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‘‘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).

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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

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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


Moving median with trend and seasonality