Trend Series Analysis II

Measurement of Seasonal Trend:

(a)    Simple Average Method,

(b)   Link Relative Method,

(c)    Ratio to Moving Average Method, and

(d)   Ratio to Trend Method.

(a)   Simple Average Method: Under this method, the average ( ) of all the monthly or quarterly values for each year are found out.  Each monthly or quarterly value (yi) is divided by the corresponding average and the results are expressed as percentage:

Then the mean index or seasonal index (Si) is calculated for each month or quarter.  If the mean of all seasonal indices is not equal to 100, then they will be adjusted.

Example 8:

Take data from Example 1, and calculate the four seasonal indices by the ‘simple-average method’.

Solution:

Year/Quarter

y

Mean

I

II

III

IV

2003

219

357

645

513

433.5

2004

549

640

701

590

620

2005

657

394

543

600

548.5

Now the above observed values are converted to indices using the following formula:

Year/Quarter

I

II

III

IV

Total

2003

50.52%

82.35%

148.79%

118.34%

 

2004

88.55

103.23

113.06

95.16

 

2005

119.78

71.83

99.00

109.39

 

Season Index (Si)

86.28%

85.80%

120.28%

107.63%

400.00

(b)   Link Relative Method: Under this method, the data for each month or quarter are expressed in percentage, known as ‘Link Relatives’.  An appropriate average of the link relatives is taken, usually a median is taken.  Convert these averages into a series of chain indices.  The chain indices are adjusted for the fraction of the effect of the trend.  The adjusted chain indices are further reduced to the same level as the first month or quarter.

Example 9:

Take the data from Example 1, and calculate seasonal indices by using ‘link relative method’.

Solution:

The observed values are converted into price relatives or link relatives using the following formula:

Where Pn is the value of current year

            Po is the value of base year

and then the link relatives are converted into chain indices (chaining process) using the following formula:

= (L.R. × C.I. of preceding year) ÷ 100

 

Where L.R. is the link relative

            C.I. is the chain index

Year

Quarter

Total

I

II

III

IV

2003

163.01%

180.67%

79.53%

 

2004

107.02%

116.58

109.53

84.17

 

2005

111.36

59.97

137.82

110.50

 

Median

(link relative)

109.19

116.58

137.82

84.17

 

Chain Index

100

116.58

160.67

135.24

512.49

Adj. C. I.

78.05%*

90.99%

125.40%

105.56%

400.00

* Adjusted chain index for QI: 100 ÷ 512.49 × 400 = 78.05, and so for other quarters.

(c)   Ratio to Moving Average Method: A 12-month or 4-quarter moving average centred is computed.  The observed values are divided by the corresponding centred moving average and the results are expressed in percentage:

The monthly or quarterly averages of these percentages are found out.  The adjusted values are the indices of the seasonal variations.

Example 10:

Take data from Example 1 and calculate seasonal indices using ‘ratio to moving average method’.

Solution:

Quarter

y

4-Quarter

Moving

Total

8-Quarter

Moving

Total

4-Quarter

Moving

Average

(Centred)

Ratio to Moving Average

 

I

II

III

IV

2003

I

219

 

 

 

 

 

 

 

II

357

 

 

 

 

 

1734

 

III

645

3798

475

 

 

135.8

 

2064

 

IV

513

4411

551

 

 

 

93.1

2347

2004

I

549

4750

594

92.4

 

 

 

2403

 

II

640

4883

610

 

104.9

 

 

2480

 

III

701

5068

634

 

 

110.6

 

2588

 

IV

590

4930

616

 

 

 

95.8

2342

2005

I

657

4526

566

116.1

 

 

 

2184

 

II

394

4378

547

 

72.0

 

 

2194

 

III

543

 

 

 

 

 

 

 

IV

600

 

 

 

 

 

 

Mean Seasonal Index (total = 410.5)

104.3

88.5

123.2

94.5

Adjusted Seasonal Index (Si) (total = 400)

101.6%*

86.2%

120.1%

92.1%

* Adjusted seasonal index for QI: 104.3 ÷ 410.5 × 400, and so on for other three quarters.

(d)   Ratio to Trend Method: An average for each year is found out and a straight line is fitted by least squares method.  The trend values for each month or quarter are calculated on the assumption that the data correspond to the middle of the month or quarter.  Each original value is divided by the corresponding calculated trend values and expressed in percentage.  A mean of these percentages are calculated for each month or quarter.  The adjusted values are the indices of seasonal variation.

Example 11:

Take data from Example 1 and calculate seasonal indices using ‘ratio to trend method’.

Solution:

Quarters

y

x

x

x2

 

2003

I

219

–11

 

 

 

 

455

48.13%

 

II

357

–9

469

76.12

433.5

–8

–3468

64

 

III

645

–7

484

133.26

 

 

 

 

 

IV

513

–5

498

103.01

 

 

 

 

2004

I

549

–3

512

107.23

 

 

 

 

 

II

640

–1

527

121.44

620

0

0

0

 

III

701

1

541

129.57

 

 

 

 

 

IV

590

3

556

106.12

 

 

 

 

2005

I

657

5

570

115.26

 

 

 

 

 

II

394

7

584

67.47

548.5

8

4388

64

 

III

543

9

599

90.65

 

 

 

 

 

IV

600

11

613

97.88

 

 

 

0

1602

0

920

128

 

 

Now arranging the above calculated values in last column as follows:

Year/Quarter

I

II

III

IV

Total

2003

48.13

76.12

133.26

103.01

 

2004

107.23

121.44

129.57

106.12

 

2005

115.26

67.47

90.65

97.88

 

Seasonal Index (Si)

90.21

88.34

117.83

102.34

398.72

Adj. S.I.

90.50%*

88.62%

118.21%

102.67%

400.00

* Adjusted seasonal index for QI: 90.21 ÷ 398.72 × 400, and so on.

Measurement of Cyclical Variation:

The cyclical and random components of a time series are first isolated from the time series using the multiplicative model:

yi = Ti + Si + Ci + Ri

Where Ti:         Secular trend

            Si:         Seasonal variation

            Ci:        Cyclical variation

            Ri:        Random variation

This can be done by dividing yi by the product of Ti and Si:

The Random component Ri will now be separated from the time series by using the smoothing technique, moving average.  These moving averages show the indices of cyclical variation.

Example 12:

Take data from Example 1 and isolate cyclical component from the time series.

Solution:

Quarters

y

*

Si**

Ci × Ri =

×100

3-Quarter

Moving

Average

(Ci)

2003

I

219

455

86.28

392.57

55.79%

 

II

357

469

85.80

402.40

88.72

85.10

 

III

645

484

120.28

582.16

110.79

98.41

 

IV

513

498

107.63

536.00

95.71

110.26

2004

I

549

512

86.28

441.75

124.28

120.51

 

II

640

527

85.80

452.17

141.54

124.52

 

III

701

541

120.28

650.71

107.73

115.95

 

IV

590

556

107.63

598.42

98.59

113.30

2005

I

657

570

86.28

491.80

133.59

103.60

 

II

394

584

85.80

501.07

78.63

95.86

 

III

543

599

120.28

720.48

75.37

81.74

 

IV

600

613

107.63

657.77

91.22

* As calculated in the previous example

** As calculated in Example

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