the second part of the specific work is analyzed the trend of the time- series as well as the decomposition of them. They are been received various time-series as the shares that negotiate in the Athens exchange stock market, , but also macroeconomic sizes, the inflation in some countries of European Union. The analysis is concentrated in 2-3 timeseries data, because the scope of this paper is to be presented the methodology and not the reliability test of that method. Some measures of error that will be used are the MdAD (median absolute deviation) or the alternative (MdAPE) Median Absolute Percentage Error (Armstrong and Collopy: 1992; Jarrett:1993,.p. 43-45).
2 Electronic copy available at: http://ssrn.com/abstract=969012
2. Moving median with trend and seasonality
First way In this part is presented an alternative method of time-series forecasting that takes into consideration, as other methods do, the trend and the seasonality (decomposition). Initially is examined the methodology real examples of time-series as prices from shares that negotiates in the Athens exchange stock market, the inflation in some countries of European Union, as well as other seasonal data, as monthly, four-month periods and daily data. The results are presented in the consecutively tables in the text. Somehow thus the analysis begins with the share of telecommunication company “COSMOTE” during the period 02 January 2007 – 25 February 2007 (Table 1-Appendix). The forecasting will be made for the period 26 February 2007- 1 May 2007. In column (1) of Table 1 are the actual prices and in column (4) of the same table F presents the predicted prices. Firstly is presented the methodology of moving median with trend and seasonality with two ways and then is presented the improved method of moving median . The analysis is: 1st STEP Because the first example is concerning the Exchange stock market, and more specific the share prices of :COSMOTE”, so the median five periods is the appropriate measure for that case, because stock market is based in five working days ,from Monday to Friday, with the exception of the holidays. Provided that there are 40 periods is obtained the median of five first periods and the median of five last periods. The mathematical type of median for the period 02-08 /6Jan / 2007 is (n + 1) /2, where n = the observations which in that case are 5 days. So it is (n + 1) /2 = (5+1) /2 = 6/2= 3. The third period of date 02-08 /Jan / 2007 is the
2
third price and this is the price 22.98. Similarly for the median of five last periods, is the period 21-27/ Feb /2007.6 The median is the price 23.06.
2nd STEP Consequently the trend results from:
Χ1 − Χ 2 (1) 10
,where Χ1 = the median of the last five periods Χ2 = the median of the first five periods and the 10 it results from the sum of periods, after is examined the first five periods and the five last ones so it is 5+5=10. Χ1 − Χ 2 23.06 − 22.98 0.08 = = = 0.008 10 10 10 The 0.008 is the trend of the time-series.
3rd STEP The formula for the trend smoothing is. Trend smoothing = Actual price + 3*trend. So for example for the period 2 January 2007 will be:
22.8 + 3*0.008 = 22.824
where 22.8 is the actual price , the 0.008 results from relation (1) and number 3 results as follows: There are five working days so there are five periods Consequently the numerator is the sum of five periods, first, second, third and so on: The denominator is five, because is been found the moving median of five periods.
1+ 2 + 3 + 4 + 5 (2) 5
3
, where the result of relation (2) is number 3. For example the trend smoothing for the next period will be: 22.24 + 3*0.008 = 22.264
The results are presented in column (2) with the letter S in table 1.
4th STEP
The next step is to divide column (1) at column (2). Column (3) =
Column(1) (3) Column(2)
5th STEP From column (3) can be found the seasonal indicators, which is the aggregation of the data which respond in the first day of season and more specific is sum of dates 2- 9-16-23-30 January and 6-13-21 February 2007 where can be find the seasonal indicator for the first day of the season and with the same way can be found the other indicators. When all of seasonal indices have been found then the sum of them (because there are five working days in the week) must be equal to five. If they are not equal to five then the following relation must be applied to obtain the adjusted seasonal indicators
5/sum of seasonal indicators (4) Then each of the five indicators is multiplied with the result of the relation (4) (see table 2).
4
6th STEP Thus the final forecasting is Forecasting = Actual price of previous period+ trend*the adjusted seasonal indicator (5) For example the forecasting of period 28/ Feb/2007 is: Forecasting = 22.2 + 0.008*0.999996 = 22.208. The forecasting of period 01/ May/2007 is: Forecasting = 22.00 + 0.008*0.999996 = 22.008 (Table 2).
Second way Now in this part is examined the forecasting moving median of trend and seasonality applying the second way for the same prices of “COSMOTE” enterprise.
1st STEP It is exactly the same with the first way
2nd STEP As in the previous way is been calculated the moving median of the first five periods and of the last five periods. Then the following formula is applied. Χ1 − Χ 2 23.06 − 22.98 0.08 = = 0.0054 = 15 15 15
, where five is the sum of 1+2+3+4+5. because is reported to five working days, so 0.0054 is the trend.
3rd STEP The smoothing is being made as: Moving median of first five periods – 3*trend
(1)
5
,where number 3 is obtained from the relation
1+ 2 + 3 + 4 + 5 =3 5
22.98 – 3*0.0054 = 22.9638 So the smoothing is S = the result of (1) + (trend * period 1), where period 1 is in column (5). For example 2 February 2007 will be: S= 22.9638 + (0.0054*1) = 22.9692 (Column 2, Table 4).
4th STEP The next step is to divide column (1) at column (2). Column (3) =
Column(1) Column(2)
5th STEP It is exactly the same with the first way (Table 5).
6th STEP It is exactly the same with the previous way So the forecasting will be for the date 28/ Feb/2007. Forecasting = 22.2 + 0.0054*0,996579= 22.205
In tables 7-9 and 10-12 are presented the results for the first and second way respectively for the share prices of “National Bank of Greece”. The trend is -0.158.
6
3. Improved moving median method with trend and seasonality
The only difference in this method, in relation with the first method, is the second step, where is the finding of the trend. The steps for the improved method are:
Χ1+ Χ2+ Χ3+ Χ4+ Χ5+ Χ6+ Χ7+ Χ8 - X8 (1) n
and n= number of variables which in this case are 8. So for the “National bank of Greece is X1 = 36.26, X2 =36.2, X3 = 36.96, X4 = 36.7, X5 = 34.7, X6 = 33.62, X7 = 35.32 and X8 = 34.68. So the result from relation 1 is 0.87 and then 0.87 is divided by 15 which 15 is the sum of 1+2+3+4+5, because is reported to five working days. The trend is 0.058. Then can be applied either the first way either the second (Tables 13-15 for the 1st way and Table 16-18 for the 2nd way).
7
4. Differences between simple moving median method with trend and seasonality and improved method and between the seasonality. Now the last example concerns the improved moving median for the inflation rate of Belgium (Wirtz) . and the simple moving median and a major difference and also a problem that can be appear . So in table 29 there are the inflation rates for Belgium during the period January 2004 – December 2005. The one dilemma that can be arise is what seasonality presents that data. The graph which was created with the help of Minitab software (Graph 1) can show a picture about the behaviour of data. So someone can say that there is a 5-period seasonality. Of course someone can say that there is logically , without the help of Graph 1 , 12-period seasonality . The tables 19-21 report the results with 5-period seasonality and the tables 22-24 report the results with 12period seasonality. But before analysis go to that point first should be noted the difference between the simple and improved method. With 12- period seasonality the trend for the simple moving median method is
Χ1 − Χ 2 2.7 − 2.05 0.65 = = 0.027 = 24 24 24
, where 2.05 is the average of the 6th and 7th price , because it is already known that moving median of 12 periods = (n+1)/2 = 13/2 = 6.5. Similarly for the 2.7 number. So the trend with the simple method and more specifically with the 1st way is positive. Let’s see the improved method. Χ1 + Χ 2 2.05 + 2.7 4.75 − 2 .7 = − 2 .7 − X2 = 2 2 2
= −0.325
8
and -0.325 is divided with 78 which is the sum of 1+2+3+4+5+6+7+8+9+10+11+12 = 78. So it is -0.325/78 = -0.0041, which is the trend. Notice that in that case the trend is negative. Also the relation 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 =6.5 . The following step are known. 12
5. Conclusion
The methods that are proposed here do not constitute an original work, but it could be said that they can constitute an alte6rnative method of forecasting, an alternative way timeseries estimation . These methods constitute an alternative opinion with regard to the estimation of forecasting. This work does not aim to
present a method of estimation or
correction of other econometric models because it is a very simple method and in many times not so reliable, but it can be used as an alternative tool.
References Armstrong J.S. and Collopy F.,1992, “Error Measures For Generalizing About Forecasting Methods: Empirical Comparisons”, Reprinted with permission from International Journal of Forecasting, 8 : 69-80.
Jarrett J., 1993“Methods of forecasts for economic and enterprising decisions”, Gutenberg , 1st Edition- Athens. Wirtz C., 2006, Statistics in focus,Economy and Finance, Harmonized Indices of Consumer Prices, EUROSTAT: SOURCES: The shares prices was been obtained by the website of newspaper «ELEFTHEROTIPIA» www.enet.gr , which data are being prepared by company ALPHA TRUST.
9
Appendix TABLE 1 (Results from the moving median method with trend and seasonality-1st way). Date
S (2)
(3)
2/1/2007
Actual price1 (1) 22.8
22.824
0.9989
3/1/2007
22.24
22.264
0.9989
4/1/2007
22.98
23.004
5/1/2007
23
23.024
8/1/2007
23.6
9/1/2007
F (4)
Period (5)
Date
(2)
(3)
F (4)
Period (5)
30/1/2007
Actual price (1) 23.4
1
23.424
0.9990
23.388
21
22.808
2
31/1/2007
23.66
23.684
0.9990
23.408
22
0.9990 0.9990
22.248
3
1/2/2007
23.5
23.524
0.9990
23.668
23
22.988
4
2/2/2007
23.5
23.524
0.9990
23.508
24
23.624
0.9990
23.008
5
5/2/2007
23.4
23.424
0.9990
23.508
25
23.02
23.044
0.9990
23.608
6
6/2/2007
23.18
23.204
0.9990
23.408
26
10/1/2007 11/1/2007
22.6
22.624
0.9989
23.028
7
7/2/2007
23.1
23.124
0.9990
23.188
27
22.92
22.944
0.9990
22.608
8
8/2/2007
23
23.024
0.9990
23.108
28
12/1/2007
23.1
23.124
0.9990
22.928
9
9/2/2007
23
23.024
0.9990
23.008
29
15/1/2007
23.34
23.364
0.9990
23.108
10
12/2/2007
22.8
22.824
0.9989
23.008
30
16/1/2007
23.62
23.644
0.9990
23.348
11
13/2/2007
22.5
22.524
0.9989
22.808
31
17/1/2007
23.86
23.884
0.9990
23.628
12
14/2/2007
22.6
22.624
0.9989
22.508
32
18/1/2007
24
24.024
0.9990
23.868
13
15/2/2007
22.8
22.824
0.9989
22.608
33
19/1/2007
24.2
24.224
0.9990
24.008
14
16/2/2007
22.6
22.624
0.9989
22.808
34
22/1/2007
24
24.024
0.9990
24.208
15
20/2/2007
22.88
22.904
0.9990
22.608
35
23/1/2007
23.4
23.424
0.9990
24.008
16
21/2/2007
22.32
22.344
0.9989
22.888
36
24/1/2007
23.6
23.624
0.9990
23.408
17
22/2/2007
22.6
22.624
0.9989
22.328
37
25/1/2007
23.2
23.224
0.9990
23.608
18
23/2/2007
23.06
23.084
0.9990
22.608
38
26/1/2007
22.9
22.924
0.9990
23.208
19
26/2/2007
22.8
22.824
0.9989
23.068
39
29/1/2007
23.38
23.404
0.9990
22.908
20
27/2/2007
22.2
22.224
0.9989
22.808
40
1. Source: www.enet.gr
TABLE 2 Seasonal and adjusted seasonal indicators
initial seasonal indicators 0.998959 0.998958 0.998966 0.998963 0.998966 Sum = 4,994812 5/4.99812 =1,001039
adjusted seasonal indicators. 0.999996 0.999996 1.000003 1.000001 1.000004 Sum = 5
TABLE 3 Actual and forecasting prices
Actual prices Forecasting prices 22.00
22.208
22.00
22.008
10
TABLE 4 (Results from the moving median method with trend and seasonality-2nd way). Date
2/1/2007
Actual price1 (1) 22.8
S (2)
(3)
F (4)
Period (5)
Date
22.9692
0.992634
1
30/1/2007
Actual price (1) 23.4
(2)
(3)
F (4)
Period (5)
23.0772
1.013988
23.38538
21
3/1/2007
22.24
22.9746
0.968026
22.80538
2
31/1/2007
23.66
23.0826
1.025015
23.40538
22
4/1/2007
22.98
22.98
1
22.24542
3
1/2/2007
23.5
23.088
1.017845
23.66542
23
5/1/2007
23
22.9854
1.000635
22.9854
4
2/2/2007
23.5
23.0934
1.017607
23.5054
24
8/1/2007
23.6
22.9908
1.026498
23.00542
5
5/2/2007
23.4
23.0988
1.01304
23.50542
25
9/1/2007
23.02
22.9962
1.001035
23.60538
6
6/2/2007
23.18
23.1042
1.003281
23.40538
26
10/1/2007
22.6
23.0016
0.98254
23.02538
7
7/2/2007
23.1
23.1096
0.999585
23.18538
27
11/1/2007
22.92
23.007
0.996219
22.60542
8
8/2/2007
23
23.115
0.995025
23.10542
28
12/1/2007
23.1
23.0124
1.003807
22.9254
9
9/2/2007
23
23.1204
0.994792
23.0054
29
15/1/2007
23.34
23.0178
1.013998
23.10542
10
12/2/2007
22.8
23.1258
0.985912
23.00542
30
16/1/2007
23.62
23.0232
1.025922
23.34538
11
13/2/2007
22.5
23.1312
0.972712
22.80538
31
17/1/2007
23.86
23.0286
1.036103
23.62538
12
14/2/2007
22.6
23.1366
0.976807
22.50538
32
18/1/2007
24
23.034
1.041938
23.86542
13
15/2/2007
22.8
23.142
0.985222
22.60542
33
19/1/2007
24.2
23.0394
1.050375
24.0054
14
16/2/2007
22.6
23.1474
0.976352
22.8054
34
22/1/2007
24
23.0448
1.04145
24.20542
15
20/2/2007
22.88
23.1528
0.988217
22.60542
35
23/1/2007
23.4
23.0502
1.015176
24.00538
16
21/2/2007
22.32
23.1582
0.963805
22.88538
36
24/1/2007
23.6
23.0556
1.023612
23.40538
17
22/2/2007
22.6
23.1636
0.975669
22.32538
37
25/1/2007
23.2
23.061
1.006027
23.60542
18
23/2/2007
23.06
23.169
0.995295
22.60542
38
26/1/2007
22.9
23.0664
0.992786
23.2054
19
26/2/2007
22.8
23.1744
0.983844
23.0654
39
29/1/2007
23.38
23.0718 1.013358 22.90542 1. Source: www.enet.gr
20
27/2/2007
22.2
23.1798
0.95773
22.80542
40
TABLE 5 Seasonal and adjusted seasonal indicators
initial seasonal indicators 0.998569 0.99842 1.004696 1.002525 1.005025 Sum = 5,009235
adjusted seasonal indicators. 0.996728 0.996579 1.002844 1.000676 1.003172 Sum = 5
5/5,009235=0,998156 TABLE 6 Actual and forecasting prices
Actual prices Forecasting prices 22.00
22.205
22.00
22.005
11
TABLE 7 (Results from the moving median method with trend and seasonality-1st way for “National Bank of Greece�). Date
Actual price1 (1)
S (2)
(3)
F (4)
Period (5)
Date
Actual price (1)
S (2)
(3)
F (4)
Period (5)
1/11/2006
36
35.526
1.01334
2/11/2006
35.66
35.186
1.01347
35.842
1 2
29/11/2006
35.9
35.426
1.01338
35.042
21
30/11/2006
34.62
34.146
1.01388
35.742
22
3/11/2006
36.26
35.786
1.01325
35.502
3
1/12/2006
34.7
34.226
1.01385
34.462
23
6/11/2006
35.92
35.446
1.01337
36.102
4
4/12/2006
34.3
33.826
1.01401
34.542
24
7/11/2006
36.2
8/11/2006
36.2
35.726 35.726
1.01327
35.762
5
5/12/2006
33.72
33.246
1.01426
34.142
25
1.01327
36.042
6
6/12/2006
34.1
33.626
1.01410
33.562
26
9/11/2006
36.04
35.566
1.01333
36.042
7
7/12/2006
34.2
33.726
1.01405
33.942
27
10/11/2006
36.2
35.726
1.01327
35.882
8
8/12/2006
33.62
33.146
1.01430
34.042
28
13/11/2006
36
35.526
1.01334
36.042
9
11/12/2006
34.1
33.626
1.01410
33.462
29
14/11/2006
36.4
35.926
1.01319
35.842
10
12/12/2006
34.16
33.686
1.01407
33.942
30
15/11/2006
36.9
36.426
1.01301
36.242
11
13/12/2006
34.6
34.126
1.01389
34.002
31
16/11/2006
36.98
36.506
1.01298
36.742
12
14/12/2006
34.96
34.486
1.01374
34.442
32
17/11/2006
36.96
36.486
1.01299
36.822
13
15/12/2006
35.32
34.846
1.01360
34.802
33
20/11/2006
36.6
36.126
1.01312
36.802
14
18/12/2006
35.6
35.126
1.01349
35.162
34
21/11/2006
37.1
36.626
1.01294
36.442
15
19/12/2006
34.7
34.226
1.01385
35.442
35
22/11/2006
37.1
36.626
1.01294
36.942
16
20/12/2006
34.78
34.306
1.01382
34.542
36
23/11/2006
37.1
36.626
1.01294
36.942
17
21/12/2006
34.6
34.126
1.01389
34.622
37
24/11/2006
36.7
36.226
1.01308
36.942
18
22/12/2006
34.68
34.206
1.01386
34.442
38
27/11/2006
36.44
35.966
1.01318
36.542
19
27/12/2006
34.9
34.426
1.01377
34.522
39
28/11/2006
35.2
34.726
1.01365
36.282
20
28/12/2006
35.08
34.606
1.01370
34.742
40
1. Source: www.enet.gr
TABLE 8 Seasonal and adjusted seasonal indicators
initial seasonal indicators 1.013468 1.013537 1.013525 1.013548 1.013616 Sum = 5,067694 5/5,067694=0,986642
adjusted seasonal indicators. 0.99993 0.999998 0.999986 1.000009 1.000076 Sum = 5
12
TABLE 9 Actual and forecasting prices
Actual prices
29/1/2006 2/1/2007 3/1/2007 4/1/2007 5/1/2007
Forecasting prices 34,922 34,742 35,762 36,142 36,562
34,9 35,92 36,3 36,72 36,94
TABLE 10 (Results from the moving median method with trend and seasonality-2nd way for the “National Bank of Greece�) Date
Actual price1 (1)
S (2)
(3)
1/11/2006
36
35.84
1.004464
2/11/2006
35.66
35.735
0.997901
3/11/2006
36.26
35.63
6/11/2006
35.92
35.525
F (4)
Period (5)
Date
Actual price (1)
S (2)
(3)
F (4)
Period (5)
1
29/11/2006
35.9
33.74
1.064019
35.09512
21
35.89531
2
30/11/2006
34.62
33.635
1.029285
35.79531
22
1.017682
35.55489
3
1/12/2006
34.7
33.53
1.034894
34.51489
23
1.011119
36.15475
4
4/12/2006
34.3
33.425
1.026178
34.59475
24
5/12/2006
33.72
33.32
1.012005
34.19493
25
7/11/2006
36.2
35.42
1.022021
35.81493
5
8/11/2006
36.2
35.315
1.02506
36.09512
6
6/12/2006
34.1
33.215
1.026645
33.61512
26
9/11/2006
36.04
35.21
1.023573
36.09531
7
7/12/2006
34.2
33.11
1.032921
33.99531
27
10/11/2006
36.2
35.105
1.031192
35.93489
8
8/12/2006
33.62
33.005
1.018634
34.09489
28
11/12/2006
34.1
32.9
1.036474
33.51475
29
13/11/2006
36
35
1.028571
36.09475
9
14/11/2006
36.4
34.895
1.043129
35.89493
10
12/12/2006
34.16
32.795
1.041622
33.99493
30
15/11/2006
36.9
34.79
1.06065
36.29512
11
13/12/2006
34.6
32.69
1.058428
34.05512
31
16/11/2006
36.98
34.685
1.066167
36.79531
12
14/12/2006
34.96
32.585
1.072886
34.49531
32
15/12/2006
35.32
32.48
1.087438
34.85489
33
17/11/2006
36.96
34.58
1.068826
36.87489
13
20/11/2006
36.6
34.475
1.061639
36.85475
14
18/12/2006
35.6
32.375
1.099614
35.21475
34
21/11/2006
37.1
34.37
1.07943
36.49493
15
19/12/2006
34.7
32.27
1.075302
35.49493
35
22/11/2006
37.1
34.265
1.082737
36.99512
16
20/12/2006
34.78
32.165
1.0813
34.59512
36
21/12/2006
34.6
32.06
1.079226
34.67531
37
23/11/2006
37.1
34.16
1.086066
36.99531
17
24/11/2006
36.7
34.055
1.077668
36.99489
18
22/12/2006
34.68
31.955
1.085276
34.49489
38
27/11/2006
36.44
33.95
1.073343
36.59475
19
27/12/2006
34.9
31.85
1.095761
34.57475
39
28/11/2006
35.2
33.845
1.040035
36.33493
20
28/12/2006
35.08
31.745
1.105056
34.79493
40
1. Source: www.enet.gr
TABLE 11 Seasonal and adjusted seasonal indicators
initial seasonal indicators 1.050413 1.048503 1.052701 1.054087 1.052325 Sum = 5,25803 5/5,25803=0,950927
adjusted seasonal indicators. 0.998865 0.997049 1.001042 1.00236 1.000684 Sum = 5
13
TABLE 12 Actual and forecasting prices
Actual prices
29/1/2006 2/1/2007 3/1/2007 4/1/2007 5/1/2007
Forecasting prices 34.98 34.80 35.82 36.20 36.62
34.9 35.92 36.3 36.72 36.94
TABLE 13 (Results from the improved moving median method with trend and seasonality-1st way for “National Bank of Greece�). Date
Actual price1 (1)
S (2)
Actual price (1)
S (2)
(3)
F (4)
Period (5)
1/11/2006
36
36.174
0.99519
2/11/2006
35.66
35.834
0.995144
36.058
29/11/2006
35.9
36.074
0.995177
35.258
21
30/11/2006
34.62
34.794
0.994999
35.958
22
3/11/2006
36.26
36.434
0.995224
3
1/12/2006
34.7
34.874
0.995011
34.678
23
6/11/2006
35.92
36.094
36.318
4
4/12/2006
34.3
34.474
0.994953
34.758
24
7/11/2006
36.2
8/11/2006
36.2
0.995216
35.978
5
5/12/2006
33.72
33.894
0.994866
34.358
25
0.995216
36.258
6
6/12/2006
34.1
34.274
0.994923
33.778
26
9/11/2006
36.214
0.995195
36.258
7
7/12/2006
34.2
34.374
0.994938
34.158
27
10/11/2006
36.2
36.374
0.995216
36.098
8
8/12/2006
33.62
33.794
0.994851
34.258
28
13/11/2006
36
36.174
0.99519
36.258
9
11/12/2006
34.1
34.274
0.994923
33.678
29
14/11/2006
36.4
36.574
0.995243
36.058
10
12/12/2006
34.16
34.334
0.994932
34.158
30
15/11/2006
36.9
37.074
0.995307
36.458
11
13/12/2006
34.6
34.774
0.994996
34.218
31
16/11/2006
36.98
37.154
0.995317
36.958
12
14/12/2006
34.96
35.134
0.995048
34.658
32
17/11/2006
36.96
37.134
0.995314
37.038
13
15/12/2006
35.32
35.494
0.995098
35.018
33
20/11/2006
36.6
36.774
0.995268
37.018
14
18/12/2006
35.6
35.774
0.995136
35.378
34
21/11/2006
37.1
37.274
0.995332
36.658
15
19/12/2006
34.7
34.874
0.995011
35.658
35
22/11/2006
37.1
37.274
0.995332
37.158
16
20/12/2006
34.78
34.954
0.995022
34.758
36
23/11/2006
37.1
37.274
0.995332
37.158
17
21/12/2006
34.6
34.774
0.994996
34.838
37
24/11/2006
36.7
36.874
0.995281
37.158
18
22/12/2006
34.68
34.854
0.995008
34.658
38
27/11/2006
36.44
36.614
0.995248
36.758
19
27/12/2006
34.9
35.074
0.995039
34.738
39
35.2
35.374
0.995081
36.498
20
28/12/2006
35.08
35.254
0.995064
34.958
40
28/11/2006
(3)
F (4)
Period (5)
Date
1 2
35.718
0.995179
36.374 36.374
36.04
1. Source: www.enet.gr
TABLE 14 Seasonal and adjusted seasonal indicators
initial seasonal indicators 0.995145 0.995121 0.995125 0.995117 0.995093 Sum = 4.975602 5/4.975602=1.004903
adjusted seasonal indicators. 1.000025 1.000001 1.000005 0.999997 0.999973 Sum = 5
14
TABLE 15 Actual and forecasting prices
Actual prices
29/1/2006 2/1/2007 3/1/2007 4/1/2007 5/1/2007
Forecasting prices 35.138 34.958 35.978 36.358 36.778
34.9 35.92 36.3 36.72 36.94
TABLE 16 (Results from the improved moving median method with trend and seasonality-1st way for “National Bank of Greece�). Date
Actual price (1)1
S (2)
(3)
1/11/2006
36
36.144
0.996016
2/11/2006
35.66
36.202
0.985028
3/11/2006
36.26
36.26
6/11/2006
35.92
7/11/2006
36.2
8/11/2006
F (4)
Period (5)
Date
Actual price (1)
S (2)
(3)
F (4)
Period (5)
1
29/11/2006
35.9
37.304
0.962363
35.25848
21
36.0581
2
30/11/2006
34.62
37.362
0.92661
35.9581
22
1
35.71807
3
1/12/2006
34.7
37.42
0.927312
34.67807
23
36.318
0.989041
36.31785
4
4/12/2006
34.3
37.478
0.915204
34.75785
24
36.376
0.995162
35.9775
5
5/12/2006
33.72
37.536
0.898338
34.3575
25
36.2
36.434
0.993577
36.25848
6
6/12/2006
34.1
37.594
0.90706
33.77848
26
9/11/2006
36.04
36.492
0.987614
36.2581
7
7/12/2006
34.2
37.652
0.908318
34.1581
27
10/11/2006
36.2
36.55
0.990424
36.09807
8
8/12/2006
33.62
37.71
0.891541
34.25807
28
13/11/2006
36
36.608
0.983392
36.25785
9
11/12/2006
34.1
37.768
0.902881
33.67785
29
14/11/2006
36.4
36.666
0.992745
36.0575
10
12/12/2006
34.16
37.826
0.903083
34.1575
30
15/11/2006
36.9
36.724
1.004793
36.45848
11
13/12/2006
34.6
37.884
0.913314
34.21848
31
16/11/2006
36.98
36.782
1.005383
36.9581
12
14/12/2006
34.96
37.942
0.921406
34.6581
32
17/11/2006
36.96
36.84
1.003257
37.03807
13
15/12/2006
35.32
38
0.929474
35.01807
33
20/11/2006
36.6
36.898
0.991924
37.01785
14
18/12/2006
35.6
38.058
0.935414
35.37785
34
21/11/2006
37.1
36.956
1.003897
36.6575
15
19/12/2006
34.7
38.116
0.910379
35.6575
35
22/11/2006
37.1
37.014
1.002323
37.15848
16
20/12/2006
34.78
38.174
0.911091
34.75848
36
23/11/2006
37.1
37.072
1.000755
37.1581
17
21/12/2006
34.6
38.232
0.905001
34.8381
37
24/11/2006
36.7
37.13
0.988419
37.15807
18
22/12/2006
34.68
38.29
0.90572
34.65807
38
27/11/2006
36.44
37.188
0.979886
36.75785
19
27/12/2006
34.9
38.348
0.910087
34.73785
39
28/11/2006
35.2
37.246
0.945068
36.4975
20
28/12/2006
35.08
38.406
0.913399
34.9575
40
1. Source: www.enet.gr
TABLE 17 Seasonal and adjusted seasonal indicators
initial seasonal indicators 0.961317 0.955015 0.954518 0.950978 0.945259 Sum = 4,767087 5/4,767087=1,048859
adjusted seasonal indicators. 1.008286 1.001675 1.001155 0.997442 0.991443 Sum = 5
15
TABLE 18 Actual and forecasting prices
Actual prices
29/1/2006 2/1/2007 3/1/2007 4/1/2007 5/1/2007
Forecasting prices 35.13848 34.9581 35.97807 36.35785 36.7775
34.9 35.92 36.3 36.72 36.94
TABLE 19 (Results from the improved moving median method with trend and seasonality-2st way of 12period for the inflation rate of Belgium). Date
Actual price (1)1
S (2)
(3)
Jan 2004
1.4
2.01583
0.694503
Feb 2004
1.2
2.01166
0.596522
March 2004
1
2.00749
April 2004
1.7
2.00332
May 2004
2.4
June 2004
F (4)
Period (5)
Date
Actual price (1)
S (2)
(3)
F (4)
Period (5)
1
Jan 2005
2
1.96579
1.017403
1.896952
13
1.39685
2
Feb 2005
2.3
1.96162
1.1725
1.99685
14
0.498134
1.196566
3
March 2005
2.8
1.95745
1.430432
2.296566
15
0.848591
0.996301
4
April 2005
2.4
1.95328
1.228702
2.796301
16
1.99915
1.20051
1.695761
5
May 2005
2.3
1.94911
1.180026
2.395761
17
2
1.99498
1.002516
2.395743
6
June 2005
2.7
1.94494
1.388218
2.295743
18
July 2004
2.1
1.99081
1.054847
1.995645
7
July 2005
2.7
1.94077
1.3912
2.695645
19
Aug 2004
2
1.98664
1.006725
2.095541
8
Aug 2005
2.9
1.9366
1.49747
2.695541
20
Sep 2004
1.8
1.98247
0.907958
1.995619
9
Sep 2005
3
1.93243
1.55245
2.895619
21
Oct 2004
2.7
1.9783
1.364808
1.795538
10
Oct 2005
2.2
1.92826
1.140925
2.995538
22
Nov 2004
2.3
1.97413
1.16507
2.695797
11
Nov 2005
2.3
1.92409
1.19537
2.195797
23
Dec 2004
1.9
1.96996
0.964487
2.295686
12
Dec 2005
2.8
1.91992
1.458394
2.295686
24
TABLE 20
initial seasonal indicators 0.855953 0.884511 0.964283 1.038647 1.190268 1.195367 1.223024 1.252097 1.230204 1.252867 1.18022 1.21144 Sum = 13,47888
adjusted seasonal indicators. 0.762039 0.787464 0.858484 0.924688 1.059674 1.064213 1.088835 1.114719 1.095228 1.115404 1.050728 1.078523 Sum = 12
12/13,47888=0,890282
16
TABLE 21 Actual and forecasting prices
Actual prices Jan 2004
Forecasting prices 2.797 2.797 2.797 2.196 2.596
2.8 2.8 2.2 2.6 2.8
Feb 2004 March 2004 April 2004 May 2004
TABLE 22 (Results from the improved moving median method with trend and seasonality-2st way of 5period for the inflation rate of Belgium). Date
Actual price (1)1
S (2)
(3)
Jan 2004
1.4
0.84
1.666667
Feb 2004
1.2
0.8
1.5
March 2004
1
0.76
April 2004
1.7
May 2004
2.4
June 2004 July 2004
F (4)
Period (5)
Date
Actual price (1)
S (2)
(3)
F (4)
Period (5)
1
Jan 2005
2
0.36
5.555556
1.869505
13
1.28874
2
Feb 2005
2.3
0.32
7.1875
2.027171
14
1.315789
1.169505
3
March 2005
2.8
0.28
10
2.279765
15
0.72
2.361111
1.027171
4
April 2005
2.4
0.24
10
2.73482
16
0.68
3.529412
1.679765
5
May 2005
2.3
0.2
11.5
2.28874
17
2
0.64
3.125
2.33482
6
June 2005
2.7
0.16
16.875
2.269505
18
2.1
0.6
3.5
1.88874
7
July 2005
2.7
0.12
22.5
2.727171
19
Aug 2004
2
0.56
3.571429
2.069505
8
Aug 2005
2.9
0.08
36.25
2.679765
20
Sep 2004
1.8
0.52
3.461538
2.027171
9
Sep 2005
3
0.04
75
2.83482
21
Oct 2004
2.7
0.48
5.625
1.779765
10
Oct 2005
2.2
0
0
2.88874
22
Nov 2004
2.3
0.44
5.227273
2.63482
11
Nov 2005
2.3
-0.04
-57.5
2.169505
23
Dec 2004
1.9
0.4
4.75
2.18874
12
Dec 2005
2.8
-0.08
-35
2.327171
24
TABLE 23
initial seasonal indicators adjusted seasonal indicators. 1.629511 2.781497 0.762387 -0.67927 0.505874 Sum = 5
11.33333 19.34545 5.302444 -4.72436 3.518382 Sum = 34,77525
5/34,77525=0,14378
17
TABLE 24 Actual and forecasting prices
Actual prices Jan 2004
Forecasting prices 2.780 2.735 2.689 2.170 2.627
2.8 2.8 2.2 2.6 2.8
Feb 2004 March 2004 April 2004 May 2004
GRAPH 1 . SEASONALITY GRAPH FOR BELGIUM INFLATION RATES
3,0
C1
2,5
2,0
1,5
1,0 3
6
9
12
15
18
18 Index
21
24
27
30
33